Figure 5 Research cooperation and academic collaboration: industrial and research partners to investigate novel concepts

The work presented in this manuscript has been established in collaboration with the team MIAM (Modélisation et Identication en Automatique et Mécanique) at the UHA and the IA (Institute of Automation), an institute of the FHNW. It has been employed for industries, namely zmb watercut and Jet Cut Power, and funded by the InnoSuisse.

The three years of research has been an intensive experience. Managing the diverging interests of industry and academics has been quite challenging but oered many opportunities to evolve personal competences. I spent most of the time with the IA, where the test bench has been located. However, the collaboration with the MIAM has always been extraordinary. I had the chance to meet kind and always interested people at the universities, during the formations and at the conferences. The rich exchange helped truly to evolve this research. Eventually, I look back at many interesting discussions and I am very happy for all the support I've received at the very dierent assignments. I would like to thank all the people who contributed to this work.

I address my thanks to all jury members for evaluating and verifying the rather large manuscript. I am very grateful for the favorable reviews and for the motivating feedbacks from Luc Dugard, concerning the robust control, as well as from Eric Bideaux, regarding the modelling of uid ow systems. Equally great thanks to Didier Theilliol who took the double-roles as exterminator and president. I have been happy to hear about the great interests for enhancing the work presented. In any case, thanks for their personal participation. It has been a great pleasure to meet these specialists.

I would like to express my sincere reconnaissance to Michel Basset, a great patron, and Rodolfo Orjuela, a great supporter. Their steady care and understanding enabled this manifold manuscript and various other publications. Both relied on my person and promoted my PhD at the UHA. All the unconstrained discussions improved my scientic work. Their theoretical and, even more, their human competences taught me a balanced approach of working. I am also very thankful for all the delightful friendships and enriching encounters with the MIAM. The only thing I regret is not to have spent more time at the UHA during this period.

I am equivalent grateful for the employment at the FHNW. The rich research work, the chance to participate for lecturers and to supervise students allowed me to attain complementary competences. I have especially to thank Roland Anderegg for his constant benevolence during all the years and for supporting the various conferences as well as to Pascal Schleuniger for the many helpful advises, for all organizational eorts and for the detailed revision of my manuscript. I like to thank Sebastian Gaulocher, who initiated the collaboration between UHA and FHNW, for its theoretical advises and Thomas Besselmann for the revision of a paper. Here, I must thank Sandro Born, Martin Hunziker and Markus Birchmeier as steady supporters of the test bench and for supplying all needs at the laboratory.

Thanks to the industries Toni Zaugg, Patrick Baumann and Sascha Niggli from zmb waterjet as well as Peter Reinbold, Mathias Kennel and Daniel Lustenberger from Jet Cut Power for the fruitful cooperation by establishing a test bench and building the high-pressure pump prototypes. This allowed to validate the work as presented in this manuscript. Also, the very best to Pascal Zeugin, Jean-Paul John, Daniel Begert, Jan Gebhardt and all the students involved for deriving measurements at the test bench and realizing experiments.

Finally, my reconnaissance to Mélanie Gaugler and Betty Labigang at the administration of the UHA as well as to Silvia Rüegger and Hanny Gerber at the administration of the FHNW to satisfy all needs for my PhD and the related industrial project. The remarkable benevolent of the MIAM and IA has enabled this transnational collaboration. i ii iii

Motivation

In the recent years, increasing the energy eciency has become a major objective for industry. The large energy demand for waterjet machining motivates to investigate novel concepts for high-pressure generation. Conventional high-pressure pumps for waterjet machining are driven by means of hydraulic drives. The latest developments of electrical direct drives with high-torque servo motors allow for replacing these hydraulic drives. This signicantly improves the energy eciency of high-pressure pumps.

In contrast to hydraulically driven pumps, electrically driven pumps can be exibly combined and spatially distributed, taking the individual demands of entire waterjet facilities into account. This results in scalable high-pressure networks of decentralized pumps and interconnected work stations, which enables industry to improve the overall energy eciency of waterjet machining.

However, these decentralized pumps require a control and managing strategy, which has to be easily adapted to the various waterjet facilities in compliance with Industry 4.0. This aims at reducing the installation time for high-pressure pumps and simplifying the maintenance of waterjet facilities. The managing enables an optimal allocation of the available pumps with respect to the time-varying demands of waterjet machining. It will reduce wear and degradation, when every pump collaborates within a high-pressure network to run at a desired operating point. The subsequent work addresses these challenges to establish a exible and sustainable waterjet machining by means of Industry 4.0.

Energy ecient high-pressure pump

In most cases, a conventional high-pressure pump supplies a single work station as shown in the illustration of Figure 1. A high-pressure pipe directly interconnects a hydraulically driven pump and a work station. The pump pressurizes water by means of two pressure intensiers, where the work station holds a cutting head producing a waterjet. Abrasive added to the waterjet allows for cutting of hard and brittle materials. Introducing an example for waterjet cutting, a typical operating pressure of 400 MPa and its water consumption of about 5 l/min result in an energy demand of about 33.3 kW per work station.

Pressure intensifiers

Oil chiller Work station

High-pressure pipe Cutting head

Hydraulic drive

Figure 1 Common setup for waterjet machining: a hydraulically driven high-pressure pump supplies a single work station.

Insofar as waterjet machining is an energy intensive process, any increase of eciency attains signicant energy savings. The hydraulic drive can be identied as a major source of losses with its energy eciency of typically below 70%. The above example results in a drive input power of at least 47.6 kW. Thus, replacing the hydraulic drive aims at improving the energy eciency of high-pressure pumps. Moreover, it will enhance the utilization of waterjet machining and yields new applications in elds, where hydraulics is proscribed to avoid contamination, e.g. food industry and medical treatments.

xii For these reasons, dierent manufacturers investigate energy ecient high-pressure pumps. They intent to replace the hydraulic drive system by electric servo drives. The illustration of Figure 2 compares a hydraulically driven pump (a) to an electrically driven pump (b). The novel pump design introduces modular piston pumps. Its pumping units consist of an independent electric drive which displaces a piston by means of a spindle shaft. In this case, combining two pumping units allows for a continuous pressure generation. Using electric servo drives, with its eciency of above 90%, reduces the energy demand of waterjet machining. With respect to the previous example, this decreases the expected drive input power to 37.0 kW. That corresponds to energy savings of up to 10.6 kW (22.2%), which results in an annual saving of 33.9 MWh for a two-shift pump operation. Moreover, implementing an electric drive further reduces the pump size by 40% and decreases the noise level. Apart from eciency concerns, waterjet machining is subject to ongoing enhancements in cutting quality. Waterjet machining requires to attenuate pressure uctuations for more than 20 dB by means of the available high-pressure pumps. These uctuations originate from work stations when switching on and o a cutting head with respect to the cutting process. This switching causes a varying water consumption and consequently induces pressure uctuations, which propagate across the high-pressure piping. Since the hydraulic drive of a conventional high-pressure pump acts as an attenuator, pressure uctuations become suciently rejected as long as the pump remains at its predened operating point. Additional high-pressure attenuators, installed to the piping system, reduce pressure uctuations and improve cutting quality. However, these attenuators are costly and subject to fast degradation.

Energy ecient high-pressure pumps come along with a major drawback. Pressure uctuations cause a time-varying load cycle acting on the spindle shafts and instantaneously aecting the electric drives. Thus, eective disturbance rejection by means of high-pressure pumps becomes a challenging task for control and involves high performance motion dynamics. In fact, an electric drive has to reproduce the disturbance rejection characteristics of a hydraulically driven pump. Whereas the novel pump design reduces the drive system complexity, its control design must meet enhanced requirements. Overcoming these diculties allows for an electrically driven pump to run at any pressure range without additional high-pressure attenuators. This thesis provides a low-level control design for energy ecient high-pressure pumps that aims at robust disturbance rejection. It is further expected to deal with the degradation of electrically driven pumps and to reduce the wear due to pressure uctuations.

xiii Energy ecient waterjet facilities

A hydraulically driven pump is dicult to scale with respect to the increased water consumption of interconnection work stations. Scaling the size of a pressure intensier will, indeed, allow for a higher pump rate, but equally increase the load acting on the pistons. This will degrade its live cycle and deteriorate the energy eciency what is limiting the maximal size of a single highpressure pump. The illustration of Figure 3 presents exemplarily a waterjet facility with several work stations. The pump size, its noise level and heat dissipation require a hydraulically driven pump to be located in a remote machine room. Interconnecting all work stations of a facility results in long high-pressure pipes, which further decreases the overall energy eciency.

On the other hand, an electrically driven pump is modular. This allows for scaling a waterjet facility with respect to the individual demands. Moreover, the reduced pump size and minimal noise level enable the distribution of high-pressure pumps at dierent locations in a waterjet facility, as shown in the illustration of Figure 4. Decentralized pumps supply several work stations according to their water consumption. The high-pressure piping interconnects pumps and work stations to realize a scalable high-pressure network. This reduces the overall piping lengths and aims to optimize the energy eciency of entire waterjet facilities. xiv This new concept of decentralized high-pressure pumps requires a control concept that provides a exibility and scalability taking the wide variety of high-pressure networks into account to realize future waterjet facilities. Pressure uctuations will propagate through the high-pressure network, causing time-varying load cycles on the decentralized pumps. This load will vary for every pump. It depends on the network topology, the pump locations and the switching state of each work station. As a consequence, each pump will obtain a dierent pump rate.

For example, an inferior pump runs close to its lower saturation, while an other superior pump will reach its upper saturation, providing the major uid ow. Neither of the two pumps will operate on an energy ecient operating point and the superior pump will suer a decrease of its life cycle. Even worse, this signicantly deteriorates the capability to attenuate pressure uctuations. Every pump, if running close to saturation, has insucient margins to increase its pump rate when needed. Instead of all interconnected pumps to contribute for disturbance rejection, only the remaining inferior pumps respond as desired. This will reduce the overall available control eort and consequently degrade the control performance for disturbance rejection.

Moreover, it becomes possible that distributed pumps counteract against each others due to their coupling over the high-pressure network, since a pump itself may induce pressure uctuations. Neighbouring pumps will compensate these additional uctuations, which however increases the overall energy demands of a waterjet facility. As a consequence, decentralized pumps should cooperate to balance the overall control eort, whereas each pump should provide robust performance and stability for its local region. Hence, this thesis investigates a distributed highlevel managing that ensures a exible and scalable application to unique high-pressure networks of various waterjet facilities.

Objectives

The energy ecient high-pressure pumps are modular due to their electric drives and exible to deal with the individual water consumption of dierent facilities. Enabling sustainable waterjet machining requires to deal with complex high-pressure networks of arbitrary topology, for which the number of interconnected pumps and work stations are most likely dierent and vary for every facility. Independent work stations induce pressure uctuations when switching their cutting heads with respect to an unknown switching pattern and the coupling of decentralized pumps may degrade the overall energy eciency if a global managing strategy is missing. However, industry dislikes to establish a centralized pump supervisor. In a typical waterjet facility, each high-pressure pump is locally controlled without any command from supervisory level. Thus, a pump operates autonomously and serves the demand of a work station. This autonomy should remain unchanged when introducing high-pressure networks with interconnected pumps. All of this requires a control design that reduces the engineering eort when setting up future waterjet facilities and that guarantees a reliable and energy ecient pump operation even for high-pressure networks of arbitrary topology and varying water consumption.

In order to deal with the previous objectives, this thesis aims to employ a hierarchical distributed control design that allocates the control task to two layers: a low-level control that has to reject any local pressure uctuations by means of the available high-pressure pumps and a high-level managing that has to balance the pump rates to converge to a global average value. The hierarchical allocation of disturbance rejection and pump rate balancing should guarantee a reliable and energy ecient pump operation, while the physical distribution among all available pumps ensures its scalability. This requires each decentralized pump to execute exactly the same algorithm, while exchanging information. xv Any ecient evaluation and verication of a hierarchical distributed control design requires a modelling methodology that is exible to describe high-pressure networks of arbitrary topology, while taking a wide range of operation into account. Waterjet machining involves a wide pressure range from 40 to 400 MPa and includes non-linearities, whereas high-pressure pumps as well as work stations are subjects to parameter variations. Besides describing the pressure propagation along a piping, using the governing equations from uid dynamics, high-pressure network of various topology will be represented by means of graphs. It is further required to identify and classify the expected range of parameter variations. Modelling and control design must then consider the most prevalent varying parameters.

This thesis provides a framework for modelling of high-pressure networks and employs a robust control design for decentralized pumps. The modelling methodology and control design are essential for establishing decentralized pumps and enhancing energy ecient waterjet facilities. A test bench has been built in the laboratory at the FHNW, in order to validate the modelling methodology and to verify the low-level control design. The veried modelling framework has been used to model more complex topologies, which allows for evaluating the high-level managing by means of simulations. The main contribution is given when applying the graph-based modelling methodology to high-pressure networks and evaluating the hierarchical distributed control design for decentralized pumps. This framework contributes to the latest research work and developments in waterjet machining with a specic emphasis to recent industrial perspectives.

Work context

The manufacturer Jet Cut Power GmbH produces conventional hydraulically driven high-pressure pumps, where Zaugg Maschinenbau AG (zmb) engineers automated work stations, e.g. for waterjet machining. The University of Applied Sciences and Arts Northwestern Switzerland (FHNW) developed in collaboration with the above mentioned industrial partners an energy ecient highpressure pump. This electrically driven pump challenges the major cause of energy loss by replacing the hydraulic drives of conventional high-pressure pumps. It is highly scalable and exible to equip various waterjet facilities. Consequently, a research cooperation has been established to investigate novel concepts for energy ecient waterjet machining.

The diagram of Figure 5 provides insight into the collaboration between industrial and research partners. This research has been founded by Swiss Innovation Agency (Innosuisse). It aims to develop a hierarchical distributed control design applied on high-pressure networks used for waterjet machining. This design procedure should reduce engineering eorts when setting up future waterjet facilities and guarantee a reliable pump operation even for high-pressure networks of arbitrary topology. The electric drive has been particularly developed in collaboration with other third parties from the industry.

This thesis has originated from an academic collaboration between the Institut de Recherche en Informatique, Mathématiques, Automatique et Signal (IRIMAS) from the Université de Haute-Alsace (UHA) in Mulhouse, France and the Institute of Automation (IA) from the FHNW in Windisch, Switzerland. Both institutes have specic knowledge in modelling and control of dynamic systems. These complementary competences become united to advance the research activities for future waterjet facilities. xvi Part I: Waterjet machining The investigations of Chapter 1 introduce typical waterjet facilities and overview conventional waterjet machining. This allows for classifying waterjet machining with respect to common waterjet applications, manufacturing processes, and work station setups. The basic principle of high-pressure generation is then explained when comparing hydraulically and electrically driven pumps. It classies the high-pressure pumps with respect to drive system and pumping chamber design. These classications allow for deriving specications for future energy ecient and scalable waterjet facilities. The system description of Chapter 2 species use cases and introduces network topologies, while presenting the high-pressure test bench. This test bench has been explicitly built for these subsequent investigations. Selected high-pressure networks will be subject for modelling, where use cases aim at verifying the modelling methodology, validating the low-level control design and evaluating the high-level managing. Apart from that, this section discusses the novel design of an electrically driven pump, as installed on the test bench. It introduces the system components, its control structure and the concept of model-based synchronization by means of camming. xvii A rst contribution in this part provides the state of the art for waterjet machining. This allows to classify the various waterjet applications with respect to the work station setup, cutting head conguration and manufacturing process. That further involves to establish a high-pressure test bench and to introduce the design of a novel electrically driven high-pressure pump [START_REF] Niederberger | Hochdruckwasserstrahlschneiden mit energieezienten hochdruckpumpen[END_REF]. Network topologies and use cases are dened, which consider common waterjet applications. Extending these specications contributes to the design of future energy ecient and scalable waterjet facilities. Another contribution is given when applying the model-based synchronization to the pump prototypes [Niederberger, 2018]. This eventually allows for considering a high-pressure pump as a continuous source of uid ow that facilitates the low-level control design.

Part II: Modelling framework

The Chapter 3 is dedicated to derive a graph-based modelling methodology. The governing equations describing uid dynamics has been approximated by means of nite dierences, while considering uid ow at equilibrium. This yields a lumped parameter model that represents a piping section by means of homogeneous segments. Mapping these segments to a graph allows for a exible and scalable representation of high-pressure networks with arbitrary topologies. A simulation study veries the computational eorts and resulting simulation accuracy due to approximations and spacial discretization. In contrast to established modelling approaches, the graph-based modelling methodology employs a pressure-dependent bulk modulus and relates this to a pressure-dependent uid density. This improves the simulation accuracy for a wide pressure range from 40 to 400 MPa. Pressure-dependent parameters as well as uid ow characteristics have been identied using the high-pressure test bench. This allows for scaling a high-pressure network model without revising its parametrization. The implementation of the graph-based modelling methodology in MATLAB Simulink provides a toolbox that simplies the modelling of future waterjet facilities and avoids cost intensive experiments at a test bench. Chapter 4 shows the application of this toolbox to model various networks topologies by means of graphs. These high-pressure network models reduce the time expanses for validating the low-level control design and evaluating the high-level managing. Experimental studies validate the graph-based modelling by means of measurements using a test bench.

In this part, a rst contribution is given by proposing a graph-based modelling methodology for high-pressure networks, which includes the most prevalent parameter variations to deal with the wide pressure range of waterjet machining. This graph-based modelling has been published in [START_REF] Niederberger | A graph-based modelling methodology for high-pressure networks applied on waterjet machining[END_REF]. Another contribution is given when introducing a toolbox by implementing the graph-based modelling into MATLAB Simulink and providing an experimental parameter identication for specic high-pressure components. This toolbox is valuable for industry and allows for modelling of various waterjet facilities without the need for an individual parametrization. Applying this toolbox to model dierent high-pressure networks and validating the obtained network models, using the measurement data from a test bench, demonstrate the benets of a graph-based modelling but also reveal its limitations. Promising results have been presented in [START_REF] Niederberger | Extended validation of a graph-based modelling methodology for highpressure networks[END_REF].

Part III: Control design

The Chapter 5 is dedicated to the robust low-level control design for decentralized high-pressure pumps. It considers the unknown switching of work stations and the interaction between neighbouring pumps as exogenous disturbances to the network subsection, while taking non-linearities xviii and uncertain parameters into account to obtain robust performance and stability for a wide range of operation. Introducing a linearised perturbed plant model to describe a network subsection for control design allows for a sensitivity analysis of parameter variations by means of simulations and for comparing dierent complex uncertainty descriptions. The control requirements for a high-pressure pump are specied with respect to the needs for waterjet machining. An electrically driven high-pressure pump is investigated to dene the generalized plant for H ∞ controller synthesis. A signal-based approach introduces shaping functions on the input channel and weighting functions on the output channel considering the expected signal dynamics and control requirements. The investigations of Chapter 6 propose to aggregate shaping functions by merging their singular values with respect to the Euclidean vector norm. That reduces the dimension of the H ∞ optimization problem for controller synthesis and results in a reduced-order state feedback controller. Employing a structured controller synthesis yields a suboptimal robust PI controller, suitable for industrial use. Robustness analysis allows for determining the permissible network subsection size for which a high-pressure pump guarantees desired performance and stability. This leads to an iterative procedure that maximizes the uncertainty range, while synthesizing a quasi-optimal PI controller. The proposed ∆-K iteration preserves the order of a generalized plant. It is therefore suitable when using unstructured uncertainties and employing a structured controller synthesis. Simulations verify the reduced state feedback controller and the quasi-optimal PI controller, using detailed high-pressure network models. A preliminary PI controller has been implemented to the electrically driven pump for experimental validation on the test bench. The robust control of a single high-pressure pump is an important milestone for the industrial implementation. Simulations of decentralized pumps, interconnected by means of a high-pressure network, show that these pumps will run at arbitrary pump rates if any high-level managing is missing. Many approaches become available to manage electrically driven pumps.

The investigations of Chapter 7 present the application of average consensus to balance distributed pumps without explicitly implementing a control law. This causes each pump to converge to the time-varying average pump rate, while considering information from neighbouring pumps. Simulations reveal the ability of the high-level managing to balance interconnected pumps to a common pump rate, which corresponds to the average overall water consumption. That provides a perspective for further developments towards energy ecient waterjet facilities.

The contribution here provides suitable shaping and weighting functions, when considering a signal-based approach to introduce a robust control design for decentralized high-pressure pumps. This requires to verify dierent system descriptions and to introduce a suitable uncertainty model, considering the varying parameters. A major contribution is given, when merging input channels by means of joint shaping functions [START_REF] Niederberger | Robust control design for electrically driven high-pressure pumps using H ∞ approach with joint shaping functions[END_REF]. This reduces the dimension of the generalized plant, implemented for control design. Another contribution is provided, when applying H ∞ controller synthesis to derive controllers of dierent structures and comparing these controllers by means of a detailed simulation study. An extended PI controller has been found suitable for implementation on a high-pressure pump with respect to industrial needs. The robust control design reduces the eort for installing a pump in a facility. Another contribution is given when proposing the ∆-K iteration. This gives a practical use and provides a straight forward procedure for a quasi-optimal H ∞ controller synthesis. The concept of distributed balancing by means of average consensus gives eventually a perspective to the managing of interconnected pumps [Niederberger et al., 2019a].

The hierarchical distributed control design for electrically driven high-pressure pumps provides a versatile framework, including modelling of high-pressure networks and control design of decentralized pumps. This aims at the implementation of Industry 4.0 capabilities to enhance Distributed pumps interconnected to a high-pressure network are subject for subsequent investigations. Applying modelling and control design requires to introduce a set of dierent indexes. It is outlined that a high-pressure network interconnects N pumps, denoted by n ∈ {1, ..., N }, with M cutting heads, denoted by m ∈ {1, ..., M }. Each pump n is assigned to an agent i ∈ {1, ..., N }, subject for managing. On the other hand, a high-pressure network N kj holds nodes k ∈ J interconnected to other nodes j ∈ J \k by means of branches. The network topology will be represented by a graph G = (V, E) that consists of nodes v ∈ V and branches e ∈ E. It is also useful to distinguish between physical variables used for modelling (see Chapter 3), such as pressures p(t) and uid ows Q(t), and logical variables used for control (see Chapter 5), e.g. control variable u(t), system state x(t) and measured variable y(t). (Pa), (Pa) p max , p 0 Maximal pressure, initial pressure (Pa), (Pa)

p x
Pressure at position x (Pa) p C , p H , p N Pressure in a pumping chamber, cutting head, network (Pa), . . . piping with nozzle when increasing the uid ow velocity to v ∞ = 3 m/s. . . . . 3.8 Simulated pressure trend and uid ow velocity for boundary condition (B-2):

piping with nozzle when increasing the uid ow velocity to v ∞ = 4.8 m/s. . . . 3.9 Simulated pressure trend and uid ow velocity for boundary condition (B-3):

piping with nozzle when increasing the uid ow velocity to v ∞ = 6.8 m/s. . . . 3.10 Simulated pressure trend and uid ow velocity for boundary condition (B-4):

piping with nozzle when increasing the uid ow velocity to v ∞ = 9.6 m/s. . . . 3.11 A piping section of constant cross section S: homogeneous segments of length ∆x along the position coordinate x with pressure state p(t) and uid ow velocity v(t). 3.12 Simulated pressure trend and uid ow velocity for a piping segment of 0.5 m: simulation using nite dierences with 2100 elements. . . . . . . . . . . . . . . . xxix 3.13 Simulated pressure trend and uid ow velocity for a piping segment of 1 m: simulation using nite dierences with 4200 elements. . . . . . . . . . . . . . . . 3.14 Simulated pressure trend and uid ow velocity for a piping segment of 2 m: simulation using nite dierences with 8400 elements. . . . . . . . . . . . . . . . 3.15 Simulated pressure trend and uid ow velocity for a piping segment of 5 m: simulation using nite dierences with 21000 elements. . . . . . . . . . . . . . . . 3.16 Simulated pressure trend and uid ow velocity for a piping segment of 10 m: simulation using nite dierences with 42000 elements. . . . . . . . . . . . . . . . 3.17 Error between distributed and lumped parameter model for a piping segment of 0.5 m: simulation using nite dierences with 2100 elements. . . . . . . . . . . . . 3.18 Error between distributed and lumped parameter model for a piping segment of 1 m: simulation using nite dierences with 4200 elements. . . . . . . . . . . . . . 3.19 Error between distributed and lumped parameter model for a piping segment of 2 m: simulation using nite dierences with 8400 elements. . . . . . . . . . . . . . 3.20 Error between distributed and lumped parameter model for a piping segment of 5 m: simulation using nite dierences with 21000 elements. . . . . . . . . . . . . 3.21 Error between distributed and lumped parameter model for a piping segment of 10 m: simulation using nite dierences with 42000 elements. . . . . . . . . . . . 3.22 Modelling framework for high-pressure networks: assigning a network section to k ∈ J segments (a) and representing the segments by means of graph (b). . . . . the symmetric and asymmetric network topologies used for modelling. . . . . . . 4.6 Network assignment for topologies (e), (f) and (g): corresponding graph networks for the symmetric, asymmetric and the decentralized distributed network topologies.104 4.7 Experimental setup to verify the modelling methodology: pressure gauges measure the pressures p x (t) at positions x, when applying an input uid ows Q(t). The estimated uid ows Q(t) are used as model inputs and the simulated pressure trends px (t) have been compared to measurements. . . . . . . . . . . . . . . . . . 4.8 Overview of measurement R-1: measured pump output pressure and corresponding reference displacement ow rate, for a cutting head congured with a 0.25 mm nozzle and position trajectories optimized for 200 MPa. . . . . . . . . . . . . . . 4.9 Step response for operating pressure around 100 MPa, considering varying parameters (1a/b), constant bulk modulus (2a/b) and constant uid density (3a/b): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow. . . . . . . . . . . . . 4.10 Step response for operating pressure around 200 MPa, considering varying parameters (1c/d), constant bulk modulus (2c/d) and constant uid density (3c/d): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow. . . . . . . . . . . . . 4.11 Step response for operating pressure around 300 MPa, considering varying parameters (1e/f), constant bulk modulus (2e/f) and constant uid density (3e/f): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow. . . . . . . . . . . . . 4.12 Step response for operating pressure around 400 MPa, considering varying parameters (1g/h), constant bulk modulus (2g/h) and constant uid density (3g/h): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow. . . . . . . . . . . . . 4.13 Steady-state of measured and simulated pump output pressure (1i/j) and cutting head input pressure (2i/j) around 100 MPa: the proposed modelling methodology is thereby compared to simulations with constant bulk modulus and constant uid density. . . . 5.4 Separation of a high-pressure network into network subsections for control design: high-pressure pumps and concatenated cutting heads to introduce uid ow sources and pressure sinks, respectively. . . . . . . . . . . . . . . . . . . . . . . . 5.5 Schematic diagram of the high-pressure system: a single high-pressure pump and a cutting head interconnected to a network subsection. The high-pressure pump consist of two linear actuators. Each linear actuator is controlled by motion control, whereby both plungers are synchronized by means of camming. . . . . . . . . . . 5.6 Overview of all system components relevant for control design: isolated network subsection represented as a decentralized control loop. . . . . . . . . . . . . . . . 5. 7 Step responses for the hard and soft constraints: considering a second order model to represent the control objectives for d(t) → y(t) and r(t) → y(t). . . . . . . . . 5.8 Singular values of weighting function W 1 and inverse weighting function W -1 2 : derived from the hard and soft constraints to specify minimal and minimal bounds for desired performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Overview of the plant model evaluation process: dierent linearised and approximated models to represent the high-pressure system subject to control design. . .

Motivations

La technologie au jet d'eau sous très haute pression est largement utilisée dans de nombreuses applications industrielles telles que la manufacture et l'usinage (découpe, décapage, nettoyage de surface, etc.). La production conventionnelle des hautes pressions nécessaire à ces applications est réalisée à l'aide de pompes hydrauliques centralisées qui présentent l'inconvénient de n'orir qu'un faible rendement. L'avènement récent des systèmes électriques à entraînement direct pilotés à l'aide de servomoteurs à grand couple a ouvert la possibilité d'électrier les pompes à haute pression dans la perspective d'améliorer leur rendement. Cette nouvelle génération de pompes électriques haute pression présente d'autres avantages, à savoir, une interconnexion en réseau plus aisée ainsi qu'une distribution spatiale recongurable respectant les demandes individuelles des postes de travail de l'installation. Cette nouvelle technologie, mise à la disposition des industriels, permet d'améliorer le rendement global de l'installation d'usinage grâce à la mise en place de réseaux de distribution haute pression évolutifs interconnectant des pompes électriques décentralisées à des postes de travail. L'interconnexion aisée des pompes électriques ore la possibilité de réduire le temps nécessaire à leur mise en place et simplie la reconguration globale de l'installation.

L'interconnexion décentralisée des pompes requiert cependant une stratégie évoluée de commande couplée à un mécanisme de gestion facilement adaptables aux diérentes topologies de l'installation. La stratégie de commande bas niveau vise à réduire les uctuations de la pression générée par chaque pompe au sein du réseau. Le mécanisme de gestion haut niveau, quant à lui, alloue d'une façon optimale la contribution individuelle de la pression générée par chaque pompe en fonction des demandes des diérents postes de travail. L'objectif est de maintenir les performances globales de l'installation grâce à la collaboration des diérentes pompes, chaque pompe travaillant dans un point de fonctionnement donné. Le travail présenté dans le cadre de cette thèse vise à relever ces dés en proposant une technologie au jet d'eau recongurable et écologiquement durable en accord avec l'industrie 4.0.

Rendement des pompes à haute pression

Dans bon nombre d'applications industrielles, une pompe hydraulique conventionnelle à haute pression alimente un seul poste de travail. L'interconnexion entre la pompe et le poste de travail, tous deux souvent déportés, est réalisée au moyen d'un tuyau résistant aux hautes pressions. La pompe charge en pression l'eau à l'aide de deux intensicateurs de pression et le poste de travail équipé d'une tête de coupe génère le jet d'eau. L'usinage de matériaux durs et cassants est rendu possible grâce à l'abrasif (sable, etc.) ajouté au jet d'eau.

L'usinage à jet d'eau étant un procédé énergivore, un gain en ecacité aussi faible soit-il se traduit directement par un gain d'énergie globale. Le système à entraînement hydraulique représente à lui seul la principale source de perte d'ecacité en raison de son faible rendement généralement inférieur à 70%. Remplacer l'entraînement hydraulique par un entraînement électrique beaucoup plus ecace représente un moyen d'améliorer le rendement. Cette électrication élargit par ailleurs le champ des applications de la technologie au jet d'eau. Elle permet, à titre d'exemple, de l'étendre à l'industrie alimentaire et/ou pharmaceutique où l'utilisation des systèmes à entraînement hydraulique est proscrite en raison des risques de contamination. Ces raisons justient l'intérêt accru accordé à l'électrication des pompes à haute pression. l Cette nouvelle génération de pompes électriques est constituée de pistons modulables. Chaque piston est déplacé à l'aide d'un système vis-écrou actionné par un système à entraînement électrique. Cet ensemble constitue une unité de pompage. L'idée est de combiner judicieusement deux unités de pompage électriques an de générer une pression de charge continue. L'utilisation d'actionneurs électriques, dont le rendement est supérieur à 90%, réduit la consommation d'énergie lors de l'usinage par jet d'eau rendant le procédé industriel plus écoresponsable. Leur utilisation permet par ailleurs, de réduire jusqu'à 40% les dimensions de la pompe tout en réduisant le niveau de bruit.

Il convient de remarquer que le procédé d'usinage par jet d'eau requiert une atténuation des uctuations de la pression, d'au minimum 20 dB, obtenue au moyen des pompes. Ces uctuations proviennent essentiellement de l'ouverture et/ou la fermeture de la buse de coupe de chaque poste de travail en fonction de la tâche d'usinage. Or, ces cycles de commutation génèrent une consommation variable d'eau qui induit des uctuations indésirables de la pression, lesquelles se propagent à travers les tuyaux. L'usage des pompes hydrauliques permet de rejeter susamment ces uctuations. Ces pompes agissent en eet comme des atténuateurs de pression à condition toutefois que leur point de fonctionnement prédéni ne soit pas modié. L'ajout d'atténuateurs de pression tout au long du tuyau contribue aussi à réduire ces uctuations et à améliorer la qualité de l'usinage. Ces atténuateurs sont néanmoins coûteux et se dégradent très rapidement au l du temps.

Les pompes électriques, quant à elles, présentent un inconvénient majeur en matière de rejet des uctuations de pression. Les uctuations induisent en eet des cycles de charge variables qui impactent instantanément l'actionneur électrique. Leur atténuation, au moyen de pompes électriques, représente un dé à relever. L'objectif est de reproduire, à l'aide d'une stratégie de commande, les mêmes caractéristiques de rejet que celles oertes par une pompe hydraulique. La nouvelle génération de pompes électriques permet de réduire dès sa conception la complexité du système à entraînement mais nécessite en revanche une stratégie de commande évoluée. Relever ces dés permettra aux pompes électriques de travailler dans un point de fonctionnement quelconque tout en évitant le recours aux atténuateurs additionnels. Cette thèse propose une stratégie de commande bas niveau pour une pompe électrique haute pression qui permet un rejet robuste des perturbations. Cette stratégie serait par ailleurs en mesure de réduire les dégradations de l'entraînement électrique dues aux uctuations de la pression.

Rendement du procédé d'usinage par jet d'eau

La modication, en fonction de la consommation d'eau des postes de travail, des capacités de charge des pompes classiques à entraînement hydraulique s'avère délicate. Augmenter les dimensions des intensicateurs de pression permet d'augmenter le débit de la pompe mais accroît la sollicitation des pistons. Elle conduit à une dégradation de leur cycle de vie et diminue le rendement énergétique du procédé, limité par les dimensions maximales de chaque pompe. Les dimensions de chaque pompe, le niveau de bruit généré et la dissipation de la chaleur nécessaire au bon fonctionnement de l'installation obligent à déporter la pompe dans une salle des machines. L'interconnexion des postes de travail nécessite alors le recours à de longs tuyaux qui dégradent à leur tour le rendement global de l'installation.

Les pompes électriques présentent l'avantage d'être modulables. Elles facilitent la reconguration des installations suivant les demandes de chaque station de travail. La réduction des li dimensions et du niveau de bruit des pompes électriques rend possible leur décentralisation. Les pompes décentralisées fournissent les diérents postes de travail en fonction de leurs besoins en eau. Le réseau ainsi mis en place, entre les pompes et les postes de travail, parce qu'il présente l'avantage d'être recongurable et de réduire la longueur des tuyaux, améliore le rendement global de l'installation.

Ce nouveau concept d'installation décentralisée des pompes exige toutefois le développement d'une stratégie de commande en mesure de s'adapter à une grande variété de topologies du réseau. Les uctuations de la pression dans le réseau sont également à l'origine de cycles de charge variables au niveau de chaque pompe décentralisée. La charge appliquée à chaque pompe varie en fonction de la topologie globale du réseau, du positionnement des pompes et des commutations des têtes de coupe de chaque poste de travail.

Objectifs généraux

La nouvelle génération de pompes électriques haute pression présente l'avantage d'être modulable, versatile et capable de répondre à la demande individuelle de consommation d'eau de diérentes installations. La mise en place d'un usinage à jet d'eau écologiquement durable se heurte au problème posé par l'interconnexion de pompes en réseau de topologie inconnue. En eet, le nombre de pompes décentralisées et de postes de travail peut considérablement varier d'une installation à l'autre.

Les postes de travail, en fonction du motif d'usinage, génèrent des uctuations de pression provoquées par l'ouverture et la fermeture des buses de coupe. L'atténuation de ces uctuations passe par la mise en place d'une stratégie de commande bas niveau. D'autre part, le couplage des diérentes pompes décentralisées peut dégrader le rendement global si aucun mécanisme de gestion haut niveau n'est utilisé. Toutefois, le recours à un mécanisme centralisé peut s'avérer très compliqué voire impossible. En eet, dans les installations conventionnelles d'usinage à jet d'eau chaque pompe haute pression est localement pilotée sans qu'aucun ordre ne soit envoyé par un mécanisme haut niveau. Chaque pompe opère de façon autonome et répond à la demande d'un poste de travail. Cette notion d'autonomie doit être préservée dans un contexte de mise en réseau de pompes électriques décentralisées.

L'ensemble de ces raisons justie le développement d'une démarche de synthèse de commande en mesure de réduire l'eort d'ingénierie nécessaire à la mise en place d'une future installation. Cette démarche doit aussi garantir le fonctionnement économe et able de chaque pompe quelles que soient la topologie du réseau et la consommation en eau. Ces travaux de recherche proposent une stratégie de commande hiérarchisée et distribuée visant à découper en deux niveaux le problème de commande. La commande bas niveau doit rejeter les uctuations locales de pression lii au moyen des pompes disponibles. La commande haut niveau gère la coopération entre les pompes an d'équilibrer les taux de charge de chaque pompe vers une valeur moyenne globale. L'allocation hiérarchisée en rejetant les perturbations et en équilibrant les taux de charge des pompes favorise le fonctionnement performant, able et économe de chaque pompe. La distribution spatiale des pompes disponibles facilite l'évolution de l'installation. Toute évaluation et vérication d'une stratégie hiérarchisée de commande passe par une phase de modélisation qui permet de décrire un réseau haute pression de topologie quelconque. Dans le contexte d'usinage à jet d'eau, le réseau est soumis à de fortes variations dans un domaine de pression relativement large qui allant de 40 à 400 MPa fait apparaitre des phénomènes non linéaires. D'autre part, les pompes électriques haute pression ainsi que les postes de travail sont soumis à des variations paramétriques. La modélisation et la synthèse des lois de commande doivent prendre en compte les variations les plus pertinentes de ces paramètres.

Contexte de travail

Ces travaux de thèse s'inscrivent dans le cadre d'une coopération entre universitaires et industriels. Ils associent les compétences complémentaires de l'Institut de Recherche en Informatique, Mathématiques, Automatique et Signal (IRIMAS) de l'Université de Haute-Alsace (UHA, France) et de l'Institute of Automation (IA) de l'University of Applied Sciences and Arts Northwestern Switzerland (FHNW, Suisse) pour répondre aux besoins de deux industriels suisses, Jet Cut Power GmbH fabricant de pompes hydrauliques haute pression et Zaugg Maschinenbau AG (zmb) spécialisé dans l'automatisation des postes de travail. Ces travaux de thèse ont été nancés par la Swiss Innovation Agency (Innosuisse).

Plan de la thèse

La synthèse des stratégies de commande hiérarchisées et distribuées dédiée aux pompes électriques haute pression est développée dans cette thèse. Cette synthèse passe par une phase de modélisation du réseau haute pression suivie d'une phase de conception des lois de commande. Ces deux tâches, modélisation et conception, sont essentielles à la mise en place des pompes décentralisées en vue d'améliorer le rendement énergétique global de l'installation. Un banc d'essais a été spéciquement construit dans le laboratoire (Institute of Automation, Suisse) de la FHNW an de valider la méthodologie de modélisation proposée ainsi que la stratégie de commande bas niveau. Après avoir fait l'objet d'une validation exhaustive, cette démarche de modélisation a été employée an d'évaluer en simulation le mécanisme de gestion haut niveau proposé.

Contributions

Le classement établi à partir des applications (procédés industriels) et de la conguration des postes de travail (tête de coupe simple, tête de coupe multiple, regroupement des stations, etc.) constitue la première contribution de ce chapitre. Ce dernier introduit également quelques exemples d'applications parmi les plus répandues ainsi que les principes de génération des hautes pressions. Ces principes sont illustrés à travers la présentation d'une nouvelle génération de pompes électriques haute pression et du banc d'essais utilisé.

Les fondements de la génération des pressions sont exposés de façon à présenter les solutions technologiques de génération des hautes pressions au moyen de pompes à pistons. Les principes de fonctionnement des pompes classiques et des nouvelles pompes sont brièvement expliqués. Une dernière contribution porte sur la comparaison entre la conception classique et la nouvelle conception de pompes hautes pressions employées par la suite. Ces travaux comprennent deux conceptions de chambres de pompage associées à deux systèmes d'entraînement.

Conclusions

Une pompe hydraulique haute pression, employée dans la découpe des contours, est présentée an d'illustrer une application courante de la technologie à jet d'eau haute pression. Diérents cas d'utilisations sont dénis à partir de la classication des applications de l'usinage à jet d'eau et des postes de travail. Ces cas d'utilisation sont mis à prot lors de la validation de la méthodologie de modélisation et de la vérication des stratégies de commande.

Les postes de travail permettent de dénir les congurations des têtes de coupe. L'application industrielle permet quant à elle de spécier les cycles d'ouverture/fermeture des têtes de coupe. D'autre part, la notation standard utilisée lors de la description des circuits hydrauliques a été liv adaptée à la description des réseaux haute pression qui relient des pompes décentralisées aux têtes de coupe des diérents postes de travail. L'étude de la conception des chambres de pompage a permis d'établir une similitude entre les pompes hydrauliques et électriques. Les pompes électriques à pistons couplés ne parviennent pas à atteindre les hautes pressions nécessaires à l'usinage à jet d'eau alors que celles à pistons à simple ou à double eet permettent la génération de pressions allant jusqu'à 400 MPa. Bien qu'adaptées aux applications d'usinage à jet d'eau, ces pompes ont fait l'objet de peu de travaux.

Chapitre 2 Description du banc d'essais Le Chapitre 2 s'appuie sur la description du banc d'essais pour introduire les cas d'utilisation, spécier les topologies des réseaux et dénir les séquences de commutation d'ouverture/fermeture de la tête de coupe (patterns). Ce banc d'essais, spécialement construit pour ces recherches, permet la conguration des diérentes topologies des réseaux à modéliser. Quant aux cas d'utilisation, ils servent à vérier la qualité du modèle obtenu, à valider la stratégie de commande bas niveau proposée et à évaluer le mécanisme de gestion haut niveau.

Le concept proposé pour améliorer l'ecacité énergétique des installations à jet d'eau fait intervenir des pompes décentralisées au sein des réseaux haute pression. Les pompes hydrauliques conventionnelles, n'orant qu'un faible rendement, ne sont pas en mesure d'opérer dans l'intervalle de fonctionnement souhaité. Les pompes électriques utilisées dans cette thèse permettent de contourner ces dicultés en réduisant la complexité du système d'entraînement et en orant la possibilité d'opérer dans un point de fonctionnement quelconque. Ce chapitre présente donc une nouvelle conception de pompe électrique haute pression comportant deux pistons à eet simple indépendants. Il expose également les composants du système, sa structure de commande et le concept de synchronisation des deux pistons nécessaire au bon fonctionnement de la pompe. La synchronisation est réalisée au moyen d'un modèle de vilebrequin virtuel servant à calculer les trajectoires de référence nécessaires aux déplacements des deux pistons. Ces trajectoires sont obtenues à partir de la modélisation de la pression générée à l'intérieur d'une chambre de pompage.

Ce chapitre comporte trois sections. La Section 2.1 présente un état de l'art des pompes haute pression. La Section 2.2 expose la conguration du banc d'essais ainsi que les topologies des réseaux. Puis, elle décrit les cas d'utilisation ainsi que quelques séquences de commutation de têtes de coupe représentatives des applications industrielles. La Section 2.3 expose en détail la pompe électrique haute pression équipant le banc test, son principe de fonctionnement ainsi que la solution mise en ÷uvre pour gérer la synchronisation des pistons.

Contributions

Le chapitre débute par la présentation du banc d'essais qui permet d'interconnecter deux pompes électriques à deux postes de travail. Diérentes topologies de réseaux haute pression peuvent être alors congurées pour les besoins de la validation expérimentale des modèles mathématiques et des stratégies de commande. Ces topologies de réseaux haute pression et plusieurs cas d'utilisation sont dénis sur la base des classications établies au cours du chapitre précédent. Les congurations envisagées sont représentatives des cas d'utilisation rencontrés dans les applications industrielles. Ces topologies et ces cas d'utilisation serviront de l conducteur tout au lv long du manuscrit an d'illustrer la méthodologie de modélisation et la démarche de synthèse des lois de commande.

Le banc test est équipé de pompes électriques multi-pistons dont la conception est fondée sur la combinaison de deux unités de pompage. La description du principe de fonctionnement d'une unité de pompage constitue la première contribution de ce chapitre. Cette pompe électrique est modulable mais requiert un mécanisme de synchronisation du déplacement des pistons an de générer un ux constant. La présentation du mécanisme de synchronisation des pistons à l'aide d'un modèle constitue la deuxième contribution de ce chapitre. L'utilisation de ce mécanisme permet de considérer chaque pompe électrique comme une source continue de ux, dans un large domaine de fonctionnement, à condition que les uctuations de la pression qu'elle engendre soient minimisées. Pour ce faire, la synthèse d'une loi de commande bas niveau pour chaque pompe et d'un mécanisme de gestion haut niveau de plusieurs pompes est requise.

Conclusions

Un banc test a été construit dans le laboratoire pour réaliser l'identication paramétrique, la validation des modèles mathématiques et la vérication de la stratégie de commande. La description des topologies des réseaux et des congurations des têtes de coupe a fait appel à la notation standard. Les applications industrielles présentées au cours du Chapitre 1 ont servi à dénir les topologies des réseaux ainsi que les congurations des têtes de coupe employées lors de la validation. Plusieurs congurations reliant deux pompes haute pression à deux têtes de coupe indépendantes (deux postes de travail) ont été présentées. Ces cas d'utilisations seront employés ultérieurement pour eectuer des mesures sur le banc test et des simulations numériques.

Ce chapitre expose également la pompe électrique à deux pistons ainsi que le mécanisme nécessaire à la synchronisation des déplacements des pistons. Les déplacements précis des pistons passent par une phase de génération de trajectoires comportant un déphasage. Ces trajectoires sont calculées à partir des relations liées à la génération des hautes pressions au moyen de chambres de pompage couplées. La synchronisation apportée par ces trajectoires évite des interférences entre les unités de pompage et permet de produire un ux constant d'eau. Le suivi de ces trajectoires est réalisé à l'aide d'une commande en position des servomoteurs.

Ces trajectoires de mouvement sont ensuite améliorées par l'introduction d'une accélération croissante prenant en compte le jerk maximum admissible. Il est ainsi possible de maximiser le ux de sortie généré par la pompe. Cependant, le calcul des trajectoires faisables constitue un problème d'ordre élevé dont la résolution requiert une procédure d'optimisation. Les trajectoires issues de cette optimisation ont été implantées sur le banc test à l'aide de splines cubiques générant les références de position pour la commande de mouvement. L'introduction de trajectoires adaptatives minimisant les uctuations de pression au moment de la commutation entre les unités de pompage ouvre d'intéressantes perspectives. Ces trajectoires peuvent être dénies, par exemple, à l'aide de tables de correspondance (look-up tables). La trajectoire à suivre, selon le point de fonctionnement de la pression désiré, peut par exemple être générée par le biais d'une interpolation cubique. lvi Chapitre 3 Méthodologie de modélisation des réseaux haute pression Les futures installations à jet d'eau haute pression combinent plusieurs pompes décentralisées an d'élargir les réseaux haute pression de topologie arbitraire. Une démarche de modélisation exible et évolutive est indispensable à la conception de ces réseaux et/ou au développement d'une architecture appropriée de commande pour ces pompes. Cette démarche de modélisation doit à la fois prendre en compte les non linéarités intrinsèques au procédé et les variations des paramètres dans un domaine relativement grand (variant de 40 et 400 MPa). Elle doit aussi offrir un certain degré de exibilité en vue de la modélisation d'un réseau de topologie quelconque connectant un nombre quelconque de pompes et de postes de travail décentralisés.

Des environnements de simulation numérique, tels que Matlab et sa boîte à outils Simscape, Modelica ou AMESim, permettent de modéliser des systèmes hydrauliques standard grâce à certains composants hydrauliques de base disponibles dans les librairies. Malheureusement, les composants spéciques à la modélisation des réseaux haute pression ne sont pas toujours disponibles. La modélisation d'une installation à jet d'eau donnée devient alors complexe et très coûteuse en temps. Le manque de composants validés dans les intervalles d'application envisagés limite par ailleurs la réutilisabilité du modèle obtenu. An de pallier les dicultés mentionnées, une démarche de modélisation de futures installations recongurables doit orir divers composants haute pression accompagnés d'une procédure de modélisation des diérentes topologies du réseau. Les paramètres variables des composants doivent être décrits au moyen de modèles mathématiques appropriés et expérimentalement identiés à l'aide du banc d'essais. Le paramétrage de ces composants doit être valable dans un large domaine de fonctionnement.

Le Chapitre 3 est dédié au développement d'une méthodologie de modélisation fondée sur la notion de graphes. Les équations à dérivées partielles gouvernant la dynamique des uides sont approchées au moyen de diérences nies en considérant le ux de uide à l'équilibre. Cette démarche aboutit à l'obtention d'un modèle à paramètres localisés où chaque section de tuyauterie est représentée par un segment homogène. La connexion de ces segments au moyen d'un graphe ore une représentation exible et évolutive d'un réseau haute pression de topologie arbitraire. Une analyse en simulation permet d'évaluer le temps de calcul et la précision des prédictions en fonction de la discrétisation spatiale et des approximations eectuées. L'originalité de l'approche proposée, au regard des méthodologies présentes dans la littérature, réside dans l'utilisation d'un module d'élasticité isostatique dépendant de la pression et relié à la densité de uide elle aussi dépendante de la pression. Ce couplage améliore la précision du modèle dans un large domaine de fonctionnement allant de 40 à 400 MPa.

Ce chapitre comporte quatre sections. La Section 3.1 propose un état de l'art des approches de modélisation des systèmes haute pression et de la théorie des graphes. La Section 3.2 expose les principes fondamentaux associés à la dynamique d'un uide. Le recours à ces principes permet d'obtenir un modèle à paramètres distribués. Les uctuations de pression dans un tuyau sont étudiées au moyen de simulations numériques réalisées à l'aide de ce modèle. La Section 3.3 décrit la démarche de modélisation générique des réseaux haute pression réalisée au moyen de graphes. Elle introduit les intervalles des variations paramétriques qui seront pris en considération pour obtenir un modèle à paramètres localisés. Enn, la Section 3.4 présente les paramètres dépendant de la pression (associés à certains composants) et leur procédure d'identication expérimentale.

lvii

Contributions

Ce chapitre propose une méthodologie de modélisation qui permet de modéliser, à l'aide de graphes, des réseaux haute pression de topologies diverses. Les composants du modèle sont considérés comme des segments homogènes, chaque segment représentant un état local de la pression à partir du principe de conservation du moment et de la continuité du uide. La méthodologie de modélisation proposée introduit, dans les équations gouvernant la physique du système, une densité de uide ainsi qu'un module d'élasticité isostatique qui dépendent de la pression. Elle permet ainsi d'améliorer la précision du modèle dans un large domaine de fonctionnement. Les segments sont par la suite interconnectés à l'aide de graphes tout au long du trajet parcouru par le uide. Chaque n÷ud du graphe représente un état de la pression et chaque branche alloue un ux de uide aux interconnexions. Les graphes orent le degré de exibilité nécessaire à la modélisation des réseaux de topologies diérentes. Cette démarche conduit à l'obtention de modèles à paramètres localisés qui permettent d'approximer le caractère non linéaire des réseaux haute pression.

La principale contribution du chapitre porte sur le développement d'une démarche de modélisation, à l'aide de graphes, de réseaux haute pression. L'originalité de l'approche proposée réside dans la prise en compte dans les équations gouvernant la dynamique du uide de deux paramètres variant, à savoir, le module de compressibilité et la masse volumique. Les équations ainsi modiées sont alors associées à un graphe. La exibilité et la versatilité inhérentes aux graphes rendent possible la modélisation des réseaux à topologie variable. Ce chapitre aborde également l'identication, dans un large domaine de fonctionnement, de paramètres physiques et de caractéristiques du uide dépendant de la pression. Cette procédure d'identication permet la modélisation de réseaux haute pression dans un large intervalle sans avoir à réitérer l'étape d'identication paramétrique initiale. D'un point de vue pratique, une boîte à outils de simulation facilitant la description des réseaux haute pression est développée. Ce simulateur prend en compte la dépendance des caractéristiques du uide (module de compressibilité, masse volumique) à la pression. Tout déplacement de ux de uide est modélisé comme une excitation externe et les déplacements des pistons sont modélisés comme une modication du volume du uide.

Conclusions

Le comportement dynamique du uide, représentant certains phénomènes de propagation dans une section donnée d'un tuyau, est présenté. Puis une méthodologie de modélisation d'un réseau haute pression est développée à partir d'un graphe. L'idée est d'associer des sections de tuyau à des segments homogènes. Ces segments sont alors interconnectés à l'aide d'un graphe de façon à reproduire la topologie du réseau. Chaque segment représente un état local de la pression et ses interconnexions sont caractérisées en considérant un ux de uide stationnaire.

Cette méthodologie de modélisation est dérivée des équations fondamentales de la dynamique des uides en approximant un modèle distribué à l'aide de diérences nies. Cette démarche de modélisation conduit à un ensemble d'équations diérentielles qui représentent le réseau haute pression sous la forme d'un modèle à paramètres localisés. L'erreur introduite par l'approximation proposée est évaluée par le biais de diverses simulations en considérant la même section de tuyau à modéliser. La segmentation du réseau en sections doit satisfaire le compromis classique lviii précision-complexité du modèle. La proposition d'une procédure optimale de segmentation ouvre d'intéressantes perspectives. Toutefois, les résultats obtenus montrent qu'une précision acceptable pour les applications industrielles est atteinte en considérant des segments d'un mètre de long.

Les paramètres intervenant dans le modèle ont fait l'objet d'une identication, dans un large domaine de fonctionnement allant de 0 à 400 MPa, à l'aide des mesures issues du banc d'essais. Considérer ce domaine de fonctionnement permet d'enrichir les données disponibles dans la littérature jusque-là limitées à 100 MPa. Un outil de modélisation générique, sous la forme d'une boîte à outils, est proposé an de modéliser des réseaux haute pression de diverses topologies et de complexité croissante. Cet outil épargne le recours à des essais expérimentaux spéciques pour valider chaque modèle. Les modèles validés servent, par exemple, à tester les stratégies de commande bas niveau proposées par la suite. L'extension de ce travail à l'identication d'autres composants du réseau constitue une perspective à envisager.

Chapitre 4 Application et validation de l'approche de modélisation

Le développement des futures installations passe par une phase de modélisation du réseau haute pression envisagé. Cette phase de modélisation peut être eectuée à l'aide de la démarche de modélisation développée. Les paramètres dépendant de la pression et les caractéristiques du ux de uide ont été identiés à partir des mesures issues du banc d'essais. L'utilisation de cette boîte à outils Matlab/Simulink simplie la modélisation des futures installations et évite les coûts engendrés par des campagnes d'essais spéciques. Le Chapitre 4 illustre l'utilisation de cette boîte à outils lors de la modélisation des réseaux haute pression de topologies diverses et compte tenu des cas d'utilisation spéciés. Les modèles obtenus sont soumis à une validation exhaustive à partir des mesures issues du banc test. Cette validation montre que les modèles restent précis dans un large domaine de fonctionnement. Ils peuvent être facilement adaptés an de prendre en compte les modications du réseau sans remettre en question l'identication paramétrique. Le temps nécessaire à la mise au point d'un modèle pour les futures installations industrielles est ainsi réduit.

Ce chapitre comporte trois sections. La Section 4.1 présente les objectifs visés lors de la démarche de modélisation. Au cours de la Section 4.2, cette démarche est appliquée à la modélisation de composants souvent rencontrés puis à la modélisation de réseaux complexes. Enn, les modèles obtenus sont validés expérimentalement au cours de la Section 4.3.

Contributions

La démarche de modélisation est illustrée en considérant diverses congurations du banc d'essais. Dans un premier temps, elle est appliquée à une pompe électrique haute pression, composée de deux pistons indépendants interconnectés, reliée à une tête de coupe simple. Dans un deuxième temps, elle est appliquée aux congurations plus complexes des réseaux précédemment dénis. D'autre part, un modèle à paramètres localisés prenant en compte les incertitudes paramétriques fait l'objet d'une discussion. Le modèle de premier ordre ainsi obtenu sert de point de départ à des travaux de recherche sur le développement de stratégies de commande robustes bas niveau.

Le nombre important d'expériences conduites pour valider les modèles proposés constitue une contribution importante de ce chapitre. En eet, tous les modèles obtenus sont validés exlix périmentalement à partir des mesures issues du banc test. Diérentes congurations du banc test, combinant deux pompes électriques et deux têtes de coupe, ont été mises en ÷uvre. Ces congurations correspondant aux cas d'utilisation déjà dénis reproduisent certaines applications industrielles.

Les performances des modèles obtenus sont comparées à celles obtenues à partir des approches de modélisation proposées dans la littérature. Ces résultats montrent qu'une compressibilité de uide variable permet de réduire les erreurs de modélisation et améliore la précision des prédictions. La précision de la simulation est évaluée à partir de divers résultats expérimentaux qui considèrent des points de mesure placés à diérents endroits du réseau. Diérents cycles de commutation de têtes de coupe, agissant comme des perturbations, sont également utilisés lors de ces validations.

L'approche de modélisation est concrétisée sous la forme d'une boîte à outils qui permet d'élaborer facilement en Matlab/Simulink un modèle de réseau. Cette boîte à outils représente un atout pour les industriels du secteur qui cherchent à modéliser de futures infrastructures haute pression en un temps réduit sans avoir besoin d'un paramétrage individuel.

Conclusions

Ce chapitre ore un cadre de modélisation, matérialisé par une boîte à outils, à la description des réseaux haute pression généralisés. La boîte à outils développée permet aux industriels partenaires du projet de concevoir et d'optimiser les futures installations à jet d'eau. An d'illustrer les performances de cette boîte à outils, 6 topologies de réseau ont été étudiées. Chaque modèle obtenu a fait l'objet d'une validation expérimentale en considérant 2 pressions de fonctionnement, 2 buses de diamètre dièrent et 2 séquences de commutation des têtes de coupe.

Dans ce chapitre, les diérents modèles sont validés expérimentalement à l'aide des données issues du banc d'essais. Ces résultats montrent qu'une modélisation considérant une densité de uide constante génère une erreur statique de la pression alors qu'un module d'élasticité isostatique constant est à l'origine d'un mauvais transitoire de la réponse. La prise en compte d'une compressibilité variable de uide permet d'améliorer la précision du modèle dans un domaine de fonctionnement compris entre 100 MPa et 400 MPa. Les diérents cas d'utilisation sélectionnés, représentatifs des congurations industrielles, montrent les aptitudes du modèle à caractériser la propagation de la pression en diérents points du réseau. Les résultats de validation en diérents points du réseau prouvent l'excellente précision obtenue grâce à la démarche de modélisation proposée. Une augmentation des résidus est cependant constatée quand le banc test opère à un faible taux de pompage. Une prise en compte des pertes occasionnées par les frictions pour de petites vitesses du uide permettrait d'améliorer le modèle dans ce cas de fonctionnement.

La méthodologie proposée rend possible la modélisation exible des réseaux de diverses topologies avec un eort raisonnable. Cet avantage est particulièrement utile à la modélisation des réseaux de diverses topologies, à l'optimisation d'une future installation et à l'évaluation des stratégies de commande. Par ailleurs, les modèles proposés de complexité croissante seront par la suite utilisés an d'évaluer le mécanisme haut niveau dédié à la gestion des pompes décentralisées.

Les perspectives envisagées portent principalement sur la validation des modèles en considérant d'autres topologies, un dysfonctionnement du clapet anti-retour ayant nui à la nalisation lx de toutes les expériences de validation initialement prévues. Remarquons que la commutation du clapet anti-retour est très sensible à tout diérentiel de la pression. L'introduction d'un modèle de commutation comportant un mécanisme d'hystérésis peut éventuellement réduire les erreurs résiduelles de simulation. Il a également été constaté des pertes de pression et/ou des dépassements quand les trajectoires des pistons de la pompe ne sont pas correctement synchronisées. Les modèles obtenus, en reproduisant ce comportement, peuvent servir à optimiser les trajectoires et à améliorer ainsi la génération de la pression.

Chapitre 5 Stratégies de commande robustes bas niveau

Le fonctionnement des pompes électriques couplées requiert le développement d'une stratégie de commande bas niveau en mesure de limiter l'impact de certains phénomènes indésirables. En eet, toute uctuation de la pression impacte directement la qualité de la coupe par jet d'eau. Or, les têtes de coupe introduisent naturellement des uctuations de la pression lors d'un cycle inconnu de commutation d'ouverture/fermeture nécessaire à la tâche d'usinage. Les uctuations engendrées par une consommation variable d'eau doivent par conséquent être compensées à l'aide des pompes haute pression disponibles. Chaque pompe, placée à un endroit quelconque du réseau, doit rejeter les uctuations locales de pression sans pour autant déstabiliser l'ensemble du réseau. Bien évidemment, la compensation des uctuations de pression doit être réalisée en limitant l'eort de commande. Ces pompes doivent donc opérer dans un large domaine de fonctionnement an d'atténuer les perturbations exogènes. Cependant, le large domaine de fonctionnement de la pression et les congurations variables des postes de travail induisent des comportements non linéaires, des incertitudes paramétriques importantes ainsi que des variations paramétriques aectant le fonctionnement global de l'installation. Pour toutes ces raisons, l'utilisation des lois de commande robustes bas niveau s'impose et nécessite la proposition d'une description mathématique appropriée à leur synthèse.

Le Chapitre 5 est dédié à la conception d'une stratégie de commande robuste bas niveau pour chaque pompe haute pression décentralisée. Lors de la conception de cette stratégie de commande, les cycles inconnus de commutation des têtes de coupe et les interactions entre les pompes voisines sont considérés comme des perturbations exogènes agissant dans une sous-section du réseau. Les non linéarités et les incertitudes paramétriques doivent aussi être prises en compte an de garantir la stabilité et les performances robustes dans un large domaine de fonctionnement. An de mener une analyse de sensibilité de la variation des paramètres, un modèle perturbé linéarisé décrivant une sous-section de réseau est proposé. Plusieurs sortes de description des incertitudes sont envisagées puis comparées an de choisir le modèle de synthèse le plus pertinent. Le modèle sélectionné sert de point de départ à la synthèse d'une loi de commande robuste H ∞ . Cette dernière doit satisfaire un cahier des charges formalisé à partir des besoins exprimés par les industriels partenaires du projet. Lors d'une synthèse H ∞ classique, les exigences en termes de rejet de perturbation et d'eort de commande visant à éviter des saturations sont traduites sous la forme de gabarits fréquentiels. D'autres gabarits fréquentiels peuvent aussi être introduits an de traduire les dynamiques des signaux d'entrée, par exemple, le bruit de mesure et la commutation des perturbations. L'étude proposée dans ce chapitre permet, après avoir déni le modèle de synthèse, de dresser la forme de ces gabarits pour poser le problème H ∞ . Une pompe électrique haute pression est utilisée comme cas d'étude pour dénir un schéma d'interconnexion standard. Tous les composants du système traité sont alors combinés pour construire ce schéma qui dénit le problème d'optimisation H ∞ à résoudre. lxi Ce chapitre est découpé en quatre sections. La Section 5.1 établit un état de l'art des techniques de commande robuste et met l'accent sur les approches dites décentralisées, distribuées et en réseau. La Section 5.2 expose les principes fondamentaux associés à la théorie de la synthèse du régulateur H ∞ . La Section 5.3 présente le système à commander ainsi que son modèle non linéaire. Une étude des variations paramétriques conduit à la proposition d'un modèle linéarisé de complexité réduite. Ce modèle de premier ordre est issu d'une linéarisation, autour d'un point de fonctionnement, d'un modèle de pompe haute pression interconnectée à une sous-section inconnue du réseau. Le point de fonctionnement, autour duquel la linéarisation est eectuée, est considéré comme un paramètre incertain additionnel. La Section 5.4

Contributions

Ces travaux visent à eectuer une synthèse de commande robuste H ∞ pour une pompe électrique haute pression. Les réseaux haute pression étant aectés par un grand nombre de perturbations, la stratégie de commande à concevoir doit orir un degré élevé de robustesse. La commande H ∞ est une technique qui se révèle bien adaptée à la résolution de ce type de problème de commande. La formalisation du problème H ∞ et l'obtention d'une représentation du système généralisée interconnectant une pompe décentralisée à des têtes de coupe gurent parmi les principales contributions de ce chapitre.

Une description du système assortie d'un modèle des incertitudes qui englobe les variations paramétriques est nécessaire à la synthèse d'une commande H ∞ adaptée aux applications à jet d'eau. Dans cette perspective, ce chapitre propose une analyse exhaustive des variations des paramètres et des descriptions possibles des incertitudes. La dernière contribution du chapitre est liée à la dénition appropriée des gabarits et des fonctions de pondération nécessaires à la synthèse H ∞ . D'un point de vue méthodologique, une fusion des signaux d'entrée au moyen des gabarits joints est développée en vue de réduire la dimension du problème d'optimisation à résoudre lors de la synthèse H ∞ .

Conclusions

Une discussion portant sur la synthèse d'une stratégie de commande robuste et décentralisée pour les pompes électriques haute pression est proposée dans ce chapitre. L'objectif est de synthétiser un régulateur bas niveau pour chaque pompe installée dans un réseau de topologie inconnue. Pour ce faire, le réseau est découpé en portions, en sous-sections. La pression dans chaque sous-section est alors régulée par une pompe haute pression associée. En émettant l'hypothèse d'une synchronisation parfaite des pistons de chaque pompe, un modèle de premier ordre est obtenu an de décrire la génération de la pression dans une sous-section. Ce modèle représente les principales incertitudes paramétriques. Les autres éléments présents dans l'installation ont été modélisés à l'aide de fonctions de transfert établies à partir des valeurs nominales fournies par les constructeurs. lxii Dans un deuxième temps, la sensibilité paramétrique du modèle a été évaluée. Cette étude montre qu'une tête de coupe avec une buse de section décroissante est à l'origine d'une augmentation du gain entre le signal d'entrée et celui de sortie en basses fréquences. Une buse complètement fermée engendre un phénomène d'intégration pur. L'augmentation de la pression de fonctionnement diminue le gain entre l'entrée et la sortie. D'autre part, l'augmentation de la section de la buse n'aecte pas le gain entre la perturbation et la pression de sortie en basses fréquences. En conclusion, une pompe haute pression est en capacité de réguler la pression dans une sous-section du réseau en rejetant les perturbations induites par le séquencement inconnu d'ouverture/fermeture des têtes de coupe.

Dans cette étude, les incertitudes paramétriques sont initialement modélisées sous la forme d'incertitudes structurées et non structurées. La forme des incertitudes non structurées estnalement retenue à l'issue de l'étude comparative réalisée. L'incertitude inverse additive fournit des résultats satisfaisants dans l'intervalle des incertitudes paramétriques étudié. Elle permet d'introduire un signal associé aux perturbations en utilisant une transformation linéaire fractionnelle. Les paramètres incertains peuvent ainsi être séparés du modèle nominal du système an de proposer des gabarits appropriés à la synthèse de la loi de commande. Par ailleurs, une incertitude inverse multiplicative de sortie ore des résultats similaires. L'une des pistes envisagées pour améliorer la modélisation des incertitudes consiste à utiliser une description des incertitudes à l'aide des facteurs premiers. Choisir une incertitude complexe non structurée présente des avantages lors de la synthèse du contrôleur. Elle permet de simplier la procédure visant à maximiser la dimension du sous-réseau, mais introduit un niveau de conservatisme supplémentaire lors de la conception de la loi de commande.

Chapitre 6 Synthèse des lois de commande robustes bas niveau

Les exigences à respecter par le régulateur dans le domaine temporel ont été étudiées au cours du Chapitre 5. Des fonctions de pondération appropriées dans le domaine fréquentiel ont donc pu être déduites. Le système augmenté obtenu prend en compte les perturbations agissant sur le système, la description des incertitudes paramétriques et les exigences à respecter par le régulateur. Ce système augmenté est le point de départ de la synthèse H ∞ du régulateur. Les travaux présentés au cours de ce chapitre portent principalement sur l'agrégation de fonctions de pondération en fusionnant leurs valeurs singulières à l'aide de la norme euclidienne. Cette fusion permet, en diminuant la dimension du problème d'optimisation H ∞ , de réduire le temps de calcul. Cette démarche a été utilisée avec succès lors de la synthèse d'un retour d'état d'ordre réduit. An de satisfaire les contraintes industrielles, la même démarche a également été appliquée à la synthèse d'un régulateur de type PI suivant une synthèse H ∞ structurée.

Il convient de remarquer que la longueur de chaque sous-section, dans laquelle la pompe doit atténuer les uctuations de pression, est inconnue car la topologie de l'installation est a priori inconnue. Il faut par conséquent déterminer la longueur maximale de la sous-section pour laquelle la pompe garantit le niveau d'atténuation souhaité. La longueur admissible de la sous-section est considérée comme un paramètre incertain qui dépend d'une part, de l'eort maximum que la pompe peut fournir et d'autre part, des performances de rejet de perturbation. Ces deux aspects étant nécessaires au maintien d'un niveau de qualité de l'usinage à jet d'eau, une analyse de robustesse est proposée pour établir la dimension de la sous-section dans laquelle la pompe garantit les performances désirées. La structure du régulateur doit être compatible avec les exigences de mise en ÷uvre formulées par les industriels. Cet aspect est pris en considération grâce lxiii à une synthèse structurée du régulateur. La structure du régulateur retenue, de type PI, permet de répondre aux exigences du cahier des charges en matière de stabilité et de performances dynamiques tout en maximisant la largeur de la sous-section. La synthèse est réalisée en appliquant une procédure itérative qui maximise l'intervalle de l'incertitude et synthétise à chaque itération un régulateur PI quasi-optimal. La procédure ∆-K proposée préserve l'ordre du système généralisé. Elle est alors en adéquation avec une description non structurée des incertitudes et avec une synthèse structurée du régulateur.

La mise en ÷uvre des futures installations à jet d'eau haute pression nécessite une commande robuste bas niveau pour les pompes haute pression, chaque pompe stabilisant une portion du réseau. Les portions correspondent à des sous-sections qui peuvent être alors interconnectées en vue de former un réseau haute pression de pompes décentralisées. Cependant, le couplage de multiples pompes dans le réseau peut conduire à une augmentation de la consommation d'énergie et limiter la stabilité globale du réseau si les pompes interfèrent les unes avec les autres. La stabilité et les performances robustes des pompes haute pression distribuées doivent être alors validées au moyen de simulations suivies de validations expérimentales. Les modèles des réseaux haute pression proposés ont permis de tester le régulateur par retour d'état réduit ainsi que le régulateur quasi-optimal PI. D'un point de vue expérimental, un régulateur PI a été implanté dans le banc test an de piloter l'une des pompes disponibles.

Ce chapitre comprend quatre sections. La Section 6.1 présente les objectifs de la synthèse d'un régulateur robuste. Le système généralisé perturbé et les fonctions de pondération sont combinés au cours de la Section 6.2 an d'obtenir un système généralisé. En appliquant une transformation linéaire fractionnaire inférieure, ce système est transformé en forme standard P-K. Un régulateur par retour d'état réduit est alors synthétisé à partir des fonctions de pondération jointes. Ses performances sont ensuite comparées à celles obtenues à l'aide d'un régulateur de type PI synthétisé à partir d'une démarche H ∞ structurée. Une synthèse sous-optimale H ∞ est également étudiée à partir d'une procédure itérative ∆-K qui permet d'ajuster les fonctions de pondération. Cette procédure aboutit à la synthèse d'un régulateur qui vérie le cahier des charges tout en maximisant l'intervalle des incertitudes liées à la taille inconnue d'une soussection du réseau. Dans la Section 6.3, les performances du régulateur sont évaluées en simulation en considérant diérentes topologies modélisées à l'aide de la méthodologie par graphes proposée. Le comportement des régulateurs obtenus, en présence de non linéarités et de variations paramétriques, peut ainsi être évalué. Lors de la phase de validation expérimentale menée dans la Section 6.4, un régulateur structuré de type PI est implanté dans le banc test. L'étude expérimentale est aussi reproduite à l'aide de modèles de validation en boucle fermée an de tester les stratégies de commande. Elle permet de vérier la bonne implantation du régulateur ainsi que la démarche de modélisation au moyen de la boîte à outils développée. La boîte à outils ainsi validée permet de mener des études poussées en simulation, au-delà des limites physiques du banc test, qui serviront à la conception de futures installations.

Contributions

Une contribution de ce chapitre porte sur la synthèse H ∞ d'une loi de commande robuste pour une pompe haute pression qui tient compte des variations paramétriques et des perturbations exogènes. Lors de la synthèse, des gabarits fréquentiels sont utilisés an de dénir les intervalles de variation des signaux d'entrée. Les performances dynamiques et la pénalisation de l'eort de commande sont quant à elles prises en compte en introduisant des fonctions de transfert. lxiv Cette démarche conduit à la construction d'un modèle généralisé englobant la modélisation non structurée des incertitudes et la description fréquentielle des signaux d'entrée. An de diminuer l'ordre de ce modèle, une fusion des signaux d'entrée par le biais de gabarits joints est introduite. Si cette démarche permet de réduire la dimension du problème H ∞ à résoudre, elle peut cependant augmenter le degré de conservatisme de la solution. Enn, la résolution du problème d'optimisation conduit à un premier régulateur par retour d'état. Ce régulateur sert par la suite de point de comparaison avec les autres régulateurs structurés synthétisés.

La synthèse d'un régulateur structuré constitue une autre contribution du chapitre. L'approche proposée conduit à l'obtention d'un régulateur de type PI aisé à implanter dans les applications industrielles. Ce régulateur doit garantir la stabilité et permettre de réguler la pression dans une sous-section d'une longueur donnée. L'approche H ∞ est également mise à prot pour déterminer la longueur maximale de la sous-section du réseau associée à la pompe. L'analyse des variations paramétriques et le modèle d'incertitudes qui en résulte conduisent à l'obtention de marges robustes de performances dans lesquelles la stabilité et les performances sont garanties. La mise en ÷uvre d'une procédure itérative de type ∆-K permet d'obtenir un régulateur quasi optimal de type PI tout en maximisant les limites de l'intervalle des incertitudes. Cette procédure se révèle très pratique et fournit un moyen d'eectuer une synthèse quasi optimale d'un régulateur H ∞ .

Plusieurs campagnes de simulation sont menées à l'aide de modèles de validation complexes an d'établir une comparaison entre les performances obtenues avec la stratégie par retour d'état et celles obtenues avec le régulateur structuré de type PI. Une première synthèse du régulateur PI est testée en conditions réelles grâce au banc d'essais. Ce dernier a été conguré pour une seule pompe puis pour l'interconnexion de deux pompes. Toutes les congurations proposées utilisent deux têtes de coupe indépendantes an de générer des uctuations de pression. L'approche de synthèse du régulateur bas niveau est validée avec succès. Toutefois, les expériences menées montrent l'apparition de couplages dynamiques entre les pompes interconnectées et révèlent la nécessité de mettre en place un mécanisme de gestion haut niveau an de coordonner les eorts fournis par les pompes.

Conclusions

Dans ce chapitre, la démarche de synthèse H ∞ d'un régulateur pour une pompe haute pression est validée. Les fonctions de pondération introduites permettent de prendre en compte les performances souhaitées et l'eort de commande. Le système interconnecté est mis en forme à l'aide d'une transformation linéaire fractionnaire de façon à séparer le régulateur des autres composants du système. Un système généralisé nécessaire à la synthèse H ∞ de la loi de commande est ainsi obtenu. L'utilisation des gabarits joints permet de regrouper plusieurs signaux d'entrée du système généralisé réduisant la dimension du problème H ∞ à résoudre et l'eort de calcul. L'application de cette démarche au cas spécique de la commande d'une pompe haute pression conduit à l'obtention d'un régulateur par retour d'état. Les résultats observés mettent en évidence des similitudes entre les performances du régulateur d'état d'ordre réduit (plus simple à implanter) et celles du régulateur d'état de plein ordre.

Par ailleurs, la procédure d'optimisation formulée conduit également à l'obtention d'un régulateur de type PI. Ce dernier a été modié en régulateur par retard de phase en introduisant un ltre passe bas. La structure PI de complexité réduite est simple à implanter par les industriels. Les performances des régulateurs par retour d'état et PI sont comparées à partir de simulations lxv numériques considérant diérentes topologies de réseaux. Les modèles mathématiques de validation sont issus de la méthodologie de modélisation proposée. Le régulateur PI n'est cependant pas en mesure de fournir le même niveau de performances que le régulateur par retour d'état. Les performances obtenues sont proches à condition de limiter la longueur de la sous-section du réseau. D'autre part, la structure par retard de phase introduit un pôle additionnel qui judicieusement placé permet d'obtenir des performances similaires à celles fournies par le régulateur par retour d'état. Les propriétés du régulateur par retour d'état sont alors approchées dans une bande de fréquences donnée sans impacter la longueur de la sous-section.

Chapitre 7 Perspectives. Vers la proposition d'un mécanisme de gestion haut niveau

La première partie de ce manuscrit de thèse est consacrée à la proposition d'une méthodologie de modélisation, basée sur des graphes, pour des réseaux haute pression. Elle rend possible la modélisation du comportement des pompes décentralisées interconnectées à des postes de travail spatialement distribués dans une installation. Au cours de la deuxième partie, une stratégie de commande robuste bas niveau pour une pompe haute pression est élaborée à l'aide de cette modélisation. La commande bas niveau permet à une pompe de réguler la pression dans une portion de sous-section du réseau. Toutefois, l'interconnexion entre les pompes dans un réseau requiert un mécanisme haut niveau de gestion. Enn, le Chapitre 7 expose quelques travaux préliminaires dédiés à la conception de ce mécanisme de gestion haut niveau des pompes décentralisées dans la perspective d'optimiser la consommation d'énergie des installations à jet d'eau.

La mise en ÷uvre d'une stratégie de commande bas niveau pour une seule pompe électrique haute pression suscite l'intérêt des industriels. La solution proposée est facilement implantable car seules sont nécessaires des informations locales. Toutefois, les simulations réalisées montrent que des pompes décentralisées connectées dans un réseau présentent des taux de fonctionnement lxvi arbitraires si aucun mécanisme de gestion n'intervient. En eet, chaque pompe atteint un taux de pompage diérent suivant la topologie du réseau et le cycle d'ouverture/fermeture des têtes de coupe. Ce comportement peut conduire à la détérioration prématurée des pompes et dégrader leurs capacités de rejet des perturbations.

Ce chapitre propose un court état de l'art des approches classiques de commande hiérarchisée et montre les limites de l'utilisation directe de ces approches dans un contexte d'usinage à jet d'eau. Les travaux préliminaires exposés dans ce chapitre initient les travaux de recherche dédiés à la commande de plusieurs pompes interconnectées. Ces travaux ont pour objet la conception d'un mécanisme de gestion en mesure d'équilibrer les taux de fonctionnement des pompes haute pression sans qu'aucune connaissance a priori sur leur position dans le réseau ne soit nécessaire. Ce chapitre présente quelques résultats préliminaires en simulation sur la gestion des pompes distribuées à l'aide d'une approche dite par consensus moyen. Le mécanisme de gestion des pompes combiné à une commande distribuée ore un bon compromis entre la quantité d'informations requise sur les sous-sections voisines et les eets de retard de transmission, de perte de paquets, etc.

Diérentes approches peuvent être envisagées pour la gestion des pompes électriques en fonction des informations disponibles sur les pompes voisines. Ce chapitre présente l'utilisation de l'approche par consensus moyen pour équilibrer les taux de fonctionnement des pompes distribuées. L'équilibrage des pompes couplées dans une installation assure la convergence, dans un temps de réaction acceptable, de toutes les pompes vers un taux moyen de pompage variant dans le temps. Dans l'approche proposée, aucun système centralisé de gestion faisant appel à une communication entre les pompes n'est considéré. Par conséquent, les pompes peuvent continuer à opérer, avec un taux de pompage sous-optimal, et à alimenter les postes de travail même si la communication est défaillante. Les simulations montrent les capacités du mécanisme de gestion proposé à équilibrer des pompes interconnectées vers un taux de pompage moyen. Ce taux reète la consommation moyenne d'énergie de l'ensemble de l'installation.

Le chapitre est composé de quatre sections. La Section 7.1 propose un état de l'art des méthodes de gestion distribuées, en particulier de la commande prédictive (Model Predictive Control, MPC ), de l'allocation de commande (Control Allocation) et de la commande coopérative (Cooperative Control ). La formulation du problème proposée dans la Section 7.2 souligne la nécessité d'introduire un mécanisme de gestion. La méthode de consensus moyen distribué est adoptée an d'estimer le taux moyen de pompage des agents interconnectés, c'est-à-dire des pompes. Enn, les objectifs associés à une gestion par consensus moyen et à la spécication du canal de communication à l'aide d'une description par graphes sont introduits. Une approche intégrée, développée au cours de la Section 7.3, permet à chaque pompe haute pression de converger vers un taux moyen. Deux algorithmes de gestion haut niveau sont évalués en simulation dans la Section 7.4. Cette étude fournit les premières simulations pour équilibrer les charges des pompes distribuées à l'aide d'une approche par consensus moyen dynamique.

Contributions

Dans ce chapitre, la méthode par consensus moyen dynamique est appliquée à la gestion de pompes électriques haute pression. Chaque pompe peut être assimilée à un agent qui fournit une estimation locale du taux moyen de pompage en échangeant l'estimation locale avec les pompes voisines. Chaque pompe disposant d'une commande bas niveau robuste, une description haut lxvii niveau des agents (les pompes) à l'aide de dynamiques discrètes peut être proposée. Dans ce chapitre, les algorithmes sont développés en temps continu en considérant un moyen de communication parfait (les délais de transmission sont par exemple négligés) pour le partage des estimations locales entre les agents voisins. L'approche par consensus moyen dynamique utilisée pour estimer le taux moyen de pompage des pompes distribuées constitue l'une des contributions de ce chapitre. Les pompes haute pression connectées dans un réseau restent ottantes si aucune action de rétroaction n'est donnée par un algorithme de gestion. La commande bas niveau proposée maintient les pressions de sortie autour d'une référence globale en présence de perturbations occasionnées par l'ouverture/fermeture des têtes de coupe. Ceci conduit à la génération de diérents taux de ux pour chaque pompe en fonction de sa position dans le réseau.

La proposition d'un algorithme d'équilibrage distribué considérant le réseau haute pression comme moyen de communication constitue une autre contribution du chapitre. Il permet d'ajuster directement la référence de la pression pour chaque agent-pompe de façon à garantir la convergence de chaque pompe vers une valeur moyenne. Cette approche fait appel à la stratégie de commande bas niveau déjà synthétisée ainsi qu'au couplage entre les pompes pour équilibrer leurs taux individuels et ce, sans avoir à ajouter de stratégie de commande.

Les résultats de simulation présentés dans ce chapitre combinent tous les résultats des chapitres précédents. La méthodologie de modélisation par graphe permet de représenter des installations à jet d'eau complexes impossibles à reproduire à l'aide du banc d'essais. La stratégie de commande bas niveau, appliquée à chaque pompe, permet de placer chaque pompe (chaque agent) dans une position arbitraire du réseau. Grâce à cette commande robuste, les uctuations de pression sont atténuées. Enn, le mécanisme de gestion haut niveau par consensus dynamique moyen ajuste le fonctionnement de chaque pompe de façon à ce qu'elle opère à un taux moyen de pompage variant dans le temps.

Conclusions

Un réseau de pompes haute pression distribuées peut être assimilé à un système sur-actionné. Dans ce chapitre, plusieurs solutions au problème de commande de ce type de systèmes sont d'abord exposées. L'équilibrage du taux de charge au moyen d'une allocation de commande adaptative est une solution possible pour réaliser la synchronisation des pompes. La procédure d'adaptation est nécessaire car la distribution optimale de l'eort de commande peut varier en fonction du cycle d'ouverture/fermeture des têtes de coupe et du nombre de pompes disponibles. Toutefois, la mise en place d'une telle procédure d'optimisation nécessite de connaître les informations sur le fonctionnement de toutes les pompes. Or, en pratique, cette connaissance peut s'avérer très compliquée à obtenir dans les installations industrielles. Une stratégie coopérative MPC (Model Predictive Control ) permet également d'aborder la gestion des régulateurs distribués. Cette approche est fondée sur l'utilisation de modèles locaux appropriés pour eectuer le suivi des trajectoires de chaque agent. Fournir un modèle pour chaque installation à jet d'eau constitue la première limite de cette approche. La deuxième limite est liée au besoin d'estimer les séquences de commutation des têtes de coupe qui aectent le comportement dynamique du réseau de distribution de pression. Cette discussion met en lumière les limites des approches classiques et justie le choix d'une approche par consensus moyen dynamique. Cette dernière permet en eet de garantir un eort moyen de commande des pompes distribuées en présence d'une consommation variable d'eau. Une distribution équilibrée au moyen d'un consensus moyen constitue par conséquent une piste intéressante à explorer pour traiter le problème de commande lxviii de futures installations à grande échelle. L'approche par consensus est adaptée à des topologies de communication inconnues et à un nombre variable d'agents. Si la communication entre un agent est rompue, les agents restant ne convergent pas forcément vers une moyenne globale mais l'installation continue toujours à opérer. Bien évidemment, en pratique, d'autres contraintes doivent être considérées an de garantir le suivi, avec une erreur nulle, d'un signal de dynamique arbitraire. Ces contraintes sont liées par exemple aux limites de la bande passante de la communication, à la synchronisation, à la propagation des retards, aux restrictions de la topologie du réseau, etc. L'introduction d'informations a priori sur la dynamique du signal et sur la topologie du réseau doit permettre d'améliorer l'approche proposée.

Dans ce chapitre, une topologie en anneau est utilisée an de modéliser le canal de communication entre les agents (les pompes). Cette topologie requiert une communication bidirectionnelle et conduit à l'obtention d'un graphe fortement connecté qui garantit une convergence en temps ni. Bien que rarement implémentée en pratique, cette topologie en anneau constitue un point de départ pour ces travaux. L'algorithme de commande doit être vérié en considérant des graphes fortement connectés et déséquilibrés. Des études en simulation, à l'aide de la méthodologie de modélisation proposée, permettent de vérier l'algorithme de gestion haut-niveau développé. Les perspectives envisagées portent principalement sur l'évaluation d'une procédure qui conduit au réglage optimal des paramètres de la stratégie de commande par équilibrage distribué. La convergence des pressions et de l'erreur statique doivent être aussi évaluées en présence de retards, de perte de paquets d'information et de perte d'agents. Enn, l'algorithme proposé d'équilibrage distribué utilise le réseau de distribution de pression comme un moyen additionnel de communication. Il s'avère donc essentiel de compléter ces travaux par une étude approfondie sur la stabilité globale du réseau en considérant diérentes topologies du réseau de distribution.

Introduction Objectives

This work motivates the concept of energy-ecient waterjet facilities. To design these future waterjet facilities requires to develop a graph-based modelling for high-pressure networks and a hierarchical distributed control design for high-pressure pumps. That requires to dene suitable use cases for modelling and to specify requirements for control design. This requires to investigate conventional waterjet machining and to derive specications for future facilities with respect to industrial needs.

State of the art

The overview in [START_REF] Molitoris | A review of research on water jet with slurry injection[END_REF] allows for understanding the past and continuing developments of waterjet machining. Investigating the inventions given by various patents will enrich this overview. Waterjet machining found early industrial applications in 1930, applying a pressure of about 10 MPa for burring of castings. A system to cut hard materials has been rst proposed in [Schwacha, 1961], generating a high-pressure of 690 MPa. This system found application in the aerospace industry to cut steel and later to cut composite materials, which are sensitive to thermal stress. The investigations in [Rice, 1965] about pulsed waterjets at 340 MPa aimed at introducing waterjet machining to other industries.

Progress to improve the durability of waterjet nozzles has been obtained with the research for ceramics [START_REF] Chadwick | Nozzle for producing uid cutting jet[END_REF], while nozzles with reduced inner diameters (about 0.051 mm) that operates at pressures of 4800 MPa has been proposed in [Franz, 1973]. The experiments in [Hashish, 1984] rst introduced abrasive particles to waterjet cutting. This increases the material removal rate and consequently improves the productivity of waterjet machining. These developments, see also [START_REF] Hashish | Method and apparatus for forming a high velocity liquid abrasive jet[END_REF], marks the breakthrough of conventional waterjet machining in various industrial applications. The cutting head design as known today and commonly used for abrasive waterjet cutting has been introduced in [Chalmers, 1991] and [Xu, 1998]. Since then, the design of hydraulically driven pumps has undertaken great progress. Waterjet cutting with increased high-pressures of about 700 MPa is investigated in [START_REF] Susuzlu | Initial research on the ultra-high pressure waterjet up to 700 MPa[END_REF], where an improved eciency of 30% is reported in [Hashish, 2009], when waterjet machining with a pressure of 600 MPa.

Advancements of multi-axis machining that allows for cutting of complex contours and surfaces by means of waterjets are overviewed in [Folkes, 2009]. The text book [Nee, 2015] reviews recent process technologies used for waterjet machining. Its shows the entire range of common applications, from surface cleaning to the cutting of 2-dimensional contours and complex 3-dimensional shapes, whereas cutting quality and process eciency are of interest.

Contribution

The main research eort still lies in terms of improving the cutting process of waterjet machining. Most activities aim to increase the material removal rate or enhancing the cutting quality. Despite economizing the waterjet machining in terms of production time, its sustainability is sparsely investigated. The design of electrically driven high-pressure pumps and the concept of decentralized pumps in high-pressure networks arise increased interest from the industry. Enabling these future research activities, this chapter reviews the common applications for waterjet machining and discusses the principles of high-pressure generation. This chapter is structured as follows: Section 1.2 introduces the high-pressure components used for conventional waterjet machining, discusses common waterjet applications and classies waterjet machining by distinguishing dierent work station setups as well as manufacturing processes. It further outlines typical waterjet facilities when introducing the principal symbols to present them. The principles of high-pressure generation is given in Section 1.3 when classifying high-pressure pumps with respect to the drive systems. This will compare the most common hydraulically driven pumps with the recent electrically driven high-pressure pumps.

Conventional waterjet machining

A waterjet facility consists of high-pressure pumps that supply cutting heads, generating a waterjet. Cutting heads are commonly installed in work stations used for waterjet machining. Where high-pressure pumps represents a uid ow source, a cutting heads can be considered as a pressure sink.

Conventional high-pressure pumps pressurizes water by means of pistons which are typically displaced with a hydraulic drive. Dierent pump sizes are available with respect to the desired operating pressure and pump rate. The properties of a typical hydraulically driven pump are given in Table 1.1, which operates in the same power range as the investigated electrically driven pump. Where the electric drive operates on a wide pressure range, the hydraulic drive has to be adjusted for every waterjet facility that the pump runs properly at a desired pressure point. This hydraulically driven pump shows an increased frame size, weight and noise level in contrast to the electrically driven pump, discussed in Table 2.7 of Section 2.3. The image of Figure 1.1 shows exemplarily a high-pressure pump of Type JCP 22 LC with two hydraulically driven pressure intensiers. Each intensier pressurizes water with its single acting piston. Each cutting head consists of a nozzle holder and an pneumatic on/o-valve. The holder contains a nozzle with diamond inlet. The diamond inlet has a single cavity that shapes the waterjet, where the on/o-valve will interrupt the high-pressure water supply to the nozzle. This enables a cutting head to control the waterjet generation whenever needed. The nozzle size will determine the uid ow consumption of a cutting head with respect to the operating pressure. It is common to distinguish those nozzles with respect to the diamond inlet inner diameter. Most common nozzles are listed in Table 1.2. The image of Figure 1.2 shows as example a cutting head, which is typically used for pure water cutting. A work station holds a given amount of cutting heads and guides them over a work piece. The impact of the waterjet will cause a material removal. An operator can program the work station to realize a desired cutting contour using a terminal. A Computer Numerical Control (CNC) will then trigger the cutting heads to turn on its waterjet for cutting and to turn it o while moving to a subsequent cutting position. The image of Figure 1.3 shows exemplarily a work station which is equipped with a single cutting head, used for waterjet cutting. The work piece is positioned on a cutting bed. A 3-axis positioner will guide the cutting head over the work piece to cut a 2D-contour.

Common waterjet applications

Waterjet machining can be found in metal, composite, textile, food and many other industries, see [Folkes, 2009], [START_REF] Wang | Cutting meat with bone using an ultrahigh pressure abrasive waterjet[END_REF]. It is of rst choice for contour cutting with reduced thermal stress and for surface stripping without the use of chemicals. The manifold applications require operating pressures typically from 100 to 400 MPa. Beside various pure water applications, abrasive waterjet cutting is prevalent to achieve an increased material removal rate when machining hard and brittle materials, see [START_REF] Molitoris | A review of research on water jet with slurry injection[END_REF].

The diagram in Figure 1.4 classies waterjet machining and its typical pressure range. It basically distinguishes between surface cleaning (< 200 MPa pressure) and waterjet cutting (> 200 MPa pressure). Waterjet cutting is further separated with respect to pure water cutting and abrasive cutting. Pure water cutting is typically applied to cut organic material such as leader, food as well as plastics with a pressure up to 300 MPa. Abrasive is added to cut materials such as metal sheets or stone with a pressure up to 400 MPa. Waterjet cutting: The image of Figure 1.5 (a) depicts a work station for abrasive cutting. It consists of a 3-axis positioner which is equipped with a cutting head. The work station will move the cutting head along a desired 2D-contour. The abrasive is added within a cutting head, but after passing the diamond nozzle. The waterjet is then cutting a work piece which lies on the cutting bead. The impact of the abrasive particles will increase the material removal rate and therefore improve the eectiveness of waterjet machining.

Surface cleaning: The cleaning machine, as shown in the image of Figure 1.5 (b), provides another typical application. A high-pressure pump supplies a common rail that is located above a holding device. The common rail holds 16 nozzles which are arranged in series to cover a large surface. A cylindrical matrix is then installed on the holding device which brings the matrix into rotation. The series of nozzles will then strip o any paint from the matrix surface.

All of the investigations in this work consider the following general assumption: Assumption 1.1. Neither the cleaning nor the cutting process aects the high-pressure generation itself. The pressure generation is rather characterized by the nozzle installed and becomes disturbed when switching the on/o-valve of a cutting head.

As a consequence of Assumption 1.1, common waterjet applications will be investigated with respect to operating pressure, cutting head conguration and switching pattern.

Classifying waterjet machining

The dierent work stations hold a given amount of cutting heads that generates the waterjets. A cutting head can be equipped with nozzles of dierent inner diameters. Every work station will open and close its cutting heads with respect to the individual demands for waterjet machining. This leads to a varying overall uid ow consumption that has to be compensated by the installed high-pressure pump. The resulting switching pattern of a work station, when switching its cutting heads, is unknown to the high-pressure pump. It is therefore assumed as an exogenous disturbance.

The diagram of Figure 1.6 presents useful characteristics to classify waterjet machining. Requirements for waterjet machining vary with respect to the waterjet application, its work station setup and manufacturing process. The operating pressure is determined by the waterjet application. A cutting head conguration depends on the work station setup, while the manufacturing process denes the switching pattern for each cutting head. The cutting head conguration, its operating pressure and switching pattern determines the overall water consumption, which is limited by the available pump rate.

Typical work station setups found in industry are presented in Table 1.3. When abrasive cutting, the water demand of a single cutting head can already occupy an entire pump. This setup of single work stations with single cutting heads is widely spread in the industry. When pure water cutting, the water demand of a single cutting head is usually smaller. Hence, a highpressure pump either supplies a single work station with multiple cutting heads or it supplies several interconnected work stations with a single cutting head each. The rst case of a single work station with multiple cutting heads is often used to cut identical contours in parallel. This aims to enhance the eectiveness of waterjet machining. These cutting heads switch simultaneously and nozzles with identical inner diameter are installed. The second case of several interconnected work stations with single cutting heads will represent a cluster for cutting individual contours. These cutting heads switch independently and various nozzles might be installed. Multiple, simultaneously switching cutting heads can be considered as alike as a single cutting head with a larger nozzle installed, if the switching heads are located in near neighbourhood and if the nozzles are chosen such that an identical water consumption will result. The rst case is not given for a cluster of multiple work stations. A single high-pressure pump can supply multiple cutting heads, as long as it serves the overall water consumption. In practice, it is rarely seen to aggregate multiple pumps for increasing the pump rate.

These work station setups include dierent cutting head congurations, as specied in Table 1.4. A surface cleaning application is chosen that desires a pressure of 200 MPa. The cleaning machine is equipped with a common rail of 16 nozzles. For nozzles of 0.10 mm inner diameter each, an overall water consumption of 2.98 l/min will results. A single 0.40 mm nozzle will require an equivalent ow rate. On the other hand, a pure water cutting application is presented that desires a pressure of 300 MPa. A single work station is thereby equipped with 4 parallel cutting heads to simultaneously cut 4 identical contours. A nozzle of 0.15 mm inner diameter is installed each, resulting in an overall water consumption of 2.04 l/min. A single 0.30 mm nozzle will result in an equivalent ow rate. Eventually, an abrasive cutting application is chosen that desires a pressure of 400 MPa. A single work station is considered with a single cutting head. The cutting head holds a nozzle of 0.25 mm inner diameter, resulting in a ow rate of 1.55 l/min. Typical manufacturing processes found in industry require dierent switching patterns as specied in Table 1.5. It relates the common waterjet applications to three corresponding manufacturing processes. Any stripping process uses multiple interconnected cutting heads with a simplied design. They remain open for the entire stripping process. In some cases the on/ovalve is totally removed. Contour and cavity cutting distinguish dierent switching patterns. Both can be seen as a periodic process, but its cycle time and on/o-ratio can vary in a wide range. Cavity cutting features a short cycle time and a limited on/o-ratio, while contour cutting has a longer cycle time with an increased on/o-ratio. The on-time is dened for an open valve when cutting a contour or cavity. The o-time is given for a closed valve when moving to a subsequent cutting position. The overall water consumption will vary due to the individual switching behaviour of all cutting heads. Cutting head conguration, operating pressure and switching pattern give the overall water consumption, which denes the desired pump rate (see Figure 1.6). The switching patterns for multiple interconnected work stations with independent cutting heads can superimpose, which results in an arbitrarily varying water consumption. Details will be given in Chapter 2.2, when providing specic use cases for waterjet machining.

Typical waterjet facilities

Industrial installations for waterjet machining typically consist of decentralized high-pressure pumps, which are remotely installed to supply single work stations for waterjet machining. A single work station setup can be depicted as a high-pressure pump with a cutting head directly interconnected by a piping, as shown in the diagram of Figure 1.7. This diagram uses the standard symbols according to ISO 1219. However, it adapts them for high-pressure systems to illustrate network topologies and its congurations for dierent use cases. A high-pressure pump is considered as a single uid ow source for any principle of pressure generation and drive system. Hence, the control signal u n (t) = [0, 1] allows for adjusting the desired input uid ow of a pump n within 0% and 100% of its maximal pump rate possible. A cutting head is considered as a single pressure sink. Its symbol consists of an on/o-valve and a nozzle. The on/o-valve is switching with respect to the signal d m (t) ∈ {0, 1} that controls the uid ow to pass a cutting head m. The nozzle is interchangeable and denes the resulting water consumption with respect to the operating pressure. The various work stations installed in a waterjet facility are not taken into account in this work, since the material removal itself will not inuence the high-pressure pump and its pressure generation, referring to Assumption 1.1. But the cutting head switching pattern is unknown to the high-pressure pump and will directly aect the pressure generation. It is usually given by the work station with respect to the manufacturing process and is therefore considered as a disturbance signal d(t). Pressure gauges can be installed in the high-pressure network. They measure the pressures p x (t) at positions x, used for control and validation.

The interconnections between pumps and cutting heads are realized by high-pressure piping sections. Hence, the total of interconnected piping of a cutting application is referred to a highpressure network. These networks will vary for dierent waterjet facilities. However, the network topology of every application can be assumed as static, since waterjet machining is interrupted to change any piping installation.

Recall the conventional waterjet facility and the future waterjet facility, as shown in Figure 3 and4 of the General introduction part, and representing these examples by applying the above standard symbols provides the diagrams in Figure 1.8. It can be seen that the rst application (a) consist of two independent high-pressure networks, while the second application (b) spans a single network with increased complexity. The conventional waterjet facility is composed of a work station setup with two simultaneously switching cutting heads and a cluster of two work stations with independent cutting heads, where a high-pressure pump supplies the cutting heads for each work station setup. On the other hand, the future waterjet facility represents a cluster of interconnected work stations with independent as well as parallel cutting heads, where several pumps supply the entire network. These interconnected pumps are no longer located in a remote machine room. They are distributed in the waterjet facility, reducing the overall piping length.

Principles of high-pressure generation

A pump is an electro-mechanical machine with the task of moving a uid. It is dedicated to perform a hydraulic work. Referring to the text book [START_REF] Will | Hydraulik[END_REF], the hydraulic work P can be understood as the amount of uid dV which will be moved within a period dt under load, hence a pump induces a displacement uid ow Q = dV /dt. The uid ow is referred to the uid passing through a dened cross section S with a speed v such as Q = S v. Speaking of a load can relate to the dierential pressure ∆p between the pump intake and outtake or equivalently to the actual pumping head h. Both are related in terms of ∆p = -g h, where denotes the uid density and g ≈ 9.81 m/s 2 is the local acceleration of gravity. The hydraulic work is then dened as P = Q ∆p . Multiple pumps can be set up in parallel to increase the ow rate, while each pump is loaded with the same pressure. On the other hand, multiple pumps can be set up in serial to increase the overall pumping head, while each pump must move the same amount of uid. Considering future waterjet facilities, high-pressure pumps will be combined in parallel to cover the increase water consumption of several interconnected work stations.

Particularly for waterjet machining, high-pressure is generated when displacing water by means of reciprocating pistons. The water has to meet rm requirements to preserve sensitive high-pressure components, e.g. seals. Table 1.6 gives the requirements provided by the manual of a hydraulically driven high-pressure pump (Type JCP 22 LC).

Classifying high-pressure pumps

Hydraulically driven high-pressure pumps, used for waterjet machining, pressurize water by means of pressure intensiers. A pressure intensier increases the pressure, provided from a hydraulic drive, with respect to its piston cross sections. The illustration of Figure 1.9 shows a pressure intensier (a) and a piston pump (b). The pressure intensier consists of two chambers separated by a double-acting piston of dierent cross sections. This piston is oating and can move axially between the chambers. The low-pressure side has a wide cross section S 1 , where the high-pressure side features a signicantly smaller cross section S 2 . The hydraulic pressure p 1 on the low-pressure side acts on the cross section S 1 of the piston that causes a force F = S 1 p 1 . This force induces a water pressure p 2 = F S -1 2 on the high-pressure side with respect to the cross section S 2 . That yields the working principle S 1 p 1 = S 2 p 2 of a pressure intensier, as given in [START_REF] Will | Hydraulik[END_REF]. Hence, a hydraulic pressure of about 28 MPa applied on the low-pressure side is intensied with respect to the cross section ratio S 1 /S 2 . Considering a ratio of 1 : 12.5 induces a water pressure of about 350 MPa into the high-pressure side. a pressure intensier pressurizes water (high-pressure chamber) by means of hydraulics (lowpressure chamber), whereas a piston pump is directly driven by an electro-mechanical actuator.

A hydraulically driven high-pressure pump is straight forward to realize with standard hydraulic components. However, its energy eciency is unsatisfying due to the hydraulic drive. Electrically driven high-pressure pumps replace the hydraulic drive to overcome this downside, while the high-pressure side remains unchanged.

The piston pump holds a chamber that contains a single-acting piston with cross section S, which is directly displaced by a electro-mechanical actuator. A piston displacement pressurizes the uid within a pumping chamber. Thus, the water pressure p from the high-pressure side causes a force F = S p that acts directly on the actuator. For example, inducing a water pressure of 350 MPa with a standard piston of 15 mm diameter requires a force of up to 62 kN.

Drive system and pumping chamber design allow for classifying four types of high-pressure pumps, as shown in diagram of Figure 1.10. With respect to the drive system, hydraulically driven pumps and electrically driven pumps can be distinguished. Each can be further assigned with respect to the principle of pressure generation. It is thereby prevalent to group pressure intensiers as well as piston pumps, considering double-acting and single-acting operation. On the other hand, electrically driven pumps with single-acting pistons can be further assigned to coupled and independent piston operation. All of them feature at least two pumping chambers. This enables approximately a continuous pump output uid ow and consequently realizes a nearly steady pressure generation. However, the principle of pressure generation by reciprocating pistons will induce sinusoidal pressure uctuations, which will degrade the quality of waterjet machining. The range of pressure uctuation will vary depending on the pump design. It is common to use multiple pistons and additional high-pressure attenuators to reduce any pressure uctuations. However, such attenuators are sensitive and cost intensive components. The subsequent sections will provide detailed characteristics of dierent high-pressure pumps.

Hydraulically driven intensier pumps

Because of the high load acting at a piston when generating high-pressure, it is evident to utilize hydraulic drives. Hydraulically driven pumps with double-acting intensiers generate unsteady pressure trends. Due to the coupling of both pumping chambers with a central piston, a halfsinusoidal input uid ow will result. Consequently, large high-pressure attenuators are installed between pump and high-pressure piping. That guarantees a steady pressure trend, suciently for waterjet machining.

An alternative pump design consists of two single-acting intensiers. Independent pistons pressurize each pumping chamber. The pistons are coupled with the hydraulic drive. These pistons run in a phase-shifted operation to obtain a nearly continuous overall uid ow. This allows for minimizing pressure uctuations when switching between the pistons. Optimizing the phase-shifted operation for a specic operating point can further improved the pressure generation such as no high-pressure attenuator will be needed. However, to retract the piston of a single-acting intensier requires additional pneumatics support.

The schematic of Figure 1.11 shows the hydraulic drive to operate a single-acting intensier pump (a) and a double-acting intensier pump (b). Both pump designs are deployed for conventional waterjet machining. They are driven with hydraulic drives of identical conguration. A conventional piston pump supplies a hydraulic circuit, which is regulated to desired pressure. A asynchronous motor powers the pump at constant speed, which runs directly from energy mains. The hydraulic circuit contains a solenoid valve that applies the pressure to one of both single-acting intensier or to the double-acting intensier, respectively. Attenuators are installed at high-pressure side to compensate pressure uctuations, induced due to the switching of the solenoid valve.

Double-acting intensier

The hydraulic drive is supplying the double-acting hydraulic cylinder of a single pressure intensier to realize a reciprocating operation. This enables to displace two coupled high-pressure pistons, which are oriented in opposite directions of motion. The direction of motion will be switched by a solenoid valve, when a piston reaches its maximum stroke. This is detected by proximity sensors. The reciprocating operation induces a sinusoidal input uid ow due to the coupled high-pressure pistons, see e.g. [START_REF] Xu | Design and experiment research on abrasive water-jet cutting machine based on phased intensier[END_REF].

Single-acting intensiers

The hydraulic drive is alternately supplying two independent pressure intensiers. Each intensier consists of a single-acting hydraulic cylinder that displaces a high-pressure piston. Its control is again straight forward based on a solenoid valve and proximity sensors. The solenoid valve coordinates both pressure intensiers. The piston of one intensier will extend to generate an displacement uid ow, while the other is retracting. A proximity sensor switches the solenoid valve, when a piston reaches its maximum stroke. This results in a phase-shifted reciprocating operation of both pressure intensiers. The valve design is often optimized for a smooth switching characteristics to minimize pressure uctuations, see e.g. [START_REF] Fabien | Dynamic modelling and identication of a waterjet cutting system[END_REF]

Electrically driven piston pumps

Electrically driven pumps with coupled pistons became recently available. A asynchronous motor drives multiple pistons using a mechanical crankshaft. These pumps with coupled pistons, as described in the text book [START_REF] Michael | High Pressure Pumps[END_REF], are fast evolving in the industry and gradually achieving pressures of up to 200 MPa. This is sucient for pure water applications. Further advances in the eld of electric servo drives enable novel pump designs. In this regard, an electrically driven pump with independent pistons has been proposed in [START_REF] Niederberger | Energy-ecient high-pressure pump[END_REF]. It consists of servo motors, each driving a spindle shaft to displace a piston. The novel pump design allows for pressures of above 400 MPa and can be employed for future waterjet facilities. Its functional principle is introduced in Section 2.3. An alternative pump design with a double-acting piston is proposed in [Borgarelli, 2015]. The high-pressure chambers of both pump designs are identical to the hydraulically driven single-acting intensier pump. The progression from hydraulically to electrically driven actuators is well illustrated in [START_REF] Qiao | A review of electromechanical actuators for more/all electric aircraft systems[END_REF].

Single-acting pistons

The schematic of Figure 1.12 (a) illustrates an electrically driven pump that consists of two independent pistons, which are individually displaced by electric servo motors. The motors are driven by frequency converters. It is possible to synchronize both pistons by means of motion control, that allows for realizing a dynamic phase-shift. This aims at generating a steady pressure trend while compensating any pressure uctuation for an arbitrary operating point. A highpressure attenuator is no longer required, see e.g. [START_REF] Niederberger | Energy-ecient high-pressure pump[END_REF].

Double-acting piston

The schematic of Figure 1.12 (b) illustrates an electrically driven pump that consists of two independent pistons, coupled to a spindle shaft which is driven by a common electric servo motor. This provides a constants phase-shift of 180°between each piston and leads to a periodically altering overall uid ow, following the superposition of two sinusoidal. The constant phase-shift does not allow for compensation of pressure uctuations. It therefore requires a high-pressure attenuator to compensate pressure uctuations, see e.g. [Borgarelli, 2015].

Conclusion

In this chapter, a hydraulically driven high-pressure pump that supplies a work station for contour cutting has been introduced as a typical application of conventional waterjet machining. It is proposed to classify the dierent waterjet applications with respect to the manufacturing process and the work station setup. This allows for dening suitable use cases, useful when evaluating the modelling methodology, and for verifying the control design of future waterjet facilities. Where the work station setup denes the cutting head conguration, the manufacturing process species switching patterns applied on the work stations. On the other hand, standard symbols describing common hydraulic circuits are adapted to describe the conguration of high-pressure networks, interconnecting distributed pumps with cutting heads of various work stations.

Water supply Singel-acting pistons

Double-acting piston

High-pressure outtake

Electric drive system (a) (b) M M M M Figure 1.
12 Physical setup of a high-pressure pump with electric drives: two drives units displacing independent pistons (a) and a single drive unit displacing coupled pistons using a crankshaft (b).

The principal of pressure generation has been discussed to explain the common approaches used for high-pressure generation by means of piston pumps. The functional principles of well established and newly emerging concepts are briey explained. It compares the novel pump design, considered in the subsequent work, with the conventional approaches for pressure generation. This investigation considers two dierent pumping chamber designs in combination with two drive system concepts.

A close similarity between hydraulically and electrically driven pumps is found, regarding the pumping chamber design. Where electrically driven pumps with coupled pistons barely reach the pressure needed for waterjet machining, experimental pump designs with independent singleacting and double-acting pistons obtain pressures up to 400 MPa. These pumps are suitable for waterjet machining, but related literature is rarely found. The subsequent chapter will present an electrically driven pump with independent single-acting pistons, while introducing a highpressure test bench. It further denes all use cases, network topologies and switching patterns in detail, which has been employed for this thesis.

Introduction Objectives

The concept of energy-ecient waterjet facilities considers decentralized pumps to establish highpressure networks. These decentralized pumps require a new drive system, since hydraulically driven pumps suer from a low energy eciency. Moreover, they fail to run within the required operating range. To overcome these drawbacks, electrically driven high-pressure pumps reduce the drive complexity and provide the ability to operate at any desired operating point. The drive system complexity becomes reduced at the expanses of a demanding control strategy. To evaluate future control strategies and to verify high-pressure network model requires the development of a high-pressure test bench and to dene suitable use cases, which represent future waterjet facilities.

State of the art

In most today's applications, waterjet machining includes a remote high-pressure pump that feeds a single work station [Nee, 2015]. A high-pressure intensier pump is developed in [Olsen, 1974] to enhance the high-pressure generation. This has been a milestone for the industrial availability of waterjet machining. Recent developments relate to this pump design, using hydraulically driven pressure intensiers. Two double-acting intensiers have been combined in [START_REF] Trieb | Pressure and ow rate uctuations at high pressure intensier pumps[END_REF] with a phase shift operation to reduce pressure uctuations induced by the pump, where a single-acting pump with phased intensiers is investigated in [START_REF] Xu | Design and experiment research on abrasive water-jet cutting machine based on phased intensier[END_REF]. The modelling and identication of double-acting pump is presented in [START_REF] Fabien | Dynamic modelling and identication of a waterjet cutting system[END_REF], while machine diagnostics for a multiple phased single-acting pump is shown in [START_REF] Ferretti | Abrasive waterjet intensier model for machine diagnostics[END_REF] that further reduces pressure uctuations. The working principle of an electrically driven high-pressure pump with single-acting pistons has been rst introduced in [START_REF] Niederberger | Energy-ecient high-pressure pump[END_REF]. Experimental studies revealed energy savings of more than 30% compared to the conventional pump design.

Contribution

This chapter rst presents a high-pressure test bench capable to interconnect two electrically driven pumps with two work stations. That allows for conguring the high-pressure piping to represent various high-pressure networks. This aims at validating specic high-pressure network models by means of measurements and evaluating future control strategies, as will be shown in a subsequent chapter. For this purpose, suitable high-pressure network topologies and use cases will be introduced with respect to the investigations of the previous chapter. These topologies and use cases are then referred to in the subsequent chapters when verifying the modelling methodology and control design.

This chapter further investigates the novel pump design of an electrically driven high-pressure pump and presents the working principle of phase-shifted single-acting pistons. An electrically driven pump is modular and manages to synchronize the pistons with an adjustable phase-shift. This synchronization is realized using a virtual crankshaft (camming). It generates the reference trajectories, as required to displace the pistons by means of motion control. These trajectories are generated, taking a model into account that describes the pressure generation within a pumping chamber. It enables the electrically driven high-pressure pump to run on a wide operating range, while any pressure uctuations, induced by the pump, become minimized. This section is structured as follows: Section 2.2 presents the considered test bench setups and network topologies. It species the use cases for future waterjet facilities, subject for investigation, and introduces switching patterns to reproduce selected manufacturing processes. Section 2.3 presents the energy ecient high-pressure pump, implemented to the test bench. It investigates its working principle, considering the piston synchronization by means of model-based camming.

Test bench for scalable high-pressure networks

A preliminary test bench combines an early prototype of the electrically driven pump with a standard cutting head to verify the novel pumping technology. This setup is insucient to represent scalable high-pressure networks. Thus, an extended test bench has been deployed in collaboration with the industry. This requires to design an enhanced electrically driven pump, including its production and the installation in the laboratory, where the extended test bench is assembled. This extended test bench has been established in the laboratory of the FHNW. It is dedicated to experimental studies of the subsequent investigations, e.g. for parameter identication, model validation and controller verication. It consists of two independent high-pressure pumps and two independent cutting heads. This allows for dening high-pressure networks which correspond to common waterjet facilities, but also to realize topologies which represent future waterjet facilities with limited complexity. The test bench can be congured, taking the various cutting head congurations into account given for typical work station setups. Its cutting heads can be switched with respect to varying switching patterns as required from common manufacturing processes. Its high-pressure pumps can be regulated to a desired pump rate such as to obtain the operating pressure in compliance with dierent waterjet applications.

This section introduces dierent high-pressure networks subject for modelling and control design. Basic topologies are used to validate the proposed graph-based modelling methodology. The validated modelling methodology is then applied to model more complex topologies which are too time consuming and cost intensive for realizing on a test bench. Use cases are dened with respect to operating pressure, cutting head conguration and switching pattern. They are assigned to experiments for parameter identication and model validation.

Test bench setup

The test bench, as shown in image of Figure 2.1, is congurable to represent dierent network topologies. The high-pressure network can interconnect two independent electrically driven pumps and two independent cutting heads. The pumps can simultaneously pressurize the highpressure network at arbitrary positions, while the cutting heads realize the desired switching patterns. A cutting head reproduces the discharge ow of a work station when generating a waterjet. The waterjet is immediately discharged into a containment. This corresponds to a pressure sink, while a high-pressure pumps represents a uid ow sources.

By taking the various cutting head congurations into account, each cutting head can be equipped with a diamond nozzle of 0.1 to 0.4 mm inner diameter. This enables a cutting head to represent various work station setups, from a single cutting head conguration of up to 0.4 mm nozzle to a single work station with up to 16 synchronously switching cutting heads of 0.1 mm nozzles. Interconnecting a second cutting head allows for realizing a cluster of 2 work stations with independently switching cutting heads. Considering a second pump, allows for a test bench setup which represents complex network topologies of future high-pressure facilities. These networks are realized by interconnecting common high-pressure piping.

The control variable u n (t) = [0, 1] regulates a high-pressure pump n ∈ {1, 2} to an corresponding pump rate, where the disturbance variable d m (t) = {0, 1} switches the on/o-valve of a cutting head m ∈ {1, 2}. This aects the overall water consumption for which the pump rate has to be adjusted, where the control signal is proportional to the desired pump rate. The diagram in Figure 2.2 represents the high-pressure test bench schematically. The pressure p x (t) can be measured at dierent network positions x using 4 pressure gauges. The applied network positions will be dened later in this section, when introducing the investigated network topologies. In general, pressure measurements near to a high-pressure pump are denoted as p P n (t), where n relates to the corresponding pump and measurements near to a cutting head are denoted as p Hm (t), where m relates to the corresponding cutting head. Any other measurement positions within a high-pressure network are denoted as p N i (t), with i as a consecutive numbering. The schematic in Figure 2.3 shows the test bench setup used to derive experiments for parameter identication, model validation and controller verication. An oine computer is used to design the desired experiments. This computer employs MATLAB Simulink to evaluate the experiments in advance by means of simulations. The design of experiment is then transmitted to an online computer, if the simulations achieved the desired parameters, e.g. bounded control values considering pump saturations or bounded pressure state with respect to system limits. This computer employs Studio 5000 to parametrize the test bench with respect to a planned experiment and to execute this experiment. It stores the measurement data locally while executing an experiment. After termination of the experiment, the experimental data are sent to the oine computer, which evaluates the data by means of MATLAB. A control unit (PLC) is employed to control the test bench, while executing an experiment. It sets the required pump rates u n (t) for each high-pressure pump and the switching states for each cutting head, considered as disturbances d m (t). A digital output module is connected to the PLC that controls the on/o-valve of a cutting heads with respect to specied switching patterns. An analogue input module is connected to the PLC that measures the pressure trends p x (t) of 4 pressure gauges at a sampling rate of 250 Hz each. The pressure gauges have a wide operating range of 0 to 500 MPa. The test bench is realized using the hardware listed in Table 2.1. The displacement uid ow is evaluated for each piston by measuring the piston velocities. Each high-pressure pump provides all data needed to the PLC. More information is given when introducing the electrically driven piston pump used with the test bench. As long as any low-level control is missing, each pump is regulated to produce a steady pump rate. That causes a continuous overall uid ow. Without considering the cutting head conguration or switching state, any pressure trend could be reached. Hence, simulations are important to estimate the expected pressures for a specic experiment that all process variables remains within permitted limits, given by the test bench and high-pressure pumps.

Investigated network topologies

The test bench can be congured to represent dierent work station setups, which correspond to common waterjet facilities. Specic setups will be chosen with regards to parameter identication, model validation and controller verication. The selected setups begin with basic topologies interconnecting a single pump to a single cutting head and are gradually extended to more complex networks including two coupled pumps and two independent switching cutting heads. Complementary to experiments on the test bench, all of these high-pressure networks will be 1 pump -1 cutting head network topologies Initial topologies consider a single pump feeding a single cutting head. Pump and cutting head are directly interconnected with a high-pressure piping. This setup involves two topologies as depicted in diagrams of Figure 2.4. Topology (a) establishes a short piping section of about 3 m length between pump and cutting head, while topology (b) represents a long piping of about 19 m. The long piping is expected to increase the uid ow resistance that yields an increased pressure loss from pump to cutting head. The pressure is measured at the test bench intake p P (t), at a piping connector within the high-pressure network p N (t) and at the test bench outtake p H (t). The rst topology has been used to validate the robust low-level control design by means of simulations, see Chapter 6, while the second topology has been used to validate the graph-based modelling methodology for a wide pressure range with measurements from the test bench, see Chapter 4.

pump -2 cutting heads network topologies

Extended topologies consider a single pump feeding two cutting heads. Pump and cutting heads are connected with a more complex high-pressure network. This setup includes again two topologies as depicted in the diagram of Figure 2.5. Where topology (c) realizes a symmetric network with 6 m of piping length from the joint to a cutting head each, the topology (d) features an asymmetric network with about 11 m for cutting head 1 and 1 m for cutting head 2. The symmetric topology is expected to generate identical resistances for both uid ow paths that should result in a symmetric uid ow distribution, when assuming identical diamond nozzles for every cutting head. Hence, the inuence of conguring dierent cutting heads can be clearly distinguished. The asymmetric topology is expected to cause dierent resistances for each uid ow path that should result in an asymmetric uid ow distribution, even for identical diamond nozzles. The cutting heads will be identically congured to investigate the inuence from the network topology. The pressure is measured at the test bench intake p P (t), at the piping connector p N (t) that couples the pump to both cutting heads and at both test bench outtake positions p H1 (t) and p H2 (t). Both topologies are used to identify the model parameters for various high-pressure components (see Section 3.4), to verify the proposed high-pressure network models (Section 4.3) and to validate the derived low-level controllers (Section 6.4), using measurements from the test bench. Dierent experiments have been derived on the same setup regarding parameter identication and model validation. However, model and controller validation will employ the same measurement data. Where the high-pressure network model is validated in open-loop, the low-level controller is validated in closed-loop control.

pumps -cutting heads network topologies

Finally, networked topologies consider two pumps feeding two cutting heads. Pumps and cutting heads are connected with complex high-pressure networks. This setup provides a wide variety to distribute pumps and cutting heads in a network. It considers three topologies as depicted in the diagram of Figure 2.6. The symmetric distributed topology (e) as well as the asymmetric distributed topology (f) consider two centrally interconnected cutting heads with a piping of about 1 m length. Where topology (e) realizes a symmetric network with 6 m of piping length from the joint to a high-pressure pump each, the topology (f) features an asymmetric network with about 11 m for pump 1 and 1 m for pump 2. The decentralized distributed topology (g) features a symmetric high-pressure network, but both pumps and both cutting heads are spatially distributed. The rst two setups aim to investigate the behaviour of two interconnected pumps when applying identical disturbances at a single network position. Dierent disturbances are applied at other network positions, considering the third setup. This can aect both pumps, even for a symmetric network topology.

The pressure is measured at both test bench intake positions p P 1 (t) and p P 2 (t), at the piping connector p N (t) that couples both pumps and at both test bench outtake positions p H1 (t) and p H2 (t). These topologies are used to verify robust low-level controllers, distributed to several coupled pumps and before establishing any high-level managing. The model validation is extended with these complex high-pressure networks, again comparing open-loop simulations and measurements. The resulting validated simulation models represent future waterjet facilities. They are then employed to verify the high-level managing by means of simulations, see Section 7.4. This section described the selected topologies for conguring the test bench. Table 2.2 gives an overview of the introduced network topologies. It relates them to its utilization and assigns them to use cases, which are dened in the subsequent section.

(f) Asymmetric distributed (e) Symmetric distributed d 2 Head 2 d 1 Head 1 1 m 1 m Pump 1 u 1 Pump 2 u 2 6 m 6 m p N p P1 p P2 p H1 p H2 7 m d 2 Head 2 d 1 Head 1 1 m 1 m Pump 1 u 1 Pump 2 u 2 6 m p N p P1 p P2 p H1 p H2 7 m 6 m 1 m Pump 1 u 1 Pump 2 u 2 6 m 6 m p P1 p P2 7 m p N d 1 Head 1 1 m p H1 d 2 Head 2 1 m p H2 (g) Decentralized distributed

Use case denition

Here, 3 use cases are taken into account for measurements with the test bench and for simulation in MATLAB Simulink. Use case 1 considers a 1 pump -1 head setup that represents a basic waterjet facility. Use case 2 deals with a 1 pump -2 heads setup that represents an extended facility, while use case 3 represents a 2 pumps -2 heads setup of a future waterjet facility.

Various experiments have been derived while taking dierent switching patterns and various nozzle congurations into account, with respect to the identied manufacturing processes and work station setups. According to the waterjet applications, two dierent operating pressures have been considered for the use cases 1 and 2. Experiments for use case 3 have been limited for a single operating pressure, since more as sucient combinations become available when combining two high-pressure pumps and two cutting heads.

Use case 1

To verify the low-level control design and the graph-based modelling methodology, the test bench has been congured considering the 1 pump -1 head topology with a short piping (a) and a long piping (b). Two dierent experiments have been realized for each topology. A rst experiment at 200 MPa operating pressure, where the cutting head holds a 0.35 mm nozzle and a second at 350 MPa pressure, where a 0.25 mm nozzle has been installed. These experiments include the measurements listed in Table 2.3. Dierent measurements have been realized using the same topology and conguration, but with respect to dierent switching patterns. In a rst step, the modelling methodology has been veried considering the test bench in open-loop. Therefore, a high-pressure pump cannot compensate any pressure uctuations induced by switching a cutting head. Hence, a stripping application is chosen, where the cutting head remains open for any time. In a second step, the control design has been veried considering an application for contour cutting and another application for cavity cutting. This requires dierent switching patterns as introduced in the subsequent section.

Use case 2

To identify the parameters of high-pressure components and to validate high-pressure network models, derived with the graph-based modelling methodology, as well as to validate the low-level controllers, derived with robust control design, the test bench has been congured considering the 1 pump -2 heads topology with a symmetric network (c) and an asymmetric network (d).

All experiments have been derived at 200 MPa operating pressure as well as at 350 MPa, while dierent cutting head congurations have been evaluated. These experiments include the measurements listed in Table 2.4. Again, dierent measurements have been realized using the same topology and conguration, but with respect to dierent switching patterns. First experiments focus on a symmetric cutting head conguration, where both cutting heads hold a 0.25 mm nozzle. Subsequent experiments investigate an asymmetric conguration, where head 1 holds a 0.3 mm nozzle and head 2 a 0.2 mm nozzle. Here, the symmetric network topology combined with an asymmetric cutting head conguration allows for investigating the eects of dierent cutting head nozzles. Subsequent experiments focus on the asymmetric cutting head conguration only.

Here, the asymmetric network topology combined with a symmetric cutting head conguration allows for investigating the eects of dierent uid ow paths in a high-pressure network.

In a rst step, measurements considering an application for contour cutting have been assigned for parameter identication. The other measurements have been used for model validation. The identication data have been used to identify the discharge coecient ζ of cutting head nozzles as well as the friction loss coecient λ for high-pressure piping sections, as introduced later in this work. In a second step, an initial low-level controller has been validated with measurements considering an application for contour cutting as well as for cavity cutting. Hence, two dierent switching patterns will disturb the high-pressure pump, which operates now in closed loop control to compensate any pressure uctuations.

Use case 3

The test bench has been congured considering the 2 pumps -2 heads topology with a symmetric distributed network (e), an asymmetric distributed network (f) and a decentralized network (g) to validate high-pressure network models of increased complexity, to validate the low-level controllers with respect to these complex network topologies and to verify the high-level managing design, used to balance both high-pressure pumps to an average pump rate. All experiments have been derived at 200 MPa operating pressure only, while evaluating dierent cutting head congurations. These experiments include the measurements listed in Single cutting head (Use case 1)

The measurements of use case 1 has been derived considering a 1 pump -1 head setup. To realize the contour cutting application, the single cutting head will open for a cutting period of 60 seconds, followed by a hold period in closed position for another 60 seconds, as depicted in diagram of Figure 2.7 (a). On the other hand, to realize the cavity cutting application, the single cutting head will open for a cutting period of 0.5 seconds, followed by a hold period in closed position for another 0.5 seconds, as depicted in diagram of Figure 2.7 (b). Both switching patterns will be periodically repeated. These switching patterns realize 2 dierent states of the high-pressure test bench, where the resulting water consumption depends on the cutting head conguration and operating pressure. For an operating pressure of 200 MPa and a cutting head congured with a 0.35 mm nozzle, a maximal uid ow consumption of 2.4 l/min results. For an operating pressure of 350 MPa and a cutting head congured with a 0.25 mm nozzle, a maximal uid ow consumption of 1.6 l/min results. A closed cutting head wont require any water. However, when a pump continues to operate, the operating pressure will increase without limits. Two active cutting heads (Use case 2)

The measurements of use case 2 have been derived considering a 1 pump -2 heads setup. To realize a contour cutting application, both cutting heads will open for a cutting period of 60 seconds, followed by a hold period in closed position for another 60 seconds, as depicted in diagram of Figure 2.8 (a). On the other hand, to realize a cavity cutting application, each cutting head will open for a cutting period of 0.5 seconds, followed by a hold period in closed position for another 0.5 seconds, as depicted in diagram of Figure 2.8 (b). Both switching patterns will be repeated periodically with a 90°phase shift between both cutting heads. These switching patterns realize 4 dierent test bench states, where the resulting water consumption depends on each cutting head conguration. If cutting head 1 holds a nozzle of 0.3 mm diameter, while the cutting head 2 is congured with a 0.2 mm nozzle and combining all possible switching states, 4 dierent uid ows (Q(t) ∈ {0, 0.78, 1.61, 2.39} l/min) will result, considering a continuous operating pressure of 200 MPa. If cutting head 1 holds a nozzle of 0.2 mm diameter, while the cutting head 2 is congured with a 0.1 mm nozzle and combining all possible switching states, 4 dierent uid ows (Q(t) ∈ {0, 0.27, 1.01, 1.28} l/min) will result, considering a continuous operating pressure of 350 MPa. If both cutting heads hold a nozzle of 0.25 mm diameter and combining all possible switching states, again 4 dierent uid ows (Q(t) ∈ {0, 1.16, 1.17, 2.33} l/min) will result, considering a continuous operating pressure of 200 MPa. Eventually, if both cutting heads hold a nozzle of 0.15 mm diameter and combining all possible switching states, 4 dierent uid ows (Q(t) ∈ {0, 0.61, 0.71, 1.32} l/min) will result, considering a continuous operating pressure of 350 MPa. Active and passive cutting head (Use case 3)

The measurements of use case 3 have been derived considering a 2 pumps -2 heads setup. Only cutting head 2 will open for a cutting period of 60 seconds, followed by a hold period in closed position for another 60 seconds to realize a contour cutting application, as depicted in diagram of Figure 2.9 (a). Equivalently, the cutting head 2 will open for a cutting period of 0.5 seconds, followed by a hold period in closed position for another 0.5 seconds to realize the cavity cutting application, as depicted in diagram of Figure 2.9 (b). Both switching patterns will be repeated periodically.

Novel electrically driven pump with single-acting pistons

This section presents the novel electrically driven high-pressure pump, designed considering the concept mentioned in [START_REF] Niederberger | Energy-ecient high-pressure pump[END_REF], which is installed with the previously introduced high-pressure test bench of Section 2.2. It pressurizes water by means of pistons, which are displaced with electric drives, see the properties listed in Table 2.7. The electric drive allows for operation on a wide pressure range without any adjustments for dierent waterjet applications. Its size is reduced by 40% and its energy eciency is enhanced by 30%, compared to a hydraulically driven pump that operates in the same power range. The electrically driven pump provides an enhanced pumping performance due to the improved overall eciency. This results in an increased maximal pressure in contrast to the hydraulically driven pump, discussed in Table 1.1 of Section 1.2. The image of Figure 2.10 shows a prototype including two electrically driven pumping units. Its modular pump design is scalable to meet the increased uid ow demands of future large-scale waterjet facilities by combining a desired amount of pumping units. However, the single-acting pistons require a precise positioning and an accurate synchronization to realize a steady pressure trend. This will be taken into account when introducing the motion control and a model-based camming, both introduced later in this section. The pistons of both pumping units operate alternately with a phase shift of 180°, such as to produce a steady output uid ow. This guarantees a continuous operation with reduced pressure uctuations. The image of Figure 2.11 gives the physical setup of a pumping unit.

Servo motor

Spindle shaft Guiding Piston Pumping chamber

# # # # # #

Figure 2.11 Modular pumping unit: independent synchronous motor positioning a piston by means of a spindle shaft to pressurize water in its pimping chamber.

To position a piston, a spindle shaft transforms the rotative motion of a servo motor to a linear piston displacement. An angular motor velocity ω P results in a linear piston velocity

v P = h 2π ω P (2.1)
where h = ### m denotes the spindle pitch. Any piston displacement then induces a uid ow Q P = S P v P with respect to the piston surface S P . The pumping chamber limits the maximal available piston stroke s max = ### mm. A pressure p in the chamber acts on the piston surface and caused the force F = S P p that is transformed to a motor torque

T = h 2π F .
(2.2)

An absolute encoder measures the angular position θ P of the servo motor, which is mounted at the motor rotary shaft. Its angular velocity ω P is derived from low-pass ltering the time derivative of the position signal according to the rst-order lter

d dt ω P = a LP d dt θ P -ω P (2.3)
with lter coecient a LP chosen to obtain a bandwidth of 27.162 kHz. The angular position θ P and velocity ω P are feedback signals, eventually used for motion control.

Pump setup

The overview of Figure 2.12 shows the hardware setup of a electrically driven pump. It is realized by means of two pumping units, using the hardware mentioned in Table 2.8. Each pumping unit requires a frequency converter (FC), that has to realize the motion control task. It drives the synchronous motors to displace the pistons within the available piston stroke, using the feedback signals from absolute encoders. A power supply powers both frequency converters from mains. This setup allows for energy recuperation, considering both pumping units and its phase-shifted operation. That aims further to improve the energy eciency of high-pressure pumps.

The control architecture of an electrically driven high-pressure pump is given in diagram of Figure 2.13. Cutting heads will switch on and o with respect to the disturbance signal d(t). This results in a varying water consumption and induces pressure uctuations into the high-pressure network. These uctuations have to be compensated with the available high-pressure pumps. A pressure control has to regulate the measured pressure y(t) = p P (t) at the pump outtake to a steady reference pressure r desired for waterjet machining. The high-pressure pump will provide an output uid ow u g (t) proportional to the commanded pump rate u(t). Figure 2.12 High-pressure pump hardware setup: a PLC generates the position commands to synchronize both piston pumps by means of camming. The piston pumps are driven by independent frequency converters, which realize the motion control loops to displace the pistons, where each synchronous motor provides measurement data from encoders.

The camming follows a reference position trajectory, one for each pumping unit. This trajectory is dynamic and must be adjusted with respect to the reference pressure r. It denes the position command r gn (t) for motion control n with respect to the feedback position y gn (t) = s P n (t). The motion control consists of three feedback loops in series, regulating the piston position s P n (t), its velocity v P n (t) and the motor current i n (t). The pumping units alternately feed the high-pressure network such as to induce a continuous input uid ow u g .

High-pressure generation using single-acting pistons

The working principle of a high-pressure pump with two pumping units will be explained, while investigating the single-acting piston as depicted in the illustration of Figure 2.14. A check-valve in the intake duct passively connects the pumping chamber with the water supply. An other check-valve in the outtake duct passively connects the pumping chamber with the high-pressure piping. These check-valves restrict the uid ow to desired direction, from the water supply to the pumping chamber and from the pumping chamber to the piping. These check-valves allow for dening three pumping states for pressure generation: both check-valves will be closed to pressurize the pumping chamber (State 1), where the outtake check-valve will open to feed the high-pressure network (State 2), and the intake check-valve will open to ll the pumping chamber (State 3), while the outtake check-valve is closed.

Introducing the pressure generation by means of a piston pump, the model

p C = 1 C (Q V 1 -Q V 2 + Q P ) dt (2.4)
describes the pressure p C within a pumping chamber, as derived from [START_REF] Will | Hydraulik[END_REF].

The uid in a pumping chamber holds a capacity C. The stroke of the piston will change the chamber volume V C , providing a displacement uid ow Q P . The uid capacity C therefore denes the pressure change for any change of volume dp/dV . Any piston displacement with velocity v P induces a displacement uid ow Q P = S P v P with respect to the piston cross section S P . A check-valve at the intake duct enables an input uid ow

Q V 1 = σ(∆p V 1 ) H V 1 |∆p V 1 | (2.5)
to ll the pumping chamber, while an other check-valve at the outtake duct enables an output uid ow

Q V 2 = σ(∆p V 2 ) H V 2 |∆p V 2 | (2.6)
to feed the high-pressure piping. Considering the Heaviside-Function

σ(z) = 0, z ≤ 0 1, z > 0 ,
(2.7)

for ∆p V 1 → z and ∆p V 2 → z, respectively, represents the switching behaviour of a check-valve with respect to the pressure dierence ∆p V 1 = p S -p C over the intake duct and the pressure dierence ∆p V 2 = p C -p P over the outtake duct. The switching dynamics itself is neglected. The intake check-valve will open, if the pressure p C within the chamber decreases bellow the supply pressure p S . This is the case, when the piston moves in descending direction. The outtake check-valve will open, if the pressure within the chamber p C increases above the pressure of the piping p P . This is the case, when the piston moves in ascending direction. Therefore, an output ow Q V 2 results, depending on the pressure dierence ∆p V 2 over the outtake check-valve and an input ow Q V 1 results, depending on the pressure dierence ∆p V 1 over the intake check-valve. H V 1 and H V 2 denote to the characteristics of the check-valves and correspond to a uid ow resistance. Considering the model (2.4), pressure equilibrium within a pumping chamber will be obtained for

Q V 1 + Q P = Q V 2
and dp/dt = 0. A reciprocating piston displacement determines the passive switching of both check-valves, as depicted in Figure 2.15 (a) -(c). The corresponding pumping cycle is therefore assigned to the three pumping states:

Q V2 Q V1 Outtake pressure Intake pressure Precharge pressure (a) (b) (c)

Pumping state 1 (Pressurization)

The piston is entirely retracted, the pumping chamber is lled with water and its pressure is equal to the water supply. While the piston moves with velocity v P in ascending direction, the pressure within the chamber begins to increase. Both check-valves stay closed as long as the pressure in the chamber remains below the pressure of the high-pressure piping. This can be expressed with the conditions

σ(∆p V 2 ) = 0, ∆p V 2 < 0 (2.8a) σ(∆p V 1 ) = 0, ∆p V 1 < 0 (2.8b) that causes Q V 1 = 0 as well as Q V 2 = 0.
The piston position will determine the pressurization within a pumping chamber. This can be seen by applying conditions (2.8a) and (2.8b) for (2.5) and (2.6), while evaluating (2.4) that

p C = 1 C (S P v P ) dt = 1 C S P s P .
(2.9)

Hence, the piston position s P directly denes the resulting pumping chamber pressure before opening a check-valve.

Pumping state 2 (Hauling)

The piston continues to move and the pumping chamber is pressurized to the pressure of the high-pressure piping. The outtake check-valve will open since conditions

σ(∆p V 2 ) = 1, ∆p V 2 ≥ 0 (2.10a) σ(∆p V 1 ) = 0, ∆p V 1 < 0 (2.10b)
are met. The pumping chamber then feeds the high-pressure piping until the piston is fully extended. This can be seen by applying conditions (2.10a) and (2.10b) for (2.5) and (2.6), while evaluating (2.4) that

p C = 1 C S P s P -H V 2 |∆p V 2 | dt .
(2.11)

A pressure equilibrium will be obtained when the output uid ow Q V 2 equals to the induced uid ow Q P . Thus, the pump reaches a steady operating pressure, where it provides a continuous output ow, equal to the overall water consumption.

Pumping state 3 (Filling)

The piston is fully extended now and it begins to move in descending direction. The pressure within the pumping chamber rapidly decreases and the outtake check-valve will close. As soon as the pressure within the chamber drops bellow the supply pressure, the intake check-valve opens and the chamber will be relled with water, according to conditions

σ(∆p V 2 ) = 0, ∆p V 2 < 0 (2.12a) σ(∆p V 1 ) = 1, ∆p V 1 ≥ 0 , (2.12b)
until the piston is fully retracted. This can be seen by applying conditions (2.12a) and (2.12b) for (2.5) and (2.6), while evaluating (2.4) that

p C = 1 C S P s P + H V 1 |∆p V 1 | dt .
(2.13) Hence, the induced uid ow Q P from piston displacement becomes negative. The lling state is not of interest for modelling the high-pressure generation, thus the water supply is neglected. Introducing these 3 pumping states require the following assumptions:

Assumption 2.1. Realizing a proper pumping operation, it is desired that the pumping chamber is either connected to the water supply or to the high-pressure piping ∀ t > 0.

Assumption 2.2. An exception is given for the initial condition when assuming p C (t) = p S and p P (t) = p S . That causes all check-valves to remain open ∀ t = 0, if (2.7) holds.

Assumption 2.3. Instead of modelling the lling of a pumping chamber over a intake checkvalve, it is here assumed that the pressure p C (t) within a pumping chamber cannot drop below the supply pressure p S , such as p C (t) ≥ p S ∀ t.

Assumption 2.4. Further assuming a water supply of innite capacity allows for considering the supply pressure p S to remain constant, in particular p S ≈ 0.4 MPa.

Coupling of N pumping units

High-pressure pumps combine several pumping units to produce a continuous output uid ow.

To obtain this, every piston is synchronized with respect to the previously introduced pumping states. A position trajectory will be generated for each piston and realized by means of motion control. This trajectory will determine the pressure generation within a pumping chamber and consequently passively aecting the check-valve switching. Any positioning error of the piston displacement immediately aects the displacement uid ow, what induces pressure uctuations.

First considering the camshaft of Figure 2.16 to coupled N pistons of a high-pressure pump. It introduces a constant phase-shift of 2π/N between each pumping unit. This allows for dening the induced overall uid ow

Q P (t) = N n=1 Q V n (t)
(2.14) of a high-pressure pump as a superposition of the sinusoidal output uid ows Q V n (t) of every single pumping unit n, subject for static synchronization.

A displacement uid ow

Q P n (t) = d dt F φ(t) + 2π n N 2 N S P (2.15)
feeds its pumping chamber with respect to the camshaft angular position φ(t), where the angular speed ω(t) = dφ(t)/dt = f (u(t)) is manipulated by the control variable u(t) to generate a desired overall uid ow Q P (t). The camming function F(•) denes the phase-shifted trajectories, which map the angular camshaft position φ to the individual piston displacements s P n . A dynamic camming function will be further discussed in the subsequent section, when introducing the model-based synchronization.

The chamber pressure

p Cn (t) = 1 C (Q P n (t) -Q V n (t)) dt (2.16)
is obtained with respect to (2.4) and Assumption 2.3, while considering an output uid ow

Q V n = σ(∆p V n ) H V n |∆p V n |
(2.17) with respect to (2.6) and Assumption 2.1. This uid ow depends on the pressure dierence ∆p V n (t) = p Cn (t) -p P (t) between chamber pressure p Cn (t) and piping pressure p P (t). From check-valve switching given by (2.7) follows that

Q V n (t) ∈ R + ∀ t.

Model-based synchronization by means of camming

The model-based synchronization will be discussed for a high-pressure pump with two singleacting pistons (N = 2), as proposed in [START_REF] Niederberger | Energy-ecient high-pressure pump[END_REF]. The camming generates reference trajectories for each piston with respect to the desired operating pressure. It replaces the static coupling of a camshaft and synchronizes both pumping units to generate a continuous output uid ow. Similar trajectories are also found in [START_REF] Ferretti | Abrasive waterjet intensier model for machine diagnostics[END_REF] to predict the piston positions for single-acting intensier pumps, used for condition monitoring.

Trajectories are chosen with respect to Assumption 2.1 = 2.4, such as only one pumping chamber is coupled to the high-pressure network at a time. The plots in Figure 2.17 present the position (a), velocity (b) and acceleration (c) references for a typical pumping cycle. Such a reference trajectory represents the previously dened pumping states: Pressurization takes place in interval t 1 , t 2 and t 3 , hauling for t 4 and t 5 , lling for t 6 , t 7 and t 8 , while a piston is ascending to its maximal stroke s max and descending to its initial position s 0 = 0. The trajectory of a pumping cycle is then dened by the piston positions s(t i ) for an interval t i of a cycle i. This periodic pumping cycle can be seen as a master axis which realizes a reciprocating piston displacement with respect to the virtual camshaft angle φ = [0 ... 2π]. The pistons are considered as slave axis realizing a piston stroke s P n = [0, s max ]. To synchronize both pumping units, the slave axes are coupled to a master axis with a phase shifted of π. This concept is known by the industry as camming. It enables a dynamic synchronization of both pistons with respect to the desired operating pressure. That aims at a continuous output uid ow Q P (t) = [0, Q max ] for a given pump rate u(t) = [0, 1], when steady-state pressure is obtained. The position trajectory of a pumping cycle must full the following conditions to avoid piston collision in the pumping chamber and pressure uctuations in the high-pressure piping due to couplings between pumping units:

p P Q P v P1 ~ Q P1 v P2 ~ Q P2 Q V1 Q V2 p C1 p C2 ω = f(u) Camshaft v PN ~ Q PN Q VN p CN 2 π / N 4 π / N 2 π
1. The piston position s must be in between the available stroke for all time (0 < s < s max ).

The available stroke must be reached by the end of hauling (t 5 ). And the initial position must be reached by the end of lling (t 8 ), such as

s 1 + s 2 + s 3 + s 4 + s 5 = s 6 + s 7 + s 8 = s max (2.18)
2. A specic piston position s = f s (p) must be reached before deceleration at the end of pressurization (t 2 ) to obtain the desired operating pressure p, this requires

s 1 + s 2 = s .
(2.19) 3. A specic piston velocity ṽ = f v (Q) must be reached before hauling (t 3 ) that corresponds to the desired overall uid ow Q. This velocity must be held constant while hauling v 4 = ṽ .

(2.20)

4. The acceleration at the end of pressurization (t 3 ) and the deceleration at the end of hauling (t 5 ) are symmetric and simultaneous (regarding the two phase shifted pistons), so that the overall ow rate will stay constant and

t 5 = t 3 .
(2.21)

5. The hauling (t 3 , t 4 ) will last half if the pumping cycle. Therefore always one piston pump is continuously hauling at a time. The lling and pressurization will last for the other half the of cycle, such as

t 4 + t 5 = t 1 + t 2 + t 3 + t 6 + t 7 + t 8 = π .
(2.22)

It is further the highest available acceleration ã applied, to maximize the resulting interval t 4 for hauling and to minimize the desired peak velocity v 7 for lling. Taking the above conditions (2.18 -2.22) into account, allows for deriving the time intervals in terms of ã, ṽ, s and s max , while considering the 1-dimensional kinematics of piston displacement (v = a dt and s = v dt), hence the timing for all periods becomes

            t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8             = 1 2π                s ã s ã ṽ/ã (smax-s) ṽ -ṽ ã ṽ/ã -z -z2 -smax ã smax ã(-z- √ z2 -smax ã ) + z + z2 -smax ã -z -z2 -smax ã                , (2.23)
as derived in [START_REF] Niederberger | Energy-ecient high-pressure pump[END_REF] by solving a quadratic equation, where

z = s ã - (s max -s) 2ṽ + ṽ 2ã .
(2.24)

The corresponding velocities v(t) and positions s(t) will be derived by integration of the acceleration a(t) over the intervals t. The trajectory design can be further extended by introducing intermediate time intervals. This allows for optimizing a trajectory for limited jerk, but requires to solve a cubic equation.

A single reference value u(t), the master axis cycle frequency, will command the high-pressure pump. An increasing cycle frequency will cause a faster pumping cycle due to the assignment

s P n (t) = F φ(t) + 2π n N ,
(2.25)

φ(t) = u(t) dt mod 2π .
(2.26)

This adjusts the piston velocity to realize the desired uid ow rate for hauling. Both pistons attain a stroke within 2π-period of a full camshaft rotation at a phase shift of π. This phase shift guarantees the decoupling of both pumping chambers, assuming a perfect synchronization.

Assumption 2.5. It is assumed that a π-periodic ramp function

s(φ) := 0, φ(t) = 0 or φ(t) = π smax π φ(t), 0 < φ(t) < π
(2.27)

approximates the reciprocating operation for two pistons, where φ(t) ∈ [0, π] refers to the angular position of half a camshaft rotation. The resulting piston velocity

v(t) = d dt φ(t) d dφ s(φ) = ω(t) d dφ s(φ)
(2.28) of two interconnected pumping chambers is then proportional to the camshaft angular speed ω(t) and the high-pressure piping is continuously feed, such as

Q P ∼ u(t).
Assumption 2.6. Steady-state around desired operating pressure requires uid ow equilibrium

Q P n = Q V n .
This yields that the piping pressure p P (t) maintain at operating pressure and that the pressure within a pumping chamber p Cn (t) has reached operating pressure, when the check-valve to the high-pressure piping opens, thus

p Cn (φ(t) = 0) = p P n (t) .
(2.29)

A perfect synchronization (Assumption 2.5) allows for evaluating the two pumping units by the overall induced uid ow Q P (t) ∼ ṽ. Each pumping chamber becomes exactly pressurized to the desired operating pressure and a continuous overall uid ow will occur, if Assumption 2.6 holds. As a consequence, the piston position s = f s (p) varies for every operating pressure p, resulting in dierent position trajectories, as shown in plot of Figure 2.18. The relation between chamber pressure p C and piston position s is given in the plot of Figure 2.19. It corresponds to the function f s (p) that represents the model used to synchronize the pumping units. This model will be introduced in Section 4.2. On the other hand, the desired piston velocity ṽ to derive a steady-state uid ow Q P depends on the number of pumping units N and the piston cross section S P , such as

f v (Q) = 2 N -1 Q S -1 P .
Figure 2.19 Relation between chamber pressure and piston displacement: expected position position to pressurize a pumping chamber.

The electrically driven high-pressure pump with independent pistons is prevalent to realize a virtual camshaft by means of model-based camming. However, it is important to understand, that it is not possible to excite the high-pressure piping directly by the pistons velocities. A reference trajectory is always desired to synchronize all pumping units and to guarantee a continuous overall uid ow at the pump outtake. If synchronization fails, the uid ow will vary even when a constant pump rate is demanded (du(t)/dt = 0). This will induce undesired pressure uctuations.

Motion control

Motion control tracks the reference trajectory from camming with minimal position error. The resulting closed-loop dynamics enables a wide bandwidth of 131.8 Hz. This is required to deploy desired performance for disturbance rejection. Figure 2.20 shows the control structure, which is available with the frequency converter introduced in Section 2.3. It consists of a position control loop, followed by a velocity control loop and a torque control loop.

The position controller provides the reference for the velocity controller and the velocity controller provides the reference for the torque controller. The command value r gn for the position controller is maintained by camming, while the commanded velocity and acceleration, subject to feed forward, are derived by dierentiation of the reference position trajectory. The feedforward gains F v and F a apply these command values to the corresponding control loops.

Proportional and integral action (PI) is applied for the position and velocity loop, where only proportional action (P) is implemented to the torque loop, while following the controller tuning discussed in [START_REF] Ramasamy | PID controller tuning for desired closed-loop responses for SISO systems using impulse response[END_REF]. The position feedback is measured by absolute encoders. The velocity feedback lter F s is given by Equation (2.3). A load observer estimates the motor torque and velocity for feedback to the control loops. This observer-based control improves the noise sensitivity of the torque and velocity feedback signals, as discussed in [START_REF] Yang | Nonlinear disturbance observer based control for systems with arbitrary disturbance relative degree[END_REF], and enables an increased closed-loop bandwidth. The acceleration output from velocity loop is converted to an equivalent torque with the torque scalar K J = J, while the torque output is scaled to an equivalent motor current with the current scalar

K t = 1/(N K),
where J is the motor inertia, K its motor constant and N denotes the pole pair number.

The controller gains are tuned with respect to the reference trajectories by successive tuning, such as a closed-loop bandwidth of 131.8 Hz and an angular position error of < 0.002°has been obtained. This corresponds to a linear position error of < 6.4 µm. The obtained parametrisation of the control loops is listed in Table 2.9 Table 2.9: Parametrization of motion control: control loop bandwidth, forward gains, feedback lter and load observer.

Pressure uctuations and check-valve hysteresis

The synchronization of two interconnected pumping units by means of model-based camming and its eect on the resulting pressure trend is exemplary shown with two experiments. Both experiments have been conducted with the same position trajectories. These trajectories are optimized for an operating pressure of 350 MPa, which refers to the trajectory reference value. A nozzle of 0.3 mm inner diameter has been installed to the cutting head. Consequently, a displacement uid ow of 35.5 cm 3 /s would be ideal to generate a pressure of 350 MPa. However, the pump rate has been slightly modied so that experiment 1 would clearly show the eects of over excitation, whereas experiment 2 would cause an under excitation. Both experiments are derived by means of simulations, using a detailed simulation model, and have been veried with measurements from the test bench.

Experiment 1) Over excitation: the displacement uid ow has been raised to 37.0 cm 3 /s to obtain an increased pressure steady-state of about 380 MPa. The insucient pressurization leads to a periodic pressure loss towards 350 MPa, as can be observed in Figure 2.21 (a). However, the pressure will asymptotically rise towards the desired steady-state value, once the check-valve of the precharged pumping chamber has been pushed open.

Experiment 2) Under excitation: the displacement uid ow has been reduced to 34.5 cm 3 /s. This results in a decreased pressure steady-state of about 320 MPa. The excessive pressurization causes a periodic pressure overshoot towards 350 MPa, of Figure 2.21 (b). Complementary to the rst experiment, the pressure will asymptotically fall towards the desired steady-state value, once the check-valve is closed. The pressure generation is sensitive to any deviations of the desired displacement ow. The pumping chambers with the pressure states p P 1 and p P 2 aect the overall pump output pressure p P . The pressure dierences ∆p V 1 and ∆p V 2 over the corresponding check-valves restrict the induced uid ows Q V 1 and Q V 2 to positive directions. On the other hand, the piston displacement uid ows Q P 1 and Q P 2 follow the three states of a pumping cycle, as previously introduced.

Detailed pressure trends of both experiments are shown in Figures 2.22 (a) and (b). The three pumping states of pressurization, hauling and lling are clearly distinguished. A checkvalve becomes passively controlled by any piston displacement that pressurizes the corresponding pumping chamber. Consequently, a piston displacement indirectly aects the coupling between the pumping chambers and piping. Depending on the amount of pressurization, pressure losses or pressure overshoots can occur, whenever a check-valves opens too early or, respectively too late. A similar behaviour is reported in [START_REF] Trieb | Pressure and ow rate uctuations at high pressure intensier pumps[END_REF] for the pressure generation of intensier pumps. To avoid any interaction between pumping chambers, a precise synchronization of the pumping units is required. This is realized by means of model-based trajectories (camming) and a precise motion control.

Conclusion

This chapter presented the test bench established in the laboratory. This test bench exclusively serves for parameter identication, model validation and controller verication, as related to the subsequent research work. The standard symbols, introduced in Section 1.2, have been applied to describe network topologies and cutting head setups, which provide the test bench congurations. The previously presented waterjet applications have been taken into account, when dening use cases suitable to investigate future waterjet facilities. It considers up to two high-pressure pumps interconnected to two independent cutting heads. These use cases will be taken into account for measurements at the test bench as well as for simulations.

The design and implementation of a hierarchical control design for the novel electrically driven high-pressure pump is the major contribution of this chapter. This requires to understand the functional principle of a pumping unit, for which three pumping states have been dened.

The pump design and the underlying motion control has been presented rst, which aims at a precise displacement of pistons by means of servo motors. Providing the relation for highpressure generation with coupled pumping chambers and introducing essential assumptions allow for generating reference trajectories, which avoid interactions between pumping units and realize a continuous output uid ow. These trajectories are subject for tracking, using motion control.

The introduction of the model-based synchronization is another contribution that allows for considering a high-pressure pump as a continuous uid ow source. This becomes valuable for low-level control design (Section 5.3) and for high-level managing of multiple interconnected pumps (Section 7.2). The presented phase-shifted trajectories give a understanding for the synchronization of two pumping units. This is required to operate the pump prototypes installed in the test bench. These position trajectories have been further improved, realizing an increased acceleration by introducing a maximal jerk limit. This maximizes the achievable overall uid ow, generated by a high-pressure pump. However, this leads to a higher-order problem that requires optimization to nd a distinct solution of a feasible trajectory. These optimized trajectories have been implemented to the PLC, using cubic splines to generate the position references for motion control. Further, adaptive trajectories are required to minimize any pressure uctuations when switching between pumping units. This is eventually realized by look-up tables, which provide reference piston positions for cubic interpolation. That generate variable position trajectories, taking the desired operating pressure into account.

Part II Modelling framework 49

Introduction

Objectives Future waterjet facilities combine several decentralize pumps to span high-pressure networks of arbitrary topology. To congure these high-pressure networks and even more, to develop a suitable control architecture that deals with decentralized pumps, requires a exible and scalable modelling methodology. This modelling methodology must also consider non-linearities and varying parameters for a wide pressure range of 40 to 400 MPa, as well as various setups for waterjet machining. In addition, it must ensure the exible modelling of arbitrary network topologies that interconnect any number of decentralized pumps and work stations.

Common engineering tools, such as the Simscape toolbox from MATLAB Simulink, Modelica or AMESim, are well established to model and design very specic dynamic systems, as applied in [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF] and [START_REF] Ferretti | Abrasive waterjet intensier model for machine diagnostics[END_REF]. However, modelling becomes time consuming, since it requires to derive an individual model for every waterjet facility. These standard libraries include various hydraulic system components, but the specic high-pressure components, used for waterjet machining, are not always available. As a consequence, it misses validation and parametrisation with respect to the non-linearities, as given by a wide pressure range of various waterjet applications.

The main objective is to provide a exible modelling methodology for scalable waterjet facilities, while taking various high-pressure components into account and providing a procedure for modelling of dierent high-pressure networks. The parametrization of these high-pressure components should consider a wide pressure range. As a consequence, the varying parameters have to be described with suitable mathematical models and experimentally identied, using the high-pressure test bench of Section 2.2.

State of the art

Modelling high-pressure networks for dierent waterjet facilities is a time-consuming task. A prevalent approach describes the uid dynamics with respect to the principle of continuity and momentum conservation, as found in various text books such as [START_REF] Streeter | Fluid Mechanics[END_REF]]. This modelling by means of algebraic dierential equations is applied for very specic highpressure systems, such as for a high-pressure pump supplying a single workstation (see [Momber, 1995], [START_REF] Tremblay | Modelling and simulation of pressure uctuations in high pressure waterjets[END_REF], [START_REF] Fabien | Dynamic modelling and identication of a waterjet cutting system[END_REF] and [START_REF] Ferretti | Abrasive waterjet intensier model for machine diagnostics[END_REF]) or for common rails of a diesel engines (see [START_REF] Hountalas | Development of a fast and simple simulation model for the fuel injection system of diesel engines[END_REF]], [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF], [START_REF] Wang | Simulation and experimental study on high pressure common rail fuel system of diesel engine[END_REF] and [START_REF] Wang | High pressure common rail injection system modeling and control[END_REF]). These approaches oers a limited exibility to cope with various network topologies, since a mathematical model has been obtained for a selected high-pressure component. However, it is well established that graph-based approaches improve the modelling exibility for networked systems [START_REF] Mohammadpour | Efcient Modeling and Control of Large-Scale Systems[END_REF]. Considering distributed parameter systems, the algebraic dierential equations will be assigned to graphs to recover the system dynamics, while considering various network topologies, as proposed in [Aalto, 2008] and [START_REF] Corbet | A model for simulating adaptive, dynamic ows on networks: Application to petroleum infrastructure[END_REF] to describe the uid ow dynamics in piping networks or in [START_REF] Koeln | Experimental validation of graph-based modeling for thermal uid power ow systems[END_REF] and [START_REF] Pangborn | Graph-based hierarchical control of thermal-uid power ow systems[END_REF] to combine thermal and uid ow systems. Nevertheless, describing high-pressure networks by means of graphs seems less investigated.

High-pressure systems

Dealing with waterjet machining, a model for intensier pumps with high-pressure attenuators is presented in [START_REF] Tremblay | Modelling and simulation of pressure uctuations in high pressure waterjets[END_REF]. Experimental studies reveal the eect of bulk modulus and uid density for pressure uctuations at 153.4, 181.0 and 215.5 MPa. Both are uid characteristic parameters, which vary for a wide pressure range. The uid density aects the uid ow characteristics and causes to change the pressure steady-state. The bulk modulus denes the uid compressibility and determines the transient pressure behaviour. More than a decade later, a lumped parameter model is adopted in [START_REF] Fabien | Dynamic modelling and identication of a waterjet cutting system[END_REF] for a similar setup, where a pressure-dependent uid density is taken into account. For this purpose, model parameters have been experimentally identied with measurement data at 153.4 MPa. Other researchers have modelled high-pressure pumps for diagnostics and condition monitoring e.g. using Modelica for a multi-domain simulation of a phased intensier pump (see [START_REF] Ferretti | Abrasive waterjet intensier model for machine diagnostics[END_REF]). They also investigated the pressure generation in relation to the piston displacement, such as to develop a model-based fault detector. Nevertheless, most authors focus on the modelling of waterjets and the resulting material removal to enhance the eciency of waterjet machining, while the highpressure generation remained ignored. For example, a generalized material removal model has been proposed in [Momber, 1995] to study various particle sizes and uid ow rates for abrasive waterjet cutting.

The study of automotive applications revealed that fuel injection systems for combustion engines have a similar setup as common waterjet facilitites: a high-pressure pump supplies injectors, which are interconnected by means of piping. A direct injection engine with an operating pressure of about 50 MPa is modelled in [START_REF] Hountalas | Development of a fast and simple simulation model for the fuel injection system of diesel engines[END_REF]]. Reasonable accuracy has been realized by introducing a pressure-dependent bulk modulus. More recent studies deal with common rail injection systems. Many of them focus on the injection process, e.g. a MATLAB Simulink model is utilized in [START_REF] Wang | Simulation and experimental study on high pressure common rail fuel system of diesel engine[END_REF] to study the injection rate, while others are interested in high-pressure generation and control. A mathematical model for control design and its implementation, using MATLAB Simulink, is presented in [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF]. This modelling considers a variable bulk modulus. It is experimentally validated for a pressure range of 30 to 90 MPa and veried with multi-domain simulations in AMESim. For control design, a mathematical model has been also developed in [START_REF] Wang | High pressure common rail injection system modeling and control[END_REF] and validated using AMESim for a pressure of 150 MPa. However, the validation results reveal some discrepancies regarding the transient behaviour of pressure generation.

An accurate simulation of a high-pressure system requires to introduce pressure-dependent parameters, such as a variable bulk modulus and uid density. Consequently, the available modelling must be extended to cope with high-pressure networks of arbitrary topologies without increasing the modelling eort. A promising approach is the graph-based modelling.

Graph-based modelling

Graphs are often used to represent vehicular trac, data packet ow, power lines and many more. Various text book, see for example [START_REF] Mohammadpour | Efcient Modeling and Control of Large-Scale Systems[END_REF] and [START_REF] Ford | Flows in Networks[END_REF], provide an introduction to the general framework and its application. Improving the modelling for a trac system, it is proposed in [Mercier, 2009] to assign a hyperbolic conservation laws to a graph. The graph represents the trac network, while considering the dynamics of trac ow. In particular, the ow of data packets is often represented with graphs. By extending this, it is proposed in [START_REF] Espitia | Fluid-ow modeling and stability analysis of communication networks[END_REF] to model communication networks by means of uid dynamics and the use of uid dynamic models for data networks is validated in [START_REF] Manzo | On the validity of uiddynamic models for data networks[END_REF]. Compartment modelling is suggested in [START_REF] Mounier | Compartmental modelling for congestion control in communication networks[END_REF] and [START_REF] Bastin | Congestion control in compartmental network systems[END_REF], where partial dierential equations has been introduced in [D' Apice et al., 2008] to describe the packet evolution. More recent work deals with stability issues and the control of compartments, see [START_REF] Briat | A conservation-law-based modular uid-ow model for network congestion modeling[END_REF] and [START_REF] Bastin | Exponential stability of networks of density-ow conservation laws under PI boundary control[END_REF].

The claim of an universal model, for simulating a piping network of any topology, reaches back to [START_REF] Kralik | A universal dynamic simulation model of gas pipeline networks[END_REF], presenting a node-oriented concept of discretization in space and introducing a continuity equation to describe the uid ow. Partial dierential equations are assigned by [Aalto, 2010] to describe the uid ow along piping sections, which have been interconnected to a network. Whereas in [START_REF] Corbet | A model for simulating adaptive, dynamic ows on networks: Application to petroleum infrastructure[END_REF], the transient solution for a diusion equation is mapped to a graph, describing the uid ow of petroleum infrastructures. Each node corresponds to a storage and a branch denes the uid ow between storages.

A graph-based approach is presented in [START_REF] Koeln | Experimental validation of graph-based modeling for thermal uid power ow systems[END_REF] and [START_REF] Pangborn | Experimental validation of graph-based hierarchical control for thermal management[END_REF], modelling thermal and uid ow systems with respect to the conservation of mass and energy. Conservation laws are investigated in [START_REF] Borsche | ADER schemes and high order coupling on networks of hyperbolic conservation laws[END_REF], using graphs to represent spatially distributed physical systems as lumped parameter models. The Riemann problem for networked systems, derived from the above conservation laws, is discussed in [START_REF] Garavello | Conservation laws on complex networks[END_REF]. Numerical methods to eciently solve the Riemann problem are investigated in [START_REF] Borsche | High order numerical methods for networks of hyperbolic conservation laws coupled with ODEs and lumped parameter models[END_REF]. Such models are often applied to thermal systems, e.g. [START_REF] Ramallo-González | Lumped parameter models for building thermal modelling: An analytic approach to simplifying complex multi-layered constructions[END_REF]. Moreover, the application of conservation laws and balancing equations to graphs is described in [van der Schaft and Maschke, 2009]. A node corresponds to a system state that associates the conservation law, while each branch balances the dierences between system states. This aims at a general class of physical network systems, taking mechanics, hydraulics and heat transfer into account. This approach is further generalized to passive physical systems using Hamiltonian formulation, see [van der Schaft, 2017].

On the one hand, there are two approaches for high-pressure generation, taking a wide pressure range into account. Modelling approaches for waterjet facilities consider a variable uid density, where common rail models assume a variable bulk modulus. However, neither of those approaches have been found to perform sucient to reproduce pressure uctuations for a wide pressure range of several 100 MPa. On the other hand, numerous studies assign dynamic equations to graphs. Introducing in this way a exibility and scalability to model networked systems of various topology. However, a exible modelling methodology that considers a wide pressure range with reasonable accuracy remains unattained, especially for high-pressure networks.

Contribution

This chapter proposes a graph-based methodology that models high-pressure networks of various topologies. It assigns high-pressure components of common waterjet facilities to homogeneous segments, each representing a local pressure state with respect to the principle of momentum conservation and uid ow continuity. The proposed methodology introduces to the governing equation a pressure-dependent uid density as well as a pressure-dependent bulk modulus. This aims to improve the simulation accuracy for a wide pressure range. The segments are subsequently interconnected along the uid ow path by means of graphs. These graphs are exible to model dierent network topologies. Where each node represents a pressure state, every branch allocates a uid ow to the interconnections. This results in lumped parameter models that approximate the non-linear characteristics of various high-pressure networks.

The major contribution is given when presenting the graph-based modelling methodology for high-pressure networks. This requires to introduce both variable parameters (bulk modulus and uid density) into the governing equations of uid dynamics and to assign these modied equations to a graph. In this way, the graph-based modelling oers a exibility and scalability to cope with dierent network topologies of various waterjet facilities. This work further identies pressure-dependent uid characteristic as well as various parameters, related to the high-pressure components. In this way, various high-pressure networks can be modelled for a wide pressure range without revising the initial parameter identication.

This chapter is structured as follows: Section 3.2 provides the fundamentals to describe the uid ow dynamics. These fundamentals are used to derive a distributed parameter model. Using the distributed parameter model allows for investigating pressure uctuations in a piping by means of numerical simulations. To derive a generalized model for high-pressure networks, Section 3.3 introduces the expected parameter variations which are taken into account to derive a lumped parameter model. On this place, the proposed graph-based modelling methodology is presented. Experiments presented in Section 3.4 introduce the pressure-dependent parameters and identify specic high-pressure components.

Fluid dynamics and modelling fundamentals

This section is dedicated to the modelling principles describing a 1-dimensional uid ow. The principles of uid ow continuity and momentum conservation are shown in [START_REF] Streeter | Fluid Mechanics[END_REF] to describe the uid ow dynamics of a piping section. The resulting equations have been simplied when approximating its partial dierentials, using nite dierences, and assuming a stationary uid ow. This yields a lumped parameter model, which represents the pressure state of homogeneous segments by means of rst order dierentials.

Introducing the fundamentals of uid dynamics, the piping section of Figure 3.1 is considered. A uid segment of initial length s 1 moves with a velocity v 1 along a piping section at height h 1 and of cross section S 1 . The pipe will ascent to a higher position h 2 and widen to a cross section S 2 . This causes the uid to decelerate to a velocity v 2 and a segment of reduced length s 2 . Common literature, e.g. [START_REF] Böswirth | Technische Strömungslehre[END_REF]] and [Spurk, 2013], describe this situation by assuming an incompressible uid ow that implies a constant uid density .

p 1 Q 1 = S 1 v 1 p 2 Q 2 = S 2 v 2 s 1 = v 1 Δt s 2 = v 2 Δt S 1 S 2 h 1 h 2 Figure 3
.1 A simple piping section to introduce the Bernoulli equation: an initial uid segment is moved to a subsequent position while conserving its momentum.

A rst principle, the conservation of momentum, is given by the Bernoulli equation [Spurk, 2013]

p 1 static + 2 v 2 1 dynamic + s 2 s 1 ∂ ∂t v 1 ds transient + g h 1 head = p 2 + 2 v 2 2 + s 2 s 1 ∂ ∂t v 2 ds + g h 2 + ∆p v loss . (3.1)
Considering the left-hand side, the rst term corresponds to the static pressure p 1 of an initial uid segment, the second term describes its dynamic pressure with respect to the uid ow velocity v 1 and the third term is dedicated to its transient ∂v 1 /∂t, where the last term includes the pressure head h 1 . The right-hand side of (3.1) includes an additional pressure loss term ∆p v due to the uid ow resistance along the piping section. The applications for waterjet machining (see Section 1.2) allow for a fundamental assumption:

Assumption 3.1. The induced static pressure is assumed much greater than any pressure head, hence the pressure head g h i ≈ 0

(3.2) can be neglected, thus the transient term

s 2 s 1 ∂ ∂t v i ds ≈ 0 (3.3)
becomes small and a stationary uid ow results.

By considering Assumption 3.1, the Bernoulli equation (3.1) is then reduced to

p 1 + 2 v 2 1 = p 2 + 2 v 2 2 + ∆p v , (3.4)
which represents the characteristics of a stationary uid ow along a horizontal piping.

A second principle, the uid ow continuity, implies that the sum of all uid ows Q i passing a piping section remains unchanged such as

i Q i = 0 . (3.5)
Hence, the input uid ow Q 1 = S 1 v 1 entering to a piping section equals to the output uid ow Q 2 = S 2 v 2 leaving the pipe, where S i denotes the piping cross section at position i ∈ {1, 2}.

With respect to the uid ow continuity (3.5), any expansion of cross section S causes a decrease of the uid ow velocity v, while the uid ow Q remains unchanged. Thus, introducing Q 1 = Q 2 into the conservation of momentum (3.4) yields that any change of the dynamic pressure causes a change of the static pressure.

Distributed parameter modelling

Investigating the uid dynamics along a piping section of length L and constant cross section S requires another fundamental assumption.

Assumption 3.2. The cross section inner diameter D(x) remains signicantly smaller than the piping length L, such as D(x) L ∀ x = [0, L], whereas the cross section remains unchanged along a piping section, such as ∂D(x)/∂x ≈ 0.

The illustration of Figure 3.2 considers a uid segment ∂x that holds a nite volume ∂V = S ∂x. This piping determines a 1-dimensional uid ow path and denes the direction of positive uid ow along the position coordinate x, hence the transient behaviour of a uid segment is described by a system of two partial dierential equations [START_REF] Streeter | Fluid Mechanics[END_REF]]: The rst equation is known as equation of continuity

∂ ∂t p(x, t) + v(x, t) ∂ ∂x p(x, t) -a 2 ∂ ∂x v(x, t) = 0 , (3.6)
taking the potential energy into account and the second equation is known as equation of motion

∂ ∂t v(x, t) + v(x, t) ∂ ∂x v(x, t) - 1 ∂ ∂x p(x, t) + λ 2D v(x, t) |v(x, t)| -g sin α = 0 , (3.7)
describing the kinetic energy. Both partial dierential equations depend on the position coordinate x and time t. The coecient a is the characteristic speed of sound of the uid, λ denotes the friction loss coecient and D the cross section inner diameter. The squared uid ow velocity v(x, t) induces a uid ow resistance and α denotes the horizontal angle of a piping section that takes into account the eect of any pressure head. The speed of sound a and density are uid characteristic constants, while the diameter D, coecient λ and angle α are parameters of specic high-pressure components, such as a piping. It is further noted that the resulting uid ow resistance is proportional to the squared velocity, with respect to the uid ow direction [START_REF] Streeter | Fluid Mechanics[END_REF].

v(x,t) p(x,t) L S ∂x D x Figure 3
.2 A piping section of constant cross section S: considering a uid segment ∂x of pressure state p(x, t) and uid ow velocity v(x, t) with respect to the position coordinate x and time t.

Two assumptions have been obtained to facilitate its practical application.

Assumption 3.3. The rst order partial derivative of the pressure p(x, t) with respect to the position variable x is small compared to the corresponding time derivative, this yields

∂ ∂x p(x, t) ∂ ∂t p(x, t) .
(3.8)

Assumption 3.4. The rst order partial derivative of velocity v(x, t) with respect to the position variable x is small compared to the time derivative, that yields

∂ ∂x v(x, t) ∂ ∂t v(x, t) .
(3.9) By Assumption 3.3, it is very common to eliminate the cross term of (3.6). Likewise, the cross term of (3.7) will be eliminated with respect to Assumption 3.4. On the other hand, from Assumption 3.1 follows a horizontal piping, such as α ≈ 0 . These assumptions reduce (3.6) and (3.7) to obtain the governing system of partial dierential equations

∂ ∂t p(x, t) -a 2 ∂ ∂x v(x, t) = 0 (3.10a) ∂ ∂t v(x, t) - 1 ∂ ∂x p(x, t) + λ 2D v(x, t) |v(x, t)| = 0 , (3.10b)
that describes the dynamics of a uid segment. The dependent variables p(x, t) and v(x, t) refer to the pressure state and to the uid ow velocity, respectively. Both partial dierential equations are coupled since any dierence of uid ow velocity induces a pressure change and any pressure dierence causes a uid ow. This modelling approach has been found in dierent studies related to high-pressure systems, e.g. [START_REF] Tremblay | Modelling and simulation of pressure uctuations in high pressure waterjets[END_REF], [START_REF] Hountalas | Development of a fast and simple simulation model for the fuel injection system of diesel engines[END_REF]], [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF] and [START_REF] Gupta | Modeling and control of a novel pressure regulation mechanism for common rail fuel injection systems[END_REF]. It presents the transient characteristics of uid ow along a horizontal piping.

Investigations with nite element methods (FEM) often model an ideal uid ow without any friction losses, thus assuming λ ≈ 0. This reduces (3.10a) and (3.10b) to

∂ ∂t p(x, t) = a 2 ∂ ∂x v(x, t) (3.11a) ∂ ∂t v(x, t) = 1 ∂ ∂x p(x, t) , (3.11b)
a system of coupled partial dierential equations without any dissipative term. The principle of uid ow continuity (3.11a) indicates that a velocity gradient along the position coordinate ∂v(x, t)/∂x causes a pressure variation in time ∂p(x, t)/∂t. The momentum conservation (3.11b) shows that a pressure gradient along the position coordinate ∂p(x, t)/∂x will induce a velocity variation in time ∂v(x, t)/∂t, which means that a uid becomes either accelerated or decelerated.

Discretization in space and implementation

The system of partial dierential equations (3.11a) and (3.11b) is investigated with numerical simulations using MATLAB. This requires the discretization of a piping section with length L into n homogeneous uid segments of length ∆x, such as L = n ∆x. For that, the partial dierentials in space ∂f (x)/∂x become approximated with nite dierences. This requires that the length ∆x of a uid segment is signicantly smaller as the piping inner diameter D, such as

∆x D.
The nite dierences are obtained by Taylor expansion

f (x ± ∆x) = f (x) ± ∂ ∂x f (x)∆x + ∂ 2 ∂x 2 f (x)
∆x 2 2! ± ... , (3.12) which can be evaluated for its backward dierences f (x -∆x) when considering a preceding segment as well as for its forward dierences f (x + ∆x) when considering a subsequent segment.

Subtracting the forward dierences and the backward dierences, obtained from (3.12), yields

f (x + ∆x) -f (x -∆x) = f (x) + ∂ ∂x f (x)∆x -f (x) + ∂ ∂x f (x)∆x , (3.13)
while higher order terms have been assumed to be small. Rearranging (3.13) yields the central dierences

∂ ∂x f (x) ≈ f (x + ∆x) -f (x -∆x) 2 ∆x , (3.14)
which is the symmetric approximation of a partial dierential [START_REF] Langtangen | Finite Dierence Computing with PDEs[END_REF]. Substituting the partial dierentials in (3.11a) and (3.11b) yields

d dt p(x, t) = a 2 v(x + ∆x, t) -v(x -∆x, t) 2 ∆x (3.15a) d dt v(x, t) = 1 p(x + ∆x, t) -p(x -∆x, t) 2 ∆x .
(3.15b)

The notation f (x + n∆x) := f (n), allows for dening a nite dierence scheme

∂ ∂x    f (1)
. . . (3.16) which evaluates each uid segment n = {2 . . . N -1} of a piping with respect to the central dierences (3.14). The boundary segments n = 1 and n = N requires the forward and backward dierences, respectively. The resulting nite dierence matrix M x is useful for numerical implementation using for example MATLAB. Applying (3.16) in (3.15a) and (3.15b) results in a system of equations

f (N )    ≈ 1 2 ∆x           3 -4 1 0 • • • 0 1 0 -1 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 1 0 -1 0 • • • 0 -1 4 -3           M x •    f (1) . . . f (N )    ,
d dt p v ≈ 0 a 2 M x 1/ M x 0 • p v (3.17)
that is employed for simulations, where p = (p(1) . . . p(N )) -1 and v = (v(1) . . . v(N )) -1 . It is possible to further improve simulation accuracy when dening a nite dierence scheme of higher orders, though, this will require increased calculation eorts.

Numerical simulations

Simulations have been derived for a piping section of L = 4 m length and an inner diameter of D = 2.4 mm. This corresponds to the topology (a) Short piping and connects a single highpressure pump with a cutting head, see Section 2.2. The piping is lled with water at any time. Subsequent simulations consider the physical properties given in Table 3.1. A piping is divided, with respect to Assumption 3.2, into N = 16 667 homogeneous segments of ∆x ≤ 0.24 mm length, such as ∆x < 0.1 D. Applying (3.17) yields a system of equations with a nite dierence matrix M x of dimension N . This system is numerically solved using MATLAB R2017b and applying the solver ode45 with variable step size. The solver has been congured for a relative tolerance of 10 -4 . It has been running on a Windows 10 machine with an Intel Core i7-7600U 2.80 GHz processor and 16.0 GB RAM. The uid ow velocity and pressure states of every segment n have been set to the initial conditions v n (t < 0) = v 0 and p n (t < 0) = p 0 , respectively. The boundary condition on the rst segment n = 1 at the piping intake will be a function in time that describes an exogenous input uid ow. The input uid ow velocity v 1 (t < 0) = v 0 will be increased at 3.18) taking the maximal dynamic range of a high-pressure pump into account with τ = 7.6 ms. This boundary condition (3.18) is applied for system excitation and consequently induces a pressure uctuation that propagates along the piping. Two dierent cases will be studied with respect to the boundary condition on the last segment n = N at the piping outtake. For both cases, the output uid ow velocity v N (t < 0) = v 0 will vary as function of the corresponding pressure state p N (t) ≥ 0 ∀ t:

t = 0 to v 1 (t > τ ) = 1.1 v 0 with respect to v 1 (t) = 1 -0.1 cos π t τ v 0 , 0 ≤ t ≤ τ , (
(A) Open piping end: The cutting head is opened, though any nozzle has been removed. This case investigates the uid dynamics for a piping section without the characteristics of a nozzle. That applies the boundary condition

v N (t) = 2 ζ p N (t), t ≥ 0 , (3.19)
referring to the discharge equation [START_REF] Böswirth | Technische Strömungslehre[END_REF]. For which a discharge coecient ζ = ### has been derived, while estimating the friction loss for turbulent ow characteristics [Blasius, 1913], see (3.32)

to v N (t) = S H S 2 ζ p N (t), t ≥ 0 , (3.20)
where S H = D 2 H π/4 denotes the nozzle cross section.

These boundary conditions enable all segments to reach a bounded steady-state pressure, such as lim t→∞ p n (t) < ∞. The simulations of has been obtained with respect to boundary condition (A), initializing each segment to v 0 for p 0 = {20, 50, 100, 200} MPa and considering (3.19), as summarized in Table 3.2. Any increasing input uid ow at the piping intake induces a pressure uctuation. This uctuation propagates through the piping and reaches the piping outtake with a time delay τ d (t), where it will be reected until an equilibrium is obtained. According to [Liptak, 2018], this time delay is expected to change with respect to the piping length L and uid ow velocity v(t), such as τ d (t) = L/v(t). It is seen that the pressure steady-state will change for dierent uid ow velocities. For example, a velocity change of 10% from v 0 = 419 to lim t→∞ v n (t) ≈ 461 m/s, causes a pressure change of 50% from p 0 = 20 to lim t→∞ p n (t) ≈ 30 MPa. It is further observed for any increased operating pressure, that the sensitivity as well as the damping ratio will decrease. However, the resulting uid ow velocity v(t) approaches the characteristics speed of sound a. This is critical for the underlying modelling principle (3.1) that considers an incompressible uid ow. It is only valid for v(t) a [Spurk, 2013]. The simulations of Figures 3.7 -3.10 has been obtained with respect to the adapted boundary condition (B). An increased uid ow velocity at the piping intake propagates towards the piping outtake, where a nozzle is located. The pressure state of every segment will raise continuously and obtain steady-state for the entire piping. It is seen that the nozzle inuences the dynamic behaviour with respect to (3.20). The induced pressure trend features an increased raise time without any oscillations. The rise time for the uid ow velocity v n (t) increases, while the time delay along a piping becomes small in contrast to the transient pressure trend p n (t). The resulting uid ow velocities are far bellow any critical values for to the underlying modelling principle, such as v(t) a ∀ t holds.

As a consequence, the subsequent section will introduce a lumped parameter model. It approximates the piping dynamics described by (3.10a) and (3.10b) for future high-pressure networks by considering uid ow equilibrium.

Lumped parameter modelling

Approximating the governing equations (3.10a) and (3.10b), which describe the uid dynamics, with nite dierences assigns a specic uid volume V = S ∆x to homogeneous segments, as shown in the illustration of Figure 3.11. This results in a lumped parameter model that represents a system of algebraic dierential equations and lumps the distributed parameters to discrete pressure states, assuming a stationary uid ow [START_REF] Ford | Flows in Networks[END_REF].

Homogeneous segments

v i (t) p i (t) L S Δx D x p i-1 (t)
Figure 3.11 A piping section of constant cross section S: homogeneous segments of length ∆x along the position coordinate x with pressure state p(t) and uid ow velocity v(t).

To obtain ordinary dierential equations, the Taylor series expansion (3.12) is again used, approximating the rst order partial derivatives in x. The backward dierential

∂ ∂x v(x, t) ≈ v(x, t) -v(x -∆x, t) ∆x (3.21)
substitutes the uid ow velocity gradient in (3.10a), where the forward dierential

∂ ∂x p(x, t) ≈ p(x + ∆x, t) -p(x, t) ∆x (3.22)
gives the pressure gradient for (3.10b), in which ∆x denotes the length of a homogeneous segment.

The uid ow velocity is then substituted by the uid ow according to v(t) = Q(t) S -1 , while the notations

∆Q - i (t) := Q i-1 (t) -Q i (t)
(3.23) and

∆p + i (t) := p i (t) -p i+1 (∆x, t)
(3.24) are introduced. This assigns a uid segment i at location x = 0 for f (x ± n∆x) = f i±n . The backward dierential (3.21) refers to a uid ow relative to a previous segment, denoted by superscript + and the forward dierential (3.22) consequently indicates a pressure dierence relative to a subsequent segment, denoted with a superscript -. Applying the dierential (3.21) for (3.10a) and the dierential (3.22) for (3.10b) yields, with respect to the notations (3.23) and (3.24), the system of ordinary dierential equations

d dt p i (t) ≈ a 2 S ∆x ∆Q - i (t) (3.25a) d dt Q i (t) ≈ S ∆x ∆p + i (t) - λ 2D S Q i (t)|Q i (t)| , (3.25b)
describing the dynamics of a homogeneous segment. As an increased uid ow dierence ∆Q -(t) induces a pressure p(t), an increased pressure dierence ∆p + (t) yields a uid ow Q(t). The squared of the resulting uid ow causes the uid ow dierence to decrease until a pressure equilibrium has been obtained. For simplicity, the subscript i is omitted for the subsequent sections.

Fluid dynamic parameters

The hydraulic capacity C (see [START_REF] Will | Hydraulik[END_REF]) determines an increase of pressure dp(t)/dt with respect to the uid ow dierence. It is dened as

C = S ∆x a 2 = V K (3.26)
where V = S ∆x is the uid volume and K = a 2 corresponds to the bulk modulus. Here, the bulk modulus

K = - dp dV /V (3.27)
is expressed in [START_REF] Streeter | Fluid Mechanics[END_REF]] as a measure of the relative change in volume dV /V to a change in pressure dp. Substituting this in (3.26) yields C = dV /dp.

In the same manner, the hydraulic inductivity L (see [START_REF] Will | Hydraulik[END_REF]) determines an increase of uid ow dQ(t)/dt with respect to the pressure dierence. It is given by

L = ∆x S = m S 2 = µ S , (3.28)
where m = V denotes the uid mass within a volume section and µ = m S -1 corresponds to a mass relative to its cross section. Assuming a typical high-pressure pipe of ∆x = 1 m length and S = 4.5239 mm 2 cross section results in a hydraulic capacity of C ≈ 2 × 10 -15 m 3 /Pa and in a hydraulic inductivity of L ≈ 2 × 10 8 kg/m 4 . The physical properties for these computations are given in the above Table 3.1. The small capacity in contrast to the inductivity, leads to a far slower dynamics of (3.25a) compared with (3.25b), what is evidence of a sti behaviour. This can cause diculties for solving eciently the system of dierential equations (3.25a -3.25b) with numerical methods. Cross section S and uid density aect the dynamics for both dierential equations, but none of them can be chosen freely.

Solving for uid ow equilibrium

Assuming a stationary uid ow allows for solving the dierential equation (3.25b) for its equilibrium (dQ(t)/dt = 0). This provides an algebraic dierential equation that simplies the approximated equation of continuity and results in a lumped parameter model

d dt p(t) ≈ K V ∆Q -(t) (3.29a) Q(t) ≈ S 2 ζ |∆p + (t)| sign ∆p + (t) (3.29b)
that approximates the local pressure state of a homogeneous segment, which becomes determined by the uid ow resistance

H = S 2 ζ (3.30)
with respect to the non-linear interconnection to the neighbouring segments. The Signum function

sign(x) =      -1, x < 0 0, x = 0 1, x > 0 (3.31)
is required to recover the uid ow direction along a uid ow path. The ow resistance H requires to determine the discharge coecient

ζ = D ∆x λ , (3.32)
for every homogeneous segment, with respect to the cross section inner diameter D and length ∆x. This discharge coecient needs further approximation by estimating the friction loss coecient

λ < c 1 Re c 2 ≈ 0.0456 , (3.33)
with c 1 = 0.3164 and c 2 = 0.25 for turbulent ow characteristics [Blasius, 1913], as a Reynolds number Re > 2320 is assumed.

The resulting algebraic equation (3.29b) is also known as discharge equation, see [START_REF] Böswirth | Technische Strömungslehre[END_REF] or [Spurk, 2013]. It is found in many studies, e.g. [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF], [START_REF] Liu | Active disturbance rejection control of common rail pressure for gasoline direct injection engine[END_REF] and [START_REF] Wang | High pressure common rail injection system modeling and control[END_REF], to describe the uid ow through components, such as valves and nozzles. For example, the evaluation of (3.32) for a piping section of ∆x = 1 m and D = 2.4 mm, while considering the above approximation (3.33), yields a discharge coecient with ζ > 0.2. On the other hand, [START_REF] Susuzlu | Initial research on the ultra-high pressure waterjet up to 700 MPa[END_REF] reports an enlarged discharge coecient of ζ = [0.6, 0.7] for cutting heads, due to the small nozzle inner diameters. The friction loss coecients of specic high-pressure components, used for modelling, will be experimentally obtained in the Section 3.4.

The resulting system of algebraic dierential equations (3.29a) and (3.29b) is less demanding for solving with a numerical solver, since the fast dynamics from (3.25b) has been resolved for uid ow equilibrium. As a consequence, it becomes possible that fast changing pressure uctuations will be reproduced with reduced accuracy. Since the modelling for high-pressure networks is eventually applied to verify the control design, at least the induced uctuations due to the switching of cutting heads has to be taken into account. Hence, the subsequent section will investigate the simplied formulation by means of simulations.

Numerical simulations

Pressure uctuations in a short piping section will be investigated for dierent piping lengths (L = {0.5, 1, 2, 5, 10} m), while deriving simulations with the distributed parameter model as well as with the lumped parameter model. The distributed parameter model will introduce a discretization with nite dierences implementing N = L/∆x homogeneous segments of length ∆x = 0.24 mm. On the other hand, the lumped parameter model considers all of the piping as a single segment N = 1, resulting in a rst order model. Both models are numerically simulated using MATLAB R2017b and applying the solver ode45, where the solver has been congured for a relative error of 10 -4 .

From the previous simulations in this section, it has been observed that the pressure sensitivity and oscillation amplitudes increase for small operating pressure. The subsequent simulations consider an initial pressure of p 0 = 100 MPa. This corresponds to the smallest expected operating pressure for waterjet machining, where the largest sensitivity is expected. A sinusoidal input uid ow of amplitude v 0 will be applied at the piping intake. The initial uid ow velocity v n (t < 0) = v 0 has been derived with respect to (3.20). That results in the input boundary condition 3.34) taking the maximal dynamic range of a high-pressure pump into account with τ = 7.6 ms. The output uid ow velocity v N (t < 0) = v 0 will vary in function to the corresponding pressure state p N (t) ≥ 0 ∀ t, considering the output boundary condition (B) Nozzle contraction. This investigation aims to determine the minimal length of a homogeneous segment and guarantees an useful simulation accuracy, when applying the lumped parameter model. The plots of Figure 3.12 -3.16 show the simulation results for the distributed parameter model, when reaching oscillation around an operating point. For a piping of L > 5 m length, the sinusoidal excitation hits resonance of the piping. This results in a phase shift of about 180°between the input pressure p 1 (t) and output pressure p N (t), see Figure 3.16.

v 1 (t) = 1 -cos π t τ -0.5 π v 0 , t ≥ 0 , (
The plots of Figure 3.17 -3.21 evaluate the simulation errors, while comparing the estimated uid ow velocities and pressure states at the piping outtake for the distributed parameter model and the lumped parameter model. The 180°phase shift is observed using either of the models, but the output pressure amplitude decreases only for the lumped parameter model. This is observed for a piping of L > 2 m length and results in increased simulation errors. To aggregate the distributed parameter model to homogeneous segments of length ∆x < 1 m seems to be acceptable for practical use. Simulation eort will be reduced, while the relative error per segment remains below 0.5%.

However, the error propagation has to be considered when connecting several homogeneous segments in series. Table 3.3 compares the simulation error for a piping of L = 20 m length that is modelled with n = {4, 10, 20, 40, 100} segments. Where a distributed parameter model requires an enhanced computational eort for simulating a large piping section, the lumped parameter model reduces the simulation complexity with acceptable error and provides a exible modelling for high-pressure networks of arbitrary topologies. The investigated topologies will be approximated, considering homogeneous segments of length ∆x ≈ 1 m. Figure 3.17 Error between distributed and lumped parameter model for a piping segment of 0.5 m: simulation using nite dierences with 2100 elements.

Framework for generalized high-pressure networks

The concept of lumped parameter modelling aims at an approximative numerical simulation of the 1-dimensional uid ow along a piping with reduced computational eort. It describes a piping section by means of homogeneous segments, representing a pressure state around its operating point, while assuming a stationary uid ow. However, the wide pressure range for waterjet machining requires to introduce varying parameters and to consider exogenous inputs for excitation. This section introduces these varying parameters and presents a graph-based modelling to obtain a generalized framework, as published in [START_REF] Niederberger | A graph-based modelling methodology for high-pressure networks applied on waterjet machining[END_REF]. In contrast to other approaches, using graphs allows for representing entire waterjet facilities with reduced modelling eort and considering parameter variations improves simulation accuracy for a wide pressure range. The resulting high-pressure network models consist of a system of rst order dierential equations, representing homogeneous segments and recovering the uid ow paths of arbitrary network topologies.

Varying parameters and exogenous inputs

Measurements show that the wide pressure range for waterjet machining aects the uid density and bulk modulus, see [START_REF] Tremblay | Modelling and simulation of pressure uctuations in high pressure waterjets[END_REF]. Due to a variable uid density and bulk modulus, the introduced Bernoulli equation (3.1) becomes insucient for describing a highpressure network. Consequently, it is useful to introduce the most prevalent parameter variations to address the wide pressure range of dierent waterjet applications and to subsequently provide a generalized description for high-pressure networks.

High-pressure networks involve parameter variations, which are either related to the uid characteristics or to specic high-pressure components. The uid compressibility includes a pressure dependent bulk modulus K(p) as well as a pressure dependent uid density (p). Highpressure components dene the uid volume V (t) and the ow path, given by the Heaviside function σ(x, ∆p), cross section S(x) and discharge coecient ζ(x). These parameter variations can be classied as dependent variables:

Parametric variables: these uncertainties are subject to change with respect to any other time-varying variable, such as c = f (ϕ(t)). The dependent variable ϕ(t) is a priori unknown. This applies for a variable bulk modulus K = f (p(t)) as well as for a variable uid density = f (p(t)), which are variable with respect to the pressure state p(t).

Time dependent variables: these uncertainties are directly dependent on time t, such as c = f (t). The function f (t) ∀ t ≥ 0 is a priori known and dened. Consequently, the parameter variation itself will be a priori known. This applies for very specic segments where its volume V = f (t) varies in time. For example when a piston is displaced within a pumping chamber.

Local variables: These uncertainties can vary dependent on the position coordinate x, such as c = f (x). The function f (x) ∀ x = [0, L] is a priori known for the entire high-pressure network. This applies on the cross section S = f (x) and the discharge coecient ζ = f (x) which can vary for dierent segments.

Local parametric variables: another uncertainty considers a change with respect to a varying variable as well as its position coordinate x, such as c = f (x, ϕ(t)). The function f (x, ϕ(t)) ∀ x = [0, L] is a priori known and dened with respect to the parameter x. However, the dependent variable ϕ(t) is unknown. A switching parameter is considered when introducing the Heaviside function σ = f (x, ∆p(t)), which allows for controlling the uid ow between specic homogeneous segments of a network with respect to the dierential pressure ∆p(t) between segments, if needed.

Where (3.29a) describes the pressure dynamics of a homogeneous segment, (3.29b) denes the uid ow between interconnected segments. Considering the parameter variations for the lumped parameter model (3.29a) -(3.29b) yields

d dt p(t) ≈ K(p) V (t) ∆Q -(t) + Q ext (t) (3.35a) Q(t) ≈ σ(x, ∆p) S(x) 2 (p) ζ(x) |∆p + (t)| sign ∆p + (t) , (3.35b)
where the exogenous excitation

Q ext (t) = [0, Q max ] is introduced.
This excitation is considered as a possible external input uid ow at a network position, where a pumping unit is located.

A uid ow dierence ∆Q -(t) induces a change of pressure dp(t)/dt with respect to the bulk modulus K(p) and the segment volume V (t). On the other hand, any pressure dierence ∆p + (t) causes a uid ow Q(t) with respect to the uid density (p), the cross section S(x) along the uid ow path and its discharge coecient ζ(x). The cross section can vary between dierent interconnected segments. Thus, a change of cross section ∆S aects the discharge coecient.

Fluid compressibility

The transient behaviour of pressure generation depends on the bulk modulus K, hence the Newton-Laplace formula [START_REF] Streeter | Fluid Mechanics[END_REF]]

K = a 2 (3.36)
assigns the uid density to the bulk modulus K with respect to the speed of sound a. Since (p) is expected to vary with respect to the pressure p, K(p) will vary as well. Consequently, a linear parametric model of form

K(p) = κ 0 + κ 1 p (3.37)
is frequently proposed, see [START_REF] Hountalas | Development of a fast and simple simulation model for the fuel injection system of diesel engines[END_REF]] and [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF], to describe a pressure-dependent bulk modulus. On the other hand, [START_REF] Streeter | Fluid Mechanics[END_REF]], [Spurk, 2013] and others relates the bulk modulus directly to a change of uid density with the formula

K = dp d / , (3.38)
dual to (3.27). Combining (3.37) with (3.38) while separating the variables and integrating yields

1 κ 0 + κ 1 p dp = 1 d .
(3.39)

Referring to [START_REF] Susuzlu | Initial research on the ultra-high pressure waterjet up to 700 MPa[END_REF] and [START_REF] Fabien | Dynamic modelling and identication of a waterjet cutting system[END_REF], an explicit solution is derived and rearranged to represent a pressure-dependent uid density

(p) = 0 1 + κ 1 p κ 0 1/κ 1 (3.40)
whereby 0 denotes the initial density. In this respect, the variable compressibility aects the transient behaviour of (3.35a) as well as the steady-state given by the discharge equation (3.35b).

Its coecients κ 0 and κ 1 will be experimentally identied for a pressure range up to 400 MPa.

Fluid volume

The transient behaviour of pressure generation also depends on the uid volume

V (t) = S ∆x(t) .
(3.41)

This volume can vary with respect to the length ∆x(t) of specic high-pressure components. For example, a piston displacement changes the volume of a pumping chamber. This variable uid volume is then expressed as

V (t) = S l 0 -Q ext (t) dt , (3.42)
considering the initial length l 0 and the displacement uid ow

Q ext (t) = S P v P (t) , (3.43)
when a piston of cross section S P becomes displaced with the velocity v P (t). The volume variation (3.42) and the displacement uid ow (3.43), respectively, are a priori known periodic functions and therefore considered as an external input uid ow that excites the highpressure network. However, most components feature a constant displacement uid ow, such as

Q ext (t) = 0 ∀ t > 0.

Flow path

Eventually, a switching law determines the existence of a uid ow along a ow path. It is associated to a dependent variable, which restricts the ow direction. Modelling a check-valve, it is proposed to introduce the Heaviside function σ(x, ∆p) := 1, ∆p(t) > 0 0, ∆p(t) ≤ 0 , (3.44) where the switching behaviour depends on the pressure dierence ∆p(t). In addition, the cross section S(x) can change along the uid ow path. This causes a change of the discharge coecient (3.45) where ∆S denotes to the change of cross section along the uid ow path and S = D 2 π/4.

ζ(x) = 1 1 + ∆x/D λ -(1 -∆S/S) 2 ,
The varying parameters (3.37) and (3.42) are introduced to the dierential equation (3.35a), where the varying parameter (3.40) and the discharge coecient (3.45) are considered for the algebraic equation (3.35b). This results in the generalized description

d dt p(t) ≈ κ 0 + κ 1 p(t) S l 0 -Q ext (t) dt ∆Q -(t) + Q ext (t) Q(t) ≈ σ(t) S 2 0 1 + κ 1 p(t) κ 0 -1/κ 1 ζ |∆p + (t)| sign ∆p + (t) (3.46a) (3.46b)
for homogeneous segments. Whereby each segment represents a local pressure state p(t), the dependent variable x is not noted for simplicity. A pressure-dependent uid density (p) as well as a pressure-dependent bulk modulus K(p) enable a wide pressure range. This is subject to improve the simulation precision, as a subsequent model validation will reveal.

Graph-based modelling

The modelling of dierent waterjet facilities requires a exible approach that interconnects homogeneous segments to reproduce arbitrary network topologies. Using graphs, as described in [START_REF] Ford | Flows in Networks[END_REF], seems prevalent to map homogeneous segments and recovering the uid ow path for various high-pressure networks. A graph G = (V, E) consists of branches or edges E that interconnects nodes or vertices V. The pressure state

p(t) → p k (t) (3.47)
of a segment k is assigned to a network node and the uid ow rate

Q(t) → Q kj (t) (3.48)
between segments k and j is assigned to a network branch. The dierential scheme (3.24) becomes

∆p + (t) → ∆p kj (t) := p k (t) -p j (t) (3.49)
that couples interconnected nodes and (3.23) becomes

∆Q -(t) → ∆Q kj (t) := N j=1 Q kj (t) , (3.50)
with respect to the uid ow continuity (3.5), determining N simultaneous interconnections of a node k to any other node j = {1, . . . , N }. A possible displacement uid ow is assigned to each node Q ext (t) → Q k (t), while the sum in (3.50) conserves the uid ow at every node

Q jk (t) = -Q kj (t) , (3.51)
with respect to the uid ow direction. Therefore, the output ow rate Q kj (t) from a previous segment yields an input ow rate Q jk (t) to a subsequent segment. This results in a network description N = (G, dp/dt, Q) of homogeneous segments to represent specic high-pressure networks using lumped parameters. Dierent network sections will be consequently assigned to J → N homogeneous segments. The illustration in Figure 3.22 maps each segment, depicted in (a), to a nodek ∈ J, as shown in (b). Every node k can be interconnected to other nodes j ∈ J \k. Consequently, a node v ∈ V represents a pressure state p k (t), such as

Q kj (t) p k (t) p k-1 (t)
d dt p k (t) : V → R , (3.52)
whereas a branch e ∈ E establishes a uid ow Q kj (t), such as

Q kj (t) : E → R , (3.53)
with positive ow direction from segment k to j. With the above introduced assignments (3.47 -3.50), the network description

N kj :                d dt p k (t) = K k (p) V k (t) (∆Q kj (t) + Q k (t)) Q kj (t) = ϑ kj (t) 2 k (p) |∆p kj (t)| sign (∆p kj (t)) (3.54)
is obtained, which represents a high-pressure network using lumped parameters. It yields a system of algebraic dierential equations. These equations approximate the local pressure states of homogeneous segments, including non-linearities due to interconnections to neighbouring segments.

Each node k holds a uid volume V k (t). The uid volume and the bulk modulus K k (p) determine the transient behaviour of a node k, whereas the parameter ϑ kj (t) = σ kj (t) S kj ζ kj and the uid density k (p) determine the uid ow along a branch kj. The uid characteristic parameters K k (p) and k (p) are expected to vary according to the corresponding pressure states p k (t). Consequently, the variable uid volume (3.42) becomes

V k (t) = S k ∆x k -Q k (t) dt (3.55)
and the linear parametric model (3.37) yields (3.56) and its explicit solution (3.40) becomes

K k (p) = κ 0 + κ 1 p k (t)
k (p) = 0 1 + κ 1 p k (t) κ 0 1/κ 1 , (3.57)
whereby 0 denotes the initial uid density. The parameter ϑ kj (t) holds the interconnection cross section S kj , its discharge coecient ζ kj and the Heaviside function σ kj . The discharge coecient represents the ow resistance along the uid ow path, while the Heaviside function indicates the existence of a uid ow path between node k and j. If the cross section changes along the uid ow path, its discharge coecient ζ kj needs to be adjusted and (3.45) becomes (3.58) where ∆S kj = S k -S kj denotes the cross section dierence.

ζ kj = 1 1 + ∆x k /D k λ kj -(1 -∆S kj /S kj ) 2 ,

Describing network topologies

The symbols of Figure 3.23 are employed to derive a graph-based network description that recovers a desired network topology when interconnecting nodes by means of branches. An exogenous uid ow Q k can be applied on every node, where a node represents a pressure state p k that simulates the network pressure px at position x, see (b) and (c). The uid ow Q kj along a branch can be restricted, see (f) and (g). These symbols allow for representing dierent high-pressure components by means of graphs as illustrated in Figure 3.24. The high-pressure pump, as shown in (a), consist of two pumping chambers (piston pumps). Each chamber is represented by a node, which includes an exogenous input uid ow. Both chambers are interconnected by means of branches. These branches restrict the uid ow to positive direction with respect to the pressure dierences. Modelling a highpressure piping, see (b), an initial piping section can be subdivided, if needed, to represent the pressure propagation with desired accuracy. On the other hand, a connector is a single node that couples any amount of input and output branches, as shown in (c), and a cutting head couples a node that discharges a waterjet to atmosphere, see (d). The proposed graph-based modelling methodology is expected to model various network topologies with reduced complexity, when considering typical high-pressure components. It recovers the uid ow path of a waterjet facility, with respect to the principle of continuity and momentum conservation. A high-pressure network couples any number of high-pressure pumps and cutting heads, where the uid ow from a pumping chamber to a cutting head is usually denoted to a positive uid ow direction.

Parameter identication for waterjet machining

This chapter is dedicated to identify the parameters required by the above introduced graphbased modelling methodology. It aims at a parametrization of specic high-pressure components, independent of any network topology. This results in a parametrized simulation toolbox, capable to model dierent high-pressure networks without revising parameter identication.

The two model coecients κ 0 and κ 1 of the parametric model (3.37) are rst identied, which also dene the pressure-dependent uid density of equation (3.38). For this, the pressure generation within a high-pressure pump has been investigated. Dierent measurements have been obtained, while pressurizing water in a pumping chamber by means of piston displacement. This experiment has been derived independent of the test bench conguration. The parametric model of Section 3.3 describes a pressure-dependent bulk modulus. This is a uid characteristic parameter, that is unaected by any high-pressure component or network topology.

On the other hand, cutting head discharge coecient and piping friction loss coecient have been derived from dierent experiments at the test bench, while taking the use cases of Table 3.4 into account. The 2 basic measurements (B-3,6) have been selected for parameter identication. The other measurements from use case 1 have been used to verify the modelling methodology and control design. The 6 extended measurements (E-1,3,5,8,10,12) have been selected for parameter identication. The other measurements from use case 2 have been used for the model and controller validation. The setups for use case 1 include a single high-pressure pump interconnected to a single cutting head, where the setups for use case 2 include a high-pressure pump that supplies 2 cutting heads. Using 2 cutting heads results in 4 possible switching states when applying the switching pattern for contour cutting. E.g. installing 6 dierent nozzles would allow for investigating 12 switching states. In combination with the use cases, 4 dierent topologies have been assumed and 2 dierent operating pressures have been applied. Pressure gauges have been installed after the pump, before both cutting heads and in the centre connector that couples the cutting heads to the pump. This allows for identifying 10 piping sections of dierent lengths. Consequently, 24 dierent parametrizations for waterjet machining have been taken into account, when identifying the cutting head discharge coecient ζ for nozzles of various inner diameter and to identify the piping friction loss coecient λ for piping sections of various lengths. These identied parameters are needed to fully dene and parametrize the above derived high-pressure network models. The induced overall uid ow will be estimated by evaluating the piston displacements. The pressure will be measured on dierent network locations x using pressure gauges. Hence, the uid ow estimation Q(t) and the dierent pressure measurements p x (t), depicted in diagram of Figure 3.25, have been taken into account to identify the required model parameters.

It has to be mentioned, that the pump requires closed-loop control to obtain the desired operating pressure. It is desired to track the operating pressure, while attenuating switching disturbances. Otherwise, the pressure will exceed given bounds of the test bench, when switching a cutting head. For this reason, the high-pressure pump holds by default a PI controller (6.46), as provided in Section 6.4.

Network

Gauges Pump Controller

Encoders

p x Q n u d m y = p P Q n r
Figure 3.25 Experimental setup for parameter identication: pressure gauges measure the pressure p x (t) at dierent positions when applying an input uid ows Q n (t) and switching the cutting heads d m (t).

Identication of bulk modulus and uid density

The bulk modulus of water is often regarded as a constant K 0 = 2.08 GPa. This is true for a relatively small changes of pressure < ±10 MPa. The bulk modulus increases to 2.68 GPa for water pressurized to 100 MPa. Detailed studies have analysed uid density as well as bulk modulus [START_REF] Chen | Thermodynamic properties for natural waters covering only the limnological range 1[END_REF]] and tabular data have been collected for a pressure range of 0.1 to 100 MPa [START_REF] Schmidt | Properties of water and steam in SI-units[END_REF]. This tabular data contain information to estimate a pressure-dependent bulk modulus, but further experimental data are necessary to validate the parametric model (3.37), introduced in Chapter 3, for a pressure range up to 400 MPa. This is realized with respect to the method proposed in [B. Titurus et al., 2010], for the identication of hydraulic characteristics from steady piston displacement.

Design of experiment

As discussed in Section 3.3, the bulk modulus

K = - dp dV /V (3.59)
denes, that any variation in uid volume V (t) induces a change in pressure p(t). This relationship will be applied to experimentally determine the bulk modulus K(p) of water at high-pressure. If a pumping chamber is entirely shot, the measured piston position s P (t) denes the volume variation (3.60) where V 0 corresponds to the initial chamber volume and S P is the piston cross section. As a consequence, any positive piston displacement pressurizes the water within a pumping chamber.

V (t) = V 0 -S P s P (t) ,
The pressure increase is captured by a high-pressure gauge at the pumping chamber outtake. An external check-valve replaces the outtake check-valve, that is positioned right after the pressure gauge. One pumping unit is pressurizing the high-pressure piping to the maximal pressure allowed (400 MPa). That causes the external check-valve at the other pumping unit to be closed. This allows for pressurizing the pumping chamber without any interaction with the piping, as long as its pressure remains below maximal pressure.

Data acquisition is performed at a sampling rate of 10 Hz. This is sucient, since the piston will move with a slow velocity to avoid any dynamic eects to the pressure generation. The measured pressure p(t) requires adjustments, such as to retrieve the absolute pressure p(t) = p(t) -p g -p s , (3.61) where p g = g h is the pressure head given by the height h between piping and pressure gauge and p s is the pressure given by the water supply.

Table 3.5 discusses the design of experiment. The initial pumping chamber volume has been varied for each run by choosing a dierent initial piston position, which aects the sensitivity of pressure generation. When performing these experiments, the piston has been slowly displaced so that a nearly isothermal process of pressure generation is expected, while the uid temperature remains constant. The various runs have been assigned to two dierent pumping chambers, to evaluate volume uncertainties. In addition, all runs have been repeated for two dierent velocities, evaluating adiabatic eects. The plots in Figure 3.26 and 3.27 present the measured and adjusted pressures (3.61) in (a) as well as the derived pumping chamber volume (3.60) in (b) for a piston velocity of v P = 1 mm/s and v P = 2 mm/s, respectively. While some initial piston positions have been dedicated to the pumping chamber 1, others correspond to the chamber 2.

The experimental data shown below is processed to evaluate the bulk modulus for each data sample i. For that, a numerical approximation of (3.59) using central dierences yields

K(p i ) = - p i+1 -p i-1 V i+1 -V i-1 V i .
(3.62) Equations ( 3.59) and (3.61) are then used to derive the parameters κ 0 and κ 1 . For this, the parametric model

K(p) = κ 0 + κ 1 p (3.63)
is evaluated by means of linear regression and further applied to estimate the uid density

(p) = 0 1 + κ 1 p κ 0 1/κ 1 (3.64)
for dierent operating pressures.

Results

A robust least-square method is applied to t the parametric model (3.63). Therefore, the least absolute residual method of [Thanoon, 2015] has been examined to generate a linear regression model with tight condence intervals. Table 3.6 presents the identied model coecients. The resulting t achieves a determination coecient of r 2 = 0.9959. The prediction errors are reasonable with a sum of square error of sse = 4.0759 and a root mean square error of rmse = 0.0382.

Table 3.6: Identied coecients of the parametric model and its condence intervals describing the pressure-dependent bulk modulus.

Coecient Fitted value 95% condence bounds Unit

κ 0 ### [###, ###] (GPa) κ 1 ### [###, ###] (-)
The regression analysis of Figure 3.28 shows the estimated bulk modulus with respect to its pressure state (a), the residuals of the measurement data (b), its probability (c) and density distribution (d). The experimental data are compared to tabular data [START_REF] Schmidt | Properties of water and steam in SI-units[END_REF], which estimate the bulk modulus for pressures below 100 MPa. An increased scattering is observed for higher pressures, indicating an enlarged sensitivity and a drift is found in the measured data of higher and lower pressures, due to acceleration and deceleration of the piston.

Considering relation (3.64) yields the variation of uid density with respect to the operating pressure, as shown in Figure 3.29. It follows almost a linear relationship, with an average slope of about 0.5 kg/m 3 /MPa. Thus, the proposed modelling methodology could be further simplied. Increasing the pressure from 0.1 to 400 MPa causes to increase the bulk modulus by 86.8% and the uid density by 20.2%. While a variable bulk modulus will aect the transient behaviour of pressure generation, the variable uid density cause a shift of the resulting steady-state, see also the experimental studies of Section 4.3 for more details. Figure 3.29 Relation between chamber pressure and uid density: expected uid density, when pressurizing a pumping chamber.

Identication of cutting head discharge coecient

Measurements derived from the test bench considering topology (c) Symmetric network and (d) Asymmetric network have been used, to identify the discharge coecient ζ for nozzles of various cross sections, while taking cutting head congurations for nozzles of D = {0.1, 0.15, 0.2, 0.25, 0.3} mm inner diameter into account. These measurements are advantageous, as they provide a wide variation of uid ow velocities and operating pressures. Parameter identication has been performed, again considering the method in [B. Titurus et al., 2010].

Design of experiment

The discharge equation

Q(t) = ζ S 2 (p) p(t)
x Subs.

(3.65) obtained from Section 3.2, allows for dening a model for identifying the discharge coecient ζ.

The uid density (p) has been adjusted according to (3.64), taking the above identied pressuredependent bulk modulus into account. Introducing the substitution variable x Subs. allows for tting a linear regression

Q i = ζ x i (3.66)
that describes the relationship between uid ow and the substitution variable, with respect to the discharge coecient. The substitution variable combines the measured pressure p(t) from the pressure gauge, located right before a cutting head, with the nozzle cross section S, given by the cutting head conguration. The plots in Figure 3.30 show the experimental data used to identify the discharge coecient. Each data point corresponds to a steady-state operating point, derived from the experiments. For that, measurement data at steady-state pressure p(t) have been evaluated with respect to each cutting head switching state, when a steady input uid ow Q(t) has been obtained. The uid ow and nozzle cross section reveal a prevalent correlation, where uid ow and cross section seems unaected by the operating pressure.

Figure 3.30 Comparison of experimental data used to identify the friction loss coecient: considering the estimated uid ows, the absolute pressures from measurements and the nozzle cross sections, given from the test bench setup.

Results

Again, the robust least-square method (see [Thanoon, 2015]) is applied to t (3.66). Table 3.7 presents the identied discharge coecient. The resulting t achieves a determination coecient of r 2 = 0.9986. The prediction errors are reasonable with a sum of square error of sse = 0.0193 and a root mean square error of rmse = 0.0182.

Table 3.7: Identied discharge coecient with linear regression and its condence interval.

Coecient Fitted value 95% condence bounds Unit

ζ ### [###, ###] ( -)
The regression analysis of Figure 3.31 shows the estimated discharge coecient with respect to the substitution variable x Subs. (a), the residuals of the measurement data (b), its probability (c) and density distribution (d). The resulting probability is light-tailed for the negative part of the residuals, hence a small variance is expected. Its density becomes asymmetric in contrast to the theoretical distribution. Thus, more measurement data with enhanced variance would improve the linear regression. Nevertheless, a sucient signicance is derived regarding the dierent operating points. However, an exact model for the nozzle friction loss coecient and a reliable prediction of its parameter remains dicult. A disturbing factor is any nozzle wear, which inuences the inner cross section S that is assumed as constant.

Identication of piping friction loss coecient

Measurements derived from the test bench considering topology (c) Symmetric distributed and (d) Asymmetric distributed are also subject to identify the friction loss coecient λ for piping sections of various lengths L = {0.91, 3.80, 5.94, 6.31, 7.17, 8.08, 11.34, 11.90, 13.06, 13.11, 18.09, 18.17} m. The friction loss coecient λ has been identied for a high-pressure piping of D = 2.4 mm inner diameter, considering the method in [B. Titurus et al., 2010].

Design of experiment

Substituting the denition

∆p(t) = λ (p) L 2 D S 2 Q 2 (t)
x Subs.

. (3.68) This allows for dening a model to identify the friction loss coecient λ. The uid density (p) has been adjusted according to (3.64), taking the above identied pressure-dependent bulk modulus into account. Introducing the substitution variable x Subs. allows for tting a linear regression

Q i = λ x i (3.69)
that describes the relationship between uid ow and the substitution variable, with respect to the friction loss coecient. The substitution variable combines the estimated uid ow from a high-pressure pump with a piping section of length L, given by the distances between dierent pressure gauges. These dene the pressure dierences ∆p(t) = p x (t) -p x-1 (t) with uid ow path of positive direction from a preceding position at x -1 to position x. The plots in Figure 3.32 show the experimental data used to identify the discharge coecient. Each data point corresponds to a steady-state operating point derived from the experiments. For that, steadystate pressure p(t) has been evaluated from measurement data, when obtaining steady input uid ow Q(t) with respect to each cutting head switching state. The parameters of the experiment reveal a signicant variation within the desired pressure range. A correlation between uid ow and pressure dierence can be expected.

Results

The robust least-square method (see [Thanoon, 2015]) is applied to t (3.69). Table 3.8 presents the identied discharge coecient. The resulting t achieves a determination coecient of r 2 = 0.967. The prediction errors are reasonable with a sum of square error of sse = 8.326 and a root mean square error of rmse = 0.303.

### [###, ###] (m)
Figure 3.32 Comparison of experimental data used to identify the friction loss coecient: considering the estimated uid ows, the calculated pressure dierences from measurements and the piping lengths, given from the test bench setup.

The regression analysis of Figure 3.33 shows the estimated coecient λ with respect to the substitution variable x Subs. (a), the residuals of the measurement data (b), its probability (c) and density distribution (d). The resulting probability is acceptable, hence measurement data with sucient variance has been available for linear regression. It has to be mentioned that straight pipings have been considered only. The eect of curved pipings, junctions and connectors are subject for further investigations, which aim at improving the simulation toolbox and providing additional validated high-pressure components. For this work, any change of friction loss due to curves and diameter variations remain subject of estimation.

Conclusion

In this chapter, the uid dynamic to represent the pressure propagation along a piping section has been discussed. The main contribution has been given, when introducing a graph-based modelling methodology to describe high-pressure networks. It assigns piping sections to homogeneous segments. These segments are then interconnected by means of graphs to recover the initial network topology. Each segment is represented as a local pressure state and its interconnections are characterized, considering a stationary uid ow. This methodology has been derived from the fundamentals of uid dynamics, by approximating a distributed parameter model using nite dierences. This results in algebraic dierential equations, which represent a high-pressure network as a lumped parameter model. The error due to approximation has been investigated by means of simulations, while considering a dened piping section. At least a discretization of 1 m is required to achieve the desired simulation accuracy.

Further, a framework to describe generalized high-pressure networks has been described. It takes into account a variable uid compressibility, that relates to a variable uid volume and a pressure-dependent bulk modulus. Any displacement uid ow has been considered as an external excitation, where a piston displacement has been modelled as a change in the uid volume. The bulk modulus, however, has been represented by a parametric model, which has been further used to derive a pressure-dependent uid density. Its coecients have been experimentally identied. Model parameters have been identied by means of measurements. In particular, a pressure dependent bulk modulus has been investigated for a wide pressure range from 0 to 400 MPa. This extends the known tabular data from literature, which have been limited to 100 MPa. The parametrized simulation toolbox enables a exible modelling of various waterjet facilities with network topologies of increased complexity, but without necessarily validating each high-pressure network model with work intensive measurements. Nevertheless, validated network models will be used in Chapter 6 to verify the low-level control design. Further work should continue to identify the model parameters for other high-pressure components. On the other hand, as the segmentation of network sections constitutes a trade-o between simulation accuracy and model complexity, it would be valuable to evaluate an optimal segmentation procedure.

Introduction Objectives

The development of future waterjet facilities requires the application of the graph-based modelling methodology, which provides high-pressure network models. These models represent the coupling of decentralized pumps, needed for evaluating the low-level control design and for investigating the distributed high-level managing by means of simulations. On the other hand, the subsequent low-level control design requires a simplied model and a suitable description of the uncertainties, as taken into account for controller synthesis.

The derived high-pressure network models have to be validated by means of measurements, considering dierent network topologies and taking the previously dened use cases into account (see Section 2.2). These models should remain accurate for a wide pressure range and scalable to various network topologies, without having to revise parameter identication and model validation. This should reduce the time expanses for modelling of future waterjet facilities.

Contribution

The application of the graph-based modelling methodology is exemplary presented for a test bench setup, considering an electrically driven high-pressure pump with independent pistons interconnected to a single cutting head. This modelling procedure is then extended for the introduced network topologies of Chapter 2 to obtain high-pressure network models, which represent future waterjet facilities. On the other hand, a lumped parameter model is discussed, which takes parameter uncertainties into account. This rst-order model of a high-pressure pump seems suitable for further investigations related to robust control.

Extensive experimental studies constitute a main contribution to this chapter. The obtained high-pressure network models have been validated by measurements, derived from the test bench. It considers setups with two coupled high-pressure pumps supplying two independent cutting heads, while applying dened use cases, which correspond to typical waterjet applications. First results compare the resulting simulation accuracy to existing modelling approaches. It is shown that a variable uid compressibility improves simulation accuracy and modelling errors can be reduced. Detailed results investigate the simulation accuracy of the graph-based modelling methodology for measurements at dierent network position, while applying various switching disturbances.

This chapter is structured as follows: Section 4.2 applies the graph-based modelling methodology rst to describe common high-pressure components and second to model the introduced high-pressure networks. The derived network models will be validated in Section 4.3 by means of measurements.

Applying the modelling methodology

In this section, the topology (a) Short piping is exemplarily modelled to provide a lumped parameter model and second, to derive a graph-based modelling. The illustration of Figure 4.1 shows the analysed high-pressure network and its conguration in detail. It includes an electrically driven high-pressure pump and a cutting head. The high-pressure pump has two pistons, each represents a uid ow source. The pistons are displaced with velocities v 1 and v 2 to generate the displacement uid ow Q 1 and Q 2 . The displacement uid ow of each piston is proportional to its velocity. These induce the pumping chamber pressures p 1 and p 2 . The outtake ducts of each pumping chamber contains internal check-valves, which ensure the unidirectional output uid ow Q 13 ≥ 0 and Q 23 ≥ 0 from the pumping chambers to the connector. These outtake ducts reduce the uid ow cross sections for a rst time, before it passes the check-valves. A coupling connects the pumping chambers to the high-pressure piping, where the cross section is further reduced, while the output ows of both cylinders are merged to the overall uid ow Q 34 . At this position, a pressure gauge has been installed to measure the pump pressure p 3 . High-pressure piping sections directly interconnect the high-pressure pump with a cutting head, where a second pressure gauge measures the cutting head pressure p 8 . The cutting head includes an on/o-valve and a nozzle. It is independently controlled by a work station. An electromagnetic actuator switches the on/o-valve. If the on/o-valve is open, the nozzle will generate a waterjet. This results in the discharge uid ow Q 80 , which represents a pressure sink to the atmosphere. This nozzle reduces the uid ow cross section one last time. Such a cutting head can be equipped with nozzles of dierent inner diameter, to reproduce the discharge ow of various work stations. The uid ow from a piston pump to a cutting head is dedicated to a positive ow direction. Applying the lumped parameter modelling as well as applying the graph-based modelling is restricted to dierent constraints and assumptions. The following requirements must be met for implementation:

1. A high-pressure network is divided into segments k ∈ J of constant cross section S k .

2. The use cases are restricted to high-pressure pumps with parallel pistons. Therefore, each pumping chamber is lled from the same water supply and each high-pressure pump is considered as an independent continuous system, which generates a steady output uid ow.

3. All high-pressure components are considered to be ideal, e.g. the pressure stress will not cause any recognizable expansion and leakage is negligible.

4. Degradation of components, such as nozzle wear, is not modelled.

5. The water supply is not part of the high-pressure network. Instead, the supply pressure p S denes the lower bounds for all pressure states p k (t), such as p k (t) ≥ p S ∀t.

6. Water supply and environment are assumed to represent an innitely large pressure potential, such as p S and p 0 are invariant and ∇p(x, t) = 0. This is fullled by high-pressure applications, when a steady water supply is guaranteed and since the cutting head discharges the water against the atmosphere.

7. The initial condition p k (t = 0) = p S requires the entire high-pressure network to be lled with water.

8. Any piston displacement inuences the chamber volume V k (t) and, thereby, the dynamics of pressure generation. The piston position is determined by a trajectory, hence the volume variation is a priori known. 9. As the pressure dierence between each segment ∆p kj (t) almost disappears for a steadystate, the modelling error from discharge coecient estimation becomes small.

10. The use cases are restricted to switching non-linearities. These allow only discrete states such as closed or open, which can be well dened with a Heaviside function. The available steady-states of a specic high-pressure network can therefore be characterized by combining all possible switching states.

11. When introducing the Heaviside function σ kj (t), no switching dynamics are modelled.

Lumped parameter model

Applying the results from the previous section allows for describing dierent high-pressure components by means of lumped parameters. The above example distinguish between, pumping chambers, check-valves, piping sections and cutting heads. Each component represents either a local pressure state, with respect to (3.46a), or denes the uid ow path, considering (3.46b).

Pumping chambers and piping sections correspond to the rst, while check-valves and cutting heads relate to the second one. The lumped parameter models of individual components can be interconnected to represents a waterjet facility.

Pumping chamber

The pumping chamber is modelled as a volume V C (t) = V 0 -S P s P (t), in which an input uid ow Q P = S P v P is generated by piston displacement and an output uid ow Q V will result for an open check-valve. Any dierence between input and output ow rate will cause a change in pressure

d dt p C (t) ≈ κ 0 + κ 1 p C (t) V 0 -S P s P (t) (S P s P (t) -Q V (t)) (4.1)
within the pumping chamber, see (2.16). The dynamic of pressure generation will be aected by the chamber volume V C and the bulk modulus K(t) = κ 0 + κ 1 p C (t). The chamber volume changes with respect to the piston position s P (t) = v P (t) dt, where V 0 is the initial chamber volume for a fully retracted piston.

Check-valve

The check-valve will open, when the pressure within the chamber increases above the pressure of the piping. Depending on the dierential pressure ∆p V (t) = p N (t) -p C (t) at the check-valve, a uid ow

Q V (t) ≈ σ(∆p V ) S V 2 0 1 + κ 1 p C (t) κ 0 -1/κ 1 ζ V |∆p V (t)| sign (∆p V (t)) (4.2)
results, see (2.17), with respect to its cross section S V and the discharge coecient

ζ V = 1 1 + L/D λ -(1 -∆S V /S V ) 2 , (4.3)
that is determined by the friction loss λ along the subsequent piping of length L and the change of cross section ∆S V = S -S V . Only a positive ow can occur (Q V (t) ∈ R + ), for that, σ(∆p V ) is assigned to the Heaviside-function (3.44). It represents the switching of the check-valve dependent on the pressure dierence ∆p V (t). The switching dynamics itself is neglected.

High-pressure piping

The piping is assigned to its volume V 0 = S L that remains unchanged during pump operation. It couples the input and output uid ows of all high-pressure components. Any deviation from the uid ow equilibrium will cause a change in pressure

d dt p N (t) ≈ κ 0 + κ 1 p N (t) V 0 (Q P -Q H ) , (4.4)
with respect to its compressibility. The input uid ows from both pumping chambers, Q V 1 and Q V 2 , generate the overall pump rate of a high-pressure pump Q P , see (2.14). The generated pump rate has to correspond to the uid ow consumption of a cutting head, such as

Q H = Q V 1 +Q V 2 ,
to reach steady-state pressure. Any friction loss, from the uid ow along a pipe, and possible time delays are not considered, when describing the high-pressure piping as a lumped parameter. However, a piping section can be further assigned to multiple segments of dierent length and inner diameter and individually combined, e.g. to couple multiple cutting heads.

Cutting head

The piping pressure p N , applied on the cutting head, causes the output uid ow

Q H (t) ≈ S H 2 0 1 + κ 1 p N (t) κ 0 -1/κ 1 ζ H p N (t) d(t) (4.5)
with respect to the cross section S H ∈ {0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4} of the nozzle installed.

Only a positive uid ow can result (Q V (t) ∈ R + ), since the nozzle generates a waterjet and releases pressure. While ζ H denotes to the resistance of the nozzle, its coupling with the piping will be enabled or disabled by a on/o-valve. This is dened as a switching disturbance d(t) ∈ {0, 1}, manipulating the output uid ow, that is associated to the water consumption of a cutting head. It is controlled independently by the waterjet application.

Summarizing the above results allows for stating a simplied description for waterjet facilities by means of lumped parameters, which represent the coupling of n piston pumps with m cutting heads at a common interconnection point. This yields the lumped parameter model

d dt p N (t) = K(p N ) V C (t) + V 0 n σ n (t) Q n (t) - m H m p N (t) d m (t) d dt V C (t) = n (Q n (t) -V 0,n ) (4.6a) (4.6b)
of a waterjet facility.

Assuming perfectly synchronized pumping units allows further for implementing a simplied model (4.6a -4.6b) for low-level control design, see Section 5.3. This lumped parameter model is sucient to represent a high-pressure network subsection with limited piping length. It is further employed for model-based synchronization by means of camming, as shown bellow. In contrast to that, a graph-based modelling enables to describe complex topologies of various high-pressure networks, representing future waterjet facilities. Its application is given in the subsequent section.

Model-based synchronization

The pumping chamber model (4.1) is useful to design the trajectories for synchronizing the pistons of a high-pressure pump. If the check-valve is entirely closed, the pressure within a pumping chamber will increase relative to the piston displacement

dp C ds P ≈ κ 0 + κ 1 p C V 0 /S P -s P , (4.7)
for which (4.1) becomes independent of parameter t and Q V (t) = 0. Separating the variables of (4.7) and integrating both side results in

1 κ 0 + κ 1 p C dp C ≈ 1 V 0 /S P -s P ds P , (4.8)
where solving the integrals (4.8) and rearranging yields p C ≈ e -κ 1 (log(V 0 /S P -s P )+c) -κ 0 κ 1 , (4.9) which describes the pumping chamber pressure with respect to the piston position, taking a variable bulk modulus into account. Evaluating (4.9) for the integration constant yields

c = - log(κ 0 + κ 1 p S ) κ 1 + log V 0 S P , (4.10)
while expecting the pumping chamber pressurized to p C (s P = 0) = p S , as initial condition, and considering a fully retracted piston. The analytic solution (4.9) provides the desired piston position s = s P of Equation (2.19) for trajectory generation in Section 2.3. That requires to synchronize the pistons with respect to the pressurization position

s P ≈ V 0 S P -e -log(κ 0 +κ 1 p C )/κ 1 -c . (4.11)
The plot in Figure 4.2 shows the relative change of volume by means of piston displacement to pressurize a pumping chamber according to (4.11). The pistons need to operate alternately providing a steady input uid ow. Assuming that a steady-state uid ow equilibrium is reached, the uid ow consumption of a cutting head corresponds then to the desired piston velocity ṽ = Q H (t)/S P , further required in (2.20) for trajectory generation.

Graph-based network model

The use of graphs assigns homogeneous segments to the initial uid ow path of a high-pressure network. This results in a system of equations considering the network description (3.54). This system of equations models the pressure propagation for various waterjet facilities, considering distributed high-pressure pumps interconnected to work stations. The segmentation of highpressure components is realized according to the following procedure [START_REF] Niederberger | A graph-based modelling methodology for high-pressure networks applied on waterjet machining[END_REF]:

1. Identify the high-pressure components which are dedicated to change the cross section along its uid ow path.

2. Divide the high-pressure network along the identied components into sections n of lengths L n , wherever a cross section changes, such as S n = S n+1 .

3. Chose a propagation time τ , that adequately represents the expected dynamics of pressure generation, as induced by any exogenous uid ow Q ext . This can be done by analysing the dynamic range of pumps and cutting heads.

4. If L n > a τ , subdivide the initial segments with respect to the uid's speed of sound a, into homogeneous segments k of length ∆x k < a τ for k = L n /∆x k . Otherwise, ∆x k = L n and k = n.

5.

Simulate the derived network model with the exogenous dynamics on Q ext to investigate whether the desired dynamic range of pressure generation has been met.

6. Repeat steps 3 to 5 with varying propagation times to provide a reasonable trade-o between model complexity and accuracy.

As a result, a constant cross section is preserved for each segment. These segments are subdivided, if needed, to satisfy the underlying principles of uid ow continuity and momentum conservation. However, it has to be mentioned that not every initial segment needs further partitioning into equally spaced homogeneous segments to accomplish the desired dynamic range for simulation.

Applying this procedure on the high-pressure network topology (a) Short piping, presented at the beginning of this section, the high-pressure components are equally divided along the uid ow path, wherever a cross section changes. Consequently, the two check-valves, the connector and the nozzle separate the high-pressure network into two pumping chambers, a coupling, a high-pressure piping and a cutting head. Table 4.1 assigns the components of the investigated high-pressure network to N = 8 nodes. All nodes are therefore interconnected as schematically shown in Figure 4.3. The high-pressure piping of length L = 4.8 m is subdivided into n = 4 segments of length l = 1.2 m to obtain a propagation time τ of less than 1 ms, taking a cutting head switching dynamics with a rise time of 10 ms into account. This results in a graph network N (a) kj that consists of By applying (3.54) to the assignments (4.12) and (4.13), the uid ow path is recovered with respect to the network description

N kj : d dt p k = K k V k 2 k ϑ kj |∆p kj | sign (∆p kj ) + K k V k Q k , (4.14)
where

p k ∈ R k , Q k ∈ R k and ∆p kj ∈ R N . The corresponding matrix representation d dt    p 1 . . . p 8    =      K 1 V 1 2 1 0 . . . 0 K 8 V 8 2 8      •              -ϑ 13 0 • • • 0 0 -ϑ 23 0 . . . ϑ 13 ϑ 12 -ϑ 34 0 0 0 ϑ 34 -ϑ 45 0 . . . . . . . . . . . . . . . 0 ϑ 67 -ϑ 78 0 0 • • • 0 ϑ 78 -ϑ 80              Θ kj    |∆p 13 | sign (∆p 13 )
. . .

|∆p 80 | sign (∆p 80 )    +    K 1 V 1 0 . . . 0 K 8 V 8       Q 1 . . . Q 8    (4.15)
includes the interconnection matrix Θ kj ∈ R k×N , that maintains the uid ow continuity at each node k. It is composed by ϑ kj = ζ kj S kj σ kj , which represents the possible uid ows along the branches from a node k to an other node j, and eventually denes a network topology with N interconnections. The proposed modelling methodology directly allocates the parameters of the involved highpressure components to the corresponding segments. Regarding the above network description, Table 4.2 shows the parameters of each segment. The Heaviside functions σ(∆p 13 ) and σ(∆p 23 ) restrict the uid ow of the check-valves to its positive direction, as each piston induces the displacement ow rates Q 1 and Q 2 . The derived high-pressure network N kj from (4.12) and (4.13) has been numerically simulated in MATLAB Simulink and validated with measurements, see the subsequent chapter. In fact, a simulation toolbox has been established that reduces the expenditure of time, when modelling dierent network topologies subject to represent various waterjet facilities.

8 3 1 2 Q 1 Q 2 4 Piping 7 Pump (2 pistons) Head 0 Q 13 Q 23 Q 34 Q 67 Q 78 Q 80
Applying the same procedure to all high-pressure networks from Section 2.2, generates a network model for each topology (a) -(g). These models are dened by means of graphs and therefore straight forward to implement in MATLAB Simulink for numerical simulations. From denition (4.14) results dierent network models, which are equivalent to the matrix representation (4.15). The parameters for the pumping chambers, its coupling and all cutting heads are the same as introduced above. Only the piping sections and its interconnections vary to represent the dierent high-pressure networks. Hence, the subsequent modelling discusses the segmentation of the piping sections into homogeneous segments, while considering the dierent network topologies.

Table 4.2: Model parameters for topology (a): parameters of each segment, corresponding to (4.12), (4.13) and grouped in nodes and branches.

S k ∆x k Q k S kj ζ kj σ kj d kj k (mm 2 ) (mm) (cm 3 /s) kj (mm 2 ) (-) (-) (-) 1 ### ### Q ext (t) [13] 1.77 ### f (∆p 13 ) 1 2 ### ### Q ext (t) [23] 1.77 ### f (∆p
D H ### 1 D H (t)
Modelling 1 pump -1 cutting head setups The discretization is again chosen to guarantee a desired propagation time with respect to the high-pressure components. This results in a graph network N (b) kj that consists of [3,4] Modelling 1 pump -2 cutting heads setups branches for the asymmetric topology. Lumped parameter models of order 24 results for both high-pressure networks, when evaluating (4.14) for (4.18) and (4.19), respectively. The parameters of Table 4.5 summarizes the piping sections used to simulate the symmetric and asymmetric network topologies. For segmentation, the networks have been divided into three sections of length L. The number of segments n per section clearly distinguishes both network topologies. The symmetric network model features 6 states for both network sections interconnecting a cutting head each. Regarding the asymmetric network model, the section to head 1 features 1 state while head 2 is connected over 11 states. Modelling 2 pumps -2 cutting heads setups The parameters of Table 4.6 summarizes the piping sections, used to simulate the symmetric distribute, asymmetric distributed and decentralized distributed network topologies. For segmentation, the networks have been divided into ve sections of length L. The number of segments n per section clearly distinguishes the network topologies. The symmetric distributed network model features 6 states for both network sections, interconnecting the high-pressure pumps. Regarding the asymmetric distributed network model, the section from pump 1 features 11 states, while pump 2 is connected over 2 states. The symmetric as well as the asymmetric distributed models interconnect two cutting heads over the same connector. On the other hand, the decentralized distributed model connects each pump individually to a cutting head, using a section of 6 states each. The sections, containing a pump and its corresponding cutting head, are coupled over a piping section of 8 states.

kj ∈ {[1, 3], [2, 3],

###

It is further outlined, that the electrically driven pump, installed at the test bench, is considered as two exogenous uid ow source (Q 1 and Q 2 , one for each piston) and that a cutting head is represented by a pressure sink to atmosphere. The pressure states pP 1 (t) and pP 2 (t), pN (t), pH1 (t) and pH2 (t) are evaluated for model validation and controller verication, see Chapter 4 and 6.

Experimental studies

The graph-based modelling methodology has been applied in Section 4.2 to obtain high-pressure network models for dened network topologies. This chapter will rst verify the graph-based modelling methodology with respect to the variable bulk modulus and variable uid density, while considering a basic test bench conguration. For that, a high-pressure network has been modelled, which directly interconnects a single high-pressure pump with a cutting head, when using the graph-based modelling methodology as well as applying alternative approaches from literature. These alternative approaches consider a constant bulk modulus or a constant uid density. The simulated pressure trends, which originate form the dierent modelling approaches, will be compared to measurement data from the test bench. The experiment have been derived with respect to a wide pressure range from 100 to 400 MPa, while operating the pump in openloop control. This is only possible, if the cutting head remains open. This chapter will then validate the dierent high-pressure network models, using again measurement data from the test bench. For that, the test bench has been congured with respect to dierent network topologies: considering a single high-pressure pump that supplies a single cutting head, a single pump with two cutting heads and two pumps with two cutting heads. The cutting heads will switch independently, according to given switching patterns, whereas a constant operating pressure is obtained by means of closed-loop control. This aims to generate a varying water consumption, which results in dierent uid ow velocities and pressure losses along the high-pressure piping.

All simulation results have been obtained with Simulink R2017b using the ode23s solver with variable step size. The solver has been congured for a relative tolerance of 10 -4 and an absolute tolerance of 10 3 . It has been running on a Windows 10 machine with an Intel Core i7-7600U 2.80 GHz processor and 16.0 GB RAM.

Graph-based modelling methodology validation

The proposed graph-based modelling methodology has been veried, considering a basic highpressure network used for waterjet machining. The topology (a) Short piping has been considered, interconnecting a single high-pressure pump with one cutting head. A rst model has been derived, using the graph-based modelling methodology (3.54), which takes into account a pressure-dependent uid density = f (p(t)) as well as a pressure-dependent bulk modulus κ = f (p(t)). In contrast to that, two alternative models have been derived by using approaches from literature. More precisely, a rst approach considers a constant uid density with a variable bulk modulus, such as [START_REF] Hountalas | Development of a fast and simple simulation model for the fuel injection system of diesel engines[END_REF]] and [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF]. The system pressure p is thereby simulated using the dierential equation

dp dt = K V dV dt -Q , (4.23)
where the dierence between an output uid ow Q and the displacement volume dV /dt induces a pressure, with respect to the uid volume V and the pressure-dependent bulk modulus [START_REF] Hountalas | Development of a fast and simple simulation model for the fuel injection system of diesel engines[END_REF]]

K = 12 (1000 + p) .
(4.24)

The output uid ow is derived by the algebraic equation

Q = sign (∆p) ζ S 2 |∆p| , (4.25)
which considers the uid ow resistance ζ along a cross section S, for d /dt = 0 ∀ t > 0.

A second approach considers a constant bulk modulus with a variable uid density, such as [START_REF] Ferretti | Abrasive waterjet intensier model for machine diagnostics[END_REF]. The output uid ow (4.25) thereby considers a pressure-dependent uid density [START_REF] Fabien | Dynamic modelling and identication of a waterjet cutting system[END_REF] = 0 1 + 7.5048 p 2209.4

1/7.5048 , (4.26)
assuming dK/dt = 0 ∀ t > 0. Both approaches will be employed using the assignments (3.52) and (3.53), as dened in Section 3.3 to derive a network description, while considering a constant uid density and a constant bulk modulus, respectively.

The measurement R-1 of Table 4.7 has been selected to investigate the pressure generation for a wide pressure range from 100 to 400 MPa. This experiment is subject for a stripping process, hence the cutting head remains open for all time. It has been congured with a nozzle of 0.25 mm inner diameter. The pump rate is manually increased, until the produced overall uid ow corresponds to the resulting water consumption, such as to obtain the desired operating pressure. The high-pressure pump operates in open-loop control. This enables a pump to attain the desired operating pressure, as long as the switching state of the cutting head remains unchanged. As a result, the coupling of two pumping units to a cutting head and the pressure generation over a wide pressure range enable to investigate the eects of varying parameters for the dierent modelling approaches: the graph-based modelling (3.54), the model for constant bulk modulus (4.26) and the model for constant uid density (4.24).

The block diagram of Figure 4.7 illustrates the experimental setup, as used for model verication. For each experiment, reference position trajectories have been loaded to the test bench. These trajectories enable a periodical piston displacement, where the pump rate u(t) denes the cycle time of piston displacement and provides the desired uid ow rate Q. Encoders measure the actual piston positions and its velocities, while pressure gauges measure the pressure trends p x (t) at dierent network positions x. The PLC acquires the data with a sampling rate of 200 Hz. The measured piston positions and velocities has been evaluated to estimate the induced uid ow Q of each pumping unit. These uid ows are then used as model inputs for numerical simulations, where the simulated pressure trends px (t) have been compared to measurements p x (t) to provide the residuals ξ(t). The simulations are repeated for the dierent modelling approaches.

Network Gauges Pump

Encoders

p x Q u Q ^Model ξ + p x Figure 4.
7 Experimental setup to verify the modelling methodology: pressure gauges measure the pressures p x (t) at positions x, when applying an input uid ows Q(t). The estimated uid ows Q(t) are used as model inputs and the simulated pressure trends px (t) have been compared to measurements.

Varying operating pressures (Stairs from 0 to 400 MPa)

The measurement data shown in plots of Figure 4.8 overview the experiment, used to compare the graph-based modelling methodology with the other approaches from literature (e.g [START_REF] Ferretti | Abrasive waterjet intensier model for machine diagnostics[END_REF] and [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF]). The reference trajectories implemented to the highpressure pump have been optimized for a pressure of 200 MPa. Operating points below this pressure cause pressure overshoots and operating points above this pressure result in pressure losses. This phenomena has been already explained in Section 2.3, when introducing the modelbased synchronization. For experiment R-1, the cutting head has been congured with a nozzle of 0.25 mm inner diameter. Thus, an induced overall uid ow of Q 200 = 1.14 l/min is required to obtain a steady-state pressure of 200 MPa. However, the overall uid ow Q(t) = {0.6, 0.8, 1.0, 1.2, 1.4, 1.2, 1.0, 0.8, 0.6} • Q 200 has been varied to follow a stair.

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m/n) Figure 4
.8 Overview of measurement R-1: measured pump output pressure and corresponding reference displacement ow rate, for a cutting head congured with a 0.25 mm nozzle and position trajectories optimized for 200 MPa.

The zoomed-in plots of Figures 4.9 -4.12 present the measured and simulated pressure trends for steps, applied to increase and decrease the displacement ow rate, around an operating pressure of 100, 200, 300 and 400 MPa, respectively. The simulation data of three dierent modelling approaches have been compared, using varying parameters (3.54), constant bulk modulus (4.26) and constant uid density (4.24). Where the rst plot of each step has been obtained by the graph-based modelling methodology (1a-h), which considers a pressure-dependent bulk modulus as well as a pressure-dependent uid density, the second and third plots consider a model for constant bulk modulus with K 140 = 2.7374 GPa (2a-h) and a model for constant uid density with 270 = 1104.6 kg/m 3 (3a-h). For each simulation, transient behaviour as well as the expected steady-states is improved, when considering variable parameters.

The zoomed-in plots of Figure 4.13 -4.15 compare the measured with the simulated pressure trends, for operating points at 100, 200 and 400 MPa, and show the resulting residuals. The pressure has been obtained from the pump outtake position (1i-n) and from the cutting head intake position (2i-n). Additional to the graph-based modelling methodology, the model for constant bulk modulus and the model for constant uid density has been considered. The parametrization of the alternative models has been individually obtained, to minimize the root mean square error (rmse) between the simulated pump output pressure and the experimental data. The constant bulk modulus of minimal rmse has been found for a pressure of 140 MPa and the constant uid density of minimal rmse has been found for a pressure of 270 MPa. In contrast to that, the introduced variable parameter model has been applied, without initial parameter tuning. 4.11 Step response for operating pressure around 300 MPa, considering varying parameters (1e/f), constant bulk modulus (2e/f) and constant uid density (3e/f): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow. 4.12 Step response for operating pressure around 400 MPa, considering varying parameters (1g/h), constant bulk modulus (2g/h) and constant uid density (3g/h): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow.

(1a) (1b) (2a) (2b) (3a) (3b)
(1i) (1j) (2i) (2j)
Figure 4.13 Steady-state of measured and simulated pump output pressure (1i/j) and cutting head input pressure (2i/j) around 100 MPa: the proposed modelling methodology is thereby compared to simulations with constant bulk modulus and constant uid density. This model validation considers three specic network topologies, for which measurement data have been derived from the high-pressure test bench. The topology (b) Long piping interconnects the high-pressure pump to a single cutting head, while the topologies (c) Symmetric network and (d) Asymmetric network consider two cutting heads. Dierent measurements have been obtained for simulation and experimental validation, as listed in Table 4.9. This section provides an overview, plots extending these results are given in Appendix A.1. The basic measurements (B-4,5,7) consider a high-pressure pump, that directly interconnects a single cutting head. The rst 3 extended measurements (E-4,5,7) consider the symmetric network topology, including two cutting heads, while both cutting heads have been asymmetrically congured with nozzles of dierent inner diameter. The last 2 extended measurements (E-9,11) consider the asymmetric network topology, while both cutting heads have been symmetrically congured with identical nozzles. This allows for investigating and distinguishing the eects due to an asymmetric work station conguration and an asymmetric waterjet facility, respectively.

(1k) (2l) (2k) (2l)
Considering the symmetric network model, each cutting head holds nozzles of dierent inner diameters. Thus, a dierent uid distribution results for both cutting heads, due to the nozzles installed, and dierent pressure losses occur for each network section. Considering the asymmetric network model, both cutting heads are equipped with identical nozzles. Thus, the asymmetric setup causes dierent pressure losses along each network section. This will slightly aect the uid distribution between the cutting heads. All experiments are derived for two operating points, each taking dierent switching patterns and dierent nozzle congurations into account. Contour cutting is investigated for each basic and extended measurement. A cutting head will exemplarily open for a cutting period of 60 seconds, followed by a hold period in closed position for another 60 seconds. Regarding the extended measurements, this switching pattern will be repeated periodically for both cutting heads with a phase shift of 90 • . The measurements B-5 and E-5 investigate cavity cutting, where the switching pattern is modied to realize an on/o period of 0.5 seconds. Consequently, a large variety of measurement data for dierent operating points, switching patterns, cutting head congurations and network topologies has been obtained to validate the high-pressure network models, as derived by the graph-based modelling methodology.

The diagram of Figure 4.16 illustrates the experimental setup, as implemented for model validation. The cutting heads open and close with respect to the predened switching patterns, given by the disturbance signals

d m (t) = [d 1 (t), d 2 (t)].
The low-level control will then adjust the pump rate, using the control signal u(t), to obtain the desired reference pressure p s (t). The model-based synchronization (Q [0 : 2.45] l/min (0 ≤ u(t) ≤ 1), see Equations (2.15) and (2.26) in Section 2.3. Dierent locations have been dened at the test bench, to measure the pressure states. Variable p P (t) denotes the input pressure at the pump outtake, p N (t) gives the pressure at the network junction and p Hm (t) gives the pressure at the cutting heads, where m ∈ {1, 2} refers to the rst and second cutting head, respectively. The induced uid ows

1 (t) = f 1 (u(t)), Q 2 (t) = f 2 (u(t))) displaces both pistons such as Q 1 (t) + Q 2 (t) =
Q(t) = [Q 1 (t), Q 2 (t)]
of both pistons are estimated by measuring the piston displacements. All measured and estimated input signals ( Q and d m ) are fed into the high-pressure network models for simulation. The measured data p x (t) = [p P (t), p N (t), p H1 (t), p H2 (t)] are then compared to the simulated pressure states px (t) = [p P (t), pN (t), pH1 (t), pH2 (t)] to validate the simulations for dierent measurement positions. The measurement data have been captured with a sampling rate of 500 Hz.

Network Gauges Pump Controller

Encoders

p x Q u d y = p P Q r Model ξ + p x Figure 4.
16 Experimental setup for model validation: pressure gauges measure the pressures p x (t) at dierent positions, while applying an input uid ows Q(t). The estimated input uid ows Q(t) are used as model inputs and the simulated pressure trends px (t) have been compared to the measurements.

The derived high-pressure network models from Section 4.2 have been numerically simulated in MATLAB Simulink, using the developed toolbox to realize a straight forward implementation of the graph-based modelling methodology. The solver ode23s for variable step-size has been congured for a relative tolerance of 10 -5 and an absolute tolerance of 10 3 . This enables a simulation precision of 10 -2 MPa.

Contour cutting: asymmetric cutting heads

A continuous operating point of 200 MPa pressure is chosen. Two cutting heads will open and close alternately with a phase shift of 90°. Considering the measurement E-4, cutting head 1 holds a nozzle of 0.3 mm inner diameter, while the cutting head 2 is congured with a 0.2 mm nozzle. As a consequence when combining all possible switching states, four dierent uid ows will result: Q(t) ∈ {0, 0.78, 1.61, 2.39} l/min. A similar experiment for a 350 MPa setup (measurement E-7) is given in the Appendix A.1. The plots of Figure 4.17 show input pressure trend, induced overall uid ow and cutting head switching pattern for simulation and measurement for an entire switching cycle. .17 Overview of experiment E-4, considering a 1 pump -2 cutting head setup at 200 MPa with dierent nozzle sizes: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles.

Zoomed-in plots for every measurement position are presented in the subsequent plots, when rst opening cutting head 1, followed by the opening of cutting head 2 and then closing head 1, followed by the closing of head 2. The plots of Figure 4.18 show the zoom-in for opening and closing of cutting head 1 (a/b), as well as for opening and closing of cutting head 2 (c/d). The input and system pressures are given in the rst plot, both output pressures in the second plot and the corresponding switching pattern in the third plot.

Contour cutting: symmetric cutting heads

A continuous operating point of 200 MPa pressure is chosen. Two cutting heads will open and close alternately with a phase shift of 90°. Considering the measurement E-9, both cutting heads hold a nozzle of 0.25 mm inner diameter. When combining all possible switching states, four dierent uid ows will result: Q(t) ∈ {0, 1.16, 1.17, 2.33} l/min. A similar experiment for a 350 MPa setup (measurement E-11) is given in the Appendix A.1. The uid ow diers slightly between each nozzle, due to the asymmetric network conguration, which causes dierent pressure losses along each piping section. The plots in Figure 4.19 show input pressure trend, induced overall uid ow and cutting head switching pattern for simulation and measurement of an entire switching cycle. Zoomed-in plots for every measurement position are given in the subsequent plots, when rst opening cutting head 1, followed by the opening of cutting head 2 and the closing of head 1, followed by the closing of head 2. The plots of Figure 4.20 show the zoom-in for opening and closing cutting head 1 (a/b), as well as for opening and closing cutting head 2 (c/d). The input and system pressures are given in the rst plot, both output pressures in the second plot and the corresponding switching pattern in the third plot. On the other hand, the high-pressure network becomes frequently disturbed by the switching behaviour of check-valves. This causes the pressure to decrease or increase rapidly, since the produced overall uid ow of the pump is no longer in balance with the consumed uid ow of the cutting head, see periodic spikes in Figure 4.19. The pump will adjust its uid ow, as fast as possible, to compensate this disturbance, induced by a pumping chamber itself. Simulations will not reproduce this check-valve hysteresis, as the idealized pumping chamber model considers a simple switching function (see Chapter 3). The hysteresis appears to occur by incidence, for yet unknown uid ow conditions. The check-valves seem less sensitive for experiments using large nozzles, as it is the case for measurements at 200 MPa operating pressure. However, it becomes very sensitive and the simulations perform inaccurate, when considering the 350 MPa experiments (see Appendix A.1). The check-valve hysteresis will be discussed later in this section.

d 1 → 1 d 1 → 0 d 1 = 1 d 1 = 0 d 2 = 0 d 2 = 1 d 2 → 1 d 2 → 0 E-
A wide pressure range can be investigated for ramping up and down a pump to its operating pressure (0 MPa → 200 MPa and 0 MPa → 350 MPa), as shown in Appendix A.1. An oset between simulations and measurements is observed, for pressures below 100 MPa, which decreases for higher pressures. As long as a pump operates apart from the congured operating pressure, its pumping units are inadequately synchronized. This induce additional pressure uctuations as a consequence of the phase-shifted pump operation. In this case, the check-valve switching behaviour is well reproduced by means of simulations.

The uid ow resistance characterizes the pressure loss for a piping section, using a single parameter model. This seems insucient for a wide pressure range, especially the residuals increase for high uid ow rates. This eect scales for an increased network size, hence the simulation accuracy for a long piping becomes worse. Comparing the experiments (Table 4.10), derived from basic topologies, with the results (Table 4.11), considering extended topologies, shows an increased rmse. As a consequence, the modelling could be improved by introducing a more complex description of the uid ow resistance. It is further expected, that any wear of the nozzles inner diameter will cause a deviation of the resulting steady-state pressure. Since the model is parametrized for the nominal nozzle diameter, the residuals between measurement and simulation will increase with respect to the operating hours.

Cavity cutting: the Appendix A.1 shows, complementary to the contour cutting application, results for cavity cutting at 200 MPa operating point. These are equally applied on the topologies (b) Long piping and (c) Symmetric network. This basic topology is again considered as a reference, when switching a single cutting head. The Table 4.12 compares the rmse for the residuals between measured and simulated data of the experiments with a single cutting head. The Table 4.13 provide the rmse considering the experiments with two cutting heads. The residuals have been evaluated for a period of 3 switching cycles. A column corresponds to a zoomed-in plot, where a row provides a measurement position. Table 4.12: rmse (MPa) to compare the dierent experiments, derived for cavity cutting with a single cutting head.

Measurement

Pos. It is remarkable how precisely the simulated pressure trends follow the measured trends for every measurement position, even for pressure uctuations of about 100 MPa in amplitude. Considering the basic topology, a relative error of about 1% has been found, for a measurement position at the pump outtake, where the error reaches about 2%, at the network and cutting head positions (see Table 4.12). The errors are slightly increased for the extended topologies and reach from 1.5 -3% (see Table 4.13).

d 1 = {0, 1} d 1 = {0, 1} d 1 = 0 d 2 = 0 d 2 = {0, 1} d 2 = {0, 1} E-
The disturbing check-valve hysteresis does not occur for transient pressure trends, but seems sensitive for steady-state operation around desired operating point. However, the closing of all cutting heads again induce pressure overshoots, followed by a continuous pressure loss. The rst eect is sensitive for any mismatch of the input uid ow that may cause a decreased simulation accuracy, while the second eect is not taken into account for modelling.

Check-valve hysteresis

All simulation models are congured for an optimal phase-shifted pump operation. Thus, any pressure uctuation due to switching check-valves will disappear in simulation, when reaching the operating pressure. In reality, some hysteresis can be observed for the switching of the check-valves, causing pressure uctuations. Taking this hysteresis into account can improve the simulation accuracy. Nevertheless, the occurrence of this eect remains unpredictable.

Observing the pressure trends of measurements E-9 and E-11, as well as comparing these with simulations, derived from nominal model, show the eect of check-vale hysteresis. In the plots of Figure 4.21 (a), it is seen that the model follows the excitation given by the phase-shifted pump operation, but possible pressure losses are missing. Hence, the simulated pressure trend lies above the measurement for the time samples between 436 and 438 seconds. Alternatively, the plots of Figure 4.21 (b) show the pressure trends, when taking a possible check-valve hysteresis into account. The simulation model applies thereby possible pressure losses, but that results in a deterioration, whenever no hysteresis occurs, see time samples between 438 and 440 seconds. The same behaviour has been observed at 350 MPa pressure, using the same model, see the plots of

High-pressure network model validation (2 pump conguration)

This section investigates the simulation accuracy for more complex high-pressure networks, when applying the graph-based modelling methodology for congurations with two high-pressure pumps. The pressure generation in a high-pressure network becomes again disturbed, when switching dierent cutting heads in open and close position. Both cutting heads remain locally coupled, where two decentralized high-pressure pumps are taken into account. Both pumps are in closed-loop control, aiming at a constant reference pressure of 200 MPa. For this reason, pump 1 holds the PI controller (6.47), whereas pump 2 has by default the PI controller (6.48), see Section 6.4. The high-pressure pumps will then adjust its induced overall uid ow, such as to compensate any disturbances. Thus, a wide variety of uid ow rates from 0 to 2.45 l/min per pump will be obtained, depending on the switching state of the cutting heads. Here, the graphbased modelling methodology has been applied to model network topologies, which interconnect two high-pressure pumps with two cutting heads. This model validation considers two specic network topologies, for which measurement data have been derived from the high-pressure test bench. Both, the topologies (e) Symmetric distributed and (f) Asymmetric distributed, interconnect two cutting heads at a local position.

Dierent measurements have been obtained for simulation and experimental validation, as listed in Table 4.14. The rst 3 distributed measurements (D-1,2,3), considering the symmetric distributed network topology, include two pumps and two cutting heads, where the piping from each pump to the cutting heads have the same length. The last 3 distributed measurements (D-4,5,6) consider the asymmetric distributed network topology, where the piping from a pump to the cutting heads have dierent lengths. For all measurements, both cutting heads have been symmetrically congured, with identical nozzles of 0.2 mm inner diameter. This allows for distin-guishing the eects due to the network topology and for investigating the eects from coupling of two pumps. All experiments take dierent switching patterns into account. The measurements D-1 and D-4 investigate the stripping process, where all cutting heads remain open. The measurements D-2 and D-5 investigate contour cutting. Here, a cutting head will exemplarily open, for a cutting period of 60 seconds, followed by a hold period in closed position, for another 60 seconds. Regarding the distributed measurements, the other cutting head will remain in open position for all time. The measurements D-3 and D-6 investigate cavity cutting, where the switching pattern is modied to realize an on/o period of 0.5 seconds. Consequently, a large variety of measurement data have been obtained to validate the high-pressure network models, derived by the graph-based modelling methodology.

The experimental setup implemented for model validation remains the same as already illustrated in the diagram of Figure 4.16. The cutting heads open and close with respect to the predened switching pattern, as given by the disturbance signals

d m (t) = [d 1 (t), d 2 (t)].
A decentralized low-level control will then adjust the pump rates of each high-pressure pump, using the control signal u n (t) = [u 1 (t), u 2 (t)], to obtain the desired reference pressure p s (t). Again, dierent locations have been dened to measure the pressure states at the test bench. p P n (t) denotes the input pressures at a pump outtake, where n ∈ {1, 2} refers to the rst and second high-pressure pump, respectively. p N (t) gives the pressure at the network junction and p H (t) gives the pressure between the two locally coupled cutting heads. The induced uid ows

Q n (t) = [Q n1 (t), Q n2 (t)],
of each piston, are again estimated by measuring the piston displacements. Measured data p x (t) = [p P 1 (t), p P 2 , p N (t), p H (t)] are then compared to the simulated pressure states px (t) = [p P 1 (t), pP 2 (t), pN (t), pH (t)] to validate the simulations for dierent measurement positions. Data acquisition and simulations are identically congured, as introduced in the previous section.

Contour cutting: symmetric and asymmetric distributed pumps

A continuous operating point of 200 MPa is chosen. The second cutting head will open and close, while the rst cutting head remains open. Both cutting heads hold a nozzle of 0.2 mm inner diameter. For this conguration, two dierent uid ows will result: Q(t) ∈ {0.78, 1.56} l/min. The plots of Figure 4.23 and Figure 4.24 show the input pressure trends, induced overall uid ows and cutting head switching pattern, considering an entire switching cycle for the symmetric and asymmetric topology of measurements D-2 and D-5, respectively.

Zoomed-in plots for the switching of cutting head 2 is given in the subsequent plots, when rst closing then opening cutting head 2. The plots of Figure 4.25 and Figure 4.26 show the zoom-in for closing (a) and opening (b) cutting head 2 for the symmetric and asymmetric topology. The input pressures are given in the rst plot, the network and output pressure is shown in the second plot. Summary for 2 pump congurations Contour cutting: this section investigated the contour cutting at 200 MPa operating point, applied on the topology (e) Symmetric distributed and (f) Asymmetric distributed. Table 4.15 compares the rmse for the residuals between measured and simulated data. The residuals have been evaluated for a period of 2 seconds, when switching a cutting head. A column corresponds to a zoomed-in plot, where a row provides a measurement position. The simulated pressure trend follows the measurement with reasonable accuracy, while reproducing the varying water consumption due to the switching of cutting heads. However, the model fails for disturbances that originate from the pumping chambers. It is observed, that the checkvalve malfunction becomes tremendous for small uid ow rates. The pump rate equilibrium becomes entirely disturbed. The low-level control increases the induced uid ow to compensate any resulting pressure losses. Since, the simulation model is not capable to predict any check-valve malfunction, the increased uid ow cause the model to predict non-existing pressure overshoots, see Figure 4.23 and 4.24. Nonetheless, the simulations aim to reproduce those pressure uctuations in a high-pressure network, which are induced by the switching of cutting heads. These uctuations are well reproduced, see Figures 4.25 and 4.26. The resulting rmse are close to the values, as previously reported for a single pump conguration (see Table 4.11). This results in a relative error from about 1 -2%. But again, the deviation for an expected pressure loss will increase for long piping sections. The pressure loss will be further discussed below.

d 1 = 1 d 1 = 1 d 2 → 0 d 2 → 1 D-2 p P 1 3.
Cavity cutting: the Appendix A.2 shows, complementary to the contour cutting application, results for cavity cutting at 200 MPa operating point, again applied on the topologies (e) Symmetric distributed and (f) Asymmetric distributed. Table 4.16 provides the rmse, considering the experiments with two cutting heads. The residuals have been evaluated for a period of 3 switching cycles. A column corresponds to a zoomed-in plot, where a row provides a measurement position.

The simulated pressure trend reproduces the measured transient behaviour with desired accuracy, considering large pressure uctuations. But an oset is observed between the switching periods, which disappears whenever a switching occurs. This oset concerns the simulated pressure trends for every measurement position, with respect to the asymmetric distributed network, and causes the relative error to reach almost 5%.

9.0329

Starting up procedure A single high-pressure pump is not capable to reach the desired operating pressure for an increased water consumption of the dened cutting head conguration. The pump will run at its upper limit. When starting the second pump, both pumps will obtain an individual pump rate equilibrium to attain the desired operating pressure. The simulations allow for evaluating the pressure generation in a pumping chamber of each pumping unit. The plots of Figure 4.27 and Figure 4.28 show the estimated pressures for every pumping chamber considering the symmetric and asymmetric topology of measurements D-1 and D-4, respectively. It is shown that pressure uctuations, measured at the pump outtake, correspond to the switching between pumping units. The low-level control of each high-pressure pump is compensating these uctuations, resulting in a varying pump rate.

As soon as desired operating pressure is obtained, pressure uctuations, due to the switching between pumping units, should become small. However, the check-valves from the second pump have malfunctions, which require a large control eort u 2 (t). The locally induced uctuations become partially compensated by means of control. Nevertheless, remaining uctuations propagate along the high-pressure network and can be observed at the rst pump, which responds with control eort u 1 (t).

Conclusion

This chapter implemented a framework in MATLAB Simulink to describe generalized highpressure networks. The obtained simulation toolbox can be used by the industrial partners to design and optimize future waterjet facilities. This toolbox has been applied to the dierent network topologies, as introduced in Section 2.2. This results in high-pressure network models, which are useful for numerical simulations. So far, 6 network topologies have been investigated, using the graph-based modelling methodology. Each network model has been then experimentally validated for 2 nozzle congurations and 2 switching patterns, using measurements at 2 operating pressures.

Another contribution of this chapter validates the dierent network models with experimental data, using the high-pressure test bench. It has been found that modelling a constant uid density cause a steady-state pressure error and a constant bulk modulus is responsible for a divergent transient behaviour. Therefore, an improved model accuracy has been achieved for a desired pressure range of 100 to 400 MPa, when a variable uid compressibility is considered.

Representative use cases have been selected, showing the abilities to simulate the pressure propagation for dierent network positions. The presented validation obtained excellent results with desired accuracy, considering dierent measurement positions. Nonetheless, increasing residuals have been reported, when operating the test bench on low pump rates. To improve this circumstance, the friction losses for small uid ow velocities should be further investigated.

It has been further recognized that the check-valve switching is very sensitive to any pressure mismatches. An accurate switching hysteresis, therefore, will reduce the resulting simulation residuals. Pressure losses or overshoots have been observed, if any trajectory imperfectly synchronizes the pistons. The derived model reproduces this behaviour and can be used to improve the pressure generation by means of optimized position trajectories. The check-valve malfunction complicates to accomplish all experiments and interferes the model validation process.

Using the proposed modelling methodology, it is possible to exibly model various network topologies with reasonable eort. This becomes advantageous when modelling various network topologies, to optimize entire waterjet facilities, and when evaluating novel control strategies for energy ecient waterjet machining. High-pressure network models of increased complexity have been introduced in Chapter 7, which allow for evaluation of distributed high-level managing. Further work should continue to validate the simulation toolbox, taking more complex highpressure networks into account.

Part III Control design 135

Introduction Objectives

Electrically driven high-pressure pumps need a sophisticated control strategy. In fact, a precise control of coupled pumps is mandatory to avoid undesired interactions and a wide dynamic range is required to attenuate exogenous disturbances. Cutting heads induce pressure uctuations, when switching on and o a waterjet with respect to an a priori unknown switching pattern. Any pressure uctuation will degrade the cutting quality. Thus, a varying water consumption has to be compensated by means of the available high-pressure pumps. The control objective is further to guarantee a continuous steady-state pressure for a given pressure reference. Also pressure uctuations, due to exogenous disturbances, has to be eliminated within limited control eort. Each pump, located at a dierent network subsection, requires to reject local pressure uctuations with out destabilizing the overall high-pressure network. In addition, dierent types of parameter variations will aect a waterjet facility. It includes a wide pressure range and a varying workstation conguration, which consequently introduces non-linear dynamics. A waterjet facility involves signicant uncertainties. This necessitates a suitable description for robust control design.

State of the art

Recent research work in the eld of waterjet machining intends to improve cutting quality [START_REF] Ahmed | Improving surface roughness of abrasive waterjet cutting process by using statistical modeling[END_REF], and process eciency [Averin, 2017]. Literature is found in related elds to deal with pressure generation and its control. Several papers in the automotive industry are investigating common rail injection systems: including high-pressure pump, common rail and injectors. This conguration is comparable to high-pressure systems for waterjet machining. In [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF], a non-linear system model is used to derive a sliding mode controller, whereby pressure uctuations are compensated with an additional solenoid valve. The design of a regulation mechanism to attenuate pressure pulsation more eciently and its control with a time-varying model-based compensator is presented in [START_REF] Gupta | Modeling and control of a novel pressure regulation mechanism for common rail fuel injection systems[END_REF]. However, a high-pressure pump for waterjet machining has to attenuate pressure uctuations without additional components, such as valves. A promising pressure disturbance rejection control is recently discussed in [START_REF] Wang | High pressure common rail injection system modeling and control[END_REF]. It suggests an extended state observer-based controller and compares it with a PID control design. On the other hand, an extended state estimator, based on a non-linear pressure control algorithm, has been applied in [START_REF] Kemmetmüller | Nonlinear pressure control of self-supplied variable displacement axial piston pumps[END_REF] to deal with unknown loads at axial piston pumps. Even though a hydraulic unit is used to control a piston pump, the experimental setup is similar to the high-pressure setup for waterjet machining. An even more exotic application for pressure generation is the so called free piston engine. For that, a mathematical model has been derived and the concept of trajectory-based non-linear feedforward control has been introduced in [START_REF] Li | Precise piston trajectory control for a free piston engine[END_REF]. However, the robust control of electrically driven high-pressure pumps seems poorly investigated.

Regarding a future waterjet facility as a high-pressure network that interconnects distributed high-pressure pumps with work stations, while considering an unknown network topology, motivates to investigate decentralized and robust control. The state of the art with respect to decentralized (a), distributed (b) and networked controls (c), as compared in the diagrams of Figure 5.1, distinguishes between large-scale and networked systems. A large-scale system includes numerous coupled system components, which are spread over a large spacial domain, e.g. power, water and trac systems. A networked system involves collaborating or interacting participants, which are interconnected by means of a communication channel, e.g. vehicle platoons and ight formations. Decentralized control aims to reduce the complexity of controller synthesis for large-scale systems. In contrast to centralized control design, a system is decomposed into subsystems (see [Bakule, 2014]). A decentralized controller K is then synthesized for each subsystem Σ. The most relevant classes of decomposed systems are the so called disjoint subsystems, overlapping subsystems and symmetric composite systems, see [Bakule, 2008] for details.

Σ 2 Σ 1 Σ 3 u 3 u 1 u 2 y 3 y 1 y 2 K 3 K 1 K 2 (a) Decentralized control (b) Distributed control Σ 2 Σ 1 Σ 3 K 3 K 1 K 2 Σ 2 Σ 1 Σ 3 K 3 K 1 K 2 (c) Networked / cooperative control Figure 5
.1 Diagram for decentralized, distributed and networked control: a system is separated in subsystems to reduce the complexity of control design. Whereas decentralized control considers an independent controller per subsection, distributed control takes into account the state of neighbouring subsections and networked control enables a data exchange between controllers.

As a result of decomposition, the whole system is controlled by several independent controllers, whereby couplings between subsystems may degrade the control performance and aect stability. This drawback is addressed, when introducing robust control to large-scale systems. Thus, the concept of robust decentralized control for networked subsections as well as the concepts of distributed and networked control are prevalent starting points for further investigations.

Decentralized robust control: decentralized robust control of networked systems is discussed in [Iftar, 2013]. Subsystem models are used for local control design and a robustness bound is introduced, taking any coupling between interconnected subsystems into account. Robust decentralized control for large-scale systems is applied in [START_REF] Benlatreche | Robust decentralised control strategies for large-scale web handling systems[END_REF], considering overlapping subsystems. The degradation of performance in contrast to centralized control is investigated. Both approaches obtain stability regarding the overall system. The design of decentralized robust PI controllers is presented in [START_REF] Marquez | Decentralized robust PI controller design for an industrial utility boiler -an IMC method[END_REF]. Its extension to gain-scheduling is interesting to improve the robust control design for high-pressure pumps, meeting advanced performance requirements. A design framework for overlapping controllers is given in [START_REF] Swarnakar | A design framework for overlapping controllers and its industrial application[END_REF]. It improves performance in contrast to simple PI control. Overall stability is obtained for non-linear systems under load variation. On the other hand, a synthesis for distributed control of networked systems is proposed in [START_REF] Viccione | LPV synthesis with integral quadratic constraints for distributed control of interconnected systems[END_REF], when describing time-varying uncertainties by means of a linear parameter-varying (LPV) system.

Distributed and networked controls: whereas some distributed control schemes allow for feedback values of neighbouring subsystems to be used for control, others consider communication links between decentralized controllers. These approaches aim at an improved performance in the case of strongly coupled subsystems. A distributed robust control for electrical power systems is discussed in [START_REF] Dekker | Distributed H ∞ -based control of electrical power systems[END_REF]] and compared with model predictive control. Each controller exchange its subsystem state with neighbouring controllers. Interpreting the subsystems as uncertainties and improving the control performance, by employing an exchange of information, is presented in [START_REF] Demir | A decomposition approach to decentralized and distributed control of spatially interconnected systems[END_REF]. A distributed control approach, generating a hierarchical architecture, is proposed in [START_REF] Tang | Distributed/hierarchical control architecture design[END_REF]. This architecture represents the interconnection of subsystems, as employed for control.

Networked control considers communication channels between controllers or between controllers and subsystems, respectively. In both cases, control design concerns reliability and determinism, due to transport delays and packet loss. This allows for dealing with serious stability concerns. Many papers can be found to challenge these concerns, e.g. [START_REF] Annaswamy | The arbitrated networked control systems approach to designing cyberphysical systems[END_REF], [START_REF] Yu | Event-triggered output feedback control for networked control systems using passivity: Achieving L 2 stability in the presence of communication delays and signal quantization[END_REF], [START_REF] Ma | Decentralized networked control for vehicle-string velocity and spacing[END_REF] and [START_REF] Shu | Decentralized control of networked discrete event systems with communication delays[END_REF], while others consider event-based communication to improve network eciency, e.g. [START_REF] Stöcker | Distributed control of interconnected systems with event-based information requests[END_REF], [START_REF] Zhang | Event-triggered control over noisy feedback channels[END_REF], [START_REF] Sigurani | Experimental evaluation of two complementary decentralized event-based control methods[END_REF] and [START_REF] Abara | Separation in coupled eventtriggered networked control systems[END_REF]. Networked control is excluded of the subsequent research work. Despite promising results in distributed control, it is in some instances conceivable for large-scale systems to manage the decentralized controllers with respect to global objectives and system constraints, e.g. optimizing the energy eciency by reducing the overall control eort, balancing the load of actuators to avoid local actuator saturation etc.

Contribution

Since high-pressure networks include large plant perturbations, decentralized control requires to implement robustness for disturbances from neighbouring network sections. Taking the robust decentralized approach presented in [START_REF] Marquez | Decentralized robust PI controller design for an industrial utility boiler -an IMC method[END_REF] and [Iftar, 2013] into account, H ∞ synthesis seems favourable to deal with these perturbations and disturbances. This work aims to employ H ∞ controller synthesis to obtain a robust control design for electrically driven highpressure pumps. A contribution here is given, when introducing the detailed problem formulation and obtaining the interconnection system that represents a network subsection, interconnecting a decentralized high-pressure pump to cutting heads. This involves to derive a perturbed plant model, suitable for robust control design. The challenge is to handle the non-linear characteristics of high-pressure generation and to deal with the parameter uncertainties for dierent waterjet applications. Another contribution is given, when analysing the expected parameter variations and evaluating dierent uncertainty descriptions. The investigated parametric uncertainties will be lumped to unstructured uncertainties. This is expected to enhance the controller synthesis.

This chapter is structured as follows: Section 5.2 introduces the fundamental principles from the theory of H ∞ controller design. Section 5.3 presents the investigated high-pressure system and its non-linear plant model. A linearised model with reduced complexity is derived with respect to the varying parameters. For that, the high-pressure pump, interconnected to an unknown network subsection, will be described as a rst order model and linearised on its operating point. The operating point is then considered as an additional uncertain parameter. Wide parameter variations cause a high complexity. Employing an unstructured uncertainty description lumps the parameter variations for various waterjet facilities, as shown in Section 5.4. Transforming the uncertainty model into standard ∆=M form, by using upper linear fractional transformation, describes a perturbed plant model. This model will be implemented in the subsequent chapter for H ∞ controller synthesis. Further, weighting functions are dened to specify requirements on performance for disturbance rejection and to penalize the control eort with respect to actuator saturations. Shaping functions are dened to consider the dynamic range of exogenous input signals, such as measurement noise and switching disturbance.

H ∞ control design fundamentals

This section introduces notations and concepts valuable for robust control design and established in literature, such as [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF] and [START_REF] Gu | Robust Control Design with MATLAB[END_REF]. All subsequent investigations refer to linear, time-invariant dynamic systems Σ, described by the continuous-time state space description

Σ := ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) ,
(5.1) where x(t) ∈ R n is the state vector, u(t) ∈ R m corresponds to the input vector and y(t) ∈ R p denotes the output vector. Hence, the outputs y(t) are measured to control the states x(t) to desired values, using the available inputs u(t).

The transfer function representation

G(s) = C (sI -A) -1 B + D (5.2)
is derived, when applying Laplace transform to (5.1). This requires for the initial conditions x(t = 0) = 0 and introduces the Laplace operator s = jω. The H ∞ controller synthesis is a frequency domain approach, thus, notation of the formalism will be given in terms of transfer functions.

Linear fractional transformation (LFT)

The linear fractional transformation (LFT) is a valuable concept for robust control design. It allows for transforming a perturbed plant into the ∆=M conguration and transforms a feedback control system into P=K conguration. The rst conguration is useful for robustness analysis, where the second conguration will be introduced for H ∞ controller synthesis.

∆=M conguration

The dynamic system (5.1) can be rearranged in a standard ∆=M conguration [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF], including the interconnection system M and the uncertainty block ∆. The block diagram of Figure 5.2 shows the interconnection system M and the uncertainty block ∆.

Norms and properties

The L 2 signal norm and the H ∞ system norm is briey recapitulated as they will nd excessive use in the subsequent sections. Discussing the small gain theorem is essential to understand requirements on stability and performance for the loop transfer functions of an interconnected systems. Especially, its formulation by means of H ∞ allows for evaluating the control design in terms of frequency domain specications.

L 2 -norm

The Euclidean norm (L 2 -norm) applied on a vector signal z(t) is dened by [START_REF] Gu | Robust Control Design with MATLAB[END_REF]

z(t) 2 = ∞ -∞ |z(t)| 2 dt 1/2 , (5.7)
where the squared of the L 2 -norm is referred to the signal energy z(t) 2 2 . This norm will be used to dene the H ∞ system norm and for controller synthesis in stacked notation, it is particularly referred to the control design with joint shaping functions, as presented in Section 6.2.

H ∞ -norm
Considering a linear and stable system G(s), with input channel d(t) and output channel e(t), that describes the transfer function d(t) → e(t) by means of e = G(s) d. The H ∞ -norm of G(s) is obtained by the maximum L 2 -norm of the input and output signals [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF], such as

G ∞ = max d =0 e 2 d 2
.

(5.8)

Thus, the H ∞ -norm is also called induced L 2 -norm. It can be interpreted as the maximal largest singular value

G ∞ = max ω σ (G(jω)) ,
(5.9) while evaluating the system for all frequencies ω. Since H ∞ corresponds to the linear space of all stable linear systems, this is a crucial measure for robust stabilization.

In other terms, the H ∞ -norm of a stable transfer function G(s) is the maximal peak value of the corresponding magnitude

G(s) ∞ = sup ω |G(jω)| .
(5.10)

Small gain theorem

Considering the stable loop transfer function L(s) The small gain theorem is often dened in terms of the spectral radius [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF] ρ (L(jω)) = max i |λ i (L(jω)) | (5.11) that is the maximum eigenvalue magnitude. The closed-loop system is then stable if ρ (L(jω)) < 1 ∀ ω (5.12) and since ρ(L) ≤ L , the small gain theorem states that the closed loop system is stable, if (5.13) where L has to satisfy the triangle inequality [START_REF] Desoer | Feedback Systems: InputOutput Properties[END_REF]. This is true for any induced norms, such as the H ∞ -norm, and thus for the largest singular value σ(L).

L(jω) < 1 ∀ ω ,
Internal stability is not necessary guaranteed, when interconnecting stable systems. Applying the small gain theorem (5.13), a feedback conguration, interconnecting the time-invariant, linear and stable systems G 1 and G 2 in closed-loop path with G 1 in the forward path and G 2 in the backward path, is internally stable, if and only if [START_REF] Desoer | Feedback Systems: InputOutput Properties[END_REF]. With respect to the multiplicative property

G 1 G 2 ∞ < 1 and G 2 G 1 ∞ < 1
G 1 G 2 ≤ G 1 • G 2 (5.14)
of the triangle inequality and considering the H ∞ -norm, yields G 1 ∞ • G 2 ∞ < 1 as a sucient condition of stability for an interconnected system.

The small gain theorem is essential for H ∞ controller synthesis. It is more conservative than the spectral radius, but very useful to evaluate the control requirements for stability and performance in terms of largest singular values σ.

Robust stability and performance analysis

Structured singular values are a general concept for analysing stability and performance of dynamic systems [START_REF] Gu | Robust Control Design with MATLAB[END_REF]. Even though introducing unstructured uncertainties to simplify the robust control design for high-pressure pumps (see Section 5.4), its properties for stability and performance will be derived, using the denitions obtained from structured singular values. Further, the condition for internal stability is briey introduced.

Structured singular values

For a perturbed plant G p = F u (M, ∆) robust stabilization can be obtained, referring to [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF], if det (I -M(jω)∆(jω)) = 0 ∀ ω .

(5.15) The nominal feedback system M is internally stable for ∆ : σ(∆) < 1 (5.16) where ∆ is the nominal set of structured uncertainty.

To allow a closed-loop system to be robustly stable, all the uncertainties in ∆ must be small, that I -M∆ wont become singular for any frequency ω. This eventually denes the structured singular value µ ∆ (M) of an interconnection system M with respect to the uncertainty ∆.

The robust stabilization criterion (5.15) and internal stability (5.16) give the inverse structured singular value [Doyle, 1987] µ -1

∆ (M) = minimize ∆ (σ(∆) : det (I -M(jω)∆(jω)) = 0) .
( 5.17) This corresponds to the smallest magnitude for σ(∆) to make I -M∆ singular and is subject for evaluation to obtain the structured singular value [START_REF] Packard | The complex structured singular value[END_REF] µ ∆ (M) = sup ω µ ∆ (M(jω)) , (5.18) which corresponds to a frequency dependent stability margin. 2011], is undesirable for high-pressure pumps, since availability and reliability of communication is not guaranteed for any waterjet facility. Nevertheless, the energy eciency of coupled and robustly controlled pumps can be improved by implementing a high-level managing, which copes with restricted means of communication. A perspective on that approach is given in Chapter 7.

The illustration of Figure 5.4 shows exemplarily the separation of a high-pressure network in network subsections, with respect to the locations of the high-pressure pumps. The cutting heads within a network subsection become concatenated, see head 3 and 4. The size of each subsection remains an unknown parameter for the low-level control design. To guarantee robust performance and stability requires to specify the upper and lower bounds of this parameter. Recall the derived lumped parameter model of Section 4.2 that describes the pressure generation in a pumping chamber, interconnected to a network subsection with respect to (4.6a -4.6b). Extending this model with a possible interconnection to neighbouring subsections yields the network subsection model t) from the high-pressure pump of section i given by two pumping chambers n = {1, 2} and the varying chamber volumes

d 1 Head 1 Pump 1 u 1 d 2 Head 2 Pump 2 u 2 Pump 3 u 3 represent by u 1 Q 1 Subsection 1 Source 1 p 1 Sink 1 d 1 u 3 Q 3 Subsection 3 Source 3 p 3 Sink 3+4 d 3+4 u 2 Q 2 Subsection 2 Source 2 p 2 Sink 2
d dt p i N (t) = K(p i N ) V i C (t) + V i 0 Q i P (t) -H * i p i N (t) d * i (t) - j H ij |p i N -p j N | sign p i N -p j N (5.23) with input uid ows Q i P (t) = n Q i V n (
d dt V i C (t) = n Q i P n (t) -V i 0,n .
(5.24)

Whereas H * i and d * i (t) concatenates all cutting heads of a network subsection to describe an overall output uid ow, the interconnection matrix H ij weights the couplings to neighbouring subsections j that assigns an intermediary uid ow.

This network subsection model (5.23 -5.24) is useful to represent entire waterjet facilites, subject for low-level control design. It allows for modelling the coupled network subsections (see Figure 5.4), representing a complex high-pressure network. The resulting model can be implemented for decentralized and distributed control design, see control schemes of Figure 5.1 (a) and (b). A rst robust design approach will consider the interconnection to neighbouring subsections as exogenous disturbances (see also [START_REF] Marquez | Decentralized robust PI controller design for an industrial utility boiler -an IMC method[END_REF] and [Iftar, 2013]). This will result in isolated network subsections and is based on the following assumption: Assumption 5.1. It is assumed that an electrically driven high-pressure pump manages to attenuate any pressure uctuations at its corresponding network subsection i by means of robust low-level control, such that the pressure dierence to a neighbouring subsection j remains smaller than a desired percentage overshoot M P , such as 100 × |p i N -p j N | / r < M P .

From Assumption 5.1 follows that p i N |p i N -p j N |, for which the coupling term to neighbouring subsection in (5.23) is neglected (H ij = 0 ∀ j) and results in the isolated subsection description

d dt p N (t) = K(p N ) V C (t) + V 0 Q P (t) -H * p N (t) d * (t)
(5.25) 5.26) where the amplitude and dynamic range of pressure uctuations by means of switching cutting heads, as well as induced uctuations from neighbouring subsections are included in a possible exogenous input and output uid ow, this occurs from |d(t)| < 1.

with d dt V C (t) = n (Q P n (t) -V 0,n ) , (
The isolated network subsection considers a single high-pressure pump and a cutting head, which become interconnected with a piping of bounded size, as illustrated by the diagram of Figure 5.5. A high-pressure pump denes the maximal uid ow for a network subsection, taking the actuator dynamics and saturation into account. This gives the maximally allowed water consumption in a network subsection. The high-pressure pump consists of two pistons, driven by independent linear actuators, see Section 1.3. The pistons alternately supply the high-pressure piping, such as to induce a continuous uid ow. This is realized by synchronization with phaseshifted reference trajectories, given by means of camming, and a precise motion control, as introduced in Section 2.3. Thus, the high-pressure pump represents a uid ow source, dedicated to a network subsection. A cutting head represents the overall water consumption of a network subsection, with respect to the expected switching dynamics. The cutting head can be equipped with nozzles of dierent size. The uid ow consumption will vary, with respect to the installed nozzle and the operating pressure. In addition, an on/o-valve will interrupt this uid ow, according to an unknown switching pattern that is given by the waterjet application.

The subsequent robust low-level control design deals with the pressure control, which has to track a reference pressure r(t) for the measurement signal y(t) that corresponds to a subsection pressure state x(t). Usually, a constant operating point is required for waterjet machining, where the on/o-valve switching is considered as a disturbance d(t). This aects the water consumption, with respect to an unknown switching pattern, for which the pump rate has to be adjusted, by means of control, to attenuate any pressure uctuation. The control signal u(t) denes the output uid ow u g (t) of a high-pressure pump, for which u(t) ≈ u g (t).

Consequently, each isolated network subsection, with its corresponding uid ow source and pressure sink, corresponds to a decentralized control loop, as depicted in Figure 5.6. This control loop is subject for robust control design.

Low-level control objectives

The control objective is rst specied with respect to the acceptable steady-state error and by means of time domain constraints, considering the closed-loop system. These constraints will then be used to derive specications in frequency domain for an open-loop system. The industry denes constraints on pressure uctuations, with respect to the waterjet application, such as to guarantee a desired cutting quality. Measurements with hydraulically driven high-pressure pumps reveal that waterjet machining with reasonable cutting quality, requires pressure uctuations of less than 10% when switching a cutting head, see also [START_REF] Tremblay | Modelling and simulation of pressure uctuations in high pressure waterjets[END_REF] and [START_REF] Trieb | Pressure and ow rate uctuations at high pressure intensier pumps[END_REF]. This further implies a settling time of τ s < 0.5 s and a damping ratio of ς > 0.5 to limit high frequencies acting on the pump actuators, see [START_REF] Xu | Design and experiment research on abrasive water-jet cutting machine based on phased intensier[END_REF] and [START_REF] Niederberger | Energy-ecient high-pressure pump[END_REF]. The control signal must be limited, such that the actuator remains within saturation bounds, i.e. u g = [0, 5.447 /min] considering the pump prototype installed at the test bench.

The hard constraints are dedicated to disturbance rejection. It requires to obtain a desired reference pressure r(t) = [40, 400] MPa → x(t) with minimal pressure uctuations |e(t)| < 0.1 r(t) when applying a disturbance step d(t) ∈ {-1, 1}, even for large plant perturbations. Pressure uctuations refer to the error, e(t) = r(t)-y(t), between desired and measured pressure state x(t) → y(t). The soft constraints are dedicated to reference value tracking. It asks for any steady-state error lim t→∞ e(t) = 0 to disappear and no overshoot for a reference value step, considering the undisturbed plant.

Time domain

Although, the high-pressure pump frequency response for u(t) → y(t), as well as the switching valve response for d(t) → y(t) will be represented using second order models and the highpressure network will be approximated with a rst order model, the time domain specications for closed-loop performance is typically dened in terms of a unit step response, applied on a second order system [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]

G(s) = ω 2 n s 2 + 2 ς ω n s + ω 2 n .
(5.27)

The poles of this model are found at s = -σ ± jω. Its natural frequency ω n = √ σ 2 + ω 2 and damping ratio ς = σ/ω n follow from the real part σ and imaginary part ω.

The imaginary part further denes the rise time τ r = π/(2 ω) (5.28) for y(t) to rst cross the steady-state value, if ς < 1. The real part gives the settling time τ s = -ln ε/σ , (5.29) where ε = 100 × (r(t) -y(t))/r(t) is the percentage steady-state error to the desired reference value. The percentage overshoot

M P ≈ -e σπ/ω = -e ςπ/ √ 1-ς 2
(5.30) depends on the real and imaginary part. It can be expressed in terms of the damping ratio. This allows for obtaining the damping ratio

ς ≈ ln M P π 2 + (ln M P ) 2 ,
( 5.31) with respect to a specied overshoot, when rearranging (5.30). Combining the real part σ = -ln ε/τ s and imaginary part ω = π/(2 τ r ) denes the natural frequency 5.32) by terms of steady-state error, settling time and rise time, where (5.31) gives an approximation ω n ≈ π 2 + (ln M P ) 2 ln ε τ s ln M P , (5.33) considering steady-state error, overshoot and settling time. Hence, (5.31) and ( 5.32) dene constraints on the dominant closed-loop poles for r(t) → y(t) and d(t) → y(t), using time domain specications and assuming a second order model. Therefore, unit steps on the scaled reference input r(t) and on the scaled disturbance input d(t) are considered to specify time domain control objectives for the investigated highpressure system. The hard constraints for disturbance rejection give objectives at the scaled plant output y(t) with respect to an input step on disturbance d(t), as listed in Table 5.1. This yields a settling time of ≤ 0.5 seconds, enables an overshoot of ±10% and requires all steady-state error to disappear. Rise time does not need specication.

ω n = - ln ε τ s 2 + π 2 τ r 2 (

Requirements Objectives Conditions

Settling time (2% e(t)): 0.98 ≤ y(t) ≤ 1.02 t ≥ 0.5 seconds Overshoot:

0.9 ≤ y(t) ≤ 1.1 t ≥ 0 seconds Steady-state error: e(t) = 0 t → ∞
On the other hand, the soft constraints for reference value tracking provide objectives to the scaled plant output y(t) with respect to an input step on r(t), as shown in Table 5.2. This yields a rise time of ≤ 10 seconds, no overshoot and no steady-state error. The settling time does not need any specication.

e(t) = 0 t → ∞
For both cases, no steady-state error is desired, thus lim t→∞ e(t) = 0. To obtain this constraint for a plant, without a pole at s = 0, the closed-loop system requires integral action by means of control. The objectives on disturbance rejection and reference tracking requires step responses as approximated with the two following second order models G Hard (s) = 53.7053 s 2 + 8.6651s + 53.7053 (5.34) and 5.35) when applying (5.31) and (5.32) to the specications of Table 5.1 and 5.2. The plot in Figure 5.7 shows the corresponding step responses of the systems (5.34) and (5.35). Since reference tracking and disturbance rejection are complementary control design requirements, it is not possible to satisfy all objectives given by the hard constraints and soft constraints, using a controller with one degree of freedom. Nevertheless, introducing an additional degree of freedom to the reference input r(t), may rectify this limitation. Figure 5.7 Step responses for the hard and soft constraints: considering a second order model to represent the control objectives for d(t) → y(t) and r(t) → y(t).

G Sof t (s) = 1.9465 s 2 + 2.7904s + 1.9465 , (

Frequency domain

The time domain specications become more comprehensive, when considering its representation in frequency domain. These frequency domain specications are useful to dene objectives with respect to specic frequencies. It further allow for introducing constraints to noise attenuation in addition to reference tracking and disturbance rejection, see [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. Reference tracking and disturbance rejection typically concerns performance at the low frequencies, where noise requires attenuation at high frequencies, where T (jω) + S(jω) = 1 has to hold for every frequency, with sensitivity function S(s) (5.61) and complementary sensitivity functionT (s) (5.62). Both functions will be discussed in the subsequent section, when introducing the interconnection system for control design. (5.37)

Consequently, |S(jω)| 1 and |T (jω)| ≈ 1 for low frequencies and |T (jω)| 1 and |S(jω)| ≈ 1 for high frequencies. To derive these weighting functions by terms of time domain specications, the second order models (5.34) and (5.35), considering the hard and soft constraints, have been substituted in (5.36) and (5.37) with respect to L(s) = G(s)/(G(s) -1). This denes a lower bound |W 1 (s)| to guarantee performance and an upper bound |W 2 (s)| for possible noise attenuation, as depicted in the plot of Figure 5.8. When the resulting open-loop system remains within these bounds, the specied control constraints will be met. 2 : derived from the hard and soft constraints to specify minimal and minimal bounds for desired performance.

However, these control constraints will not guarantee closed-loop stability and performance for large plant perturbations. Referring to the next section, robust stability and performance require to introduce a representative uncertainty description. The following list summarizes the requirements for robust control design:

The controller K(jω) must stabilize the perturbed plant G p (jω). A hard constraint is to retrieve sucient rejection for input disturbances. Thus, the singular values of the sensitivity function and perturbed plant σ(G p (jω) S(jω)) should become small for low frequencies.

A rst soft constraint is to guarantee a satisfying reference tracking with zero steady-state error. This indicates that σ(T (jω)) ≈ σ(S(jω)) ≈ 1.

A second soft constraint is employed to obtain a reasonable noise attenuation, for measurement noise on plant output. The singular values of the complementary sensitivity function σ(T (jω)) should become small for high frequencies.

In addition, the control eort, needed for an eective disturbance rejection, should be minimal. This requires the sensitivity function, perturbed plant and controller σ(K(jω) G p (jω) S(jω)) to be small with a roll-o for high frequencies.

Finally, robust stability should be enabled for any destabilizing uncertainty. Assuming an inverse additive uncertainty model, the largest singular value of the sensitivity function and the nominal plant should be smaller than the largest singular value of the uncertainty σ(∆ ia (jω)) = σ(G n (jω) S(jω)) -1 .

First order plant model

The high-pressure system of Figure 5.5 can be described in a simplied meaner by means of a non-linear model. This model of rst order becomes aected by various varying parameters. Section 5.4 will discuss and classify these parameters to dene the uncertainty range and to analyse its sensitivity with respect to the input and output channels. This section investigates the linearisation of the rst order model around an operating point, with respect to dierent approximations and simplications to provide a model for control design.

The diagram of Figure 5.9 presents this model evaluation process. The non-linear model provides the starting point of investigation. An approximated model has been obtained, when linearising it with respect to independent parameters. These parameters are then substituted after linearisation by the dependent variables. On the other hand, an extended model results, when rst introducing the dependent variables and then linearising around operating point. It is further simplied by its evaluation on equilibrium. Figure 5.9 Overview of the plant model evaluation process: dierent linearised and approximated models to represent the high-pressure system subject to control design.

These dierent linear models has been compared with respect to various time and parameterdependent variables. The simplied approximated model has been nally identied to provide best approximation of the non-linear model. It is further simplied by assuming a disturbed input channel. This allows for separating the uncertainties and provides the perturbed plant for control design.

Non-linear plant

To model the high-pressure system of Figure 5.5 for control design, assumes a perfect synchronization of both pumping units without any interactions. This is obtained by means of motion trajectories, as described in Section 2.3, causing one of the pumping units to the pumping state of hauling, while the other is retracting and pressurizing. This enables a high-pressure pump to generate a steady overall uid ow. As a consequence, only one pumping chamber at a time is interconnected to the corresponding network subsection. Thus, the normalized control variable u(t) of the pressure controller refers to the desired piston velocity v P (t), while hauling. This piston displacement generates a steady uid ow Q P (t) = S P v P (t) that corresponds to the eective control value u g (t) → Q P (t) of a high-pressure pump. Referring to the derived lumped parameter model (5.25 -5.26) allows for introducing the non-linear plant model

d dt x(t) = K δ V δ u g (t) -H δ x(t) d(t)
( 5.38) that describes a common pressure state x(t) → p N (t) of a network subsection and represents all time-varying and parameter-varying variables in terms of parametric uncertainties.

This non-linear plant model represents the pressure, established in a high-pressure network subsection, and its interconnected pumping chamber, where the network subsection couples a high-pressure pump to a corresponding cutting head. A high-pressure pump induces an input uid ow u g (t), where the nozzle of a cutting head causes an output uid ow, with respect to the pressure state. This output uid ow is aected by the on/o-valve. This valve has an unknown switching behaviour of discrete states d(t) ∈ {0, 1}. Where the bulk modulus K δ and uid volume V δ aect the input gain as well as the system dynamics, any variation of the ow resistance H δ will change the feedback gain. The disturbance input can be considered as an variable gain, in particular, applied on the system feedback path. This rst order model is nonlinear in x(t) and describes a bilinear relation between x(t) and d(t), as depicted in the block diagram of Figure 5.10. Notice that this model is subject to dierent parameter variations, as listed in Table 5.3. These uncertainties deal with parameter variations, given by the high-pressure network conguration and the pump prototype (see Section 2.2 and 2.3, respectively). The subscript δ labels the uncertain characteristics of the corresponding parameters. H δ ∈ [H min , H max ] is the varying uid ow resistance of a cutting head nozzle, dening the output uid ow. K δ ∈ [K min , K max ] denotes a varying uid bulk modulus and V δ ∈ [V min , V max ] represents a varying uid volume, both dening the dynamics of pressure generation. Table 5.3: Uncertainty range of varying parameters obtained from test bench setup and highpressure pump design.

x K δ V δ u ∫ + • × H δ d

Nominal value Min value Max value Relative range

Flow resistance

H δ ×10 -9 ### 0 ### 100% Fluid volume V δ (cm 3 ) ### ### ### 37% Bulk modulus K δ (GPa) ### ### ### 28%
Operating point x δ (MPa)

220 40 400 82%
Nozzle section S δ (mm 2 ) 0.0653 0.0050 0.1260 92%

The installed nozzle strongly aects the uid ow resistance

H(t) = S δ ζ 2 (t) -→ H δ , (5.39)
where S δ = [S min , S max ] denotes an uncertain nozzle cross section, ζ represents its uid ow coecient and (t) is the uid density.

It has been shown in Section 3.3, that it is useful to describe the variable bulk modulus

K(t) = κ 0 + κ 1 x(t) -→ K δ (5.40)
by means of the pressure state x(t), where κ 0 and κ 1 denote the identied coecients of a linear model. Also the variable uid density

(t) = 0 1 + κ 1 x(t) κ 0 1/κ 1 (5.41)
can be modelled with respect to the pressure state and the same parameters κ 0 and κ 1 , as given in Section 3.4. Introducing (5.39) and ( 5.40) into (5.38) results in an extended plant model

d dt x(t) = κ 0 + κ 1 x(t) V δ   u g (t) -S δ ζ 2 0 1 + κ 1 x(t) κ 0 -1/κ 1 x(t) d(t)   , (5.42)
which introduces dependent variables on x(t), improving in this way the accuracy over a wide pressure range. The remaining parametric uncertainties correspond to the unknown cutting head conguration, specied by the nozzle cross section S δ , and the unknown high-pressure network topology, given by the uid volume V δ .

Linearised plant

Both rst order models, the non-linear plant model (5.38) and the extended plant model ( 5.42) can be linearised around their operating points (x 0 , u 0 , d 0 ) by the rst order Taylor expansion

d dt x(t) = f (x 0 , u 0 , d 0 ) + ∂ ∂x f (x 0 , u 0 , d 0 )∆x(t) + ∂ ∂u f (x 0 , u 0 , d 0 )∆u(t) + ∂ ∂d f (x 0 , u 0 , d 0 )∆d(t) , (5.43)
where ∆x(t), ∆u(t) and ∆d(t) refer to the corresponding deviations around the operating points with respect to

x(t) = x 0 + ∆x(t) , (5.44a)
u(t) = u 0 + ∆u(t) , (5.44b)
d(t) = d 0 + ∆d(t) .
(5.44c)

The expansion (5.43) holds for small deviation around the operating points. It gives the state space description

∆ ẋ(t) = A ∆x(t) + B 1 ∆u(t) + B 2 ∆d(t) ∆y(t) = C ∆x(t) + D 1 ∆u(t) + D 2 ∆d(t) , (5.45)
with separated control input u(t) and disturbance input d(t), whereby

A = ∂ ∂x g(x 0 , u 0 , d 0 ); B 1 = ∂ ∂u g(x 0 , u 0 , d 0 ); B 2 = ∂ ∂d g(x 0 , u 0 , d 0 ) , C = ∂ ∂x h(x 0 ) = 1; D 1 = ∂ ∂u h(x 0 ) = 0; D 2 = ∂ ∂d h(x 0 ) = 0 .
(5.46)

The linearisation of the non-linear plant model (5.38) according to (5.45) yields the following approximated model

∆ ẋ(t) = - K δ H δ d 0 2V δ √ x 0 ∆x(t) + K δ V δ ∆u(t) - K δ H δ √ x 0 V δ ∆d(t) , (5.47)
while considering independent parameters. As a consequence of linearisation, the operating pressure range

x δ = [x min , x max ] -→ x 0 (5.48)
is taken into account, extending the set of uncertain parameters. The block diagram of Figure 5.11 shows that the control input ∆u as well as the disturbance input ∆d become scaled with the bulk modulus K δ and uid volume V δ , where the disturbance becomes additionally aected by the ow resistance H δ and operating point x δ . Varying the ow resistance and operating point will change the system dynamics as well as the disturbance input scaling. Introducing the dependent variables (5.39) and (5.40) in (5.47) results in the extended approximated model

H δ 2 x δ Δx Δd K δ V δ Δu + H δ x δ ∫ + +
∆ ẋ(t) = - (κ 0 + κ 1 x 0 )S δ ζ d 0 V δ 1 2x 0 0 1 + κ 1 x 0 κ 0 -1/κ 1 ∆x(t) + κ 0 + κ 1 x 0 V δ ∆u(t) - (κ 0 + κ 1 x 0 )S δ ζ V δ 2x 0 0 1 + κ 1 x 0 κ 0 -1/κ 1 ∆d(t)
(5.49) that describes the bulk modulus K δ and uid ow resistance H δ with respect to the operating point x 0 .

In contrast to that, the linearisation of the extended plant model (5.42) according to (5.45) yields the linearised extended model

∆ ẋ(t) = -   (κ 0 -x 0 + 3κ 1 x 0 )S δ ζ d 0 V δ 1 2x 0 0 1 + κ 1 x 0 κ 0 -1/κ 1 - κ 1 u 0 V δ   ∆x(t) + κ 0 + κ 1 x 0 V δ ∆u(t) - (κ 0 + κ 1 x 0 )S δ ζ V δ 2x 0 0 1 + κ 1 x 0 κ 0 -1/κ 1 ∆d(t) , (5.50)
taking the dependent variables into account, such as a variable bulk modulus K(t) and a variable uid ow resistance H(t), for linearisation with respect to the pressure state x(t).

Here, the resulting extended approximated model (5.49) and the linearised extended model (5.50) become dierent in terms of its coecient A. Where for the extended approximated model, this coecient depends on the operating points x 0 and d 0 , such as A(x 0 , d 0 ), it becomes aected by x 0 , d 0 and u 0 , with A(x 0 , u 0 , d 0 ), considering the linearised extended model. The other coefcients remain conditioned by x 0 , such as B 1 (x 0 ) and B 2 (x 0 ) .

Whereas the operating pressure x 0 can be measured and is subject to control, the resulting induced uid ow u 0 and switching disturbance d 0 are time-varying parameters with unknown behaviour and are dicult to determine. Insofar as the induced uid ow must follow the overall water consumption of the cutting heads to obtain a steady operating pressure, the required uid ow can be determined by assuming equilibrium on operating point. This enables to dene the expected input uid ow

u 0 = S δ ζ 2x 0 0 1 + κ 1 x 0 κ 0 -1/κ 1 d 0 (5.51)
by means of pressure state and disturbance.

Substituting (5.51) in ( 5.50) allows for obtaining the evaluated extended model

∆ ẋ(t) = - (κ 0 -x 0 + κ 1 x 0 )S δ ζ d 0 V δ 1 2x 0 0 1 + κ 1 x 0 κ 0 -1/κ 1 ∆x(t) + κ 0 + κ 1 x 0 V δ ∆u(t) - (κ 0 + κ 1 x 0 )S δ ζ V δ 2x 0 0 1 + κ 1 x 0 κ 0 -1/κ 1 ∆d(t) (5.52)
that is closely related to the extended approximated model (5.49).

The dierent linearised plant models will be compared on equilibrium. The on/o-valve is thereby assumed as open (d 0 = 1), while the operating pressure x 0 is subject for variation. The Bode plots of Figure 5.12 show the frequency responses of the approximated model (5.47), the extended approximated model (5.49) and the evaluated extended model (5.52). It can be seen that all plant models have a similar dynamic behaviour, where the characteristics range from a rst order lag to an integrator, due to the varying parameters (see parameter range of Table 5.3). In general, the varying parameters cause an increased gain uncertainty for low frequencies, which is expected for all proposed plant models.

Considering the evaluated extended model (5.52) as a reference, the approximated model (5.47) features smaller deviations than the extended approximated model (5.49). The simplied linearisation, applied to derive the approximated plant model, seems satisfactory, instead of applying the exact linearisation, taking all time-varying and parameter-varying parameters into account. Thus, the approximated plant model is of rst choice and will be further simplied for low-level control design.

Simplied model

The linearised model basically allows for distinguishing two cases. A rst case requires at least one cutting head of a network subsection to be open, thus d(t) = 1, where another case assumes all cutting heads to be closed, hence d(t) = 0. The resulting simplied approximated model will be subject for control design in the subsequent section. It aims at robust control for the case where |d(t)| = 0. Before investigating the plant uncertainties, it is useful to dene a common input channel. This input channel combines the control and disturbance input to a disturbed control input (5.53) which assumes the parameters upper bounds H δ = H max and x δ = x max , dening the maximum disturbance

u d (t) = ∆u(t) -d(t) ,
d(t) = H max √
x max ∆d(t) (5.54) and applies the available disturbance range, for any permissible cutting head conguration. Introducing (5.53) and (5.54) further simplies the approximated model (5.47) to obtain the simplied approximated model

∆ ẋ(t) = - K δ H δ 2V δ √ x δ ∆x(t) + K δ V δ u d (t) .
(5.55) This simplication will not change the model dynamics, if a continuously changing value d(t) = [-1, 1] is introduced, but allows for establishing an operating point of d 0 = 1. The Block diagram in Figure 5.13 illustrates the corresponding model structure. It can be seen that a signal from control input ∆u(t) to output ∆x(t) possess the same dynamics as a signal from ∆d(t) to ∆x(t). However, the disturbance input d(t) will be bounded by H max and x max . This simplied model is useful, as it takes the full disturbance range into account and isolates the uncertain system dynamics to the perturbed plant G p (s).

Complementary to that, it is also possible to assume d 0 = 0. This would lead to the integral behaviour

∆ ẋ(t) = K δ V δ u d (t)
( 5.56) and corresponds to a fully closed cutting head. It represents the parameter conguration, when input to state stability is not guaranteed. Using Laplace transformation with s = d/dt allows for rewriting the simplied approximated model (5.55) to obtain the perturbed plant 5.57) in transfer function form. This corresponds to a rst order lag, from input u d (t) to output ∆x(t), with coecients

H δ 2 x δ Δx Δd K δ V δ u d Δu + H m a x x m a x ∫ + G p + d
G p (s) = b 0 s + a 0 (
a 0 = K δ H δ 2V δ √ x δ and b 0 = K δ V δ , (5.58)
where M δ = b 0 /a 0 denotes an uncertain gain and τ δ = 1/a 0 is an uncertain time constant. It is not the standard form of a rst order lag chosen, since this may cause division by zero for the given parameter ranges with

a δ = [a min , a max ] -→ a 0 and b δ = [b min , b max ] -→ b 0 .
(5.59)

Applying the values of Table 5.3 with respect to denition (5.58), gives the uncertainty ranges for lumped parameters, as listed in Table 5.4. Table 5.4: Uncertainty range of lumped parameters used for transfer function description.

Nom. value Min value Max value

Rel. range

a δ ( √ Pa/m 3 ) ×10 ### ### ### ### [-100, +225] % b δ (Pa/m 3 ) ×10 ### ### ### ### [ -43, +93] %
The Bode plots of Figure 5.14 compare the simplied approximated model in its state space representation (5.55) to the corresponding transfer function representation (5.57), with lumped parameters and again applying the parameter range of Table 5.3. The frequency responses for the perturbed plants show undesired dierences, insofar as the uncertain parameters of the state space description have been lumped to the transfer function coecients a δ and b δ . Especially, a broaden time constant variation as well as a smaller gain for low frequencies is observed.

The perturbed plant in transfer function form (5.57), derived from the simplied approximated model, is desirable for control design in frequency domain. However, it is not allowed to lump the various uncertain parameters, since this changes the dynamic perturbation range. The subsequent section combines the simplied approximated model (5.55) for nominal parameter values with the system components of a high-pressure system (see Figure 5.5) to derive an interconnection system, suitable for robust control design. The sensitivity to parameter variations will be discussed in Section 5.4, when introducing the uncertainty modelling.

Interconnection system

This section introduces the interconnection system for controller synthesis, as illustrated in the block diagram of Figure 5.15. It represents the dynamics of dierent system components, by means of transfer functions. The actuator G a (s) and pressure gauge G s (s) interconnect the nominal plant G n (s) to the controller K(s). The nominal plant is derived from the perturbed plant (5.57), considering the nominal parameter values of Table 5.3 with respect to the coefcients (5.58). The resulting interconnection system is than scaled for unit input and output signals. The subscript w corresponds to weighted signals. It further includes shaping functions W r (s), W d (s), W n (s) and W p (s) to specify input signals and weighting functions W e (s) and W u (s), taking the control requirements into account. Since the perturbed plant G p (s) contains known non-linearities and all parameter variations, the other components are regarded as linear and time invariant (LTI). The high-pressure pump limits the available dynamic range of the control signal u g (t). It is aected by the bandwidth of the linear actuator G a (s) in closed-loop position control. The control signal becomes disturbed. Introducing the dynamic range of the on/o-valve, W d (s) allows for reproducing the disturbance signal d(t), as induced from a cutting head. Further, the bandwidth of the pressure gauge G s (s) denes the dynamic range of the measured process signal y g (t). This measurement signal is aected by noise n(t) shaped by W n (s).

The interconnection system of Figure 5.15 allows for deriving the closed-loop transfer functions

y = G n G a K 1 + G s G n G a K r + G n 1 + G a KG s G n d - G n G a K 1 + G s G n G a K n + G n 1 + G a KG s G n p, (5.60a)
e = 1 1 + G s G n G a K r - G s G n 1 + G a KG s G n d - 1 1 + G s G n G a K n - G s G n 1 + G a KG s G n p, (5.60b)
u = K 1 + G s G n G a K r - KG s G n 1 + G a KG s G n d - K 1 + G s G n G a K n - KG s G n 1 + G a KG s G n p , (5.60c)
which describe the eects on reference value r(t), disturbance d(t) and noise n(t) on process value y(t), control error e(t) and control value u(t). Here, the process value y(t) = x(t) corresponds to the system pressure state. The denominator is the same for each input channel, in addition, the transfer functions from the noise input are the same as the transfer functions from the reference input, but with a change of sign. Introducing the sensitivity function (5.61) and the complementary sensitivity functions

S(s) = 1 1 + G s (s) G n (s) G a (s) K(s)
T y (s) = G n (s) G a (s) K(s) 1 + G s (s) G n (s) G a (s) K(s)
and (5.62) in (5.60a -5.60c), provides the simplied matrix notation

T u (s) = K(s) G s (s) G n (s) 1 + G a (s) K(s) G s (s) G n (s)
  y e u   =   T S KS   r +   G n S -G s G n S -T u   d +   -T y -S -KS   n +   G n S -G s G n S -T u   p .
(5.63)

For control design, it is useful to keep in mind that S(jω) + T (jω) = I, ∀ω [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF].

Eective disturbance rejection requires the H ∞ -norm from disturbance d(t) to control error e(t) to be small, where acceptable control eort requires the H ∞ -norm from disturbance d(t) to control variable u(t) to be limited. Hence, the control design specications on performance and control eort, for any insignicant plant perturbation, can be evaluated from the H ∞ -norm of the selected transfer functions

G n S G s G n S T u ∞ (5.64)
in stacked notation, where the perturbation p(t) to the measurement output y(t) must remain bounded to obtain robust stability, that is given for G n S ∞ < 1. However, to guarantee robust stability and performance for the expected uncertainties requires to consider the perturbed plant G p (s). For this reason, the perturbation input p(t) is introduced. However, this requires an adequate uncertainty modelling, that will be investigated in Section 5.4. The evaluation of the desired loop transfer functions by means of H ∞ controller synthesis, further requires an adequate signal scaling for the interconnection system of Figure 5.15. This scaling will be introduced rst.

Signals scaling phase

The interconnection system will be scaled to bound the signal amplitudes to unit gain | • | ≤ 1 with respect to the maximal values of Table 5.5. This represents the maximal input and output values, as expected for a network subsection. Table 5.5: Signals scaling to bound the amplitudes of an interconnection system to unit gain.

Maximal system pressure

p max = 400 MPa Maximal input uid ow Q u,max = 4.766 /min Maximal output uid ow Q d,max = 2.426 /min A scaling phase is applied to the following signals, where is used to mark unscaled values.

Reference value r(t): The unscaled reference value r(t) denotes the desired pressure. The scaling is given by r

(t) = R r r(t), whereby R r = 1/p max , if |r(t)| < |p max | and |r(t)| < 1 holds.

Control value u(t):

The unscaled control value ũ(t) corresponds to the uid ow, generated by a high-pressure pump. The scaling is given by u

(t) = R u ũ(t), whereby R u = 1/Q u,max . Insofar as |ũ(T )| < |Q u,max |
and |u(t)| < 1 holds, the maximal output uid ow is given by the limited piston velocity v max and its cross section surface S P , such as Q u,max = S P v max .

Disturbance value d(t):

The unscaled input disturbance d(t) denotes the unknown uid ow consumption of a high-pressure network. The scaling is given by d

(t) = R d d(t), whereby R d = 1/Q d,max . Insofar as |r(t)| < |Q d,max
| and |d(t)| < 1 holds, the maximal output uid ow is limited by the generated uid ow, such as Q d,max < Q u,max . A typical value is found, when considering a use case with maximal uid ow of Q d,max = H max √ p max .

Measurement value y(t):

The unscaled measurement value ỹ(t) denotes the measured system pressure. The scaling is given by y

(t) = R y ỹ(t), whereby R y = 1/p max , if |ỹ(t)| < |p max | and |y(t)| < 1 holds.
Noise value n(t): The unscaled noise value ñ(t) denotes the measurement noise. The scaling is given by n(t) = R n ñ(t), whereby R n = 1/p max , identical to the reference and measurement value.

The scaling is directly implemented to the corresponding system components and weighting functions, where needed.

System blocks denition

The actuator G a (s) of the high-pressure pump, the nominal plant G n (s), representing a network subsection, the pressure gauge G s (s) and the controller K(s), describe system components of the feedback control loop. To design an augmented interconnection system requires to specify each system block as a transfer function.

Pump actuator G a : A high-pressure pump generates an input uid ow to supply a network subsection. The dynamics from the pump control signal u(t) to the induced overall uid ow u g (t) is approximated with a second order transfer function

G a (s) = 1 ### s 2 + ### s + 1 R -1 u , (5.65)
which scales the output u(t) with R -1 u = 7.944 10 -5 that the output corresponds to the induced overall uid ow of a high-pressure pump.

This transfer function (5.65) originates from the desired actuator dynamics in closed-loop position control. It consists of an inner current loop, a velocity loop and an outer position loop, as discussed in Section 2.3. Whereas the inner control loop has proportional gain, the two outer control loops feature additional integral action. Load estimation is used to improve the disturbance rejection. The frequency response of Figure 5.16 represents the expected closed-loop dynamics for a pumping unit. Its actuators have been tuned to obtain a bandwidth of ### Hz (ω = ### rad/s) for the outer position loop and a damping coecient ς > ###, considering a second order model (5.27). This aims at a fast response with a rise time τ r = ### seconds and an overshoot of < 20%, where steady-state error is eliminated. The verication of this parametrization requires identication by means of measurements. The identication procedure for a continuous-time model from sampled data is performed with respect to [START_REF] Garnier | Continuous-time model identication from sampled data: Implementation issues and performance evaluation[END_REF]. To identify these model coecients, for a pumping unit in operation, reveals some diculties. The actuator stroke is limited and only the forward direction is of interest for identication. This since one pumping chamber is only interconnected to the highpressure network, when its pumping unit is in the state of hauling. The identication data have been obtained from experiments with sinusoidal excitation, relating to [do [START_REF] Prado | On-line identication of induction motors using discrete models for sinusoidal signals[END_REF]. Therefore, a piston trajectory of limit bounds is implemented for camming, where a sinusoidal is super-positioned for excitation. This excitation is only applied for the piston stroke in ascending direction, as presented in plot (a) and (b) of Figure 5.17. The pumping states of pressurization and lling remain unmodied, where the state of hauling excites the pumping unit for a dened frequency. The camming shaft angular velocity dφ(t)/dt is then varied, by adjusting the control value u(t) = [0, 1] as dened in (2.25) and (2.26) (see Section 2.3). This causes the excitation frequency to shift for the frequency range of interest, as shown in plot (c).

The resulting displacement velocities of each piston s P 1 and s P 2 are then measured by absolute encoders. Only the measurement data for hauling will be used for identication. The reference piston velocities from trajectory and the measured velocities are then evaluated for continuous-time transfer function estimation, using the instrument variable (IV) method [START_REF] Gilson | Identication of Continuous-time Models from Sampled Data[END_REF]. The frequency responses of Figure 5.18 (a) show the identied second order transfer functions and compare that to the expected second order model from controller tuning. Whereas the pumping unit 1 matches the reference transfer function with respect to frequency and damping ratio, the pumping unit 2 obtains an increased damping. The step responses of Figure 5.18 (b) compare the identied and tuned dynamics for both pumping units. It seems acceptable to apply the control tuning model (5.65) for describing the pump actuator dynamics. The identied bandwidth of a high-pressure pump is about 8 times faster as the desired closed-loop dynamics for pressure control. This is fundamental to obtain desired performance for disturbance rejection.

Nominal plant G n : A network subsection interconnects a high-pressure pump to cutting heads. It contains all uncertainties due to varying parameters, e.g. the unknown subsection size. These parameters are investigated in a subsequent section to derive an adequate uncertainty description for control design. This allows for separating the uncertainties from the nominal plant by introducing a perturbation input. The simplied approximated model (5.55), evaluated for nominal parameter values (see Table 5.3), gives the nominal plant. It approximates the dynamics from the disturbed control input u d (t) to the measured system state y(t) with a rst order model

G n (s) = ### s + ### R r , (5.66)
which represents the expected nominal characteristics for a network subsection, where R r = 2.5 10 -9 scales the output y(t) with respect to the maximal system pressure. The frequency response of Figure 5.19 shows the nominal dynamics of pressure generation for a network subsection. Pressure gauge G s : The pressure gauge measures the system state of a network subsection subject for control. It limits the dynamics from the measured system state y(t) to the real measurement signal y g (t), represented by a second order model G s (s) = 1 6.747 10 -8 s 2 + 5.714 10 -4 s + 1 .

(5.67)

With respect to manufacturer specications, an over-damped second order model has been considered, with a wide sensor bandwidth of 3.85 kHz and a damping coecient of 1.1, such as to obtain a response time of < 1 ms for a step response from 10 -90%. The frequency response of Figure 5.20 represents the approximated characteristics of a pressure gauge. No scaling is needed as the pressure gauge is located in the scaled feedback path of the interconnection system. Controller K: A controller will close the feedback loop of the interconnection system. Its parameters are subject for controller synthesis, it provides the characteristics from the control error e(t) to the control value u(t). Two concepts to specify a controller will be investigated. They are basically distinguished with respect to the control structure.

On the one hand, the state feedback controller

K SF = ξ(t) = A ξ(t) + B e(t) u(t) = C ξ(t) + D e(t) , (5.68)
with the controller states ξ(t) ∈ R n , the control output values u(t) ∈ R q and the error input values e(t) ∈ R p have been considered. This form is suitable for standard H ∞ controller synthesis, as provided in [START_REF] Doyle | Statespace solutions to standard H 2 and H ∞ control problems[END_REF]. Measuring a single system state and controlling this by means of a single control variable yields q = p = 1 and consequently,

A ∈ R n×n , B ∈ R n×1 , C ∈ R 1×n
and D = 0. This requires to tune (n + 2 × n) control parameters, where n will be equal to the order of the overall interconnection system, which corresponds to the number of system poles. Hence, reducing the order of the interconnection system, will equally reduce the dimension of the resulting state feedback controller.

On the other hand, the continuous time PI controller

K P I (s) = K P + K I 1 s
( 5.69) in parallel form has been considered, where K P is a proportional gain and K I denotes the integral gain. This form requires a structured H ∞ controller synthesis, as available in [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF]. It is standard for implementation in industrial applications and features a slim structure with 2 parameters independent of the augmented interconnection system. However, the reduced degree of freedom for H ∞ optimization limits the PI controller to perform competitive to a state feedback controller, in particular, a missing capability for noise rejection has been observed that is tremendous. Thus, considering a low-pass lter as an additional tuning parameter, enables the PI controller to obtain a precise approximation, with almost identical stability and robustness properties, comparable to a state feedback controller.

The extended PI controller

KP I (s) = K P + K I 1 s 1 1 + τ f s (5.70)
introduces a rst order low-pass lter with the tunable time constant τ f . The H ∞ synthesis to obtain robust controller parameters is discussed in Chapter6.

Input signal shaping functions

Shaping functions will dene the dynamic range of the input channels. As required to build the augmented interconnection system, each weighting function is related to a corresponding input channel, as denoted by its subscript. The shaping function W r is applied on the reference signal r(t), where W d shapes the disturbance signal d(t), W p relates to the perturbation signal p(t) and W n denes the dynamics of the noise signal n(t).

Reference shaping function W r : The scaled reference signal r w (t) = [0, 1] provides the desired operating point for waterjet machining. Any change of the operating point should be gradually applied on the high-pressure network. This is realized with a reference shaping function

W r (s) = 1 10 -4 s + 0.6 s + 0.6 , (5.71)
that denes the dynamic range for any changes on the reference value r(t). This can be considered as a pre-lter for control design. It is preferably a low-pass lter, where a cut-o frequency of 0.6 Hz has been selected with respect to the predened soft constraints. This enables a response time of > 5 s to reach 95% of the desired reference value. The frequency response of Figure 5.21 shows the dynamic range for the reference signal.

W d (s) = 1 10 -4 s + 2.209 10 2 s + 2.209 10 2 R -1 d , (5.72)
which is again a low-pass lter, as shown with the frequency response in Figure 5.22 (a). It is scaled with R -1 d = 4.044 10 -5 that the output corresponds to the maximal water consumption of the cutting heads, allowed for a network subsection.

A cut-o frequency of 221 Hz has been chosen, as specied by the manufacturer, taking the on/o-valve switching dynamics into account. This enables a response time of < 10 ms considering the interval from 10 -90%, when switching the on/o-valve, see step response of Figure 5.22 (b). Perturbation shaping function W p : The inverse additive uncertainty model W ia (s) of Equation ( 5.133), dened later in the subsequent section, further extends the interconnection system with a perturbation input p(t). The dynamic range of this additional input signal is given by the perturbation shaping function

W p (s) = W ia (s) R -1 r , (5.73)
which is scaled with R -1 r = 4.0 10 8 to obtain a scaled input perturbation p w (t) = [0, 1]. The frequency response of Figure 5.23 shows the expected dynamic range for the plant perturbation. The investigations to derive a suitable uncertainty model is subject for the subsequent section. Noise shaping function W n : To investigate the noise dynamics of the measurement signal y n (t) = y g (t) + n(t) from a pressure gauge, the pressure has been measured at dierent operating points r(t) ∈ {100, 200, 300, 400} MPa. Measurement data have been derived from 2 dierent pressure gauges of the same type. The high-pressure pump will run in open-loop. This allows for generating a continuous input uid ow. A pressure equilibrium will be obtained, when the resulting output uid ow at the cutting head corresponds to the generated input uid ow. Data acquisition has been realized with a sampling rate of 200 Hz, when steady-state pressure near desired operating point has been reached. The experiment has been repeated to obtain 2 measurements for each operating point. That gives a total of 16 measurements.

Assuming Gaussian distribution allows for considering that the mean value

ȳn = 1 M M i=1 y n,i (5.74)
of a measurement with M samples reproduces the actual value μ(y) ≈ ȳn with increased precision for a large number of samples M → ∞. This mean value provides then the expected noise n = y n -μ(y) that corrupts the measurement signal. The plots of Figure 5.24 show selected measurements for dierent operating points. Small variance around expected value with 3 σ < 0.2 MPa has been observed for each measurement.

Figure 5.24 Measured noise for dierent operating points: residuals, mean values and 3 σ interval for dierent measurements reveal similar variance around expected value.

The Box-plot of Figure 5.25 shows the deviations for each measurement, where the plots of Figure 5.26 present the corresponding density function, when combining all measurements for an operating point. These show that the deviation due to noise resemble Gaussian distribution for an increasing numbers of samples. The variation remains similar for any operating point. As a consequence, additive Gaussian noise becomes acceptable to represent the measurement noise. Applying Fast Fourier Transformation (FFT) [START_REF] Frigo | FFTW: an adaptive software architecture for the FFT[END_REF]] to the noise n(t) and investigating its amplitude spectrum A(f ), as shown in plot of Figure 5.27, allows for evaluating the noise dynamics with respect to the frequency f . This aims to specify an adequate shaping function, see frequency response of Figure 5.28, which approximates the noise dynamics. For this, the maximum magnitude |A max (f )| = max|A k (f )| for each measurement k denes a frequency dependent upper bound. This bound rise with > 100 dB per decade from about -160 dB for low frequencies (< 1 Hz) to about -100 dB at high frequencies (> 3 Hz) for every operating point. Resonance peaks of -80 dB occur at high frequencies (74 Hz), where a maximal magnitude of -140 dB is observed at low frequencies. As a consequence, a second order transfer function has been considered. This provides a trade-o between accuracy and complexity. This results in the noise shaping function

W n (s) =
1.3 10 -4 s 2 + 1.186 10 -5 s + 2.704 10 -7 s 2 + 9.121s + 20.8 , (5.75) that represents the expected worst case frequency response, taking into account that the feedback loop is already scaled. It introduces a small magnitude for low frequencies and expects a high magnitude for high frequencies. This covers the observed signal dynamics and represents the observed additive noise of the measured signal. Eventually, a signal to noise ratio SNR > 60 dB has been observed. Table 5.6 shows details for each measurement. Compared to the pump actuator model, the expected disturbance dynamics seems to be slightly faster. However, the frequency response of Figure 5.29 shows for a frequency range of interest, that the magnitude for which the control value acts on the nominal plant is larger as the disturbance magnitude. In other terms, for frequencies f < ### Hz where the damping reaches > 60 dB, the open-loop gain from control input to plant output (u g (t) → y(t)) is higher as the gain from disturbance input to plant output (d w (t) → y(t)). This guarantees a feasible and eective disturbance rejection by means of the pump actuators. Figure 5.29 Scaled frequency responses of the pump and valve actuators with plant in series: dynamic range from the control signal to the pressure state and from the disturbance signal to the pressure state.

Output signal weighting functions

Output weighting functions are selected and shaped with respect to the control system design requirements. A performance weight applied on the error signal and a control eort weight applied on the control signal will be specied. The performance weight has typically a small magnitude at low frequencies. This aims at good disturbance rejection, whereas the control eort weight has a small magnitude at high frequencies. This attenuates shuttering on the actuator due to signal noise and avoids actuator saturations.

Performance weighting function W e : The condition for nominal performance |W e GS| < 1 requires the largest singular values σ(GS) ≤ |W e | -1 to become smaller as the inverse performance weight [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. This can be evaluated using the H ∞ -norm 5.76) and applying the small gain theorem (5.13) yields

GS ∞ < 1 W e ∞ (
W e GS ∞ < 1 , (5.77)
where W e is a weighting function applied on the error signal e(t). It is used as performance weight with respect to disturbance rejection. This requires a controller K(s), employed to the interconnection system of Figure 5.15, to hold the weighted error signal |e w (t)| < 1 bounded for a unit input on the shaped disturbance signal |d w (t)| = 1.

A signal from input channel d w (t) to output channel e w (t) relates to the transfer function

e w (t) = -W e G s G n S W d d w (t) , (5.78)
which evaluates nominal performance for disturbance rejection (d w (t) → w w (t)) and provides the performance criterion

W e G s G n S W d ∞ < 1 , (5.79)
related to (5.20b). This criterion will be satised if

|S(jω)| < 1 |W e (jω) G s (jω) G n (jω) W d (jω)| ∀ ω .
(5.80)

Considering the control objective (5.36), given by the weighting function

|S(jω)| -1 > |W 1 (jω)|, denes a lower bound |W e (jω) G s (jω) G n (jω) W d (jω)| < |W 1 (jω)| (5.81) to select a performance weight |W e (jω)| < |W 1 (jω)| |G s (jω) G n (jω) W d (jω)| .
(5.82)

The performance weighting function W e (s) = 1 10 -5 s + 221 s + 0.221 (5.83) has been specied to attenuate frequencies of < 10 Hz with > 20 dB. That should restrict the overshoot for pressure uctuations, due to exogenous disturbances, below 10%. Increasing the bandwidth for disturbance rejection would further improve performance, but noise attenuation becomes worse.

The frequency response of Figure 5.30 compares the weighting function W e and the lower bound, given by the control requirements. Reference value tracking is also specied by the performance weighting function W e and the high magnitude for low frequencies causes integral action, eliminating steady-state error. requires the largest singular values σ(T ) ≤ |W u | -1 to become smaller as the inverse performance weight [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. This can be evaluated using the H ∞ -norm

T ∞ < 1 W u ∞ (5.84)
and again applying the small gain theorem (5.13) yields

W u T ∞ < 1 , (5.85)
where W u is a weighting function applied on the control signal u(t). It is used as control eort weight, taking the actuator saturation into account by penalizing the control eort. Thus requires a controller K(s), employed to the interconnection system of Figure 5.15, to hold the weighted control signal |u w (t)| < 1 bounded for a unit input on the shaped disturbance signal |d w (t)| = 1.

A signal from input channel d w (t) to output channel u w (t) relates to the transfer function

u w (t) = -K G s G n S W d d w (t) , (5.86)
which penalizes the control eort for a disturbance signal (d w (t) → w w (t)) and provides the control eort criterion

K G s G n S W d ∞ < 1 .
(5.87)

Approximating T ≈ G n G s K S, criterion (5.87) will be satised if

|T (jω)| < 1 |W u (jω) W d (jω)| ∀ ω .
(5.88)

Considering the control objective (5.37), given by the weighting function

|T (jω)| -1 > |W 2 (jω)|, denes an upper bound |W u (jω) W d (jω)| < |W 2 (jω)| (5.89) to select a performance weight |W u (jω)| < |W 2 (jω)| |W d (jω)| .
(5.90)

The control eort weighting function W u (s) = 1 10 5 s + 8.281 10 4 s + 8.281 10 7

(5.91) has been considered, taking the actuator dynamic range into account.

The frequency response of Figure 5.31 compares the weighting function W u and the upper bound, given by the control requirements. Noise attenuation for high frequencies is desired, avoiding these frequencies to act on the actuators. Robustness weighting function W y : The condition for robust stability |W p GS| < 1 requires the largest singular values σ(GS) ≤ |W p | -1 to become smaller as the inverse perturbation shaping function. This condition holds particularly for an inverse additive uncertainty description [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. It can be evaluated, using the H ∞ -norm

GS ∞ < 1 W p ∞ (5.92)
and applying the small gain theorem (5.13) yields

W p GS ∞ < 1 .
(5.93)

For instance, W p = W ia is a shaping function in inverse additive form, applied on the system output state y(t). It is used as robustness weight with respect to the introduced plant perturbations. This requires a controller K(s), employed to the interconnection system of Figure 5.15, to hold the measured system state |y(t)| < 1 bounded for a unit input on the shaped perturbation signal |p w (t)| = 1.

A signal from input channel p w (t) to output channel y(t) relates to the transfer function

y(t) = -G n S W ia p w (t) , (5.94)
which provides robust stability for the perturbed plant (p w (t) → y(t)) and yields the robustness criterion

G n S W ia ∞ < 1
(5.95) related to (5.22a). This will be satised if

|S(jω)| < 1 |G n (jω) W ia (jω)| ∀ ω .
(5.96)

Considering the control objective (5.36), given by the weighting function

|S(jω)| -1 > |W 1 (jω)|, denes an upper bound |G n (jω) W ia (jω)| < |W 1 (jω)| (5.97) to select a robustness weight |W ia (jω)| < |W 1 (jω)| |G n (jω) | .
(

The singular values of Figure 5.32 compares the shaping function W p and the upper bound, given by the control requirements. Reference value tracking is not explicitly subject of control design, insofar as low dynamics for the reference value is desired. This is taken into account by the input weighting function on r(t). On the other hand, noise attenuation for high frequencies is desired. The measurement noise is not tremendous, as measurements reveal. Since the suppression of high frequencies is already specied by the control eort weighting function, noise attenuation is not explicitly considered.

First order functions are used to weight the output channels with respect to performance and control eort. This signal-based approach describes the robust control design problem, subject for H ∞ controller synthesis. Higher order weighting functions could be applied to sharpen the control requirements, if needed. But wherever possible, rst order weighting functions have been implemented. This aims at a low order interconnection system, since the order of the resulting generalized plant will dene the controller order.

It is important to select weighting functions with respect to dened bounds. Otherwise, a feasible controller will comply with S(s) + T (s) = 1, but fails to meet the control objectives. The proposed performance weight W e on e(t) and control eort weight W u on u(t) are taken into account for the H ∞ controller synthesis of Section 6.2, which are in compliance with the lower bound (5.81) and upper bound (5.89), respectively. The robustness weight W p on y(t) will be considered to evaluate robust performance for the resulting close-loop system. The subsequent section is dedicated to obtain a suitable uncertainty model that describes the robustness weight.

Uncertainty modelling

The introduced plant models contain varying parameters, which can be described as structured uncertainties of real parameters. Another approach is to lump these parametric uncertainties to an unstructured uncertainty description of complex values, representing the varying parameters with respect to the frequency domain. Thereby, an additive form, a multiplicative form and its inverse representations are compared, when approximating all the uncertain parameters with a shaping function. The diagram of Figure 5.33 summarizes the model evaluation process of this section. It applies both approaches to the simplied approximated model (5.55), that is linearised and of rst order. The obtained unstructured uncertainty descriptions are then compared to the structured description to evaluate an adequate uncertainty model for control design. Linear fractional transformation (LFT) is used, to separate the uncertainty and the nominal plant. For the structured uncertainty case, robust stabilization has to consider the concept of structured singular values, where the unstructured uncertainty case can be evaluated with respect to the small gain theorem [START_REF] Gu | Robust Control Design with MATLAB[END_REF]. Before introducing the uncertainty modelling, this sections rst classies the varying parameters and analyses the plant input to output sensitivity due to parameter variations.

Evaluating parameter variations and input to output sensitivity

When formulating the control design problem (see Section 5.3), it is recognized that dierent uncertainties aect the behaviour of a high-pressure system. This section classies and veries these parameter variations, which will aect the resulting closed-loop performance and stability. It is therefore distinguished between parameter-varying and time-varying uncertainties:

Two classes of parameter-varying uncertainties are considered. The rst class depends on the network conguration. These parameters will not change during operation, but are dierent for every waterjet facility. The second class considers varying parameters conditioned by the pressure state. These parameters will only change, when the operating point for waterjet machining varies.

Also time-varying uncertainties distinguish two classes. Where the variation in time follows an a priori known function or it is driven by an unknown function. In both cases, the variation will occur during pump operation. The diagram of Figure 5.34 classies these uncertainties, subject for further investigations.

Range and type of parameter variation

The variables related to the plant models (5.47) and (5.52) can be assigned to two groups, as shown in Figure 5.35. Each group is related to a parameter-varying as well as to a timevarying uncertainty. A rst group depends on the time-varying pressure state p(t) and on the parameter-varying nozzle cross section S H . The second group is dedicated to a time-varying camming angle φ(t) and a parameter-varying uid volume V 0 , that relates to the size of a highpressure network subsection. Hence, group 1 depends on the measurable pressure state, but unknown nozzle cross section and group 2 considers a measurable camming angle, but includes the unknown network volume.

The overall uid volume

V δ = V 0 + V C (s(φ))
(5.99) deals with an unknown parameter-varying uncertainty and a periodic time-varying uncertainty. It is composed from the uid volume V 0 of a high-pressure network and the displacement volume V C in the pumping chamber. The network contributes to an unknown volume that is constant in time for a specic waterjet facility, but changes when the network conguration becomes adjusted. On the other hand, the pumping chamber volume changes with respect to the piston position s(φ) and the alternating operation of two coupled pumping units, see the model-based synchronization introduced in Section 2.3. This follows a periodic function in time with respect to the camming angle φ(t), as shown in plot of Figure 5.36. Considering the alternating operation of pumping units results in an unsteady periodic function s δ . It shows a periodic discontinuity, when the maximal stroke s(φ(t) = π) = s max is reached. The bulk modulus K δ will vary with respect to the pressure state p(t), as already dened in (3.37). The pressure state is assumed to follow the reference pressure p 0 by means of control. The cutting process requires a constant pressure. It will only vary, if an on/o-valve disturbs the pressure generation. The pressure state will also aect the uid density (p) as given in (3.40).

The uid ow resistance

H δ = S H 2 (p) ζ (5.100)
is parameter-varying as well as time-varying. Since the uid density (p) characterizes the uid resistance, it becomes a function of the pressure state p(t). On the other hand, it will depend on the nozzle inner cross section S H installed at the cutting heads and its uid ow coecient ζ. The nozzles are typically referred to by its inner diameter D H ∈ {0.08, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4} mm, such as S H = D 2 H π/4. A nozzle is only changed, if waterjet machining is not in progress. But a waterjet application can include several cutting heads, which switch dierent waterjets on and o with respect to an unknown switching pattern. Hence, the work station conguration denes a discrete set of possible overall cross sections, while the cutting process will cause this parameter to vary during operation. The plots of Figure 5.37 show the uid density with respect to the pressure state (a) and the pumping chamber volume dependent on the piston displacement (b). Extending this, the plots of Figure 5.38 present the uid ow resistance for dierent nozzles installed (a) and the variation of the overall system volume (b), for possible network subsections of dierent size. The permissible size of a network subsection is unknown and subject of specication for robust control design. Respectively, the requirements on performance and stability will dene the maximally permissible subsection size. For the uncertainty analysis in this section, an overall piping length L = [5,75] m is assumed, based on the experiences at the high-pressure test bench. For a typical piping with an inner diameter of 2.4 mm 2 results an expected volume of V 0 = [21.69, 341.7] cm 3 . On the other hand, the pumping chamber is well specied with a piston diameter of ### mm and its stroke s P = [0, ###] m, hence, the resulting displacement volume V P = [0, ###] cm 3 causes a varying pumping chamber volume

V C = [###, ###]cm 3 .

Input to output sensitivity

The eect of a parameter variation on the frequency response will be investigated with respect to the control signal (u → y) and disturbance signal (d → y). Both characteristics will be compared with respect to the approximated model and the evaluated extended model. This is useful to evaluate the demands for robust control design and to nd an adequate uncertainty description. Investigating each parameter independently, allows for evaluating its sensitivity for a specic frequency range. Applying the maximal control signal u max and the maximal disturbance signal d max , then evaluating the system gain and time constant τ , give a measure for input to output sensitivity of each parameter.

The frequency response from control signal u(t) to pressure state y(t) is investigated. The plots of Figure 5.39 show the eects, when varying a single parameter at once, while the other uncertain parameters remain on nominal value. It compares the approximated with the evaluated extended model. The ow resistance and the nozzle cross section cause an increasing magnitude (a) for low frequencies, while the pressure state will decrease the magnitude (b). The uid volume and the bulk modulus cause a varying magnitude for high frequencies, where the variation due to an uncertain uid volume (c) is larger than the variation due to an uncertain bulk modulus (d). It is further recognized, that a minimal nozzle cross section S = 0 cause integral action.

On the other hand, the frequency response from disturbance signal d(t) to measured pressure state y(t) has been investigated. The plots of Figure 5.40 compare again the approximated with the evaluated extended model. Basically, the variation of magnitude, considering an uncertain pressure state, uid volume and bulk modulus, are comparable to the eects observed for the control signal. However, the variation at low frequencies with respect to the ow resistance and nozzle cross section becomes compensated. This avoids integral action. However, a decreasing magnitude at high frequencies is observed.

Scaling u g (t) = u(t)/u max with the maximal input uid ow u max = S max ζ 2 p max / max , required to obtain equilibrium at desired steady-state pressure, and d g (t) = d(t)/d max for d max = 1 allow for comparing the aect of disturbance and control signal to the pressure state. For this, the plots in Figure 5.41 show the frequency responses for u(t) → y(t) and d(t) → y(t).

The pressure state has a major eect on decreasing magnitudes at low frequencies, considering the control channel as well as the disturbance channel. This nding might be useful to dene an adequate uncertainty representation for robust control design. Evaluating each uncertain parameter reveals for all frequencies that the control channel is superior to the disturbance channel. The increasing magnitude for low frequencies only aects the control channel. It occurs when varying the ow resistance and nozzle cross section, respectively. This enables the control channel to stabilize the system, since the disturbance channel is not aected by the integral action. disturbance inputs, reveals that all uncertain parameters, considered for control design, have a perceptible impact on the system gain as well as on the time constant. The investigated parameter variation has to be taken into account for robust control design. A possible approach is discussed, when introducing the structured uncertainty description. (%)

Approx.

H δ ### ∞ [-75, + ∞]
Extend.

S δ ### ∞ [-75, + ∞] x δ
1395.8 1600.0 [-7.6, +5.9]

Structured parametric uncertainty description

Using the framework of H ∞ requires to separate the uncertainties from the plant model to employ additional perturbation input channels. That is realized by applying upper linear fractional transformation (ULFT) [START_REF] Gu | Robust Control Design with MATLAB[END_REF]. This section will introduce the structured uncertainty description to the perturbed plant model. This means that each real-valued uncertain parameter directly introduces an independent perturbation.

Parametric uncertainties

Starting point is the simplied approximated model (5.55)

d dt x(t) = - K δ H δ 2V δ √ x δ x(t) + K δ V δ u d (t) .
(5.105)

Additive and inverse additive uncertainty descriptions have been chosen in compliance with the model structure, since K δ and H δ stay in the numerator and V δ and x δ in the denominator.

Dening the parametric uncertainties of additive form

K δ = K 0 + δ K , H δ = H 0 + δ H (5.106)
and the parametric uncertainties of inverse additive form (5.107) where K 0 , V 0 , H 0 , √ x 0 denote the corresponding nominal parameter values and δ K , δ V , δ H , δ √

1 V δ = 1 V 0 + δ V , 1 √ x δ = 1 √ x 0 + δ √ x ,
x give the uncertainty range. It has to be pointed out that the operating point √ x 0 and its uncertainty range δ √ x correspond to the square root values of √ x δ .

Introducing (5.106) and ( 5.107) in (5.105) gives the perturbed plant

d dt z(t) = K 0 + δ K V 0 + δ V w(t) - 1 2 H 0 + δ H √ x 0 + δ √ x z(t) , (5.108)
assuming the input w(t) = u d (t) and output z(t) = x(t).

The block diagram of Figure 5.42 illustrates the structure of the perturbed plant (5.108), where e n (t) denotes an output to the n-th uncertain parameter and d n (t) is a perturbation input from the n-th uncertain parameter.

d dt z(t) = - KH 2V √ x z(t) + E d(t) + K V w(t) , (5.109
)

with E = [ 1 V -1 V -K 2V √ x K 2V √
x ] and to describe the output channels

e(t) =            -H 2 √ x -KH 2V √ x 1 H √ x            z(t) +            0 0 -1 2 √ x 1 2 √ x 1 V -1 V -K 2V √ x K 2V √ x 0 0 0 0 0 0 1 √ x -1 √ x            d(t) +           1 K V 0 0          
w(t) , (5.110) where the subscripts indicating a nominal parameter is omitted for simplicity. The structure of the perturbed plant with parametric uncertainties also implies e 2 = dz/dt. The perturbed plant model considers parametric uncertainties. This enables an accurate uncertainty representation, since the parameter range of each uncertainty is individually implemented with respect to the structure of the plant. However, the size of the optimization problem for control design will increase with the amount of uncertain parameters.

Applying upper linear fractional transformation (ULFT)

The state space description (5.109) can be rearranged in the standard ∆=M conguration, as presented in block diagram of Figure 5.2 (see the control design fundamentals in Section 5.2). This conguration includes the interconnection system M and the uncertainty block ∆, which contains the parametric uncertainties.

Considering the perturbed plant model in the state space form (5.109) and applying the partitioning from the augmented interconnection system (5.3) ( 5.111) with 5.112) for which the common denominator

M 11 =             -H 2V √ x H 2V √ x -s 2 √ x s 2 √ x s V -s V Ks 2V √ x -Ks 2V √ x 1 V -1 V -K 2V √ x K 2V √ x H V √ x -H V √ x s √ x -s √ x             D -1 (s), M 12 =            s Ks V KH V √ x K V            D -1 (s), M 21 = 1 V -1 V -K 2V √ x K 2V √ x D -1 (s), M 22 = K V D -1 (s) , (
D(s) = s + KH 2V √ x
( 5.113) has been introduced. This separates all real-valued uncertainties in form of a diagonal matrix 5.114) where each diagonal entry corresponds to a specic uncertain parameter, implemented in the plant model. The matrix M 22 gives the transfer function for the nominal plant

∆ =     δ K 0 0 0 0 δ V 0 0 0 0 δ H 0 0 0 0 δ √ x     , (
G n (s) = K V s + KH 2V √ x
( 5.115) and applying upper linear fractional transformation (5.4) denes the perturbed plant

G p (s) = M 22 + M 21 ∆ (I -M 11 ∆) -1 M 12
( 5.116) by means of M and ∆, where the exogenous input w(t) for z = G p (s) w holds the control signal u d (t) and the output z(t) corresponds to the pressure state x(t).

Unstructured complex uncertainty description

In contrast to the above structured uncertainty description, unstructured uncertainties will approximate the dierent varying parameters in the frequency domain. This results in a complexvalued function that shapes an additional perturbation input. The error frequency response of the perturbed plant denes the unstructured uncertainty, where ULFT again separates the uncertainties from the nominal plant.

Starting point is once more the simplied approximated model ( 5.55)

d dt x(t) = - K δ H δ 2V δ √ x δ x(t) + K δ V δ u d (t) .
(5.117) However, the parametric uncertainties will be lumped to a transfer function of minimal realization, that represents plant perturbations. For this, the transfer function of the nominal plant

G n (s) = K V s + KH 2V √ x 0 (5.118)
is obtained, when applying Laplace transformation on (5.117) and taking the nominal parameter values into account. The block diagram of Figure 5.43 represents the structure of the nominal plant (5.118). It reveals that any variation of the parameters K, V , H and x 0 will change the feedback gain and, thus, aects the plant output x(t). Variations on K and V also aect the input gain on the plant input u d (t). This requires to investigate dierent complex uncertainty descriptions to nd an adequate representation for the varying parameters.

Complex uncertainties

The objective is to nd an adequate structured uncertainty description. It is required to verify, whether to represent the parametric uncertainties of the simplied approximated model (5.55) by means of the plant input or by using the plant output channel. The block diagrams of Figure 5.44 show various structures to represent the perturbed plant by means of dierent unstructured uncertainty descriptions. The additive uncertainty as well as the input multiplicative uncertainty depend on the plant input u d (t), which corresponds to a uid ow Q(t). Where the additive form aects the plant output x(t), the multiplicative form aects the plant input. On the other hand, the inverse additive description and the inverse multiplicative description depend on the plant output x(t), which corresponds to the pressure state p(t). Where the additive form aects the plant input, the multiplicative form aects the plant output. The additive and input multiplicative uncertainty are in most cases interchangeable (W a = W m G n ). The same holds for the inverse additive and inverse output multiplicative uncertainty (W im = G n W ia ). (5.122) will be obtained, for which ∆ is norm bounded such as |∆(jω)| ≤ 1 ∀ ω [START_REF] Gu | Robust Control Design with MATLAB[END_REF].

Considering the error frequency response of the perturbed plant yields the weighting functions W a (s) and W m (s), respectively.

In the same manner, the inverse additive uncertainty representation

G p (s) = G -1 n (s) + ∆ ia (s) -1
( 5.123) and the inverse output multiplicative representation

G p (s) = G -1 n (s) (I + ∆ im (s))
-1

(5.124) can be obtained, where the inverse additive uncertainty block

∆ ia (s) = W ia (s) ∆
( 5.125) and the inverse multiplicative uncertainty block

∆ im (s) = W im (s) ∆
( 5.126) introduce a weighting function W ia (s) and W im (s).

Approximating the weighting functions

These weighting functions represent the upper bounds of the error frequency response. Introducing (5.121) to (5.119) yields (5.128)

Evaluating (5.127) for the simplied approximated model (5.55), the second order weighting function W a (s) = 2.605 10 9 s 2 + 6.886 10 13 s + 5.191 10 14 s 2 + 3.943 s + 2.989 10 -3

(5.129) has been obtained to represent the frequency response of an additive uncertainty, whereas the rst order weighting function W m (s) = 1.1 s + 8.315 s + 8.398 10 -4

(5.130) represents the frequency response of an input multiplicative uncertainty, where both take the parameter range of Table 5.3 into account. The plots of Figure 5.45 compare the error frequency responses of the additive form (5.127) (a) and input multiplicative form (5.128) (b). It is observed that the error of the additive uncertainty description is large at low frequencies, but becomes small for high frequencies, while the multiplicative uncertainty description represents a constant error for high frequencies.

The plots of Figure 5.46 show the frequency response of the perturbed plant, described by means of additive uncertainty (5.119) (a) and the perturbed plant, using multiplicative uncertainties (5.120) (b). It results in identical behaviour. However, comparing these frequency response from complex uncertainties with the error frequency response derived by real parametric uncertainties (5.116), reveal undesired behaviour at low frequencies. This is evidence that the introduced perturbation channel, with respect to plant input u d (t), provides an inadequate choice. The upper bounds of the error frequency response will also dene the weighting functions for the inverse representations. Introducing (5.125) to (5.123) yields

|W ia (jω) ∆(jω)| = 1 |G p (jω)| - 1 |G n (jω)| = |G n (jω) -G p (jω)| |G p (jω) G n (jω)| ≤ |W ia (jω)|
( 5.131) and introducing (5.126) to (5.124) yields

|W im (jω) ∆(jω)| = G -1 p (jω) G -1 n (jω) -I = G -1 p (jω) -G -1 n (jω) G -1 n (jω) ≤ |W im (jω)| .
(5.132)

Evaluating (5.131) for the simplied approximated model (5.55), the inverse additive weighting function

W ia (s) =
4.562 10 -10 s + 3.442 10 -9 s + 5.454 10 4

(5.133) has been obtained to represent the frequency response of an additive uncertainty, whereas the inverse multiplicative weighting function W im (s) = 5.233 10 -1 s + 3.959 s + 3.956 (5.134) represents the frequency response of an input multiplicative uncertainty, again with respect to the parameter range of Table 5.3. The plots of Figure 5.47 compare the error frequency response of the inverse additive (5.131) (a) and input multiplicative form (5.132) (b). For the inverse additive uncertainty description, a constant error for low frequencies is observed. It increases for high frequencies. The inverse multiplicative uncertainty description represents a constant error for low frequencies, as well as for high frequencies. A smaller error occurs for high frequencies than for low frequencies.

The plots of Figure 5.48 show the frequency response of the perturbed plant, described by means of inverse additive uncertainty (5.123) (a) and the perturbed plant, using inverse multiplicative uncertainties (5.124) (b). It results in identical behaviour. Comparing these frequency responses from complex uncertainty with the error frequency response derived by real parametric uncertainties (5.116), reveal an excellent t for the frequency range of interest. This is evidence that the introduced perturbation channel, with respect to plant output x(t), provides an adequate choice. Some variance has been observed for high frequencies, which seems small.

The additive as well as the multiplicative uncertainty representations are unsatisfying to represent the varying parameters of the simplied approximated model. However, the resulting dynamics of the inverse uncertainty descriptions represent the varying parameters with desired accuracy. These inverse uncertainties of complex values perform more conservative, as a large perturbation is given for low frequencies. Nevertheless, it is simpler to formulate a perturbed plant using unstructured than structured uncertainties. That becomes advantageous for the subsequent controller synthesis, when implementing robust stability. A weighting function of order 1 is sucient for both unstructured uncertainty descriptions in inverse form. Low order weighting functions are desired for control design, since the total order of the generalized plant, used for controller synthesis, will dene the order of the resulting controller. More complex uncertainty descriptions, e.g using co-prime factors [START_REF] Gu | Robust Control Design with MATLAB[END_REF], could further improve the accuracy for unstructured uncertainties. The inverse output multiplicative form match the investigated parametric uncertainty with desired accuracy. In contrast to the inverse additive form, its weighting function is favourable with respect to the resulting coecients of the numerator and denominator. Consequently, its transfer function is used for control design. It will shape the dynamics on the perturbation input p(t).

Applying upper linear fractional transformation (ULFT) Also the perturbed plant, considering unstructured uncertainties, can be rearranged to the standard ∆=M conguration, as shown in the block diagram of Figure 5.2 (see the control design fundamentals in Section 5.2). Rearranging the perturbed plant (5.123) in inverse additive uncertainty form, with respect to the conguration (5.3), allows for dening the interconnection transfer function matrix 5.135) where the uncertainty block ∆ = ∆ ia (s) corresponds to the inverse additive uncertainty description. Applying upper linear fractional transformation F u (M, ∆), according to (5.4), denes the perturbed plant

M = M 11 M 12 M 21 M 22 = -G n G n -G n G n , (
G p (s) = G n (s) -G n (s)∆ ia (s)(I + G n (s)∆ ia (s)) -1 G n (s) , (5.136)
which describes the transfer function w(t) → z(t) with respect to the inverse additive uncertainty. That is realizable if I + G n (s)∆ ia (s) is invertible. Also the perturbed plant (5.124) in inverse multiplicative form yields a interconnection transfer function matrix

M = M 11 M 12 M 21 M 22 = -I G n -I G n ,
(5.137) with respect to (5.3), where the uncertainty block ∆ = ∆ im (s) contains the inverse multiplicative uncertainty. Applying the upper linear fractional transformation (5.4), denes the perturbed plant 5.138) which describes the transfer function w(t) → z(t) with respect to the inverse multiplicative uncertainty, if (I + G n (s)∆ im (s)) is again invertible.

G p (s) = G n (s) -G n (s)∆ im (s)(I + G n (s)∆ im (s)) -1 G n (s) , (

Robust stabilization

Considering the feedback system of Figure 5.49, allows for introducing the requirements for robust stabilization [START_REF] Gu | Robust Control Design with MATLAB[END_REF] G p (jω 5.139) for the inverse additive uncertainty description (5.123) and (5.140) for the inverse multiplicative uncertainty description (5.123). The maximal singular values 5.141) and 5.142) are bounded by the corresponding weighting functions W ia (jω) and W im (jω), respectively, for which the matrix inequalities

) (I + K(jω) G p (jω)) -1 ∞ < 1 ∆(jω) ∞ (
(I + K(jω) G p (jω)) -1 ∞ < 1 ∆(jω) ∞
σ (∆(jω)) ≤ σ (W ia (jω)) ∀ ω ∈ R (
σ (∆(jω)) ≤ σ (W im (jω)) ∀ ω ∈ R (
W ia (jω) G p (jω) (I + K(jω) G p (jω)) -1 ∞ < 1
( 5.143) and

W im (jω) (I + K(jω) G p (jω)) -1 ∞ < 1
(5.144) can be derived from (5.139) and ( 5.140), when substituting (5.141) and ( 5.142) as a strengthened formulation for robust stability (5.22a) with respect to the small gain theorem (5.13).

Eventually, solving the optimization problems derived from (5.143) and (5.144), respectively, yields a suboptimal H ∞ stabilizing controller K(s).

The H ∞ problem formulation, considering the control requirements in form of shaping and weighting functions, is subject for the subsequent chapter. The interconnection system, depicted in the block diagram of Figure 5.15, needs to be rearranged to obtain a generalized plant, which denes the optimization problem for H ∞ controller synthesis.

Conclusion

This chapter discussed the robust decentralized control design for an electrically driven highpressure pump. It considers the implementation of a single controller for every pump, installed in a high-pressure network of unknown topology. This requires for separating a high-pressure network into subsections. Each subsection is then controlled by the assigned high-pressure pump.

A contribution of this chapter is given, when dening the H ∞ optimization problem for this specic high-pressure system. For this, a decentralized pump supplying a high-pressure subsection has been considered. Assuming a perfect synchronization of the involved pumping unit, enables a rst order model to describe the pressure generation within a subsection. This plant model contains all prevalent parametric uncertainties. The other system components have been modelled with transfer functions of nominal values. The plant model has been linearised, to obtain an exogenous input disturbance. The sensitivity of the plant model, subject to parameter variation, has been investigated. It has been found that a cutting head with decreasing nozzle cross section, causes an increased gain from control input to pressure output, in particular at low frequencies. Hence, a closed cutting head results in integral action. Further, an increasing operating pressure will decrease the input to output gain. On the other hand, an increasing nozzle cross section does not aect the gain from disturbance input to pressure output for low frequencies. Summarizing this result, the high-pressure pump is capable of stabilizing its corresponding network subsection for every disturbance induced by an unknown switching of cutting heads.

These uncertain parameters have been lumped to unstructured uncertainties, which have been compared to the structured uncertainty description. The inverse additive uncertainty provided satisfying results by reproducing the investigated parametric uncertainty range. It allows for introducing an perturbation input channel to the interconnection system, by using linear fractional transformation. The uncertain parameters are thereby separated from the nominal plant model. Specifying an input shaping function on the perturbation channel, while assuming a nominal plant for control design. Alternatively, the inverse output multiplicative uncertainty obtains identical results. Though, coprime factor uncertainties could further improve the uncertainty modelling [START_REF] Gu | Robust Control Design with MATLAB[END_REF]. The choice of an unstructured complex uncertainty is advantageous for controller synthesis (see the subsequent chapter). It will simplify the procedure to maximize the permissible network subsection size, but introduces additional conservativism to the control design. Eventually, all system components have been combined to build an interconnection system. Shaping functions have been dened to shape the corresponding input channels, with respect to the investigated signal dynamics.

Introduction Objectives

Electrically driven high-pressure pumps require a robust control design, as they are installed to future waterjet facilities at unknown network locations. Establishing high-pressure networks, aims to improve the energy eciency of waterjet machining. The previous chapter introduced the control design requirements in time domain and derived adequate weighting functions in frequency domain. The obtained interconnection system considers the expected plant perturbations, by implementing an uncertainty description, and includes the control requirements. This interconnection system is useful for H ∞ controller synthesis. However, because of the unknown topology of high-pressure networks for future waterjet facilities, it becomes impossible for industry to specify the network subsection size, for which a high-pressure pump must be installed to guarantee an eective attenuation of pressure uctuations. The permissible subsection size is unknown and therefore subject for investigations. This requires a robust control design procedure that maximizes the possible uncertainty range, while obtaining robust performance with respect to the dened control requirements. The permissible size of a network subsection is a design parameter that depends on the available control eort of an electrically driven pump and the attainable performance for disturbance rejection, needed to maintain the desired cutting quality for waterjet machining. The robust control design is further required to employ a structured controller that is straight forward for implementation on an industrial high-pressure pump. In this particular case, the controller preferably provides a PI structure, which obtains the desired requirements on stability and performance while maximizing the permissible subsection size. This aims at a robust control design to synthesize a quasi-optimal PI controller. Realizing future waterjet facilities requires then to combine robustly controlled high-pressure pumps, each stabilizing a network subsections. The interconnected subsections establish a high-pressure network with decentralized pumps, which stabilize a network subsection by means of local disturbance rejection. However, the coupling of multiple pumps across the high-pressure network could cause an increased energy consumption and limits the overall stability, if the pumps begin to act against each other. Consequently, simulations and experimental studies are required to validate performance and stability for robustly controlled and distributed high-pressure pumps.

Contribution

This chapter employs a H ∞ controller synthesis [START_REF] Gu | Robust Control Design with MATLAB[END_REF] for electrically driven highpressure pumps, taking expected parameter variations, exogenous disturbances and control requirements into account (as introduced in Chapter 5). Shaping functions are introduced to consider the dynamic range of input signals and weighting functions are used to dene the requirements for performance and to penalize the control eort. A generalized plant for controller synthesis is obtained, when combining the unstructured uncertainty description with a signalbased design approach [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. Formulating the generalized plant for controller synthesis, it is proposed to merge input signal by introducing joint shaping functions. This may lead to a more conservative control design, but reduces the dimension of the resulting H ∞ optimization problem. The optimization problem is then solved to obtain an initial state feedback controller [START_REF] Doyle | Statespace solutions to standard H 2 and H ∞ control problems[END_REF]. This controller serves as a reference for further structured controller designs. It can be shown that the introduction of joint shaping functions will decrease the calculation cost and can further reduce the order of the resulting state feedback controller [START_REF] Niederberger | Robust control design for electrically driven high-pressure pumps using H ∞ approach with joint shaping functions[END_REF].

Another contribution is to apply a structured design approach [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF], resulting in a PI controller, suitable for industrial use as well as capable to operate within desired performance and guaranteed stability. Extending the PI controller to a lead compensator, will remarkably improves the obtained closed-loop performance and manages to recover the initial state feedback controller for the frequency range of interest. The H ∞ frame work is further used to determine the permissible network subsystem size, considering a specic high-pressure pump setup. When analysing the parameter variations and the resulting uncertainty model, for which robust performance and stability can be guaranteed, a robust performance margin will be obtained. This allows for implementing an iterative procedure, which obtains a quasi-optimal PI controller, while maximizing the permissible uncertainty range. Simulations compare the disturbance rejection of the derived PI controllers with the state feedback controller, using detailed high-pressure network models (as previously introduced in Section 4.2). Further experimental studies investigate stability and performance of a preliminary PI controller applied on the test bench. It has been congured, considering single pump setups as well as congurations with two interconnected pumps. For all congurations, two independent cutting heads will induce pressure uctuations. The experiments show dynamic couplings between interconnected pumps and reveal the need for a high-level managing.

This chapter is structured as follows: Section 6.2 combines the perturbed plant, weighting functions and shaping functions, from the previous chapter, to formulate a generalized plant. Therefore, the interconnection system is transformed into standard P=K form for control synthesis [START_REF] Gu | Robust Control Design with MATLAB[END_REF], by applying lower linear fractional transformation. A reduced robust state feedback controller has been obtained by means of joint shaping functions. Its performance and stability is compared to a PI controller, as derived from a structured H ∞ synthesis. The suboptimal H ∞ controller synthesis is then extended with the ∆-K iteration, proposed for shaping function adjustment. This allows for obtaining a robust controller that fulls the control requirements for performance and stability (as introduced in Section 5.2), while maximizing the uncertainty range with respect to the unknown network subsection size. In Section 6.3, the proposed graph-based modelling methodology, taking the topologies of various high-pressure systems into account, will be used to evaluate the robust control design. The derived robust controllers will be veried on validated simulation models, which considers non-linearities and parameter variations. For the experimental validation in Section 6.4, a structured controller will be implemented on the test bench. The experimental study is reproduced, using the validated simulation models in closed-loop control. This aims to verify the implementation of the controller and allows for validating the graph-based modelling methodology as a simulation toolbox for control design evaluation. The toolbox is useful to investigate complex high-pressure network for large-scale waterjet facilities, which exceed the capabilities of a test bench.

Controller synthesis

This section presents the robust feedback control design for an electrically driven high-pressure pump by means of H ∞ suboptimal controller synthesis [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF] . The diagram of Figure 6.1 shows the subsequent control design process. First, a state feedback controller will be obtained, which should provide superior performance and robustness with respect to the specied uncertainties. This is realized at expenses of a high-order controller, which requires an order reduction for implementation in the industrial application. Second, a structured feedback controller of standard PI structure is synthesized. It approximates the behaviour of the state feedback controller. Both control designs will be compared by means of simulations, taking the time domain specications into account. Introducing a low-pass lter to the PI control design will seriously improve performance and robustness. This extended PI controller is capable to recover the behaviour of a state feedback controller, resulting in a control law straight forward to implement into the application. The structured feedback controller synthesis is then veried to evaluate the robustness margin with respect to the size of an uncertain high-pressure network subsection. This allows for specifying the maximal subsection size, which guarantees robustness and optimizes the control design. The derived interconnection system (see Section 5.3) will be rearranged, using lower linear fractional transformation (LFT) [Safonov, 1981], to obtain a full generalised plant. It takes the dened input signal shaping functions and output signal weighting functions into account and considers the plant uncertainties as additional perturbation input. This generalized plant describes a suboptimal H ∞ optimization problem [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF], where the size of the generalized plant will dene the dimension of the optimization problem. A procedure will be introduced to combine input channels by means of joint shaping functions . This procedure lumps shaping functions of the initial interconnection system. That reduces the generalized plant and results in a simplied, but more conservative optimization problem. Finally, the suboptimal control design will be extended to derive a quasi-optimal controller, subject to nd the permissible uid volume V δ , while searching a controller K(s) that minimizes the H ∞ -norm of the generalized plant. Finding the maximal uid volume for which desired stability and performance can be guaranteed, allows for dening the maximal size of a network subsection, which can be stabilized by a high-pressure pump.

Generalized plant

The interconnection system, illustrated in the block diagram of Figure 5.15 from the previous chapter, contains the control requirements in form of shaping and weighting functions. It needs to be rearranged to obtain a generalized plant T (s), which denes the optimization problem for H ∞ controller synthesis. The interconnection system is therefore rearranged to the standard P=K conguration, as shown in the block diagram of Figure 5.3 (see the control design fundamentals in Section 5.2).

Applying the standard conguration (5.5) to the interconnection system (5.63), yields the interconnection transfer function matrix

with

P 11 = W e IW r -W e G s G n W d -W e IW n 0 0 0 , P 12 = -W e G s G n G a IW u , P 21 = IW r -G s G n W d -IW n , P 22 = [ -G s G n G a ] , (6.2)
which is partitioned with respect to the input channel w ∈ R m and output channel z ∈ R n , assuming a control variable u ∈ R q as well as a measurement variable y ∈ R p of dimension q = p = 1, m = 3 and n = 2. It concatenates all weighted exogenous inputs w = [r w d w n w ] and all weighted exogenous outputs z = [e w u w ] , as illustrated in Figure 6.2. The nominal plant G n with pump actuator G a and pressure gauge G s are thereby connected to the controller block K, by closing the lower loop with

P 11 = -G s (s) G n (s) G a (s).

Applying lower linear fractional transformation (LLFT)

Using lower linear fractional transformation (5.6), as dened in Section 5.2, provides the generalized plant T (s) = P 11 + P 12 K (I -P 22 K) -1 P 21 . (6.3) This describes the closed-loop system z = T (s) w from the exogenous input channels to the exogenous output channels w(t) → z(t) with respect to the standard conguration (5.5), including a controller of desired structure, e.g. (5.68) or (5.69).

Evaluating (6.3) for the interconnection transfer function matrix (6.2) yields the generalized plant 6.4) that denes the desired cost function for H ∞ controller synthesis. For convenience, the quasisensitivity function S = (1 + G s G n G a K) -1 and the quasi-complementary sensitivity function T u = KG s G n S have been introduced. The rst column of of the generalized plant (6.4) corresponds to the reference signal r(t), the second to the disturbance signal d(t), the third to the noise signal n(t) and the fourth to the perturbation signal p(t), whereas the rst row corresponds to the control error e(t) and the second to the control value u(t). For example, the performance for disturbance rejection is given by T 21 and the control eort for suppressing any disturbances is dened by T 22 .

T = W e SW r -W e G s G n SW d -W e SW n -W e G s G n SW p W u KSW r -W u T u W d -W u KSW n -W u T u W p , (
H ∞ suboptimal control design A generalized plant T (jω) denes the cost functions of the suboptimal H ∞ optimization problem [START_REF] Gu | Robust Control Design with MATLAB[END_REF] minimize

K T (jω) ∞ < γ , (6.5)
that is used to nd a stabilizing controller K, such as T ∞ < γ, where γ denotes the inverse robust stability margin in terms of the peak magnitude. Nominal performance and robust stability will be obtained, if γ ≤ 1 is satised. Thus, an optimization problem of dimension n × m has to be solved, minimizing the H ∞ -norm

T ∞ = sup ω σ (T (jω)) (6.6)
for the cost function of dimension 4 × 2, as given by (6.4). This requires to nd a controller K that

W e SW r W e G s G n SW d W e SW n W e G s G n SW p W u KSW r W u T u W d W u KSW n W u T u W p ∞ < 1 (6.7)
is obtained.

Using the Riccati method [START_REF] Doyle | Statespace solutions to standard H 2 and H ∞ control problems[END_REF], as implemented in Matlab, to solve for the inequalities (6.7) results in a state feedback controller of order 12, for which a peak value of γ = 0.6067 has been achieved.

The step responses of Figure 6.3 evaluates the perturbed plant (5.116), with the derived robust state feedback controller K SF in closed-loop. In contrast to the generalized plant used for control design, where an unstructured inverse additive uncertainty has been considered, the step and frequency responses for controller evaluation will be derived, using a perturbed plant with real parametric structured uncertainties (see Section 5.4). This holds for all investigations in the subsequent sections. The simulation studies in Section 6.3 will validate the derived structured and unstructured controllers, using the detailed non-linear simulation models from Chapter 4. The step response from reference input r(t) to measurement output y(t) shows a desired reference value tracking with rise time τ r < 4 second and no steady-state error exists. The resulting characteristics is given by the a priori dened input shaping function W r , see Section 5.3. Any step on the reference signal is therefore gradually applied to the system. The response from reference input r(t) to the control value u(t) reveals the wide operating range needed to obtain the reference pressure for dierent plant perturbations. This means that up to 40% of the available pump rate is required to reach pressure equilibrium. This depends on the conguration of a waterjet facility. The step response from disturbance d(t) to measurement output y(t) shows that any unit step disturbance will aect the system pressure with a maximal 9.3% overshoot, considering worst case perturbation. The controller will reject the disturbance within a settling time τ s < 0.2 seconds. Any disturbance is attenuated with zero steady-state error. The step response from d(t) to u(t) reveals that another 60% of the available pump rate is applied for disturbance rejection. The step response from noise n(t) to y(t) and to u(t) shows that the noise on the measurement channel will be attenuated and noise acting on the actuator is reasonably small (with amplitudes of 10 -4 ). Figure 6.3 Step responses of state feedback controller K SF : applying unit steps on the dierent input channels.

The control eort to follow the reference signal is less than 40%. The control eort to attenuate disturbance can reach 60%. Consequently, the pump actuator limits are sucient to supply a high-pressure network subsection without reaching saturation, but further increasing the water consumption is not permitted. The evaluation of the closed-loop system with respect to the control objectives is summarized in Table 6.1, see page 211.

The above signal-based approach to employ the H ∞ optimization problem can be considered as a generalized form of the mixed-sensitivity approach [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF], for which specic transfer functions of desired input channels to selected output channels provide the cost functions subject for optimization. Investigating the control requirements with respect to the introduced performance criterion (5.79), the control eort criterion (5.87) and the robustness criterion (5.95), requires to evaluate the H ∞ -norm (6.7) for the transfer functions p → y, d → e and d → u. Recalling the control design specications (5.64), yields the optimization problem minimize 6.8) where the rst cost function denes stability for perturbations from p(t) to y(t), the second cost function denes the dynamic range for disturbance rejection from d(t) to e(t) and the third cost function penalizes the control eort from d(t) to u(t).

K G n SW p W e G s G n SW d W u T u W d ∞ < γ , (
To nd a controller K, the largest singular values σ of each system transfer function, including the input weights, must be smaller by a factor γ as the smallest singular value of the corresponding output weight. For that, the inequalities

σ (G n (jω)S(jω)W p (jω)) ≤ γσ ( 1 ) σ (G s (jω)G n (jω)S(jω)W d (jω)) ≤ γσ W -1 e (jω) σ (T u (jω)W d (jω)) ≤ γσ W -1 u (jω) (6.9)
have to be satised for all ω. This is evaluated by applying the H ∞ -norm (6.6) for the stacked notation (6.10) where the largest singular value σ is evaluated by the Euclidean vector norm (6.11) Thus, the singular values of the inequalities (6.9) describe the cost function

sup ω σ   T 1 (jω) : T k (jω)   < γ ,
σ   T 1 (jω) : T k (jω)   = |T 1 (jω)| 2 + |T 2 (jω)| 2 + ... + |T k (jω)| 2 = k i=1 |T i | 2 .
σ(T ) = |G n SW p | 2 + |W e G s G n SW d | 2 + |W u T W d | 2 (6.12)
in terms of (6.11).

The plots of Figure 6.4 compare the singular values of the selected cost functions (6.9). The rst inequality guarantees nominal performance with respect to disturbance rejection (a), while the second inequality penalizes control eort, considering actuator saturation (b). The third inequality evaluates the perturbation to obtain robust stability for the closed-loop system (c). It is shown that the inequalities hold for the obtained controller K SF , as the γ-value has already revealed. Even more, control eort could become stronger penalized and plant perturbation can be further increased without violating the requirements on stability and performance, as given by σ(T ) < 1. In any case, the inequalities for nominal performance and robust stability have to be fullled to obtain robust performance. The plot of Figure 6.5 provides the singular values for selected cost functions and evaluates its Euclidean vector norm (6.12). The peak gain obtained is bellow the 0 dB line. This means that robust performance and stability is guaranteed. The distance between peak gain and 0 dB line corresponds to the robust stability margin by means of peak magnitude γ.

γ < 1 Figure 6.5 Frequency responses of nominal generalized plant with state feedback controller K SF : evaluating for robust performance and stability with respect to the Euclidean vector norm.

Joint shaping functions

Introducing more complex uncertainty descriptions or extending the above control design, considering an entire high-pressure network with multiple interconnected pumps, will cause the dimension of the resulting optimization problem to increase. With increased complexity, the task of nding a stabilizing controller becomes time consuming. Thus, this section aims to reduce the dimension of the optimization problem (6.5) by concatenating the generalized plant (6.4) for k of its m input channels. Where k denotes the number of selected exogenous inputs that are subject to be merged. This yields again a stacked notation of the largest singular values (6.13) whereas the Euclidean vector norm (6.11) is used to evaluate for peak value γ. This stacked notation is adopted from the mixed sensitivity approach in [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. Dening a generalized plant, composed by a weighting function matrix W , a shaping function matrix V and a system transfer function matrix U , such as T := W U V and substituting this in the stacked notation (6.13), enables to aggregate shaping functions by means of the largest singular values Although, this procedure reduces the dimension of the resulting optimization problem, it may lead to a more conservative control design. For example, merging the input vector w ∈ R m of size m = 2 with corresponding transfer functions T 1 and T 2 . The error when using a joint shaping function, due to the Euclidean vector norm (6.15), becomes maximally

σ (T (jω)) = σ   T 1 : T k   = k i=1 |T i | 2 ≤ γ,
σ   W U V 1 : W U V k   = k i=1 |W U V i | 2 = |W U | k i=1 |V i | 2 = σ W U Ṽ .

Reduced generalized plant

Combining the additively interconnected input signals r and n with respect to (6.14) results, according to denition (6.15), in The reduced interconnection system, regarding this simplication, is illustrated in the block diagram of Figure 6.7. The exogenous inputs were concatenated to w = [rn w dp w ] This provides a standard conguration with reduced interconnection function matrices 6.18) separating the controller block K. Applying lower linear fractional transformation for (6.18), see Section 5.2, results in the reduced generalized plant 6.19) that denes the reduced cost functions for H ∞ controller synthesis. The rst column of (6.19) corresponds to the joint reference and noise signal rn(t), the second to the joint disturbance and perturbation signal dp(t), whereas the rst row corresponds to the control error e(t) and the second to the control value u(t). (6.20) that is to nd a stabilizing controller K, such as T ∞ < 1. Considering the design with joint shaping functions, a reduced optimization problem of dimension n × (m -k) has to be solved, that requires to minimize the H ∞ -norm

σ (W rn ) = |W r | 2 + |W n | 2 ,
P 11 =   W e IW r -W e G s G n W d -W e IW n 0 0 0   , P 12 =   -W e G s G n G a IW u   , P 21 = IW r -G s G n W d -IW n , P 22 = [ -G s G n G a ] , (
T = W e SW r,n -W e G s G n SW d,p W u K SW r,n -W u Tu W d,p , (
W e SW rn W e G s G n SW dp W u K SW rn W u Tu W dp ∞ < 1 .
(6.21) of a reduced cost function matrix (6.19) with dimension 2×2. Ultimately, solving this considering the Riccati method [START_REF] Doyle | Statespace solutions to standard H 2 and H ∞ control problems[END_REF], results in a state feedback controller of order 9 and a peak value of γ = 0.6167. Even though, using joint shaping functions result in a more conservative control design, it reduces the optimization problem and yields a reduced order controller.

Comparing the state feedback controllers

The frequency response of Figure 6.8 (a) compares the dynamic range for the state feedback controllers K SF and KSF , derived from the full generalized plant (6.4) and from the reduced generalized plant (6.19). Again, the simulation studies in Section 6.3 will validate the derived structured and unstructured controllers, using the detailed non-linear simulation models from Chapter 4. The resulting controller from reduced control design is found to recover the frequency response of the full control design for the frequency range of interest (ω ∈ [10 -1 , 10 4 ] rad/s).

Interconnecting this controller to the perturbed plant (5.116), which contains all parametric uncertainties, enables it to evaluate the control requirements of Section 5.3. On the other hand, the step response of Figure 6.8 (b) compares the disturbance rejection for both controllers. Both closed-loop systems perform equivalent, considering a unit step input on d(t). Consequently, the eects for the perturbed plant remains unchanged, when applying joint shaping functions for control design. Joint shaping functions allow for reducing the dimension of the optimization problem, that reduces the computational eort to nd a feasible controller. It further reduces the order of the generalized plant, before synthesizing a controller. The obtained state feedback controller from a reduced generalized plant will then be of lower order. For the presented use case, reducing the full generalized plant, by means of joint shaping functions, gives a valuable approximation KSF of the initial design K SF . Only 75% of the initial calculation time is needed to solve the optimization problem T ∞ < γ instead of T ∞ < γ.

Nonetheless, robust performance and stability have been obtained for both control designs, as the singular values of the cost functions reveal. The plots of Figure 6.9 evaluate the cost functions derived from the reduced generalized plant by means of performance weight (a), control eort weight (b) and robustness margin (c). Robust disturbance rejection is expected, since the H ∞norm from disturbance d w (t) to control error e(t) and from perturbation p w (t) to measurement signal y(t) remain below 1. Also an acceptable control eort is achieved, since the H ∞ -norm from d w (t) to control signal u(t) remains below 1. The closed-loop system dynamics remains unchanged, applying the reduced controller KSF instead of the initial controller K SF . The frequency responses of Figure 6.10 compare the dynamic range of the closed-loop system for disturbance rejection (a) and for the required control eort (b). A desired damping of about -20 dB for d(t) → y(t) has been obtained with limited control eort, as required from specications in Section 5.3. This relates to the observed overshoot of < 10%, when applying a unit step on the disturbance input. The evaluation of the closed-loop system, with respect to the control requirements, is summarized in Table 6.1. A minor degradation is found for the reduced controller derived by means of joint shaping functions. The obtained state feedback controllers are not well adapted for industrial application, because of the increased complexity for implementation. In order to cope with industrial requirements, a more standard controller structure is desired. In this perspective, the structured H ∞ synthesis [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF], as implemented in Matlab, allows for optimizing a robust PI controller in parallel form (5.69). This results in a control law, which is straight forward to implement into industrial applications. It features a structure with only 2 independent parameters subject for optimization.

Considering the reduced generalized plant (6.19), yields the PI controller

K P I (s) = K P + K I 1 s (6.22)
with proportional gain K P = ### and integral gain K I = ###. However, the reduced degree of freedom, for H ∞ controller synthesis, limits the PI structure to perform competitive to a state feedback controller.

Comparing PI controller and state feedback controller

The frequency response of Figure 6.11 (a) compares the initial state feedback controller K SF with the robust PI controller K P I . This structured feedback controller is incapable to recover the dynamic range of the state feedback controller in the frequency range of interest. As a consequence, the perturbed closed-loop systems of both controllers features dierent performances. The step response of Figure 6.11 (b) compares the disturbance rejection. Considering the PI controller, any unit step on disturbance d(t) will aect the system pressure with a maximal 14.9% overshoot. It will reject the disturbance within a settling time of τ s < 0.2 seconds. Any disturbance is attenuated with zero steady-state error.

A typical PI behaviour has been observed with oscillation amplitudes below a 2% error range. Both, overshoot and settling time, are deteriorated, where the desired control requirement for overshoot is not met. The evaluation of the closed-loop system with respect to the control objectives is summarized in Table 6.2, see page 214. The limited degree of freedom for a robust PI controller, derived by the structured H ∞ synthesis, cause major degradations. A peak value of γ = 1.5610 has been found and consequently, the desired robust performance is not achieved (since γ > 1). This basically conrms the observed overshoot of > 10%. The closed-loop dynamics is remarkably aected, when applying the PI controller K P I instead of the initial state feedback controller K SF . The singular values of Figure 6.12 compare the dynamic range of the closed-loop systems for disturbance rejection (a) and for requirements on control eort (b). The desired damping of -20 dB for d(t) → y(t) is not guaranteed, considering the PI controller. Disturbance rejection with less than -12 dB relates to the observed overshoot about 15%, when applying a unit step on the disturbance channel. Also a peaking for the control eort and a reduced actuator bandwidth is observed. Thus is still within the actuator limits, but may cause actuator deterioration.

Introducing a low-pass lter to the PI controller, as an additional tuning parameter, allows for improving performance and robustness. Such an extended PI controller KP I is expected to recover the behaviour of a state feedback controller with improved precision.

PI control introducing low-pass lter

The structured H ∞ synthesis has again been used for optimizing a robust PI controller with lowpass lter (5.70). Including a rst order low-pass lter, provides an additional tunable parameter, the time constant τ f . This extended PI controller is subject for recovering the dynamic range of the state feedback controller with reduced complexity. It features a structure with 3 independent parameters subject for optimization. Considering the reduced generalized plant (6.19), yields the extended PI controller (6.23) with K P = ###, K I = ### and low-pass lter, dened by a time constant of τ f = ### seconds. This corresponds to the lead compensator

KP I (s) = K P + K I 1 s 1 1 + τ f s
KP I (s) = ###s + ### ###s 2 + s .
( 6.24) This lead compensator design, as applied in [START_REF] Salloum | Robust lead compensator design for an electromechanical actuator based on H ∞ theory[END_REF], will improve the control performance. The additional degree of freedom for H ∞ controller synthesis enables the extended PI controller to perform competitive to a state feedback controller.

Comparing extended PI controller and state feedback controller

The frequency response of Figure 6.13 (a) compares the dynamic range for the initial state feedback controller K SF with the extended PI controller KP I . The structured feedback controller approximates the dynamic range of the state feedback controller in the frequency range of interest (ω ∈ [10 -1 , 10 4 ] rad/s). Diverging behaviour is found for the high-frequency range, where the closed-loop system is already suciently attenuated.

As a consequence, the perturbed closed-loop systems of both controllers feature equivalent performances. The step response of Figure 6.13 (b) compares the disturbance rejection. Any unit step on disturbance d(t) to will aect the system pressure with a maximal 9.6% overshoot, considering the extended PI controller. Thus will reject the disturbance within a settling time τ s < 0.2 seconds. Any disturbance is attenuated with zero steady-state error. Both overshoot and settling time are remarkably improved by introducing the low-pass lter. All desired control requirements from Section 5.3 are met, since a peak value of γ = 0.6354 has been obtained. The evaluation of the closed-loop system with respect to the control objectives is found in Table 6.2. Applying the extended PI controller KP I , results in a similar closed-loop dynamics as with the initial state feedback controller K SF . The singular values of Figure 6.14 compare the dynamic range of the closed-loop systems for disturbance rejection (a) and for requirements on control eort (b). A damping of -20 dB is almost obtained, considering the extended PI controller. That allows for sucient disturbance rejection, when applying a unit step on the disturbance input. Also the control eort remains unchanged. The introduced low-pass lter enables a robust PI controller to approximate the state feedback controller with reasonable accuracy. It obtains almost identical stability and robustness margins. It has been shown that the state feedback controller can be approximated with an extended PI controller. On the other hand, it has been seen that the resulting controller obtains a large robust performance margin, since the suboptimal controller synthesis considers a predened plant perturbation with respect to the expected uncertainty range. The subsequent control design will nd a quasi-optimal controller with respect to the maximal plant perturbation allowed, while adopting the structured H ∞ controller synthesis [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF] to obtain a peak value γ → 1.

Optimal robust control design

The suboptimal H ∞ controller synthesis, as already applied in the above control design, aims to obtain a controller K(s) by minimizing the H ∞ -norm for the generalized plant T (jω). This is described by the suboptimal optimization problem (6.20), where robust stability is obtained for a peak value of γ < 1. The generalized plant includes the control requirements from Section 5.3 in terms of weighting functions and considers the expected uncertainties ∆, by describing the perturbed plant with respect to a suitable uncertainty description (see Section 5.4). The subsequent investigation aims at an optimal robust control design, which provides stability and robust performance for a maximal uncertainty range. This requires to state an extended suboptimal optimization problem maximize ∆ minimize K T (jω) ∞ : T (jω) ∞ < γ . (6.25) Sensitivity analysis by means of µ is a powerful tool to analyse robust performance for a closed-loop systems with known controller [START_REF] Gu | Robust Control Design with MATLAB[END_REF]. The evaluation of a generalized plant T (jω) using the robust performance margin µ ∆ (T (jω)) depends in the uncertainty description, as denoted by subscript ∆, see Section 5.2. This can be applied for uncertainties of any type, taking real parametric and complex uncertainties into account. The robust performance margin µ ∆ (T (jω)) is related to the spectral radius ρ(T (jω)) and the largest singular value σ(T (jω)), such as ρ(T (jω)) ≤ µ ∆ (T (jω)) ≤ σ(T (jω)) [START_REF] Gu | Robust Control Design with MATLAB[END_REF]. To investigate robust performance requires to introduce the standard ∆=M=K conguration of Figure 6.15, see [START_REF] Elisante | Robust controller synthesis for multivariable nonlinear systems with unmeasured disturbances[END_REF]. A transfer function matrix T = F ( P, K) is thereby obtained, including the perturbed plant P = F u (M, ∆). Assumption 6.1. Considering a complex unstructured uncertainty and if ∆ is a full matrix yields ρ(T (jω)) = µ ∆ (T (jω)) = σ(T (jω)). In this special case, it is appropriate to evaluate robust performance by means of the largest singular values, as stated in [START_REF] Gu | Robust Control Design with MATLAB[END_REF]. Robust performance is then given by the peak value γ ≤ 1 that is evaluated by applying the H ∞ -norm for T (jω) ∞ < γ.

The objective of nding an optimal controller that provides stability and robust performance with a maximal robust performance margin is denoted as µ-synthesis [Doyle, 1987] . This problem is solved, following an iterative procedure that aims to minimize the µ-value. The D-K iteration, as presented in [START_REF] Packard | The complex structured singular value[END_REF], adjusts a scaling matrix D for a given controller K and alternately solves the scaled H ∞ optimization problem minimize K * DT (jω)D -1 ∞ < γ , (6.26) for a stabilizing controller K * . Adjusting the matrix D(jω) for each frequency ω within the range of interest, corresponds to a convex optimization problem. The adjusted matrix is then approximated by a minimum-phase transfer function D(s). This procedure will successively atten the µ-values for a frequency range and thus, maximizing the robust performance margin. On the other hand, the µ-K iteration, proposed in [START_REF] Lin | µ-k iteration: A new algorithm for µ-synthesis[END_REF], adjusts the optimization problem at each iteration, using the normalized robust performance margin μ. This yields minimize K * μ( P, K) T (jω) ∞ < γ (6.27) to nd a controller K * that will successively reduce the peak-value of µ. However, the order of D and μ will successively increase, hence both approaches result in a high-order state-space controller, attening the µ-value for a frequency range of interest.

Quasi optimal robust PI controller Despite the D-K iteration or µ-K iteration, which lead to a high-order controller, this section proposes a quasi-optimal structured H ∞ controller synthesis, by introducing ∆-K iteration. This requires rst to synthesize a structured controller, that minimizes the peak value γ for a xed unstructured uncertainty ∆ and second to adjust ∆, minimizing |1 -σ(T (jω))| for a xed controller K. Then for the subsequent iteration, the scaled ∆ is approximated with a minimum-phase transfer function.

Since the permissible network subsection size related to a high-pressure pump is undened, a robust control design is required to nd a quasi-optimal controller that considers a maximal uncertainty range. The investigated use case desires stability and robust performance with respect to an uncertain uid volume V δ , which corresponds to the network subsection size. The optimization problem (6.25) is thereby rened in the form minimize (6.28) considering the parametric uncertainty V δ , where ε introduces an acceptable tolerance for γ → 1. The controller K * P I (s) as well as the uid volume V δ dene tunable parameters for the structured controller synthesis [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF].

K * P I T * (jω) ∞ < γ , minimize V δ {I -µ ∆ (T * (jω))} < ε ,
Referring to the parameter variations analysed in Section 5.4, an increased uid volume V δ will move the transfer function pole of the perturbed plant (5.117) to lower frequencies. As a consequence, the transfer function pole, of the corresponding unstructured uncertainty description (5.133), becomes equally moved. This allows for adjusting the uncertainty model to achieve a maximal uid volume, while remaining the nominal plant unchanged.

For example, the plot in Figure 6.16 (a) presents the eect on the peak value γ, when varying the uid volume V δ (Step 1) and iteratively solving the H ∞ optimization problem (Step 2), according to (6.28). On the other hand, the plot in Figure 6.16 (b) shows the adjusted uncertainty description, with its transfer function pole moved to a lower frequency, where an optimized controller has been found with increased network subsection size for γ ≈ 1.

Approximating the plant perturbation with a stable, minimum-phase transfer function 6.29) of amplitude M and bandwidth ω B , gives an unstructured complex uncertainty model (see Section 5.4). In this case, the uncertainty ∆ can be adjusted, using the parameters M and ω B to obtain a peak value of γ ≈ 1. Considering a constant amplitude M , requires to move the pole p = ω B towards lower frequencies, until γ → 1 is achieved. This yields the optimization problem minimize 6.30) considering the unstructured uncertainty model W p (s). In order to cope with the rened optimization problem (6.30), an iterative procedure is implemented, in terms of Assumption 6.1 and considering the peak value γ as a robust performance margin. The robust performance margin represents a worst case gain to the largest singular value. It is proposed to adjust the transfer function W p , in terms of the robust performance margin, to synthesize an optimized structured PI controller K * P I that obtains γ ≈ 1. Considering the inverse additive uncertainty W ia and moving its pole p = ω B with respect to ω B × γ, will reduce the available robust performance margin by maximal -20 dB per decade. Recalling the Euclidean vector norm, a robust performance margin of maximal √ 2 = 3 dB remains and the maximal singular value of the cost functions T * will then attain γ → 1.

W p (s) = s/M + ω B s + ω B A , (
K * P I T * (jω) ∞ < γ , minimize ω B {1 -σ(T * (jω))} < ε , (

Procedure for ∆-K iteration

This ∆-K iteration for transfer function adjustment and structured H ∞ controller synthesis follows a two step procedure to realize a quasi-optimal control design.

Initialization: obtain a rst controller K (0) P I for an initial perturbed plant P(0) by solving the optimization problem minimize 6.31) and applying the H ∞ -norm to calculate the corresponding peak value .32) First step: update the perturbed plant

K (0) P I F P(0) , K P I ∞ (
γ (0) = F P(0) , K (0) P I ∞ . ( 6
P(i) = F u M, ∆ (i) p (6.33) for ∆ (i) p = W (i)
p ∆ by adjusting the shaping function 6.34) when moving its bandwidth

W (i) p (s) = s/M + ω (i) B s + ω (i) B A , (
ω (i) B = ω (i-1) B × γ (i-1) (6.35)
with respect to the previously obtained peak value γ (i-1) .

Second step: obtain an enhanced controller K (i) P I for the updated perturbed plant P(i) by solving the optimization problem minimize K (i) P I F P(i) , K P I ∞ (6.36) and applying the H ∞ -norm to calculate the corresponding peak value .37) Terminus: whenever γ (i) ≈ 1 with desired accuracy ε is reached, such as

γ (i) = F P(i) , K (i) P I ∞ . ( 6
1 -σ F ( P(i) , K (i)
P I ) < ε (6.38) a quasi-optimal PI controller is found (K * P I = K (i) P I ) that guarantees stability and robust performance for an enhanced uncertainty range ∆ (i) p . Otherwise repeat step 1 -2.

Applying the ∆-K iteration to the generalized plant (6.19), the uncertain uid volume V δ will be increased to the largest volume possible, when adjusting the inverse additive uncertainty W ia by means of the peak value γ. Repeating this procedure will iteratively cause γ → 1 with a fast rate of convergence, see Table 6.3. Table 6.3: Optimization for uncertain parameter V δ : nding a maximal network subsection size for which γ-value remains below 1.

Iter. The quasi-optimal extended PI controller 6.39) with K P = ###, K I = ### and τ f = ### seconds, respectively, the quasi-optimal lead controller 6.40) has been found. The singular values of Figure 6.17 show exemplarily the adjusted transfer function W * p , where the peak value of the overall cost functions for nominal performance and robust stabil-

K P K I τ f γ-value V δ ×10 -5 1 ### ### ### 0.
K * P I (s) = K P + K I 1 s 1 1 + τ f s (
K * P I (s) = ###s + ### ###s 2 + s (
ity (|W e G s G n SW d | + |G n SW * p |) reaches γ ≈ 1.
Compare this to the initial cost functions of Figure 6.5, obtained before optimization. γ = 1 Figure 6.17 Singular values of the nominal generalized plant with quasi-optimal robust PI controller K * P I : evaluating for robust performance and stability with respect to the Euclidean vector norm.

Comparing the quasi-optimal PI controller to the state feedback controller

The frequency response of Figure 6.18 (a) compares the initial extended PI controller KP I to the optimized PI controller K * P I . For the open-loop system, an increased phase margin by reduced cross-over frequency is expected. This is conrmed when investigating the step responses given in Figure 6.18 (b). The settling time has deteriorated to τ s = 0.1538, while the overshoot has been slightly decreased. Nevertheless, the control objectives of Section 5.3 are still met. Table 6.4 summarizes the obtained closed-loop performance. This singular values, as shown in the plots of Figure 6.19, reach the bounds for performance (a) and control eort (b) by increasing the robustness for perturbations (c), when implementing the optimized controller. The bounds on robustness is still not fully exhausted. This gives a margin for stability. However, performance or control eort requires relaxation for further enhancing the robustness.

Finally, the resulting perturbation weight comes close to the initially dened upper bound, given by the control requirements, see the frequency response of Figure 6.20. A maximal uid volume of about 95 cm 3 has been found by applying the proposed ∆-K iteration. This corresponds to an improvement of 61% in contrast to the initial uid volume of 59 cm 3 . The obtained extended and optimized PI controller enables robust performance and stability for a network subsection of increased size. It corresponds to a piping section of 120 m length. It is recognized that a robust control design, considering all possible parameter variations, is limited to guarantee stability and performance for any plant conguration. But the varying uid volume V δ and the operating point x δ can be measured to further reduce the uncertainty range. This would then require to design a robust gain-scheduling that improves the overall robustness.

Simulation studies

This section veries the robust control design, when comparing the expected closed-loop performance given from the linearised plant model (5.116) with the achieved performance, by applying the control law to the non-linear simulation model. The simulation model is obtained, using the graph-based modelling methodology, as introduced in Section 3.3. This model is employed for control design verication. On the other hand, the linearised plant model, derived in Section 5.3 for control design, takes the expected parameter variations and non-linear behaviour into account in terms of a plant perturbation.

Low-level control design verication

The control design evaluation from the above section requires further verications. The H ∞ controller synthesis, to derive unstructured state feedback controllers and structured PI controllers, originate from a simplied and linearised plant. Even when taking all expected uncertainties into account, this perturbed plant for control design (5.116) remains a rst order approximation, describing the pressure state of a network subsection by means of the lumped parameter model (5.23 -5.23).

The linearisation, applied for control design, employs a disturbed input channel u d (t) = ∆u(t)+ d(t) (see Figure 5.13 in Section 5.3). This describes the perturbed plant for any deviation around operating points (u 0 , x 0 and d 0 ), considering the dynamics from the control variable ∆u(t) to pressure state ∆x(t) as well as from the exogenous disturbance ∆d(t) to the pressure state ∆x(t). The exogenous disturbance is thereby scaled with respect to maximal parameter values (H max , x max ), taking the maximal disturbance range into account for control design.

However, comparing the obtained performance and stability from control design (see Section 5.3) with numerical simulations from a non-linear simulation model, requires to apply a nominal disturbance step, while initializing both models to an operating point at equilibrium. Thus, the control design verication in this section considers nominal parameter values (H nom , x nom ) applied on the exogenous disturbance ∆d(t), as depicted in Figure 6.21. This nominal disturbance step requires to congure the uid ow resistance

H nom = S nom ζ 2 nom = ### (6.41)
with a nominal nozzle surface S nom = 0.0625 mm 2 , the nozzle discharge coecient ζ = ### and a nominal uid density nom = 1.0705 kg/dm 3 , as expected for an operating pressure of x nom = 200.5 MPa, and the disturbance (5.54) becomes .42) This resulting disturbance of nominal amplitude is then applied for both models, considering control design and numerical simulations.

d(t) = H nom √ x nom ∆d(t) . ( 6
H δ 2 x δ Δx Δd K δ V δ u d Δu + H nom x nom ∫ + _ G p + d Figure 6
.21 Approximated plant model: block diagram of the simplied approximated model in state space form with exogenous disturbance of nominal amplitude.

All parameter variations within the disturbed plant G p , as denoted by the subscript δ, follow the previously reported characteristics. The adjusted parameter variations, used for control design verication, are given in Table 6.5. They can be compared with the initial parametrization of Table 5.3, used for control design.

Simulation model

To verify the derived performance and stability from controller synthesis with the linearised plant, the closed-loop system is numerically simulated for its nominal values, considering a detailed non-linear simulation model. This simulation model is based on the graph-based modelling The control variable u(t) and process variable y(t) = p P (t) are simulated, applying the derived unstructured state feedback controllers and structured PI controllers. These simulations provide valuable results to verify the control design of Section 6.2.

The corresponding graph network is depicted in the diagram of Figure 6.23. Each network section represents a piping of about 1 m length, which is divided into 10 segments for numerical simulations. Table 6.6 provides the network parametrization, derived from measurements (see Section 2.2). The control design verication is eventually realized for dierent operating pressures r = {100, 200, 300, 400} MPa, using the conguration of Table 6.7. Basically, both cutting heads were congured with the same nozzle of nominal inner diameter. If only one nozzle is open, the system dynamics is expected to correspond to the nominal plant. The second nozzle is then opened to apply an additional disturbance of nominal amplitude. This will aect the system dynamics, to move apart from nominal behaviour. Consequently, it is not possible to hold the system about nominal parameters, while instantaneously applying a disturbance. The linearised plant, used for control design, represents the transient behaviour around its operating point. Thus, the process variable y from numerical simulation is scaled with respect to the operating point y nom = r to obtain ∆y = (y -y nom )R r . In the same manner, the control variable u from simulations is scaled to obtain ∆u = u -u nom , with a nominal control value u nom = {0.439, 0.609, 0.744, 0.839}, depending on the operating pressure r. This allows for comparing the results from control design with the numerical simulations. The applied control value u has been further scaled by means of camming, see Section 2.3. The scaled control value u g = u/R u corresponds to the resulting pump uid ow of a high-pressure pump.

Evaluating structured and unstructured feedback controllers

The suboptimal PI controller K P I , the quasi-optimal PI controller with low-pass ltering K * P I and the reduced suboptimal state feedback controller KSF will be implemented to the above non-linear simulation model for closed-loop simulations, when applying a disturbance step of nominal value. Cutting head 1 will thereby remain open for all time, while alternately switching cutting head 2 between open and close position. A set of nominal step responses results from the time-varying parameters, such as the uid volume in a pumping chamber that varies due to piston displacement and the bulk modulus that varies with respect to the operating point. This set is then compared to the upper and lower bounds, as expected from H ∞ controller synthesis when assuming plant perturbations. The results summarized in Table 6.8, compare the expected settling times and overshoots from control design (see Section 6.2) with the obtained values from numerical simulations. In general, an excessive settling time and overshoot is observed for simulations at low pressures. Extended and quasi-optimal PI controller K * PI An excellent match is obtained, applying the extended quasi-optimal PI controller (6.39) on the detailed simulation model. The plots of Figure Reduced suboptimal state feedback controller KSF Also an excellent match is obtained, applying the reduced state feedback controller, derived from (6.20), on the detailed simulation model. The plots of Figure 6.32 -6.35 compare the expected nominal step response from the controller synthesis with the linearised plant and the set of quasinominal responses from numerical simulations at 100, 200, 300 and 400 MPa operating pressures. It shows the scaled pressure dierences ∆y around the operating point and the corresponding control signals ∆u. The observed variations stay within expected upper and lower bounds of the nominal plant derived from control design. It is further recognized that any disturbance is rejected with expected settling time and overshoot. For the 100 and 200 MPa case, the detailed simulations reveal degradation of the settling time, which results into an increased overshoot. The decreased performance is tremendous for the 100 MPa case. The nominal response remains within expected bounds. The 400 MPa operating point holds a bulk modulus of 3.96 GPa, whereas 2.55 GPa results for 100 MPa. The overall volume for the detailed simulation model varies between 40-80 cm 3 and the ow resistance H = {0, ###, ###} 10 ### changes with respect to the cutting head switching state. Regarding the above simulation results considering these parameter variations, robust stability and nominal performance is obtained, even though the controller has been derived using a strongly simplied model. The next section will present experimental results derived from the test bench with a high-pressure pump in closed-loop control, considering the initial robust PI controller without low-pass ltering. Figure 6.32 Step responses of optimized state feedback controller at 100 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model. 6.45) and the closed loop gain remains unchanged for every operating pressure that the obtained control law becomes properly applied to each pumping unit. Typical PI behaviour can be observed from measurements, where ab excessive characteristics occurs, such as an increased overshoot and settling time. This relates to all closed-loop experiments presented in this section. Eventually reproducing the faulty controller scaling by means of simulation, results in step responses, which correspond to the measurement data. At least, these experiments allow the validation of dierent high-pressure network models in closed-loop, which have been used to verify the robust control designs. Due to a defect of the high-pressure pump, it has not been possible to repeat the experiments with correct controller scaling, hence the investigations for robust stability and robust performance remain limited to simulation studies.

Q P n (t) ∼ F (u(t)ψ(p)) (

PI controller application and validation (1 pump conguration)

This experimental study applies the initial robust PI controller (6.22), as derived in Section 6.2, to a single high-pressure pump. Deriving a robust controller, the expected disturbance of maximum amplitude and the entire uncertainty range has been taken into account, while applying the structured H ∞ synthesis. Considering the faulty scaling ψ f , used for implementation on the high-pressure pump, and recovering the controller parameters, yields 6.46) which has been eectively applied to the high-pressure pump. It is dierent from the desired robust PI controller K P I (s) and consequently does not achieve expected performance.

K f P I (s) = K P I (s) ψ ψ f = ### + ### 1 s , (
To verify performance and stability of the implemented PI controller (6.46), the high-pressure test bench has been congured to a symmetric and asymmetric network topology. For both topologies, (c) Symmetric network and (d) Asymmetric network, a single high-pressure pump is supplying two cutting heads. The cutting heads switch alternately open and close. The symmetric topology considers nozzles of dierent inner diameters, allowing for investigating the eects of asymmetric cutting head congurations. The asymmetric topology takes nozzles of identical diameters into account, hence the eect of dierent piping sections lengths can be observed. This aims at disturbance rejection due to switching cutting heads, while holding a constant reference pressure of either 200 or 350 MPa. The zoomed-in plots of Figure 6.39 compare measurements (p P (t), u(t)) and simulations (p P (t), û(t)) for each switching state of experiment E-4. The transitions (a) and (b) present the closing and opening of cutting head 1, where the transitions (c) and (d) show the closing and opening of cutting head 2. Investigating transition (b) and (c), the pump reaches upper and lower saturation, respectively. For the rst case, cutting head 1 stays open and cutting head 2 switches in open position, causing an increased uid ow consumption. The high-pressure pump is not capable for instantaneously compensating the resulting pressure loss. For the second case, cutting head 1 stays close and cutting head 2 switches in close position, causing any uid ow consumption to disappear. The pump is not able to diminish the pressure overshoot, since only positive uid ows can be induced. On the other hand, the zoomed-in plots of Figure 6.40 compare measurements and simulations for each switching state of experiment E-9, where similar behaviour has been observed.

An excellent match is observed between measurements and simulations for the investigated transient behaviour. The possible variations due to parameter uncertainties are within expected range. Actuator saturations, limiting the realisable pump rate, are precisely estimated by simulation. Solely the measured amplitude of pressure overshoot has been slightly increased, as expected by simulations for the 350 MPa operating pressure. The maximally observed settling times and overshoots are given in Table 6.10. Overall, the performance in simulation and measurement cant achieve the requirements from control design. This is due to the use of an initial PI controller in combination of a failure, when implementing the controller to the high-pressure pump. The nal extended PI controller is expected to enhance the closed-loop performance. Nevertheless, the high-pressure network models provide accurate simulations, considering the high-pressure pumps in closed-loop operation. 6.4.2 Quasi optimal robust PI controller application (1 pump conguration)

This simulation study applies the extended robust PI controller (6.39), as derived in Section 6.2, to a single high-pressure pump. Deriving a quasi-optimal robust controller, the expected disturbance of maximum amplitude has been taken into account, while maximizing the uncertainty range by applying the proposed ∆-K iteration. To verify performance and stability of the extended PI controller K * P I , simulations have been derived with respect to the symmetric and asymmetric network topology (as already introduced in Table 6.9). The 2 extended measurements (E-4 and E-7) consider the topology (c) Symmetric network, where both cutting heads have been asymmetrically congured. The other 2 extended measurements (E-9 and E-11) consider the topology (d) Asymmetric network, while both cutting heads have been symmetrically congured. These experiments aim at disturbance rejection, while tracking a constant reference pressure of either 200 or 350 MPa. An overview, considering the 200 MPa operating pressure, is provided below. Plots extending these results are given in Appendix B.2.

Contour cutting: asymmetric and symmetric cutting heads

Simulations have been obtained, when applying disturbance steps by switching the cutting heads to open and close position. This alternately switching of two cutting heads results again in 4 possible pump rates. Consequently, 8 transitions have been obtained with respect to the cutting head switching. Simulation results are shown in the plots of Figures 6.41 The zoomed-in plots of Figures 6.43 and 6.44 show the simulation results for each switching state of experiment E-4 and E-7, respectively. The transitions (a) and (b) present the closing and opening of cutting head 1, where the transitions (c) and (d) show the closing and opening of cutting head 2.

As expected, the quasi-optimal robust PI controller K * P I performs superior in contrast to the initially implemented PI controller K f P I . The possible variations, due to parameter uncertainties, are within expected range. The maximally observed settling times and overshoots are given in Table 6.11. Applying the extended PI controller K * P I , the observed closed-loop performance achieve the desired objectives for control design. Table 6.11: Achieved performance by means of simulations for dierent network topologies: providing closed-loop performance from simulations, using detailed simulation model with quasioptimal robust PI controller. Improving performance and uncertainty range (gain-scheduling)

Dierent approaches could become interesting to further increase the permissible subsection size.

Research on gain-scheduling is provided in [START_REF] Rugh | Research on gain scheduling[END_REF], where an augmented plant is considered, using linear fractional transformation to represent time-varying parameters for control design. For that, robust gain-scheduling has been employed to obtain controller gains with respect to scheduling variables. As a consequence, uncertain parameters have been considered as measurable. The resulting controller has to deal with a reduced overall uncertainty range. This enables it to guarantee performance and stability for large parameter variations. A rst critical uncertainty is the varying pumping chamber volume, which interacts with the available network subsection volume. The varying pumping chamber volume follows a periodic, non-linear function that is related to the piston displacement. Due to its time-varying property, the pumping chamber volume is undesired for applying robust H ∞ controller synthesis, which interfere with stability requirements. Since the piston displacement is well known, it is a rst candidate as a scheduling variable. Scheduling the controller gains with respect to the resulting pumping chamber volume, allows for increasing the available network subsection size. It extends the robust control design to schedule the PI gains with respect to the measured piston displacement. This allows for eliminating the time-varying uncertainty related to the displacement volume of a pumping chamber. A second uncertainty is the pressure operating point, which becomes a parameter when linearising the plant model used for control design. This parameter is considered as time constant, since the controller is assumed to track steady-state operating pressure around desired operating point. The pressure is measured for low-level control as well as available as a reference signal. The control surfaces of Figure 6.45 present the controller gains with respect to both scheduling variables. While scheduling the integrator gain, a back calculation of the controller output is recommended to realize a smooth pump operation. However, the diculty with the piston displacement consists of the unknown switching states of the check-valves, which interconnect a pumping chamber with the high-pressure network. As long as the switching is hard to predict, the error between estimated and eective chamber can become maximally. This circumstance is unfavourable, as stability is not guaranteed. Nevertheless, assuming perfect synchronization of the pumping units allows for realizing an increased subsection size of about 270 m piping length. Using the reference pressure for control design allows for increasing the permissible subsection size up to 290 m. Combining both scheduling variables leads to a maximal subsection size of 370 m, where resulting closed-loop stability and performance remain unattained.

The simulations of Figure 6.46 show the step response for disturbance rejection when applying robust gain scheduling. In contrast to the initial robust low-level controller, see Table 6.12, this approach shows the ability to reduce the resulting variation for a constant operating pressure. For any case, it would be interesting to evaluate an approach to compensate the varying parameters, such as gain scheduling or linear parameter varying control. Figure 6.46 Step responses of PI gain scheduling controller K * PI with low-pass lter and quasioptimal robust PI design: considering a reduced generalized plant derived from joint shaping functions. This experimental study applies an initial PI controller to both high-pressure pumps available at the test bench. It aims to investigate stability and performance for two distributed pumps, which are coupled over a high-pressure network and disturbed due to cutting heads switching. Pump 1 holds the controller 6.47) whereas pump 2 has by default the controller

K P I1 (s) = ### + ### 1 s , (
K P I2 (s) = ### + ### 1 s .
(6.48)

These control parameters originate from the initial test bench conguration as already used for model validation in Section 4.3. Measurements for the robust PI controller has not been applicable due to a defect of the high-pressure pump. Nevertheless, these results show the eect of running two interconnected pumps with independent decentralized controllers. Measurement data are obtained on the high-pressure test bench with respect to 2 specic network topologies, considering two pumps and two cutting heads. Both cutting heads have been symmetrically congured with identical nozzles. The experiments listed in Table 6.13 have been derived for simulation and experimental validation. Measurement D-2 considers the topology (e) Symmetric distributed, where the piping from each pump to the cutting heads have the same length. Measurement D-5 considers the topology (f) Asymmetric distributed, where the piping from a pump to the cutting heads have dierent lengths. Both experiments interconnect two cutting heads at the same network position.

(t) = [d 1 (t), d 2 (t)],
where d 1 (t) = 0 ∀t. The low-level pressure controllers of both pumps will adjust the pump rate over the control signal u(t) to obtain a desired reference pressure. The reference value r(t) and disturbance d(t) are fed into the extended high-pressure network models for closed-loop simulation. The pump output pressures p P (t) = [p P 1 (t), p P 2 (t)] as well as the pump rates u(t) = [u 1 (t), u 2 (t)] will be measured and compared with simulations (p P (t) and û(t)) to validate the PI controllers. The measurement data have been captured with a sampling rate of 500 Hz.

Applying the graph-based modelling methodology according to the test bench conguration, results in dierent high-pressure network models, as already introduced and experimentally validated in Section 4.3. The zoomed-in plots of Figures 6.49 and 6.50 compare measurements (p P (t), u(t)) and simulations (p P (t), û(t)) for each switching state of experiment D-2 and D-5, respectively. Acceptable match is observed for the investigated transient pressure trends. Increased oscillation is found for the measurements, where the simulated pump rates show discrepancies. The expected variations due to parameter uncertainties is mostly within expected range. Some additional disturbance is observed on the pressure trend due to check-valve switching hysteresis. This eect is not taken into account, when modelling a high-pressure network. This simulation study applies the extended robust PI controller (6.39), as derived in Section 6.2, to both high-pressure pumps. To verify performance and stability of the extended PI controller K * P I , simulations have been derived with respect to two distributed pumps, as already introduced in Table 6.13. However, these simulations employ two identical pumps to the corresponding high-pressure network models. The measurements D-2 considers the topology (e) Symmetric distributed and measurements D-5 considers the topology (d) Asymmetric distributed. These experiments aims at disturbance rejection due to switching cutting heads, while holding a constant reference pressure of 200 MPa.

Contour cutting: symmetric and asymmetric distributed pumps Improving performance and uncertainty range (decentralized control)

A decentralized control design for multiple pumps would require to describe the coupling between network subsections. The H ∞ optimization problem for the proposed distributed high-pressure pumps is then to nd a common controller that is distributed on several pumps which are coupled over a high-pressure network. However, the overall network size, its topology and the amount of available pumps remain unknown. The diagrams of Figure 6.55 compares the investigated robust low-level control design (a), that consider neighbouring subsystems as additional disturbances and a decentralized control design (b), taking the dynamics of neighbouring subsystems into account.

Σ n Σ 1 u y PI (a) Independent control design Σ 2 Σ 1 Σ n u n u 1 u 2 y n y 1 y 2 PI PI PI (b) Decentralized control design Figure 6
.55 Applied concept for independent control design compared to decentralized control: while the independent control design considers other network subsections as exogenous disturbances, the distributed control design takes the dynamic coupling between neighbouring pumps into account.

Since the disturbances due to check-valve malfunctions propagate to the neighbouring network subsections, it becomes necessary for a robust control design to deal with an increased disturbance amplitude. The test bench setup with two distributed pumps encounters stability concerns. It would be interesting to investigate a decentralized control design for controller synthesize. This would address the stability concerns for disturbances which originate from neighbouring pumps due to malfunctions.

Conclusion

In this chapter, the H ∞ controller synthesis for a electrically driven high-pressure pump has been validated. The specied control requirements on performance and control eort have been applied on related output channels, by specifying the corresponding weighting functions. The interconnection system has then been rearranged to separate the controller from the remaining system components, using linear fractional transformation. This provides a generalized plant, that corresponds to the required H ∞ optimization problem, needed for robust control design.

A rst contribution is provided, when introducing joint-shaping functions. This lumps multiple input channels of a generalized plant, which reduces the size of the H ∞ optimization problem with respect to the numbers of merged channels. It reduces the computational eort for controller synthesis and results further in a state feedback controller of reduced order. This approach has been applied for the robust control design of a high-pressure pump. It has been shown, that the reduced controller performs close to the full order design. The resulting dierences are negligibly small.

The derived optimization problem, considering the decentralized pump and its corresponding network subsection, has been solved to obtain a state feedback controller as well as a PI controller, using the available algorithms of MATLAB. The PI controller has then been extended with a low-pass lter, which results in a lag control law. These controllers have been compared by means of simulations, using the detailed network models implemented with the introduced simulation toolbox. It has been recognized, that the PI control design is useful to reduced the controller complexity. This control structure is straight forward for implementation in the high-pressure pump. That is an advantage for industrial use. However, the PI controller is not capable to recover the enhanced performance of a state feedback controller. It barely achieves the desired performance and stability, when limiting the permissible subsection size. On the other hand, a lag compensator introduces an additional pole, which can be placed, such as to recover the dynamics of the state feedback controller. It achieves the desired performance and stability without disclosing the permissible subsection size.

Another contribution aims at a single controller for every decentralized pump, as installed in a high-pressure network of unknown topology. This requires for separating a high-pressure network into subsections, which can be controlled by an assigned high-pressure pump. The subsection size is thereby considered as an uncertain parameter for robust controller synthesis, where the permissible uncertainty range, to guarantee robust stability and to obtain robust performance, has been taken into account when applying the ∆-K iteration. It applies H ∞ controller synthesis, while adjusting the uncertainty range ∆ with respect to minimize the robust performance margin. This procedure allows for designing a controller K that maximizes the permissible subsection size. A signicant increase of 61% has been obtained in contrast to the initial suboptimal design, while guaranteeing robust stability and desired performance.

The obtained PI controller has been evaluated on the test bench by means of measurements. These measurements have been compared to simulation results. Unfortunately, the high-pressure test bench has not been available to verify the nal optimal lag controller, due to a damaged pump. Nevertheless, the lag controller, implemented to the high-pressure network models while considering distributed setups with two pumps and two cutting heads, allows for verifying more complex waterjet facilities. These models in closed-loop control will further serve as use cases, to evaluate distributed average consensus and distributed balancing by means of numerical simulations (see Section 7.3).

Introduction Objectives

The rst part of this thesis introduced the graph-based modelling of high-pressure networks, describing decentralized pumps interconnected with distributed work stations for waterjet machining. Based in this modelling, the second part investigated the robust control design for high-pressure pumps, subject for local disturbance rejection. A robustly controlled pump manages to stabilize a network subsection, whereas the required performance for waterjet machining limits the permissible subsection size. This last chapter provides rst results to cope with the high-level managing of decentralized pumps, which aims to optimize the energy consumption of waterjet facilities.

In fact, the decentralized control of high-pressure pumps provides a solution (see the previous chapter), that is straight forward to implement, as only local information is required. Introducing managed pumps and applying distributed control, will be a trade-o between the amount of information required from neighbouring subsections and the eects of transmission delay, packet loss, etc. This managing requires to synchronize high-pressure pumps without knowledge about their positions in the network. Depending on the network topology and the cutting heads switching states, each pump will reach a dierent pump rate. This causes some of the pumps to deteriorate faster and degrade its ability to respond for disturbance rejection. The synchronization of coupled pumps in a waterjet facility further, causes all pumps to converge towards an average pump rate within acceptable time. This has to be realized without any centralized manager, whereas reliability of communication is not necessarily provided. Hence, the pumps should continue to operate at a suboptimal pump rate and supply the work stations, even though when communication fails.

This chapter provides a short overview of various managing concepts and presents preliminary simulation results, for the managing of distributed pumps by means of dynamic average consensus. The requirements from waterjet machining restrict the availability for many prevalent concepts. Nevertheless, the subsequent investigations present a starting point for ongoing research activities, considering the managing of distributed pumps within future waterjet facilities.

State of the art

The control design problem for weakly coupled decentralized systems can be solved by dividing the global problem into almost independent sub-problems [Bakule, 2008]. The demanding task for decomposition of strongly coupled systems restricts the application of decentralized and distributed control design.

On the other hand, hierarchical approaches are used to manage distributed controllers, while obtaining a globally desired behaviour of the overall system, see [Tatjewski, 2008] and [Scattolini, 2009]. The hierarchical control structure is typically organized in two layers: A low-level layer, that holds the distributed controllers, and a high-level layer, that coordinates the subsystems. This coordination is centralized and comprises an optimization procedure. A global optimization becomes computationally dicult for large-scale systems, thus dierent approaches aim at distribution of the optimization task.

A survey of various managing approaches, with emphasis on distributed algorithms, is provided in Appendix C.1. This survey summarizes control allocation (CA), model predictive control (MPC) and cooperative control, as organized in the diagram of Figure 7.1. It is dedicated to evaluate suitable concepts for the synchronization of distributed pumps in a high-pressure network.

Those approaches assigned to cooperative control, particularly aiming at average consensus and distributed synchronization, will be discussed in this subsequent sequel. Model predictive control (MPC): the managing of distributed controllers comprises an optimization procedure, which is often designed using model predictive control [Mayne, 2014]. It provides the optimal output variables for a multi-variable system, while predicting the system states for a nite horizon. Since it becomes computationally dicult to solve this optimization problem for large-scale systems, the initial problem is often decomposed into decoupled subsystems. In the framework of decentralized model predictive control (DMPC), the decentralized controllers generate their output variables, subject to a local optimization objective [START_REF] Christodes | Distributed model predictive control: A tutorial review and future research directions[END_REF].

On the other hand, non-cooperative and cooperative algorithms are distinguished. Considering non-cooperative MPC, each controller follows its local optimization criteria [START_REF] Farina | Distributed non-cooperative MPC with neighbor-to-neighbor communication[END_REF], where cooperative MPC enables distributed controllers to retrieve global objectives by sharing information, such as state and set point trajectories [START_REF] Trodden | Cooperative distributed MPC of linear systems with coupled constraints[END_REF]. Solving the optimization problem in a decentralized manner, involves to estimate the trajectories of the other subsystems and requires the subsystem states to be updated periodically by means of communication [START_REF] Ferramosca | Cooperative distributed MPC for tracking[END_REF].

Control allocation (CA): the problem of control distribution is often realized by means of control allocation [START_REF] Johansen | Control allocation -a survey[END_REF]. It manages a multi-variable system in a centralized manner. Hierarchical control distribution for over-actuated systems is considered in [START_REF] Singla | A hierarchical control distribution approach for large scale over actuated systems[END_REF]. This aims to assign the overall control eort to the available actuators, with respect to actuator limitations. This is typically solved by means of optimization, taking dierent constraints and objectives into account, such as actuator limitations and energy eciency.

Using the L 2 -norm for optimization, allows for combining all actuators while minimizing the overall control eort [START_REF] Grechi | Comparison between optimal control allocation with mixed quadratic & linear programming techniques[END_REF]. Using the L ∞ -norm, balances the control eorts while minimizing the maximal deection for every actuator [START_REF] Bodson | Control allocation with load balancing[END_REF]. The rst approach results in optimal allocation, where the second approach refers to balancing allocation. Combining both by means of mixed optimization, results in optimal load balancing [START_REF] Frost | Resource balancing control allocation[END_REF].

Cooperative control: cooperative control of distributed systems aims at consensus between multiple autonomous agents. These algorithms are fully distributed, while information is exchanged between neighbouring agents. Convergence to a consensus value depends on the network topology. A nite-time consensus protocol is considered in [START_REF] Liu | Synchronization of multi-agent systems with delayed control input information from neighbors[END_REF]. It takes the control inputs of neighbouring agents into account. This approach improves the control performance and guarantees convergence, independent of the network topology. A decentralized synchronization protocol is discussed in [START_REF] Yuan | Decentralised minimum-time consensus[END_REF], which is sharing state variables between neighbouring agents.

Optimal strategies to solve the average consensus problem are discussed in [START_REF] Delvenne | Optimal strategies in the average consensus problem[END_REF]. It compares the convergence behaviour in terms of network topology and gives strategies for which consensus is guaranteed. A protocol that guarantees consensus in nitetime is presented in [START_REF] Mirali | First-order average consensus for cooperative control problems using novel weighting strategies[END_REF]. Most approaches deploy an iterative scheme, where every agent updates its estimation using local values and the estimation from neighbouring agents. This procedure enables all agents to converge to the global average. An optimal weighting strategy is essential to obtain fast convergence. Concerning this weighting, a self-conguration procedure is discussed in [START_REF] Dung | Distributed design of nitetime average consensus protocols[END_REF]. It is shown that consensus is achieved, even for unknown topologies and varying number of agents. Dynamic average consensus is studied in [START_REF] Zhu | Discrete-time dynamic average consensus[END_REF], where all agents have to agree to the average of a time-varying signal. This work proposes a higher-order average consensus protocol, capable to eliminate steady-state error. An other consensus protocol is considered in [START_REF] Wang | Distributed dynamic average consensus for nonlinear multi-agent systems in the presence of external disturbances over a directed graph[END_REF], where the system states are subject to converge towards a time-varying input signals. The convergence depends on the initial condition of the average consensus algorithm. A method for dynamic averaging with robust initial conditions is given in [START_REF] Montijano | Robust discrete time dynamic average consensus[END_REF].

More sophisticated approaches, considering recongurable control structures, require to nd suitable control laws in an adaptive manner, e.g. see [START_REF] Wen | Decentralized adaptive backstepping stabilization of interconnected systems with dynamic input and output interactions[END_REF] and [Wang et al., 2016a] for backstepping. It takes into account a varying structure of large-scale systems, where subsystems can be added and removed. Assuming a time-varying number of sources and sinks in a system, restrict these approaches to obtain convergence in a useful time.

Contribution

Distributed high-pressure pumps can be seen as an over-actuated waterjet facility. Load balancing by adaptive control allocation would become a possible solution for synchronization. An adaptive procedure is required, since the optimal distribution of the control eort will vary with respect to the cutting heads switching states and the available high-pressure pumps. Such a global optimization requires the knowledge about all pumps involved. This information is not necessarily given for future waterjet facilities. On the other hand, cooperative MPC is prevalent for managing of distributed controllers. This requires adequate local models to track the state trajectories of each agent. Firstly, it is undesirable to provide a model for every waterjet facility.

Secondly, it would require to estimate the cutting head switching states, which aect the dynamic behaviour of a high-pressure network. In these cases, dynamic average consensus might be advantageous, tracking the average control eort of distributed pumps with respect to the time-varying water consumption of various waterjet applications. Consensus can be reached even for unknown communication topologies and varying number of agents [START_REF] Dung | Distributed design of nitetime average consensus protocols[END_REF]. Even if communication to an agent fails, the remaining agents will not converge to a global average, but the facility remains in operation.

This chapter applies dynamic average consensus to manage distributed high-pressure pumps. Each pump corresponds to a pump agent that provides a local estimation of the average pump rate, while exchanging the local estimation with neighbouring pumps. This approach is then extended to a distributed balancing algorithm, taking the high-pressure network into account as an additional means of communication. Each pump converges to a desired average pump rate, by adjusting its local reference pressure. Considering robustly controlled high-pressure pumps at low-level control (see Section 6.2), discrete dynamics will adequately describe the pump agents at high-level managing. However, the algorithms will be rst implemented in continuous time, when assuming perfect means of communication, and the local estimations will be shared between neighbouring agents without considering delays. It further requires a bidirectional communication, using a ring topology, that results in a strongly connected graph and guarantees convergence in nite time. In reality, dierent constraints exists on tracking a signal of arbitrary fast dynamics with zero error, such as limited communication bandwidth, time synchronization, propagation delays, restrictions of the network topology and others. This could be improved, using a priori information about the signal dynamics and network topology.

The chapter is structured as follows: rst, the problem formulation of Section 7.2 will present the objectives for average consensus and derives the need for distributed managing of highpressure pumps, where also the communication channel is specied, using graph description. Second, distributed average consensus is adopted to estimate the average pump rate of interconnected agents. An integrated approach is then proposed in Section 7.3, that enables each high-pressure pump to converge towards an average pump rate. This aims to improve the energy eciency of entire waterjet facilities. Both distributed algorithms will be evaluated in Section 7.4, using a detailed simulation model to represent an extended high-pressure network. This introduction provides rst simulations for load balancing of distributed pumps by means of dynamic average consensus. Improvements will be subject for future investigations.

Problem formulation

The dynamic average consensus problem [START_REF] Kia | Singularly perturbed algorithms for dynamic average consensus[END_REF] aims at agreement of N distributed agents to a changing local reference signal u i (t). It requires to track the time-varying average

ū(t) = 1 N N i=1
u i (t) (7.1) of all reference signals by means of a distributed algorithm. This algorithm depends on the local reference at an agent J i and those from direct neighbours {I j } j∈N i . Such an algorithm, of form ẋi (t) = c i (J i (t), {I j (t)} j∈N i ) , (7.2) assigns an agreement state x i (t) to each agent i that converges asymptotically to the average reference ū, such as lim t→∞ x i (t) = ū ∀ i, if a proper initialization is provided [START_REF] Cherukuri | Initialization-free distributed coordination for economic dispatch under varying loads and generator commitment[END_REF].

Distributed algorithm

Average consensus provides a distributed algorithm to track a static reference signal u i , compliant with (7.2). It updates the local agreement states x i (t) with respect to the dierence to the neighbouring states x j (t), such as [START_REF] Zhu | Discrete-time dynamic average consensus[END_REF] ẋi (t) = -N j=1 a ij x i (t) -x j (t) , (7.3) while initializing the agreement state of every agent i to its reference value x i 0 = u i . For this, a ij ∈ B N ×N denes the interconnection between agents with respect to the communication path. That denes the logical network topology, see also the subsequent section, where B = {0, 1} is chosen to obtain a weight-balanced graph.

First-order average consensus

The dynamic average consensus algorithm [START_REF] Kia | Dynamic average consensus under limited control authority and privacy requirements[END_REF] ṗi (t) = N j=1 a ij x i (t) -x j (t) (7.4a) x i (t) = u i (t) -p i (t) (7.4b) implements a change of variable to the algorithm (7.3) by introducing p i = u i -x i , with initial condition N i=1 p i 0 = 0. This replaces the derivatives on the local reference signal ui (t) and avoids steady-state tracking error due to faulty initialization of the agreement state x i 0 = u i (t = 0). As a consequence, the most basic initialization would be p i 0 = 0.

The reference signal u i (t), subject for tracking, is considered as an external time-varying input. This algorithm can be extended to reduce tracking errors and to improve the convergence rate.

Robust dynamic average consensus

The distributed algorithm presented in [START_REF] Freeman | Stability and convergence properties of dynamic average consensus estimators[END_REF]] extends (7.4a), (7.4b) to introduce robustness for initialization error and loss of agents. It does not require to know the agents positions in the network. Considering the change of variable p i = u i -x i , yields the extended dynamic average consensus algorithm ṗi (t) = α x i (t) -u i (t) + β N j=1 a ij x i (t) -x j (t) + q i (t) qi (t) = α β N j=1 a ij x i (t) -x j (t)

x i (t) = u i (t) -p i (t) , (7.5a) (7.5b) (7.5c) which eliminates steady-state error in nite time by introducing the integral term q i (t). This algorithm allows for tuning the tracking error with parameter β, while α adjusts the rate of convergence.

Taking the switching of work stations into account, yields a time-varying water consumption Q m (t). This requires the more general condition

N n=1 Q n ≥ M m=1
Q m (t) ∀ t . (7.7) The switching cutting heads induce pressure uctuations, which propagate through the highpressure networks. The low-level control (see Section 6.2) adjusts the individual pump rates to attenuate these uctuations and to guarantee a steady operating pressure. This cause an unbalanced pump operation, if any high-level managing is missing. These oating pumps reach arbitrary pump rates, depending on the network topology, pump positions and cutting head switching states. Consequently, a high-level managing is required to balance all coupled pumps to an average pump rate with respect to (7.1) and satisfying (7.7).

The opposing constraints for local disturbance rejection by means of low-level control (see Section 5.3) and for high-level managing of distributed pumps, require to employ a hierarchical distributed control strategy, see the high-level managing in the diagram of Figure 7.2. It presents the proposed control design for future waterjet facilities, which aims at an energy ecient operation by balancing the coupled high-pressure pumps without any centralized managing. At low-level control, each pump tracks a desired reference pressure r n (t), while attenuating pressure uctuations. At high-level managing, each pump has to estimate the overall time-varying water consumption ū(t), e.g. as previously presented by means of average consensus (7.3). The managing has then to provide a suitable local reference pressure r n (t) that every pump converges towards an average pump rate, such as u n (t) → ū(t).

Such an algorithm for low-level control and distributed high-level managing is required to operate on every high-pressure pump. Realizing this control hierarchy with respect to a physical distribution, requires each pump to perform local computations and to provide communication to its neighbours for sending and receiving of information.

Communication networks

Distributed algorithms, subject to solve the average consensus problem, require to establish communication channels between neighbours. It is prevalent to introduce a graph notation [START_REF] Cherukuri | Initialization-free distributed coordination for economic dispatch under varying loads and generator commitment[END_REF], when describing entire communication networks. The initial investigations in this chapter will be restricted to a ring topology. In contrast to the directed graph in the diagram of Figure 7.3 (a), an undirected graph with unit edge weights has been considered, as shown in the diagrams (b). It allows a bi-directional communication between agents.

Referring to the general graph notation in [START_REF] Bullo | Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms[END_REF] and describing these ring topologies as a static graph, allows for rewriting the interconnection matrix a ij from the dynamic average consensus problem (7.5a -7.5c) in a graph Laplacian L, such as {Lx} i = j∈V a ij (x i -x j ) , (7.8) with the variable vectors x = [x 1 . . . x N ], u = [u 1 . . . u N ], p = [p 1 . . . p N ] and q = [q 1 . . . q N ] and {x, u, p, q} ∈ R 1,N . The ring topology (Figure 7.3 (b)) spans a strongly connected, undirected and weightbalanced graph that describes the logical layer for bidirectional communication between neighbouring agents, using the matrices On the other hand, a high-pressure network representing the physical layer is also described by means of graphs, see the modelling approach in Section 3.3. It interconnects pumps, respectively its network subsections, with respect to the uid dynamics. Revisiting the introduction example of Section 1.2, which proposes the topology of a future waterjet facility, and adopting the graph Laplacian L, such as ( 7.12) This assigns the network subsections to pump agents as depicted in Figure 7.4, while considering the lumped parameter model (5.23), as used for low-level control design in Section 5.1. It is distinguished between the logical layer with the pump agents, interconnected over a communication network, and the physical layer of pumps and work stations, interconnected over the high-pressure network. The physical and logical layers are most likely not of the same topology.

A =        0 1 0 1 1 0 1 • • • 0 0 1 0 1 . . . . . . 1 0 1 0        and L =        2 -1 0 -1 -1 2 -1 • • • 0 0 -1 2 -1 . . . . . . -1 0 -1 2        . ( 7
high-pressure pump to converge towards the desired overall average water consumption (7.1), satisfying (7.7). The coupling of subsections given by the physical layer is considered to balance the uid ows between distributed high-pressure pumps. The desired reference pressures are obtained using the logical layer. This yields the integrated approach for average consensus by means of distributed balancing

d dt r i (t) = α j∈V a ij u i (t) -u j (t) d dt z i (t) = K P r i (t) -y i (t) d dt u i (t) = K I r i (t) -y i (t) + z i (t) d dt p i N (t) = K(p i N ) V * i (t)   Q i (t) -H * i p i N (t) d * i (t) - j∈V ϑ ij p i N -p j N   , (7.13a)
(7.13b) (7.13c) (7.13d) which couples the logical and physical layers, such as u i (t) = Q i (t) and p i N (t) = y i (t). The logical layer, dened by the interconnection matrix A, updates the neighbouring agents j with the local pump rate u i (t). The distributed managing (7.13a) adjusts the reference pressure r i (t) for the low-level controller (7.13b -7.13c). The physical layer, dened by the interconnection matrix Θ, distributes the change of the local pressure state p i trough the high-pressure network (7.13d). That causes each pump rate to converge to the average pump rate u i (t) → ū(t), which corresponds to the overall average water consumption.

Whereas the logical layer provides the average consensus error to adjust the reference values, the physical layer will balance to pump rates. The low-level controller applies integral action to the reference signal that is required to implement robustness for initialization error and to eliminate steady-state error.

Evaluation of average consensus and distributed balancing

The extended dynamic average consensus algorithm (7.5a -7.5c) will be investigated by means of simulations, using high-pressure network models. It implements distributed pump agents to estimate the average pump rates lim t→∞ x i (t) = ū(t). In addition, the proposed distributed managing (7.13a) is implemented to balance the pump rates to average lim t→∞ u i (t) = ū(t), including the low-level controllers (7.13b -7.13c) and the interconnection to neighbouring network subsections (7.13d).

The high-pressure network model, subject for simulations, has been derived using the graphbased modelling methodology of Section 4.2. It considers a network conguration with 2 distributed pumps interconnecting 2 independent cutting heads. The simulations are then extended for the setup of Figure 7.6, including three pumps and four cutting heads. The pumps are locally controlled, using the extended robust PI controller. This controller has been obtained from optimal robust H ∞ synthesis, see section 6.2. The pump agents are in both cases interconnected with a ring topology, using bidirectional communication. This results in a strongly connected and weight balanced graph, see Section 7.2.

Application on a 2 pump setup

Distributed average consensus has been evaluated, considering two interconnected high-pressure pumps and a high-pressure network of topology (g) Decentralize distributed, as introduction in Section 2.2. Referring to Table 7.1, the experiment D-7 has been modelled, including two cutting heads of dierent inner diameters. The alternately switching of both cutting heads simulates a contour cutting application. This results in 4 switching states. The plots of Figure 7.7 show simulations with two cutting heads, which alternately switch open and close. As expected, the missing high-level managing causes a oating operation of the distributed pumps. The varying water consumption cause the high-pressure pumps to operate at dierent pump rates, with respect to their positions in the network and the cutting head switching states. The low-level controller attenuates pressure uctuations and tracks the desired reference pressure of 350 MPa, while eliminating the steady-state error. The rst plot provides the simulated pressure trends at the outtake position of both pumps. The second plot gives the varying pump rates to track the desired operating pressure. The third plot gives the switching state for both cutting heads. The dynamic average (7.5a -7.5c) is found to estimate the overall average water consumption at each pump agent, even for time-varying pump rates due to the switching behaviour of cutting heads. The rate of convergence depends on the tuning variables α and β. The zoomed-in plots of Figure 7.8 illustrate the convergence when estimating the average pump rates for dierent α-values and for dierent switching states (a) -(d). The rate of convergence is enhanced, when increasing α and considering β = α -1 , compare doted (α = 0.1), dashed (α = 1) and full lines (α = 10), respectively. The global reference is given by solving for (7.1).

Conclusion

This chapter provides a preliminary perspective to future waterjet facilities. Dynamic average consensus has been applied to show its ability for estimating the average pump rate of distributed high-pressure pumps. The pumps within a high-pressure network remain oating, if any feedback from a managing algorithm is missing. The low-level controller holds the pump output pressures around the global reference, while switching cutting heads will disturb the pressure generation. This results in dierent uid ow rates for every pump, dependent on its position in the highpressure network. Additional control action could be implemented for each pump agent, to bring the pump rates close to average. A contribution is given, when proposing a distributed balancing algorithm. This will directly adjust the reference pressure for each pump agent, such as the individual pump rates converge towards an average value. This approach considers the existing low-level controllers and the dynamic coupling between high-pressure pumps for balancing the individual pump rates, without implementing additional control action. Finally, the simulation results presented in this section combine all work from the previous Chapters 2 to 6. The proposed graph-based modelling represents a waterjet facility of increased complexity, which is too large to be established on the high-pressure test bench. The low-level controller, applied on every high-pressure pump, allows to distribute the local pump agents at arbitrary high-pressure network positions. The robust control design, by means of H ∞ controller synthesis, attenuates pressure uctuations, where the high-level managing adjusts each pump to operate at the time-varying average pump rate.

This framework results in a hierarchical distributed control design for high-pressure pumps. Extended simulation studies could verify the high-level managing using the graph-based modelling methodology. On the one hand, a procedure to nd optimal tuning parameters for distributed balancing has to be evaluated as well as limits on convergence and steady-state error needs to be investigated, especially when dealing with time delays, packet loss and loss of agents. Further, the ring topology, as considered to describe the communication channel between pump agents, is rarely implemented for real applications. This requires to verify the proposed distributed algorithm, also considering weakly connected and unbalanced graphs. On the other hand, the proposed distributed balancing considers a high-pressure network as an additional means of communication. Hence, it becomes essential to investigate stability of the overall algorithm with respect to various network topologies.

Retrospective

The applied research of this thesis employs a design framework in compliance with industrial needs. This framework includes the robust control of electrically driven high-pressure pumps and the graph-based modelling of high-pressure networks. Improving the energy eciency of entire waterjet facilities requires a novel concept of decentralized high-pressure pumps, which supply distributed work stations used for waterjet machining. This aims at the scaling of waterjet facilities according to their individual demands and requires a exible distribution of modular high-pressure pumps. Industry and academia work on the prototype of an electrically driven high-pressure pump. This electrically driven pump needs a robust control design that it can be installed in an arbitrary position of a waterjet facility, where the design and verication of future waterjet facilities require a exible modelling methodology.

The problem formulation for this thesis requires a multidisciplinary approach with a practical orientation. This asks of extending the common understanding by providing new impulses to the eld of waterjet machining. First, the exible modelling of high-pressure networks asks for simplifying the modelling task, that any individual parametrization for dierent network topologies becomes obsolete. Second, the robust control design for decentralized pumps requires to dene the limitations to a network subsection, for which a pump can operate within desired performance. Third, the optimization of entire waterjet facilities needs a concept for balancing distributed pumps, with minimal communication demands. These objectives desire a general framework for hierarchical distributed control design of electrically driven high-pressure pumps. Thereby, the hierarchical allocation for robust control and distributed balancing should guarantee a reliable and energy ecient pump operation, while the physical distribution of the computation task among all available pumps ensures its scalability.

A high-pressure test bench has been established in the laboratory within this thesis. This includes the design of two prototypes of electrically driven high-pressure pumps. The test bench serves to validate the modelling methodology and to verify the control design. The validation work by means of measurements has been a demanding task. The properties of the test bench is close to the industrial application. It involves the common diculties, such as a fast degradations due to the high-pressures, pressure losses due to leakages, switching delays and jitter from the PLC, etc. This makes it time intensive to reproduce experimental results, but demonstrates the close relation to the practical use of this research and the eorts to comply with industrial requirements. Especially, the work with preliminary pump prototypes has complicated the experimental studies. Indeed, the malfunction of check-valves disturbed the obtained measurement data, compromising the presented research results. Even worse, the sever damage of a pump prototype has caused to terminate the experimental studies earlier as planned.

Contributions

The design framework for hierarchical distributed control design of electrically driven highpressure pumps combines the results from multiple elds of research. Thus, a general contribution is given by adopting established methods from the theory to realize practical solutions with respect to industrial constraints.

The rst part of this thesis is dedicated to merge the common knowledge in the eld of waterjet machining. A contribution is given, when classifying waterjet machining with respect to the work station setup, cutting head conguration and manufacturing process. The common setups of high-pressure pumps are presented, including novel electrically driven pumps [START_REF] Niederberger | Hochdruckwasserstrahlschneiden mit energieezienten hochdruckpumpen[END_REF]. Network topologies and use cases are dened, which consider common waterjet applications. Extending these specications contributes to the design of future energy ecient and scalable waterjet facilities. Presenting the design of the novel electrically driven high-pressure pump, provides another contribution when applying the concept of model-based synchronization by means of camming [Niederberger, 2018].

The second part provides a simulation toolbox for the exible modelling of high-pressure networks. It assigns the governing equations from uid dynamics to the versatile graph theory. This yields a graph-based modelling methodology. That simplies the modelling and introduces a spacial discretization, resulting in a lumped parameter model. A simulation study evaluates the eect of this discretization. It compares the simulation results from the lumped parameter model with the results when numerically solving the PDE's, which describe the principles of continuity and momentum conservation. A contribution is given when considering the most prevalent parameter variations to improve simulation accuracy over a wide pressure range [START_REF] Niederberger | A graph-based modelling methodology for high-pressure networks applied on waterjet machining[END_REF]. In contrast to other modelling methodologies, this modelling introduces a pressuredependent bulk modulus and relates this to a pressure-dependent uid density. The simulation toolbox is nally proposed, when implementing the graph-based modelling methodology into MATLAB Simulink and providing an experimental parameter identication for specic highpressure components. This toolbox is valuable for industry. It allows for modelling of various waterjet facilities without the need of an individual parametrization. A major contribution is given, when applying this toolbox to model dierent high-pressure networks and validating the obtained network models by means of measurements [START_REF] Niederberger | Extended validation of a graph-based modelling methodology for highpressure networks[END_REF]. The graph-based modelling methodology is useful for the simulation of more complex high-pressure networks, which exceeds the capability of a test bench. This allows for evaluating of low-level control and highlevel managing with reduced eort and cost, in contrast to time expensive experiments at an extended test bench.

The third part proposes a robust low-level control design for high-pressure pumps, capable to stabilize a network subsection. It provides a single controller for all high-pressure pumps, independent of the operating point, the high-pressure network topology and its location in the network. This robust control design enables the industry to reduce the eort for installing a pump in a facility and facilitates the application of the novel electrically driven pump. The procedure of specifying control objectives in time domain, then transforming them in frequency domain, is eective to nd suitable weighting functions. Robustness is introduced by lumping the uncertain parameters, using an unstructured uncertainty description. Comparing dierent system descriptions for varying parameters, applying dierent uncertainty descriptions and describing the generalized plant, using a signal-based approach, contribute to the robust control design for high-pressure pumps. Dening the generalized plant for a high-pressure pump employs the optimization problem for H ∞ controller synthesis. Another contribution is given, when merging input channels to reduce the dimension of the resulting optimization problem by means of joint shaping functions [START_REF] Niederberger | Robust control design for electrically driven high-pressure pumps using H ∞ approach with joint shaping functions[END_REF].

Solving the reduced H ∞ optimization problem, considering an unstructured as well as a structured approach, provides various state feedback and PI controllers. A major contribution is given by a detailed simulation study that compares the derived controllers by means of simulations. An extended PI controller has been found suitable for implementation on the high-pressure pump with respect to industrial needs. Introducing the ∆-K iteration to derive a quasi-optimal controller by adjusting the unstructured uncertainty, aims to obtain robust stability for a maximally permissible uncertainty range. This provides a practical use and provides a straight forward procedure for optimal H ∞ controller synthesis, in particular when unstructured uncertainties and structured controllers are involved. Demonstrating that the concept of distributed balancing by means of average consensus will balance the pumps to a time-varying average pump rate, gives eventually a perspective to the managing of distributed pumps and to the objective of energy ecient waterjet facilities [Niederberger et al., 2019a].

Perspectives

The hierarchical distributed control design for electrically driven high-pressure pumps provides a general framework to the industry to design and to optimize future waterjet facilities. The main objective of this research work is to follow a global approach and thereby to initiate future research activities in the eld of distributed high-pressure pumps. Improving the graph-based modelling methodology, it could be very interesting to introduce and to evaluate higher order models, which take the principles of continuity and momentum conservation into account without solving for equilibrium. Then, possible time delays due to pressure propagation along large piping could become subject for further investigations by means of measurements. This could provide improved simulation models and enhance the control design for large-scale waterjet facilities. Also, the friction loss along network subsections requires detailed evaluation by means of measurements. The current representation of friction along a piping is insucient for the dierent uid ows in a high-pressure network. Thus, an extended friction loss model is highly desired. On the other hand, the industry is interested to extend the simulation toolbox with more standardized highpressure components. This requires detailed studies for parameter identication. Check-valve malfunctions degrade the benets of an electrically driven high-pressure pump. It is of interest to understand these phenomena and improve the pump design.

Rening the low-level control design, it could become interesting to replace the simplied and linearised rst-order model with an higher order representation or to investigate the eect of time delays to the robustness. Lumping parametric perturbations to a unstructured uncertainty description and assuming a single parameter for optimization is very useful for controller synthesis, but requires further verication. Considering more complex uncertainty descriptions would further allow for reducing the conservativism for the proposed control design and could enhance performance. The obtained robust stability and performance of the proposed quasi-optimal PI controller requires verication on a test bench, considering various high-pressure network topologies while taking the permissible uncertainty range into account. On the other hand, detailed experimental studies would be of interest to investigate possible couplings between decentralized pumps and to prove the robust stability and performance for installations to various waterjet facilities.

Considering possible couplings to neighbouring pumps as exogenous disturbances is again an interesting approach for control design. Extending the control design by taking the coupling of decentralized pumps into account, would allow for improving the stability of the overall system, but increases the computational requirements. It is further desired to investigate the requirements on stability for the proposed distributed balancing of decentralized pumps. Testing the distributed balancing on the test bench would provide rst results concerning performance and stability. It is required to dene the restrictions on the network topology at physical and logical layer, with respect to the convergence. A practical benet is expected, when investigating the robustness for loss of pump agents in runtime, when considering discrete computation and when introducing event triggered communication. Thus, graph-based modelling, low-level control and high-level managing are subject for ongoing investigations and motivate future research work, considering time delays, robustness, decentralization, convergence, etc.

A.1.2 Contour cutting: single cutting head

The reference value will rst ramp up to the desired operating pressure of 200 and 350 MPa. The measurement p in is chosen as control value. At the end, the reference value will ramp down to release the pressure. In between the cutting head will switch to close position, as shown in plots of Figure A.7 and A.8. That will cause the high-pressure pump to a full stop, resulting in a pressure increase. The cutting head will then switch to open position what causes a pressure loss and the pump returns in operation. The cutting head is congured with a nozzle of 0.35 mm inner diameter, considering the 200 MPa setup of measurement B-4. Thus, a maximal uid ow rate of 2.4 l/min results for a pump rate of 100%. The cutting head is congured with a nozzle of 0.25 mm inner diameter, considering the 350 MPa setup of measurement B-7. Thus, a maximal uid ow rate of 1.6 l/min results for a pump rate of 100%. .12 shows detailed trends for the input and system pressure in the rst plot, the output pressures in the second plot, as well as the switching pattern in the third plot. This survey provides an overview of various managing algorithms which emerged from dierent domains. It aims at evaluation of a suitable concept to manage distributed pumps in a highpressure networks. This is not a straight forward task, since sever restrictions are given from waterjet machining and facilities.

Hierarchical approaches are used to manage the collaboration between distributed controllers while obtaining a globally desired behaviour of the overall system. Common hierarchical distributed architectures are reviewed in [Scattolini, 2009], while dierent multi-layer structures are discussed in [Tatjewski, 2008]. Subject to process optimization, the control structure is at least organized in two layers. A lower layer that holds the distributed controllers and an upper layer dedicated to coordinate the subsystems by generating the desired subsystem set points. This coordination is centralized and comprises an optimization procedure which is often designed using model predictive control. However, for large-scale systems it becomes computationally dicult to solve the global optimization problem.

A hierarchical control design for interconnection systems is presented in [START_REF] Jilg | Hierarchical distributed control for interconnected systems[END_REF]] that aims at decreasing the communication burden and computational eort when separating the overall system into clusters. The distributed controllers communicate on low-level within a cluster where the clusters exchange informations at high-level with reduced data rate. A proper decomposition of large-scale system suitable for hierarchical distributed control is addressed in [START_REF] Tang | Distributed/hierarchical control architecture design[END_REF] when introducing a networked decomposition method.

Hierarchical control distribution is considered for over-actuated systems in [START_REF] Singla | A hierarchical control distribution approach for large scale over actuated systems[END_REF]. It aims at applying the overall control eort on the available actuators with respect to actuator limitations, where the control distribution problem is decomposed into many decoupled small-scale problems. This control distribution problem links to control allocation discussed in the subsequent chapter.

A rarely discussed design problem is the so called recongurable control structure, e.g. plug&play-control, where subsystems can be added and removed in runtime. This requires then a method for reconguration of the control and managing task. Such a reconguration to stabilize an overall interconnection system can be realized by backstepping [START_REF] Wen | Decentralized adaptive backstepping stabilization of interconnected systems with dynamic input and output interactions[END_REF]. This technique can deal with failing actuators [Wang et al., 2016a].

C.1.1 Control allocation (CA)

Control allocation aims at managing multi-variable systems in a centralized meaner. It successfully challenges over-actuated systems with a redundant set of eectors. Recent developments in control allocation is overviewed in [START_REF] Johansen | Control allocation -a survey[END_REF]. The allocation algorithm can be considered as the mid-layer in a hierarchical control scheme. A top-layer controller commands the desired control eorts of the overall system. The mid-layer allocation combines all available actuators to reproduce the virtual control eort by generating individual set points. This is typically solved by means of an optimization problem where dierent constraints and objectives are considered, e.g. actuator limitations, energy eciency, etc. Its solution is in general not unique for an over-actuated system. Finally, low-level controllers realize the set point for each actuator. Control allocation implies many applications in automotive and ight control, e.g. [START_REF] Alberding | Nonlinear hierarchical control allocation for vehicle yaw stabilization and rollover prevention[END_REF], [Tavasoli andNaraghi, 2011] and[de Almeida, 2016], stabilizing a vehicle or vessel. It originates from the concept of ganging. However, mapping the virtual control eort for complex systems to a redundant set of eectors is no longer an intuitive task. Some allocation techniques are shortly discussed in this sequel.

Linear and non-linear allocation

Linear and non-linear eector models are distinguished that leads to linear or non-linear programming methods for solving the optimization problem [START_REF] Tjønnås | Adaptive control allocation[END_REF]. Some linear allocation approaches allow for explicit solutions of the optimization problem using static eector models. These solutions can be calculated o-line. A robust o-line control allocation is discussed in [de Almeida, 2016] where a reference governor (RG) is implemented to avoid online optimization. A sequential quadratic programming is used in [START_REF] Johansen | Constrained nonlinear control allocation with singularity avoidance using sequential quadratic programming[END_REF] to approximate non-linear programs for singularity avoidance during online optimization.

A decentralized implementation of a non-linear control allocation scheme is proposed in [START_REF] Liu | A nonlinear control allocation framework for complex decentralized cooperation control system[END_REF]. It aims at solving the optimization problem of optimal allocation in a decentralized meaner.

Dynamic and adaptive allocation

A comparison of static and dynamic control allocation is given in [START_REF] Tavasoli | Comparison of static and dynamic control allocation techniques for integrated vehicle control[END_REF]. Static allocation uses algorithms such as Simplex or Interior-point (IP) to iteratively solve the optimization problem. Dynamic allocation implements a dynamic update law which reduces the computational burdens when solving the optimization problem.

An approach for adaptive control allocation is derived in [START_REF] Tohidi | Adaptive control allocation for over-actuated systems with actuator saturation[END_REF]. Instead of solving an optimization problem, an adaption law is designed to allocate the control eorts with respect to actuator saturation. In particular, the update law in [START_REF] Tjønnås | Adaptive control allocation[END_REF], proposes an adaptive allocation for time-varying systems when introducing an algorithm that estimates unknown parameters.

Most control allocation scheme neglect actuator dynamics. Where model predictive control (MPC) is capable to handle actuator dynamics, computational burden increase when considering a large number of actuators. A model predictive algorithm was revised in [START_REF] Luo | Dynamic control allocation with asymptotic tracking of time-varying control input commands[END_REF] to track time-varying control inputs with control allocation, taking into account a dynamic eector model.

Optimal and balancing allocation

Using the quadratic norm (L 2 ) to formulate the optimization problem tends to combine all actuators while minimizing the overall control eort. On the other hand, using the supremum norm (L ∞ ) balances the control eorts while minimizing the maximal deection for every actuator (min-max criterion). The rst case results in optimal allocation where the second case refers to balancing allocation.

Dierent algorithms, such as linear programming (LP), quadratic programming (QP) and their mixed integer equivalents are compared in [START_REF] Grechi | Comparison between optimal control allocation with mixed quadratic & linear programming techniques[END_REF]. It aims at solving the optimisation task that results from L 2 -norm problem formulation. A dynamic approach is considered in [START_REF] Passenbrunner | Optimality-based dynamic allocation with nonlinear rst-order redundant actuators[END_REF], subject to optimal control allocation that results in a PI control action. How to transform an optimization problem into a linear program is well explained in [START_REF] Bouarfa | A fast active-balancing method for the 3-phase multilevel ying capacitor inverter derived from control allocation theory[END_REF].

Solving the optimization problem given by L ∞ -norm with linear programming is presented in [START_REF] Bodson | Control allocation with load balancing[END_REF]. It aims at load balancing of various actuators in model reference control. Two optimization criteria are combined using a mixed optimization approach to obtain optimal load balancing. For a high-pressure system it is very much desirable to balance the pump load while minimizing the overall control eort. It is shown in [START_REF] Frost | Resource balancing control allocation[END_REF] that dierent objectives can be combined while applying L 1 -norm for minimizing the tracking

Figure 2

 2 Figure2Energy ecient high-pressure pump: replacing hydraulically driven pressure intensiers by electrically driven piston pumps.

Figure 3 Figure 4

 34 Figure 3 Conventional waterjet facility: remote hydraulically driven pumps interconnected to work stations, using long high-pressure pipings.

  p P , p S Pressure at a pump outlet, water supply (Pa), (Pa) p, p w Perturbation channel, weighted perturbation channel (Pa), ( -) P Interconnection transfer function matrix ( -) xxiv Q, Q kj Fluid ow of a segment, branch kj (m 3 /s), (m 3 /s) Q ext , Q max Exogenous uid ow, maximal uid ow (m 3 /s), (m 3 /s) Q H , Q V Fluid ow through a cutting head, check-valve (m 3 /s), (m 3 /s) Q k , Q P Displacement uid ow of a node k, piston (m 3 /s), (m 3 /s) 1.1 Example of a conventional high-pressure pump (Type JPC 22 LC): hydraulically driven pressure intensiers with two single acting pistons. . . . . . . . . . . . . . 5 1.2 Example of a cutting head used for pure water applications: pneumatically actuated on/o-valve and nozzle holder generating the waterjet. . . . . . . . . . . . . 6 1.3 Example of a work station used for waterjet cutting: CNC work station guiding a cutting head over a work peace to cut a desired 2D-contour. . . . . . . . . . . . . 6 1.4 Common waterjet applications: classication of waterjet machining according to its operating pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Example of waterjet applications: (a) work station equipped with a single cutting head after cutting a desired 2D-couture in a steel sheet. (b) stripping machine with 16 nozzles on a common rail and holding device for matrices used to strip o paint from its surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Characteristics for waterjet machining: classication of waterjet machining with respect to waterjet application, manufacturing process and work station setup. . 9 1.7 Standard symbols (ISO 1219) used to draw diagrams of high-pressure networks: n pumps are interconnected to m cutting heads by means of piping. A cutting head is composed by an on/o-valve and a nozzle. Operating pressures are measured at positions, denoted by x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 Diagrams of high-pressure networks with dierent topologies: upgrading conventional waterjet facilities (a) to future energy-ecient facilities (b). . . . . . . . . . 12 1.9 High-pressure generation using a pressure intensier (a) and a piston pump (b): a pressure intensier pressurizes water (high-pressure chamber) by means of hydraulics (low-pressure chamber), whereas a piston pump is directly driven by an electro-mechanical actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.10 Types of high-pressure pumps: classication of high-pressure pumps with respect to drive system and pumping chamber design. . . . . . . . . . . . . . . . . . . . . 14 xxvii 1.11 Physical setup of a high-pressure pump with a conventional hydraulic drive: operating two single-acting intensiers (a) and a double-acting intensier (b). . . . . 1.12 Physical setup of a high-pressure pump with electric drives: two drives units displacing independent pistons (a) and a single drive unit displacing coupled pistons using a crankshaft (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 High-pressure test bench: two electrically driven piston pumps supply two cutting heads over a congurable high-pressure network. . . . . . . . . . . . . . . . . . . 2.2 Test bench topology: pumps and cutting heads interconnected with a congurable high-pressure network to represent various waterjet facilities. . . . . . . . . . . . . 2.3 Test bench hardware setup: hardware employed to regulate the test bench and to execute experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 1 pump -1 head topologies: single pump setup with an independent cutting head directly interconnected with (a) Short piping and (b) Long piping. . . . . . . 2.5 1 pump -2 heads topologies: single pump setup with independent cutting heads connected to (c) Symmetric network and (d) Asymmetric network topologies.2.6 2 pumps -2 heads topologies: distributed pumps with independent cutting heads connected to (e) Symmetric distributed, (f) Asymmetric distributed and (d) Decentralized distributed network topologies. . . . . . . . . . . . . . . . . 2.7 Switching pattern for a single cutting head conguration: cutting head in open position as initial condition for performing contour cutting (a) and cavity cutting in repeated cycles (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Switching pattern for a two cutting head conguration: cutting head A in open and B in closed position as initial condition for performing contour cutting with 90°phase shift (a) and cavity cutting in repeated cycles of 90°phase shift (b). . . 2.9 Switching pattern for a two cutting head conguration: cutting head A remains in open position for all time, while cutting head B performs contour cutting (a) and cavity cutting (b) in repeated cycles. . . . . . . . . . . . . . . . . . . . . . . 2.10 Novel high-pressure pump prototype: two single-acting pistons with independent electric drives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Modular pumping unit: independent synchronous motor positioning a piston by means of a spindle shaft to pressurize water in its pimping chamber. . . . . . . . 2.12 High-pressure pump hardware setup: a PLC generates the position commands to synchronize both piston pumps by means of camming. The piston pumps are driven by independent frequency converters, which realize the motion control loops to displace the pistons, where each synchronous motor provides measurement data from encoders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Hierarchical control design for modular high-pressure pumps: each pumping unit is controlled by local motion control. The units are synchronized by means of camming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Pumping unit with a single-acting piston: piston displacement pressurize water for high-pressure generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Functional principle of a piston pump illustrated at the three states of a pumping cycle: (a) pressurization of the pumping chamber, (b) hauling the pressurized uid to the piping and (c) lling the chamber from the water supply. . . . . . . . . . . 2.16 Camshaft to couple N pistons: the camshaft enables a static synchronization of multiple pistons, generating a sinusoidal output uid ow. . . . . . . . . . . . . . xxviii 2.17 Reference trajectories used for camming: example of a reference trajectory representing the desired piston position (a), velocity (b) and acceleration (c) values for a pumping cycle with a phase-shift of 180°between both pistons. . . . . . . . . . 2.18 Pressure-dependent position trajectories: desired piston displacement to obtain continuous output uid ow for dierent operating pressures. . . . . . . . . . . . 2.19 Relation between chamber pressure and piston displacement: expected position position to pressurize a pumping chamber. . . . . . . . . . . . . . . . . . . . . . . 2.20 Structure of motion control: implemented position, velocity and current control loop to the frequency converters and its load observer. . . . . . . . . . . . . . . . 2.21 Measured pump output pressure compared to simulated pressure trends and corresponding displacement uid ows using a 350 MPa trajectory: over excitation (a) causing a periodic pressure loss and under excitation (b) casing a periodic pressure overshoot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22 Simulated pumping chamber pressures for a pumping cycle and resulting pump output pressure from measurement due to check-valve switching hysteresis: experiment 1 (a) shows the pressure trend at 380 MPa and experiment 2 (b) shows the pressure trend at 320 MPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A simple piping section to introduce the Bernoulli equation: an initial uid segment is moved to a subsequent position while conserving its momentum. . . . . . . . . 3.2 A piping section of constant cross section S: considering a uid segment ∂x of pressure state p(x, t) and uid ow velocity v(x, t) with respect to the position coordinate x and time t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Simulated pressure trend and uid ow velocity for boundary conditions (A-1): piping without any nozzle installed when increasing the uid ow velocity to v ∞ = 461 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Simulated pressure trend and uid ow velocity for boundary condition (A-2): piping without any nozzle installed when increasing the uid ow velocity to v ∞ = 729 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Simulated pressure trend and uid ow velocity for boundary condition (A-3): piping without any nozzle installed when increasing the uid ow velocity to v ∞ = 1031 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Simulated pressure trend and uid ow velocity for boundary condition (A-4): piping without any nozzle installed when increasing the uid ow velocity to v ∞ = 1458 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Simulated pressure trend and uid ow velocity for boundary condition (B-1):

3 .

 3 23 Principal symbols to illustrate waterjet facilities by means of graphs: branches interconnect nodes to recover the uid ow path of a high-pressure network. . . . 3.24 Combination of symbols to illustrate typical high-pressure components by means of graphs: branches provide various functionalities to determine and to manipulate the uid ow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.25 Experimental setup for parameter identication: pressure gauges measure the pressure p x (t) at dierent positions when applying an input uid ows Q n (t) and switching the cutting heads d m (t). . . . . . . . . . . . . . . . . . . . . . . . . . . 3.26 Experiments with v P = 1 mm/s: measured pressure trends and pumping chamber volumes, while displacing the pistons from dierent initial positions s 0 . . . . . . . 3.27 Experiments with v P = 2 mm/s: measured pressure trends and pumping chamber volumes, while displacing the pistons from dierent initial positions s 0 . . . . . . . 3.28 Measurement data derived by experiment to identify bulk modulus and uid density: data samples with tted model and prediction interval (a), resulting residuals (b), corresponding regression analysis (c) and distribution of data samples (d). .3.29 Relation between chamber pressure and uid density: expected uid density, when pressurizing a pumping chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.30 Comparison of experimental data used to identify the friction loss coecient: considering the estimated uid ows, the absolute pressures from measurements and the nozzle cross sections, given from the test bench setup. . . . . . . . . . . . 3.31 Measurement data derived by experiment to identify the cutting head discharge coecient: data samples wit tted model and prediction interval (a), resulting residuals (b), corresponding regression analysis (c) and distribution of data samples (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.32 Comparison of experimental data used to identify the friction loss coecient: considering the estimated uid ows, the calculated pressure dierences from measurements and the piping lengths, given from the test bench setup. . . . . . . . . xxx 3.33 Measurement data derived by experiment to identify piping friction loss coecient: data samples with tted model and prediction interval (a), resulting residuals (b), corresponding regression analysis (c) and distribution of data samples (d). . . . . 4.1 Illustration of the investigated high-pressure network: two pistons of a high-pressure pump induce its displacement uid ows to supply a cutting head, that generates a discharge uid ow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Piston pressurization position: estimated position to pressurize pumping chamber by means of piston displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Network diagram corresponding to topology (a): scheme to assign the investigated high-pressure network to homogeneous segments of nodes and branches. . . . . . 4.4 Network assignment for topologies (a) and (b): corresponding graph network for the short and long piping network topologies used for modelling. . . . . . . . . . 4.5 Network assignment for topologies (c) and (d): corresponding graph networks for

  4.20 Zoom-in pressure trends of the 1 pump -2 cutting head setup trend at 200 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b), as well as opening of cutting head 2 (c) and closing of cutting head 2 (d). . . . . . . . . . . . . . . . 4.21 Check-valve hysteresis at 200 MPa operating pressure of the 1 pump -1 cutting head setup: considering optimal phase-shift (a) and simulated hysteresis (b). . . . 4.22 Check-valve hysteresis at 350 MPa operating pressure of the 1 pump -1 cutting head setup: considering optimal phase-shift (a) and simulated hysteresis (b). . . . 4.23 Overview of experiment D-2 for a 2 pump -2 cutting head setup with symmetric topology: pressure trend p P (t) and pump rate u(t), due to switching pattern d(t) for selected switching cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.24 Overview of experiment D-5 for a 2 pump -2 cutting head setup with asymmetric topology: pressure trend p P (t) and pump rate u(t), due to switching pattern d(t) for selected switching cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.25 Zoom-in pressure trends of a 2 pump -2 cutting head setup with symmetric topology: opening of cutting head 1 (a) and closing of cutting head 1 (b). . . . . 4.26 Zoom-in pressure trends of a 2 pump -2 cutting head setup with asymmetric topology: opening of cutting head 1 (a) and closing of cutting head 1 (b). . . . . 4.27 Start up of experiment D-1, considering a 2 pump -2 cutting head setup at 200b MPa: simulated pressure trends for each pumping chamber, compared to measured pressure at pump outtake, and resulting pump rates for the symmetric distributed network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.28 Start up of experiment D-4, considering a 2 pump -2 cutting head setup at 200 MPa: simulated pressure trends for each pumping chamber, compared to measured pressure at pump outtake, and resulting pump rates for the asymmetric distributed network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Diagram for decentralized, distributed and networked control: a system is separated in subsystems to reduce the complexity of control design. Whereas decentralized control considers an independent controller per subsection, distributed control takes into account the state of neighbouring subsections and networked control enables a data exchange between controllers. . . . . . . . . . . . . . . . . xxxii 5.2 Block diagram of standard ∆=M conguration: interconnection system M and uncertainty block ∆, containing the parametric uncertainties. . . . . . . . . . . . 5.3 Block diagram of standard P=K conguration: interconnection matrix P and controller K of desired structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 .

 5 10 Non-linear plant model: block diagram of the bilinear rst order model with varying parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Approximated plant model: block diagram of the linearised model around operating points in state space form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Frequency responses for the linearised plant models: wide frequency range on the left hand side and zoomed in on the right hand side, showing minor deviations for the introduced approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Simplied approximated plant model: block diagram of the simplied approximated model in state space form. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14 Frequency responses comparing the simplied approximated model in state space form and its transfer function representation: wide frequency range on the left hand side and zoomed in on the right hand side, showing remarkable deviations when lumping uncertain parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Block diagram of the interconnection system: considering system components, shaping and weighting functions for control design. . . . . . . . . . . . . . . . . . 5.16 Frequency response of the scaled actuator transfer function G a (s): considering the dynamic ranges of pumping units required for robust control design. . . . . . . . 5.17 Design of experiment to evaluate the actuator transfer function: velocity trajectory (a), position trajectory (b) for pump excitation and realized frequency range (c).164 5.18 Frequency and step responses of pump actuators: identied second order transfer function for both pumping units and expected second order model provided from motion control tuning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Frequency response of the scaled nominal plant transfer function G n (s): considered nominal dynamic range of a network subsection required for robust control design. 5.20 Frequency response of the pressure gauge transfer function G s (s): considered dynamic range of the measurement signal required for robust control design. . . . . 5.21 Frequency response of the scaled reference shaping function W r (s): considered dynamic range of the reference value required for robust control design. . . . . . . xxxiii 5.22 Frequency response of the scaled disturbance shaping function W d (s): considered dynamic range of the cutting head on/o-valve to shape the disturbance signal. . 5.23 Frequency response of the scaled perturbation shaping function W p (s): considered dynamic range of the inverse additive uncertainty to shape the perturbation signal.168 5.24 Measured noise for dierent operating points: residuals, mean values and 3 σ interval for dierent measurements reveal similar variance around expected value. 5.25 Box-plot of measured noise: residuals for measurements at 100, 200, 300 and 400 MPa pressures reveal evidence for additive noise phenomena. . . . . . . . . . 5.26 Histogram of measured noise: residuals for measurements at 100, 200, 300 and 400 MPa pressure reveal similar variance close to Gaussian distribution. . . . . . 5.27 Amplitude spectrum of measured noise: FFT for measurements at 100, 200, 300 and 400 MPa pressures, its upper bound and selected shaping function. . . . . . . 5.28 Frequency response of noise shaping function W p (s): considered dynamic range of the noise signal required for robust control design. . . . . . . . . . . . . . . . . . 5.29 Scaled frequency responses of the pump and valve actuators with plant in series: dynamic range from the control signal to the pressure state and from the disturbance signal to the pressure state. . . . . . . . . . . . . . . . . . . . . . . . 5.30 Frequency response of the weighting function W e (s): dynamic range of the weighting on the error signal and lower bounds from control requirements. . . . . . . . . 5.31 Frequency response of the weighting function W u (s): dynamic range of the weighting on the control signal and upper bound from control requirements. . . . . . . . 5.32 Frequency response of the shaping functions W p (s): dynamic range of the weighting on the perturbation signal and lower bounds from control requirements. . . . 5.33 Overview of the uncertainty model evaluation process: dierent approaches to describe the varying parameters for control design. . . . . . . . . . . . . . . . . . 5.34 Classication of uncertainties: time-varying and parametric uncertainties used to describe varying parameters, subject for control design. . . . . . . . . . . . . . . . 5.35 Grouping of uncertain parameters: time-varying and parametric uncertainties used to describe varying parameters, subject for control design. . . . . . . . . . . . . . 5.36 Position displacement changes the pumping chamber volume: desired piston positions of two pumping chambers to obtain a continuous overall uid ow. . . . . . 5.37 Range for known time-varying uncertainties: uid density dependent on the pressure state (a) and pumping chamber volume with respect to the piston displacement (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.38 Uncertainty range for unknown parametric uncertainties: ow resistance considering dierent nozzles (a) and overall volume for various high-pressure networks (b). 5.39 Frequency responses of approximated and extended models for u → y: varying a single uncertain parameter at once, where the others stay constant on nominal value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.40 Frequency responses of approximated and extended models for d → y: varying a single uncertain parameter at once, where the others stay constant on nominal value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.41 Frequency responses of approximated model for u → y and d → y: varying a single uncertain parameter at once, where the others stay constant on nominal value. . 5.42 Block diagram of simplied approximated model with real parametric uncertainties: additive uncertainty and inverse additive uncertainty description used to separate these uncertainties from the plant model. . . . . . . . . . . . . . . . . . . . xxxiv 5.43 Block diagram of the nominal plant: representing the simplied approximated model for nominal parameter values. . . . . . . . . . . . . . . . . . . . . . . . . . 5.44 Block diagram of unstructured uncertainty descriptions: additive perturbation from plant input to output (a), input multiplicative perturbation from plant input to input (b), inverse additive perturbation from plant output to input (c) and inverse output multiplicative perturbation form plant output to output (d). . . . 5.45 Weighting functions for additive and multiplicative uncertainties: error frequency responses in additive form, considering absolute error (a) and multiplicative form for relative error (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.46 Perturbed plant using additive (a) and multiplicative (b) descriptions: unstructured uncertainties compared to parametric uncertainty description reveal inadequate behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.47 Weighting functions to obtain the inverse additive and inverse multiplicative uncertainties: error frequency responses in inverse additive form, considering an absolute error (a) and inverse multiplicative form for relative error (b). . . . . . . . . . . . 5.48 Perturbed plant using inverse additive (a) and inverse multiplicative (b) descriptions: unstructured uncertainties show adequate behaviour compared to parametric uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  ) and distributed balancing (right) applied for the introduction example: managing of 3 interconnected pumps in a high-pressure network with 4 cutting heads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Start up and shut down ramping of experiment B-1 considering a 1 pump -1 cutting head setup at 200 MPa: start up pressure trend (a) and shut down pressure trend (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Start up and shut down ramping of measurement B-7 considering a 1 pump -1 cutting head setup at 350 MPa: start up pressure trend (a) and shut down pressure trend (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Zoom-in around 100 MPa of ramping for 200 MPa operating point: pressure trends during start up (a) and shut down (b). . . . . . . . . . . . . . . . . . . . . . . . . A.4 Zoom-in around 100 MPa of ramping for 350 MPa operating point: pressure trends during start up (a) and shut down (b). . . . . . . . . . . . . . . . . . . . . . . . . A.5 Zoom-in around 150 MPa of ramping for 200 MPa operating point: pressure trends during start up (a) and shut down (b). . . . . . . . . . . . . . . . . . . . . . . . . A.6 Zoom-in around 300 MPa of ramping for 350 MPa operating point: pressure trends during start up (a) and shut down (b). . . . . . . . . . . . . . . . . . . . . . . . . A.7 Overview of experiment B-4: 1 pump -1 cutting head setup with 0.35 mm nozzle at 200 MPa operating pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 Overview of experiment B-7: 1 pump -1 cutting head setup with 0.25 mm nozzle at 350 MPa operating pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 Switching cutting head of the 1 pump -1 cutting head setup at 200 MPa: closing cutting head pressure trend (a) and opening cutting head pressure trend (b). . . A.10 Switching cutting head of the 1 pump -1 cutting head setup at 350 MPa: closing cutting head pressure trend (a) and opening cutting head pressure trend (b).

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Conventional waterjet machining . . . . . . . . . . . . . . . . . . 1.2.1 Common waterjet applications . . . . . . . . . . . . . . . . . . . . . 1.2.2 Classifying waterjet machining . . . . . . . . . . . . . . . . . . . . . 1.2.3 Typical waterjet facilities . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Principles of high-pressure generation . . . . . . . . . . . . . . . 1.3.1 Classifying high-pressure pumps . . . . . . . . . . . . . . . . . . . . 1.3.2 Hydraulically driven intensier pumps . . . . . . . . . . . . . . . . 1.3.3 Electrically driven piston pumps . . . . . . . . . . . . . . . . . . . . 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1

 11 Figure 1.1 Example of a conventional high-pressure pump (Type JPC 22 LC): hydraulically driven pressure intensiers with two single acting pistons. 1

Figure 1 . 3

 13 Figure 1.3 Example of a work station used for waterjet cutting: CNC work station guiding a cutting head over a work peace to cut a desired 2D-contour. 2

Figure 1 . 4

 14 Figure 1.4 Common waterjet applications: classication of waterjet machining according to its operating pressure.

Figure 1 . 5

 15 Figure 1.5 Example of waterjet applications: (a) work station equipped with a single cutting head after cutting a desired 2D-couture in a steel sheet. (b) stripping machine with 16 nozzles on a common rail and holding device for matrices used to strip o paint from its surface.3

Figure 1 . 7

 17 Figure 1.7 Standard symbols (ISO 1219) used to draw diagrams of high-pressure networks: n pumps are interconnected to m cutting heads by means of piping. A cutting head is composed by an on/o-valve and a nozzle. Operating pressures are measured at positions, denoted by x.

Figure 1 . 8

 18 Figure 1.8 Diagrams of high-pressure networks with dierent topologies: upgrading conventional waterjet facilities (a) to future energy-ecient facilities (b).

  Figure1.9 High-pressure generation using a pressure intensier (a) and a piston pump (b): a pressure intensier pressurizes water (high-pressure chamber) by means of hydraulics (lowpressure chamber), whereas a piston pump is directly driven by an electro-mechanical actuator.

Figure 1

 1 Figure1.10 Types of high-pressure pumps: classication of high-pressure pumps with respect to drive system and pumping chamber design.

  .

Figure 1 .

 1 Figure 1.11 Physical setup of a high-pressure pump with a conventional hydraulic drive: operating two single-acting intensiers (a) and a double-acting intensier (b).

2. 3

 3 Novel electrically driven pump with single-acting pistons . . . . 2.3.1 Pump setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 High-pressure generation using single-acting pistons . . . . . . . . . 2.3.3 Coupling of N pumping units . . . . . . . . . . . . . . . . . . . . . 2.3.4 Model-based synchronization by means of camming . . . . . . . . . 2.3.5 Motion control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Pressure uctuations and check-valve hysteresis . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2

 2 Figure 2.1 High-pressure test bench: two electrically driven piston pumps supply two cutting heads over a congurable high-pressure network.

Figure 2 . 2

 22 Figure 2.2 Test bench topology: pumps and cutting heads interconnected with a congurable high-pressure network to represent various waterjet facilities.

Figure 2

 2 Figure 2.4 1 pump -1 head topologies: single pump setup with an independent cutting head directly interconnected with (a) Short piping and (b) Long piping.

  Figure 2.5 1 pump -2 heads topologies: single pump setup with independent cutting heads connected to (c) Symmetric network and (d) Asymmetric network topologies.

Figure 2 .

 2 Figure 2.6 2 pumps -2 heads topologies: distributed pumps with independent cutting heads connected to (e) Symmetric distributed, (f) Asymmetric distributed and (d) Decentralized distributed network topologies.

Figure 2 . 7

 27 Figure 2.7 Switching pattern for a single cutting head conguration: cutting head in open position as initial condition for performing contour cutting (a) and cavity cutting in repeated cycles (b).

Figure 2 . 8

 28 Figure 2.8 Switching pattern for a two cutting head conguration: cutting head A in open and B in closed position as initial condition for performing contour cutting with 90°phase shift (a) and cavity cutting in repeated cycles of 90°phase shift (b).

Figure 2 . 9

 29 Figure 2.9 Switching pattern for a two cutting head conguration: cutting head A remains in open position for all time, while cutting head B performs contour cutting (a) and cavity cutting (b) in repeated cycles.

Figure 2 .

 2 Figure 2.10 Novel high-pressure pump prototype: two single-acting pistons with independent electric drives.

Figure 2

 2 Figure 2.13 Hierarchical control design for modular high-pressure pumps: each pumping unit is controlled by local motion control. The units are synchronized by means of camming.

  Figure 2.14 Pumping unit with a single-acting piston: piston displacement pressurize water for high-pressure generation.

Figure 2 .

 2 Figure 2.15 Functional principle of a piston pump illustrated at the three states of a pumping cycle: (a) pressurization of the pumping chamber, (b) hauling the pressurized uid to the piping and (c) lling the chamber from the water supply.

Figure 2 .

 2 Figure 2.16 Camshaft to couple N pistons: the camshaft enables a static synchronization of multiple pistons, generating a sinusoidal output uid ow.

Figure 2

 2 Figure2.17 Reference trajectories used for camming: example of a reference trajectory representing the desired piston position (a), velocity (b) and acceleration (c) values for a pumping cycle with a phase-shift of 180°between both pistons.

Figure 2 .

 2 Figure 2.18 Pressure-dependent position trajectories: desired piston displacement to obtain continuous output uid ow for dierent operating pressures.

Figure 2

 2 Figure 2.20 Structure of motion control: implemented position, velocity and current control loop to the frequency converters and its load observer.

Figure 2 .

 2 Figure 2.21 Measured pump output pressure compared to simulated pressure trends and corresponding displacement uid ows using a 350 MPa trajectory: over excitation (a) causing a periodic pressure loss and under excitation (b) casing a periodic pressure overshoot.

Figure 2

 2 Figure 2.22 Simulated pumping chamber pressures for a pumping cycle and resulting pump output pressure from measurement due to check-valve switching hysteresis: experiment 1 (a) shows the pressure trend at 380 MPa and experiment 2 (b) shows the pressure trend at 320 MPa.

  and (3.33).(B) Nozzle contraction: A high-pressure pump cannot provide the uid ow required, if a nozzle is missing. Consequently, this case considers a nozzle of D H = 0.35 mm inner diameter with a discharge coecient of ζ = ###. It adopts the discharge equation(3.19)

Figure 3 . 3

 33 Figure 3.3 Simulated pressure trend and uid ow velocity for boundary conditions (A-1): piping without any nozzle installed when increasing the uid ow velocity to v ∞ = 461 m/s.

Figure 3 . 4

 34 Figure 3.4 Simulated pressure trend and uid ow velocity for boundary condition (A-2): piping without any nozzle installed when increasing the uid ow velocity to v ∞ = 729 m/s.

Figure 3 . 5

 35 Figure 3.5 Simulated pressure trend and uid ow velocity for boundary condition (A-3): piping without any nozzle installed when increasing the uid ow velocity to v ∞ = 1031 m/s.

Figure 3 . 6

 36 Figure 3.6 Simulated pressure trend and uid ow velocity for boundary condition (A-4): piping without any nozzle installed when increasing the uid ow velocity to v ∞ = 1458 m/s.

Figure 3 . 7

 37 Figure 3.7 Simulated pressure trend and uid ow velocity for boundary condition (B-1): piping with nozzle when increasing the uid ow velocity to v ∞ = 3 m/s.

Figure 3 .

 3 Figure 3.8 Simulated pressure trend and uid ow velocity for boundary condition (B-2): piping with nozzle when increasing the uid ow velocity to v ∞ = 4.8 m/s.

Figure 3 . 9

 39 Figure 3.9 Simulated pressure trend and uid ow velocity for boundary condition (B-3): piping with nozzle when increasing the uid ow velocity to v ∞ = 6.8 m/s.

Figure 3 .

 3 Figure 3.10 Simulated pressure trend and uid ow velocity for boundary condition (B-4): piping with nozzle when increasing the uid ow velocity to v ∞ = 9.6 m/s.

Figure 3 .

 3 Figure 3.12 Simulated pressure trend and uid ow velocity for a piping segment of 0.5 m: simulation using nite dierences with 2100 elements.

Figure 3 .

 3 Figure 3.13 Simulated pressure trend and uid ow velocity for a piping segment of 1 m: simulation using nite dierences with 4200 elements.

Figure 3 .

 3 Figure 3.14 Simulated pressure trend and uid ow velocity for a piping segment of 2 m: simulation using nite dierences with 8400 elements.

Figure 3 .

 3 Figure 3.15 Simulated pressure trend and uid ow velocity for a piping segment of 5 m: simulation using nite dierences with 21000 elements.

Figure 3 .

 3 Figure 3.16 Simulated pressure trend and uid ow velocity for a piping segment of 10 m: simulation using nite dierences with 42000 elements.

Figure 3 .

 3 Figure 3.18 Error between distributed and lumped parameter model for a piping segment of 1 m: simulation using nite dierences with 4200 elements.

Figure 3 .

 3 Figure 3.19 Error between distributed and lumped parameter model for a piping segment of 2 m: simulation using nite dierences with 8400 elements.

Figure 3 .

 3 Figure 3.20 Error between distributed and lumped parameter model for a piping segment of 5 m: simulation using nite dierences with 21000 elements.

Figure 3 .

 3 Figure 3.21 Error between distributed and lumped parameter model for a piping segment of 10 m: simulation using nite dierences with 42000 elements.

Figure 3 .

 3 Figure 3.22 Modelling framework for high-pressure networks: assigning a network section to k ∈ J segments (a) and representing the segments by means of graph (b).

Figure 3 .

 3 Figure 3.23 Principal symbols to illustrate waterjet facilities by means of graphs: branches interconnect nodes to recover the uid ow path of a high-pressure network.

  Connector coupling n inputs with m outputs : (d) Cutting head with nozzle generating the waterjet :

Figure 3 .

 3 Figure 3.24 Combination of symbols to illustrate typical high-pressure components by means of graphs: branches provide various functionalities to determine and to manipulate the uid ow.

Figure 3 .Figure 3 .

 33 Figure 3.26 Experiments with v P = 1 mm/s: measured pressure trends and pumping chamber volumes, while displacing the pistons from dierent initial positions s 0 .

Figure 3 .

 3 Figure 3.28 Measurement data derived by experiment to identify bulk modulus and uid density: data samples with tted model and prediction interval (a), resulting residuals (b), corresponding regression analysis (c) and distribution of data samples (d).

Figure 3 .

 3 Figure 3.31 Measurement data derived by experiment to identify the cutting head discharge coecient: data samples wit tted model and prediction interval (a), resulting residuals (b), corresponding regression analysis (c) and distribution of data samples (d).

  Section 3.2, into the discharge equation (3.65) yields

Figure 3 .

 3 Figure 3.33 Measurement data derived by experiment to identify piping friction loss coecient: data samples with tted model and prediction interval (a), resulting residuals (b), corresponding regression analysis (c) and distribution of data samples (d).

Figure 4 . 1

 41 Figure 4.1 Illustration of the investigated high-pressure network: two pistons of a high-pressure pump induce its displacement uid ows to supply a cutting head, that generates a discharge uid ow.

Figure 4 . 2

 42 Figure 4.2 Piston pressurization position: estimated position to pressurize pumping chamber by means of piston displacement.

Figure 4 . 3

 43 Figure 4.3 Network diagram corresponding to topology (a): scheme to assign the investigated high-pressure network to homogeneous segments of nodes and branches.

  Equivalent to the above discussed topology (a) Short piping, the topology (b) Long piping is represented as a graph, considering homogeneous segments of about 1 m length. Table4.3 assigns the high-pressure components to N = 22 nodes. All nodes are interconnected as schematically illustrated inFigure 4.4 (b). The piping section 1 of length L = 11.3 m is therefore subdivided into n = 11 segments of length l = 1.026 m, whereas the piping section 2 of length L = 7.2 m is subdivided into n = 7 segments of length l = 1.031 m.

Figure 4 . 4

 44 Figure 4.4 Network assignment for topologies (a) and (b): corresponding graph network for the short and long piping network topologies used for modelling.

Figure 4 .

 4 Figure 4.5 assigns the high-pressure networks for the topologies (c) Symmetric network and (d) Asymmetric network to graphs, again considering homogeneous segments of about 1 m length. This segmentation results in a graph network N (c) kj of kj ∈ {[1, 3], [2, 3], [3, 4], . . . , [10, 11], . . . , [16, 17], [17, 0], [10, 18], . . . , [23, 24], [24, 0]}

Figure 4 . 5

 45 Figure 4.5 Network assignment for topologies (c) and (d): corresponding graph networks for the symmetric and asymmetric network topologies used for modelling.

Figure 4 .

 4 Figure 4.6 assigns the high-pressure networks for topology (e) Symmetric distributed, (f) Asymmetric distributed and (g) Decentralized distributed to graphs, again considering homogeneous segments of about 1 m length. This segmentation results in a network N (e) kj of

Figure 4 . 6

 46 Figure 4.6 Network assignment for topologies (e), (f) and (g): corresponding graph networks for the symmetric, asymmetric and the decentralized distributed network topologies.

Figure 4 . 9

 49 Figure 4.9 Step response for operating pressure around 100 MPa, considering varying parameters (1a/b), constant bulk modulus (2a/b) and constant uid density (3a/b): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow.

Figure 4 .

 4 Figure 4.10 Step response for operating pressure around 200 MPa, considering varying parameters (1c/d), constant bulk modulus (2c/d) and constant uid density (3c/d): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow.

Figure

  Figure 4.11 Step response for operating pressure around 300 MPa, considering varying parameters (1e/f), constant bulk modulus (2e/f) and constant uid density (3e/f): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow.

Figure

  Figure 4.12 Step response for operating pressure around 400 MPa, considering varying parameters (1g/h), constant bulk modulus (2g/h) and constant uid density (3g/h): zoom-in of measured and simulated pressure at pump outtake and cutting head intake for a reference step on the induced overall uid ow.

Figure 4 .

 4 Figure 4.14 Steady-state of measured and simulated pump output pressure (1k/l) and cutting head input pressure (2k/l) around 200 MPa: the proposed modelling methodology is thereby compared to simulations with constant bulk modulus and constant uid density.

Figure 4 .

 4 Figure 4.15 Steady-state of measured and simulated pump output pressure (1m/n) and cutting head input pressure (2m/n) around 400 MPa: the proposed modelling methodology is thereby compared to simulations with constant bulk modulus and constant uid density.

Figure 4

 4 Figure 4.17 Overview of experiment E-4, considering a 1 pump -2 cutting head setup at 200 MPa with dierent nozzle sizes: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles.

Figure 4 .

 4 Figure 4.18 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 200 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b), as well as opening of cutting head 2 (c) and closing of cutting head 2 (d).

Figure 4 .

 4 Figure 4.19 Overview of experiment E-9, considering a 1 pump -2 cutting head setup at 200 MPa with nozzles of the same size: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles.

Figure 4 .

 4 Figure 4.20 Zoom-in pressure trends of the 1 pump -2 cutting head setup trend at 200 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b), as well as opening of cutting head 2 (c) and closing of cutting head 2 (d).

Table 4 .

 4 11: rmse (MPa) to compare the dierent experiments, derived for contour cutting with asymmetric and symmetric cutting heads. Measurement Pos. Fig. (a) Fig. (b) Fig. (c) Fig. (d)

Fig

  13: rmse (MPa) to compare the dierent experiments, derived for cavity cutting with asymmetric and symmetric cutting heads.

  Figure 4.22 (a) for nominal model and (b) considering check-valve hysteresis.

Figure 4 .

 4 Figure 4.21 Check-valve hysteresis at 200 MPa operating pressure of the 1 pump -1 cutting head setup: considering optimal phase-shift (a) and simulated hysteresis (b).

Figure 4 .

 4 Figure 4.22 Check-valve hysteresis at 350 MPa operating pressure of the 1 pump -1 cutting head setup: considering optimal phase-shift (a) and simulated hysteresis (b).

Figure 4 .

 4 Figure 4.23 Overview of experiment D-2 for a 2 pump -2 cutting head setup with symmetric topology: pressure trend p P (t) and pump rate u(t), due to switching pattern d(t) for selected switching cycles.

Figure 4 .

 4 Figure 4.24 Overview of experiment D-5 for a 2 pump -2 cutting head setup with asymmetric topology: pressure trend p P (t) and pump rate u(t), due to switching pattern d(t) for selected switching cycles.

Figure 4 .

 4 Figure 4.25 Zoom-in pressure trends of a 2 pump -2 cutting head setup with symmetric topology: opening of cutting head 1 (a) and closing of cutting head 1 (b).

Figure 4 .

 4 Figure 4.26 Zoom-in pressure trends of a 2 pump -2 cutting head setup with asymmetric topology: opening of cutting head 1 (a) and closing of cutting head 1 (b).

Figure 4 .

 4 Figure 4.27 Start up of experiment D-1, considering a 2 pump -2 cutting head setup at 200b MPa: simulated pressure trends for each pumping chamber, compared to measured pressure at pump outtake, and resulting pump rates for the symmetric distributed network.

Figure 4 .

 4 Figure 4.28 Start up of experiment D-4, considering a 2 pump -2 cutting head setup at 200 MPa: simulated pressure trends for each pumping chamber, compared to measured pressure at pump outtake, and resulting pump rates for the asymmetric distributed network.

Figure 5 . 4

 54 Figure 5.4 Separation of a high-pressure network into network subsections for control design: high-pressure pumps and concatenated cutting heads to introduce uid ow sources and pressure sinks, respectively.

Figure 5 . 5 Figure 5 . 6

 5556 Figure 5.5 Schematic diagram of the high-pressure system: a single high-pressure pump and a cutting head interconnected to a network subsection. The high-pressure pump consist of two linear actuators. Each linear actuator is controlled by motion control, whereby both plungers are synchronized by means of camming.

  Here, reference tracking and disturbance rejection require the open-loop gain |L(jω)| to be large for low frequencies. Introducing therefore the weighting function W 1 (s) for condition |S(jω)| |W 1 (jω)| < 1 yields |1 + L(jω)| > |W 1 (jω)| . (5.36) On the other hand, noise rejection requires the open-loop gain |L(jω)| to be small for high frequencies, hence |T (jω)| ≈ |L(jω)|. This is written as condition |T (jω)| |W 2 (jω)| < 1, considering a weighting function W 2 (s) gives |L(jω)| < |W 2 (jω)| -1 .

Figure 5 . 8

 58 Figure 5.8 Singular values of weighting function W 1 and inverse weighting function W -12 : derived from the hard and soft constraints to specify minimal and minimal bounds for desired performance.

Figure 5 .

 5 Figure 5.10 Non-linear plant model: block diagram of the bilinear rst order model with varying parameters.

Figure 5 .

 5 Figure 5.11 Approximated plant model: block diagram of the linearised model around operating points in state space form.

Figure 5 .

 5 Figure 5.12 Frequency responses for the linearised plant models: wide frequency range on the left hand side and zoomed in on the right hand side, showing minor deviations for the introduced approximations.

Figure 5 .

 5 Figure

Figure 5 .

 5 Figure 5.14 Frequency responses comparing the simplied approximated model in state space form and its transfer function representation: wide frequency range on the left hand side and zoomed in on the right hand side, showing remarkable deviations when lumping uncertain parameters.

Figure 5 .

 5 Figure 5.15 Block diagram of the interconnection system: considering system components, shaping and weighting functions for control design.

Figure 5 .

 5 Figure 5.16 Frequency response of the scaled actuator transfer function G a (s): considering the dynamic ranges of pumping units required for robust control design.

Figure 5 .Figure 5 .

 55 Figure 5.17 Design of experiment to evaluate the actuator transfer function: velocity trajectory (a), position trajectory (b) for pump excitation and realized frequency range (c).

Figure 5 .

 5 Figure 5.19 Frequency response of the scaled nominal plant transfer function G n (s): considered nominal dynamic range of a network subsection required for robust control design.

Figure 5 .

 5 Figure 5.20 Frequency response of the pressure gauge transfer function G s (s): considered dynamic range of the measurement signal required for robust control design.

Figure 5 .

 5 Figure 5.21 Frequency response of the scaled reference shaping function W r (s): considered dynamic range of the reference value required for robust control design. Disturbance shaping function W d : The switching of the on/o-valve aects the water consumption of a cutting head. The resulting output uid ow is considered as a disturbance d(t), which acts against the input uid ow from the high-pressure pump. It renes the disturbed control input u d (t) = u g (t) + d(t), where a scaled disturbance signal d w (t) = [0, 1] switches the on/o-valve with respect to an unknown switching pattern. The switching dynamics of the on/o-valve gives the frequency range for any change on the disturbance. That is represented by the disturbance shaping function

Figure 5 .

 5 Figure 5.22 Frequency response of the scaled disturbance shaping function W d (s): considered dynamic range of the cutting head on/o-valve to shape the disturbance signal.

Figure 5 .

 5 Figure 5.23 Frequency response of the scaled perturbation shaping function W p (s): considered dynamic range of the inverse additive uncertainty to shape the perturbation signal.

Figure 5 .

 5 Figure 5.25 Box-plot of measured noise: residuals for measurements at 100, 200, 300 and 400 MPa pressures reveal evidence for additive noise phenomena.

Figure 5 .

 5 Figure 5.26 Histogram of measured noise: residuals for measurements at 100, 200, 300 and 400 MPa pressure reveal similar variance close to Gaussian distribution.

Figure 5 .

 5 Figure 5.27 Amplitude spectrum of measured noise: FFT for measurements at 100, 200, 300 and 400 MPa pressures, its upper bound and selected shaping function.

Figure 5 .

 5 Figure 5.28 Frequency response of noise shaping function W p (s): considered dynamic range of the noise signal required for robust control design.

Figure 5 .

 5 Figure 5.30 Frequency response of the weighting function W e (s): dynamic range of the weighting on the error signal and lower bounds from control requirements. Control eort weighting function W u : The condition to penalize control eort |W u T | < 1

Figure 5 .

 5 Figure 5.31 Frequency response of the weighting function W u (s): dynamic range of the weighting on the control signal and upper bound from control requirements.

Figure 5 .

 5 Figure 5.32 Frequency response of the shaping functions W p (s): dynamic range of the weighting on the perturbation signal and lower bounds from control requirements.

Figure 5 .

 5 Figure 5.33 Overview of the uncertainty model evaluation process: dierent approaches to describe the varying parameters for control design.

Figure 5 .

 5 Figure 5.34 Classication of uncertainties: time-varying and parametric uncertainties used to describe varying parameters, subject for control design.

Figure 5 .

 5 Figure 5.35 Grouping of uncertain parameters: time-varying and parametric uncertainties used to describe varying parameters, subject for control design.

Figure 5 .

 5 Figure 5.36 Position displacement changes the pumping chamber volume: desired piston positions of two pumping chambers to obtain a continuous overall uid ow.

Figure 5 .

 5 Figure 5.37 Range for known time-varying uncertainties: uid density dependent on the pressure state (a) and pumping chamber volume with respect to the piston displacement (b).

Figure 5 .

 5 Figure 5.38 Uncertainty range for unknown parametric uncertainties: ow resistance considering dierent nozzles (a) and overall volume for various high-pressure networks (b).

Figure 5 .Figure 5 .

 55 Figure 5.39 Frequency responses of approximated and extended models for u → y: varying a single uncertain parameter at once, where the others stay constant on nominal value.

Figure 5 .

 5 Figure 5.42 Block diagram of simplied approximated model with real parametric uncertainties: additive uncertainty and inverse additive uncertainty description used to separate these uncertainties from the plant model. This perturbed plant includes the perturbation inputs d(t) = [d 1 d 2 d 3 d 4 ] , which provide the outputs e(t) = [e 1 e 2 e 3 e 4 ] . This allows for representing the nominal plant in state space description

Figure 5 .

 5 Figure 5.43 Block diagram of the nominal plant: representing the simplied approximated model for nominal parameter values.

Figure 5 .

 5 Figure 5.44 Block diagram of unstructured uncertainty descriptions: additive perturbation from plant input to output (a), input multiplicative perturbation from plant input to input (b), inverse additive perturbation from plant output to input (c) and inverse output multiplicative perturbation form plant output to output (d). The additive uncertainty representation G p (s) = G n (s) + ∆ a (s)

  |W a (jω) ∆(jω)| = |G p (jω) -G n (jω)| ≤ |W a (jω)| (5.127) and introducing (5.122) to (5.120) yields |W m (jω) ∆(jω)| = |G p (jω) -G n (jω)| |G n (jω)| ≤ |W m (jω)| .

Figure 5 .

 5 Figure 5.45 Weighting functions for additive and multiplicative uncertainties: error frequency responses in additive form, considering absolute error (a) and multiplicative form for relative error (b).

Figure 5 .

 5 Figure 5.46 Perturbed plant using additive (a) and multiplicative (b) descriptions: unstructured uncertainties compared to parametric uncertainty description reveal inadequate behaviour.

Figure 5 .

 5 Figure 5.47 Weighting functions to obtain the inverse additive and inverse multiplicative uncertainties: error frequency responses in inverse additive form, considering an absolute error (a) and inverse multiplicative form for relative error (b).

Figure 5 .

 5 Figure 5.48 Perturbed plant using inverse additive (a) and inverse multiplicative (b) descriptions: unstructured uncertainties show adequate behaviour compared to parametric uncertainties.

Figure 5 .

 5 Figure 5.49 Block diagram for robust stabilization of the investigated feedback system: one degree of freedom state feedback controller K(s) interconnected to the perturbed plant G p (s).

minimizeKWW

  ia (jω) G p (jω) (I + K(jω) G p (jω)) im (jω) (I + K(jω) G p (jω))

Figure 6 . 1

 61 Figure 6.1 Overview of the control design process: structured and unstructured controllers compared by means of simulations to derive a robust control design for high-pressure pumps.

Figure 6 . 2

 62 Figure 6.2 Block diagram of generalized plant for control design: considering system components, input and output weighting functions.

Figure 6 . 4

 64 Figure 6.4 Frequency responses of the nominal generalized plant with state feedback controller K SF : evaluating nominal performance (a), control eort (b) and robust stability (c).

  generalized plant T := W U Ṽ will be obtained. It approximates the optimization problem for T ∞ < γ and thereby reduces the resulting optimization problem of dimension n × (m -k + 1).

√ 2 ≈ 3

 23 dB magnitude at any ω where |T 1 (jω)| = |T 2 (jω)|.

  shaping function W rn (s) of Figure 6.6 (a) has been obtained. Repeating this procedure for the additively interconnected input signals d and p yields σ (W dp ) = |W d | 2 + |W p | 2 , (6.17) 206 which has been approximated by the joint shaping function W dp (s), shown in Figure 6.6 (b).

Figure 6 . 6

 66 Figure 6.6 Frequency response of the joint shaping functions: dynamic range of the shaping functions on the reference and noise signals (a) and dynamic range of the shaping functions on the disturbance and perturbation signals (b).

Figure 6 . 7

 67 Figure

Figure 6 . 8

 68 Figure 6.8 Closed-loop frequency responses (a) and step responses (b) from d → y: implementing controller K SF , when optimizing for the full generalized plant (independent shaping functions), and controller KSF from the reduced generalized plant (joint shaping functions).

Figure 6 . 9

 69 Figure 6.9 Frequency responses of the nominal generalized plant with reduced state feedback controller KSF , when considering joint shaping functions: evaluating nominal performance (a), control eort (b) and robust stability (c).

Figure 6 .

 6 Figure 6.10 Frequency response of the inverse weighting functions |W e (s)| and |W u (s)|, considering the reduced state feedback controller: dynamic range of generalized plant |T (s)| from d → e (a) and its dynamic range from d → u (b).

Figure 6 .

 6 Figure 6.11 Closed-loop frequency responses (a) and step responses (b) from d → y: implementing state feedback controller KSF and PI controller K P I , when optimizing for the reduced generalized plant (using joint shaping functions).

Figure 6 .

 6 Figure 6.12 Frequency response of the inverse weighting functions |W e (s)| and |W u (s)|, considering the reduced PI controller: dynamic range of generalized plant |T (s)| from d → e (a) and its dynamic range from d → u (b).

Figure 6 .

 6 Figure 6.13 Closed-loop frequency responses (a) and step responses (b) from d → y: implementing state feedback controller KSF and extended PI controller KP I with low-pass lter, when optimizing for the reduced generalized plant (using joint shaping functions).

Figure 6 .

 6 Figure 6.14 Frequency response of the inverse weighting functions |W e (s)| and |W u (s)| considering the extended PI controller with low-pass lter: dynamic range of generalized plant |T (s)| from d → e (a) and its dynamic range from d → u (b).

Figure 6 .

 6 Figure 6.15 Block diagram of standard ∆=M=K conguration: interconnection system M connecting uncertainty block ∆ and stabilizing controller K. ∆ and M can be concatenated by a upper linear fractional transformation to describe the perturbed plant P used for µ-synthesis.

Figure 6 .

 6 Figure 6.16 Achieved robustness margin for dierent uncertainty ranges (a) and resulting perturbation shaping functions (b), applying ∆-K iteration: peak values γ for dierent uid ow volumes to evaluate the maximal network subsection size and manipulating the inverse additive uncertainty to shape the perturbation signal.

  ### ### 0.999803 9.5069 6 ### ### ### 0.999961 9.5087 7 ### ### ### 0.999994 9.5091

Figure 6 .

 6 Figure 6.18 Closed-loop frequency responses (a) and step responses (b) from d → y: implementing extended PI controller KP I with low-pass lter and quasi-optimal robust PI controller K * P I , when optimizing for the reduced generalized plant (using joint shaping functions).

Figure 6 .

 6 Figure 6.19 Frequency responses of the nominal generalized plant with quasi-optimal robust PI controller K * P I , when considering joint shaping functions: evaluating robust performance (a), control eort (b) and stability (c).

Figure 6 .

 6 Figure 6.20 Frequency responses of the weighting functions: dynamic range of the weighting on the perturbation signal and lower bound from control requirements.

Figure 6 .

 6 Figure 6.22 High-pressure network topology used for control design verication: single pump interconnected to two cutting heads with identical nozzles.

Figure 6 .

 6 Figure 6.23 Corresponding graph network of the above network topologies used for numerical simulations: single pump interconnected to two cutting heads, considering a symmetric network topology.

Figure 6 .

 6 Figure 6.24 Step responses of robust PI controller at 100 MPa operating point, when opening (a) and closing (b) a cutting head: nominal plant, expected upper and lower bounds from control design compared to resulting step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.25 Step responses of robust PI controller at 200 MPa operating point, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to resulting step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.26 Step responses of robust PI controller at 300 MPa operating point, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to resulting step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.27 Step responses of robust PI controller at 400 MPa operating point, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to resulting step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.28 Step responses of extended quasi-optimal PI controller at 100 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.29 Step responses of extended quasi-optimal PI controller at 200 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.30 Step responses of extended quasi-optimal PI controller at 300 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.31 Step responses of extended quasi-optimal PI controller at 400 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.33 Step responses of optimized state feedback controller at 200 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.34 Step responses of optimized state feedback controller at 300 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model.

Figure 6 .

 6 Figure 6.35 Step responses of optimized state feedback controller at 400 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model.

Figure 6 .Figure 6 .

 66 Figure 6.37 Switching pattern considering 4 transitions for 200 MPa operating pressure of experiment E-4: desired switching states resulting in dierent pump rates for disturbance rejection by means of control.

Figure 6 .

 6 Figure 6.39 Disturbance steps at 200 MPa operating point for asymmetric cutting heads (E-4): resulting step responses from non-linear simulation model and measured step responses from experiment. Nozzle 1 closes while nozzle 2 remains open (a), nozzle 1 opens while nozzle 2 remains close (b), nozzle 2 closes while nozzle 1 remains close (c) and nozzle 2 opens while nozzle 1 remains open (d).

Figure 6 .

 6 Figure 6.40 Disturbance steps at 200 MPa operating point for symmetric cutting heads (E-9): resulting step responses from non-linear simulation model and measured step responses from experiment. Nozzle 1 closes while nozzle 2 remains open (a), nozzle 1 opens while nozzle 2 remains close (b), nozzle 2 closes while nozzle 1 remains close (c) and nozzle 2 opens while nozzle 1 remains open (d).

  and 6.42. These plots overview the experiments E-4 and E-9, representing the asymmetric and symmetric cutting heads at 200 MPa. The plots for the inverse 4 transitions as well as the complementary experiments for controller validation at 350 MPa (E-7 and E-11) are provided in the Appendix B.2.

Figure 6 .Figure 6 .

 66 Figure 6.41 Switching pattern considering 4 transitions for 200 MPa operating pressure of experiment E-4: desired switching states resulting in dierent pump rates for disturbance rejection by means of control.

Figure 6 .

 6 Figure 6.43 Disturbance steps at 200 MPa operating point for asymmetric cutting heads (E-4): resulting step responses from non-linear simulation model. Nozzle 1 closes while nozzle 2 remains open (a), nozzle 1 opens while nozzle 2 remains close (b), nozzle 2 closes while nozzle 1 remains close (c) and nozzle 2 opens while nozzle 1 remains open (d).

Figure 6 .

 6 Figure 6.44 Disturbance steps at 200 MPa operating point for symmetric cutting heads (E-9): resulting step responses from non-linear simulation model. Nozzle 1 closes while nozzle 2 remains open (a), nozzle 1 opens while nozzle 2 remains close (b), nozzle 2 closes while nozzle 1 remains close (c) and nozzle 2 opens while nozzle 1 remains open (d).

Figure 6 .

 6 Figure 6.45 Control surface obtained for gain-scheduling: Controller gains with respect to measured operating pressure and varying pumping chamber volume due to piston displacement.

Figure 6 .

 6 Figure 6.48 Switching pattern for 200 MPa operating pressure of experiment D-5, considering an initial PI controller: desired switching states resulting in dierent pump rates for disturbance rejection by means of control to reference pressure.

Figure 6 .Figure 6 .

 66 Figure 6.49 Disturbance steps at 200 MPa operating point for symmetric distributed pumps (D-2), considering an initial PI controller: resulting step responses from non-linear simulation model and measured step responses from experiment. Nozzle 2 closes (a) and opens (b) while nozzle 1 remains open.

Figure 6 .Figure 6 .Figure 6 .Figure 6 .

 6666 Figure 6.51 Switching pattern for 200 MPa operating pressure of experiment D-2, considering the quasi-optimal PI controller: desired switching states resulting in dierent pump rates for disturbance rejection by means of control to reference pressure.

Figure 7 . 1

 71 Figure 7.1 Overview of managing approaches, aiming at hierarchical distributed control for high-pressure pumps: model predictive control, control allocation and cooperative control are distinguished with emphasis on distributed algorithms.

Figure 7 . 3 N

 73 Figure 7.3 N vertices interconnected to a ring topology using edges of unit weights: considering directed (a) and undirected (b) data transmission, resulting in strongly connected and weight balanced graphs.

  .9)Considering the same network topology as a directed, strongly connected and weight-balanced graphs (Figure7.3 (a)), yields the matrices

Figure 7 . 5

 75 Figure 7.5 Application of dynamic average consensus to interconnected high-pressure pumps: local pump rate and information from neighbouring pump agents are used to estimate the average water consumption.

Figure 7 . 6

 76 Figure 7.6 Application of distributed balancing to interconnected high-pressure pumps: local pump rate and information from neighbouring pump agents are used to adjust the local pressure reference values, this cause the pump rates to converge to desired average value.

Figure 7 . 7

 77 Figure 7.7 Floating operation of 2 interconnected pumps: switching cutting heads causing the pumps to run at dierent pump rates, dependent on it position in the high-pressure network.

Figure 7 . 9

 79 Figure 7.9 Balanced operation of 2 interconnected pumps: distributed balancing causing the pumps to converge towards the average pump rates by manipulating the local reference pressures.

Figure 7 .

 7 Figure 7.10 Distributed balancing to adjust the reference pressure for each high-pressure pump: distributed pump agents estimating the desired reference pressure for dierent switching states and various rate of convergence.

Figure 7 .

 7 Figure 7.11 Distributed balancing to obtain various operating pressures for each high-pressure pump: distributed pump agents estimating the desired reference pressure for dierent switching states and various rate of convergence.

Figure 7 .Figure 7 .

 77 Figure 7.12 Distributed balancing causing each high-pressure pump to average pump rate: distributed pump agents estimating the desired reference pressure for dierent switching states and various rate of convergence.

Figure A. 2

 2 Figure A.2 Start up and shut down ramping of measurement B-7 considering a 1 pump -1 cutting head setup at 350 MPa: start up pressure trend (a) and shut down pressure trend (b).

Figure A. 3

 3 Figure A.3 Zoom-in around 100 MPa of ramping for 200 MPa operating point: pressure trends during start up (a) and shut down (b).

Figure A. 6

 6 Figure A.6 Zoom-in around 300 MPa of ramping for 350 MPa operating point: pressure trends during start up (a) and shut down (b).

Figure A. 9

 9 Figure A.9 Switching cutting head of the 1 pump -1 cutting head setup at 200 MPa: closing cutting head pressure trend (a) and opening cutting head pressure trend (b).

Figure A. 10

 10 Figure A.10 Switching cutting head of the 1 pump -1 cutting head setup at 350 MPa: closing cutting head pressure trend (a) and opening cutting head pressure trend (b).

Figure A

  Figure A.12 shows detailed trends for the input and system pressure in the rst plot, the output pressures in the second plot, as well as the switching pattern in the third plot.

Figure A. 12

 12 Figure A.12 Fast switching disturbance for a 1 pump -1 cutting head setup: zoom-in at dierent sample times.

Figure

  Figure A.14 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 350 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b).

Figure

  Figure A.15 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 350 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b).

Figure

  Figure A.17 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 350 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b).

Figure

  Figure A.18 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 350 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b).

Figure A. 20

 20 Figure A.20 Fast switching disturbance for a 1 pump -2 cutting head setup: zoom-in when switching of head 1.

Figure A. 21

 21 Figure A.21 Fast switching disturbance for a 1 pump -2 cutting head setup: zoom-in when almost simultaneously switching of head 1 and 2.

Figure A. 23

 23 Figure A.23 Overview of experiment D-3 considering fast switching disturbance for a 2 pump -2 cutting head setup with symmetric topology: cutting head 1 remains open while head 2 switches periodically.

Figure A. 24

 24 Figure A.24 Overview of experiment D-6 considering fast switching disturbance for a 2 pump -2 cutting head setup with asymmetric topology: cutting head 1 remains open while head 2 switches periodically.

Figure B. 2

 2 Figure B.2 Switching pattern A for 350 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure.

Figure B. 3

 3 Figure B.3 Inverse switching pattern B for 350 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure.

Figure B. 6

 6 Figure B.6 Disturbance steps at 350 MPa operating point, switching states (a): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains open.

Figure B. 7

 7 Figure B.7 Disturbance steps at 350 MPa operating point, switching states (b): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains closed.

Figure B. 13

 13 Figure B.13 Disturbance steps at 200 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open.

Figure B. 14

 14 Figure B.14 Disturbance steps at 200 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed.

Figure B. 17

 17 Figure B.17 Disturbance steps at 350 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open.

Figure B. 18

 18 Figure B.18 Disturbance steps at 350 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed.

Figure B. 20

 20 Figure B.20 Switching pattern A for 350 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure.

Figure B. 21

 21 Figure B.21 Inverse switching pattern B for 350 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure.

Figure B. 24

 24 Figure B.24 Disturbance steps at 350 MPa operating point, switching states (a): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains open.

Figure B. 25

 25 Figure B.25 Disturbance steps at 350 MPa operating point, switching states (b): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains closed.

Figure B. 31

 31 Figure B.31 Disturbance steps at 200 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open.

Figure B. 32

 32 Figure B.32 Disturbance steps at 200 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed.

Figure B. 35

 35 Figure B.35 Disturbance steps at 350 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open.

Figure B. 36

 36 Figure B.36 Disturbance steps at 350 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Steady-state of measured and simulated pump output pressure (1k/l) and cutting Steady-state of measured and simulated pump output pressure (1m/n) and cutting head input pressure (2m/n) around 400 MPa: the proposed modelling methodology is thereby compared to simulations with constant bulk modulus and constant uid density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Experimental setup for model validation: pressure gauges measure the pressures p x (t) at dierent positions, while applying an input uid ows Q(t). The estimated input uid ows Q(t) are used as model inputs and the simulated pressure trends px (t) have been compared to the measurements. . . . . . . . . . . . . . . . . . . . . . . 4.17 Overview of experiment E-4, considering a 1 pump -2 cutting head setup at 200 MPa with dierent nozzle sizes: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles. . . . . . . . . . . . . . . . . . 4.18 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 200 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b), as well as opening of cutting head 2 (c) and closing of cutting head 2 (d). . . . . . . . . . . . . . . . . . . . . . 4.19 Overview of experiment E-9, considering a 1 pump -2 cutting head setup at 200 MPa with nozzles of the same size: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles. . . . . . . . . . . . . . . .

head input pressure (2k/l) around 200 MPa: the proposed modelling methodology is thereby compared to simulations with constant bulk modulus and constant uid density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi 4.15

  Frequency response of the joint shaping functions: dynamic range of the shaping functions on the reference and noise signals (a) and dynamic range of the shaping functions on the disturbance and perturbation signals (b). . . . . . . . . . . . . . 6.7 Block diagram of reduced generalized plant for controller synthesis: taking joint shaping functions into account for input signals dp and rn. . . . . . . . . . . . . . 6.8 Closed-loop frequency responses (a) and step responses (b) from d → y: implementing controller K SF , when optimizing for the full generalized plant (independent shaping functions), and controller KSF from the reduced generalized plant (joint shaping functions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Frequency responses of the nominal generalized plant with reduced state feedback controller Frequency response of the inverse weighting functions |W e (s)| and |W u (s)|, considering the reduced state feedback controller: dynamic range of generalized plant |T (s)| from d → e (a) and its dynamic range from d → u (b). . . . . . . . . . . . 6.11 Closed-loop frequency responses (a) and step responses (b) fromd → y: implementing state feedback controller KSF and PI controller K P I , when optimizing for the reduced generalized plant (using joint shaping functions). . . . . . . . . . 6.12 Frequency response of the inverse weighting functions |W e (s)| and |W u (s)|, considering the reduced PI controller: dynamic range of generalized plant |T (s)| from d → e (a) and its dynamic range from d → u (b). . . . . . . . . . . . . . . . . . . 6.13 Closed-loop frequency responses (a) and step responses (b) from d → y: implementing state feedback controller KSF and extended PI controller KP I with lowpass lter, when optimizing for the reduced generalized plant (using joint shaping functions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Frequency response of the inverse weighting functions |W e (s)| and |W shaping functions). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.19 Frequency responses of the nominal generalized plant with quasi-optimal robust PI controller K * P I , when considering joint shaping functions: evaluating robust performance (a), control eort (b) and stability (c). . . . . . . . . . . . . . . . . . 6.20 Frequency responses of the weighting functions: dynamic range of the weighting on the perturbation signal and lower bound from control requirements. . . . . . . 6.21 Approximated plant model: block diagram of the simplied approximated model in state space form with exogenous disturbance of nominal amplitude. . . . . . . 6.22 High-pressure network topology used for control design verication: single pump interconnected to two cutting heads with identical nozzles. . . . . . . . . . . . . . 6.23 Corresponding graph network of the above network topologies used for numerical simulations: single pump interconnected to two cutting heads, considering a symmetric network topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.24 Step responses of robust PI controller at 100 MPa operating point, when open-Step responses of robust PI controller at 200 MPa operating point, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to resulting step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.26 Step responses of robust PI controller at 300 MPa operating point, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to resulting step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.27 Step responses of robust PI controller at 400 MPa operating point, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to resulting step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.28 Step responses of extended quasi-optimal PI controller at 100 MPa, when open-Experimental setup for controller validation: pressure gauges measure the pressure p x (t) at dierent positions, the input uid ows Q(t) vary with respect to the switching disturbance d(t). Reference pressure r(t) and disturbance d(t) are used as the model inputs. The estimated pressures px (t) have been compared to the measured pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.37 Switching pattern considering 4 transitions for 200 MPa operating pressure of experiment E-4: desired switching states resulting in dierent pump rates for disturbance rejection by means of control. . . . . . . . . . . . . . . . . . . . . . . . . 6.38 Switching pattern considering 4 transitions for 200 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means of control. . . . . . . . . . . . . . . . . . . . . . . . . 6.39 Disturbance steps at 200 MPa operating point for asymmetric cutting heads (E-4): resulting step responses from non-linear simulation model and measured step .45 Control surface obtained for gain-scheduling: Controller gains with respect to measured operating pressure and varying pumping chamber volume due to piston displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.46 Step responses of PI gain scheduling controller K * PI with low-pass lter and quasioptimal robust PI design:considering a reduced generalized plant derived from joint shaping functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Switching pattern for 200 MPa operating pressure of experiment D-5, considering an initial PI controller: desired switching states resulting in dierent pump rates for disturbance rejection by means of control to reference pressure. . . . . . . . . 6.49 Disturbance steps at 200 MPa operating point for symmetric distributed pumps (D-2), considering an initial PI controller: resulting step responses from non-linear simulation model and measured step responses from experiment. Nozzle 2 closes (a) and opens (b) while nozzle 1 remains open. . . . . . . . . . . . . . . . . . . . 6.50 Disturbance steps at 200 MPa operating point for asymmetric distributed pumps (D-5), considering an initial PI controller: resulting step responses from non-linear simulation model and measured step responses from experiment. Nozzle 2 closes (a) and opens (b) while nozzle 1 remains open. . . . . . . . . . . . . . . . . . . . 6.51 Switching pattern for 200 MPa operating pressure of experiment D-2, considering the quasi-optimal PI controller: desired switching states resulting in dierent pump rates for disturbance rejection by means of control to reference pressure. . . . . . 6.52 Switching pattern for 200 MPa operating pressure of experiment D-5, considering the quasi-optimal PI controller: desired switching states resulting in dierent pump rates for disturbance rejection by means of control to reference pressure. . . . . . 6.53 Disturbance steps at 200 MPa operating point for symmetric distributed pumps (D-Proposed hierarchical distributed control design for high-pressure pumps: each pump is robustly controlled at low-level to stabilize a network subsection and managed on high-level to balance the pump rate with respect to the overall water consumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 N vertices interconnected to a ring topology using edges of unit weights: considering directed (a) and undirected (b) data transmission, resulting in strongly connected and weight balanced graphs. . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Assignment of a high-pressure network to pump agents subject for distributed managing: each pump is controlled with a decentralized PI controller and feeds a network subsection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of distributed balancing to interconnected high-pressure pumps: local pump rate and information from neighbouring pump agents are used to adjust the local pressure reference values, this cause the pump rates to converge to desired average value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Floating operation of 2 interconnected pumps: switching cutting heads causing the pumps to run at dierent pump rates, dependent on it position in the high-pressure network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Dynamic average consensus estimating the overall average pump rate: distributed pump agents estimating the average pump rates for dierent switching states and various rate of convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Balanced operation of 2 interconnected pumps: distributed balancing causing the pumps to converge towards the average pump rates by manipulating the local reference pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Distributed balancing to adjust the reference pressure for each high-pressure pump: distributed pump agents estimating the desired reference pressure for dierent switching states and various rate of convergence. . . . . . . . . . . . . . . . . . . 7.11 Distributed balancing to obtain various operating pressures for each high-pressure

	6.34 Step responses of optimized state feedback controller at 300 MPa, when open-
	ing (a) and closing (b) a cutting head: nominal plant, upper and lower bounds
	from control design compared to step responses from numerical simulations, using
	non-linear model.

5.49 Block diagram for robust stabilization of the investigated feedback system: one degree of freedom state feedback controller K(s) interconnected to the perturbed plant G p (s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Overview of the control design process: structured and unstructured controllers compared by means of simulations to derive a robust control design for highpressure pumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Block diagram of generalized plant for control design: considering system components, input and output weighting functions. . . . . . . . . . . . . . . . . . . . . . 6.3 Step responses of state feedback controller K SF : applying unit steps on the dierent input channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Frequency responses of the nominal generalized plant with state feedback controller K SF : evaluating nominal performance (a), control eort (b) and robust stability (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Frequency responses of nominal generalized plant with state feedback controller K SF : evaluating for robust performance and stability with respect to the Euclidean vector norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 KSF , when considering joint shaping functions: evaluating nominal performance (a), control eort (b) and robust stability (c). . . . . . . . . . . . . . . xxxv 6.10 u (s)| considering the extended PI controller with low-pass lter: dynamic range of generalized plant |T (s)| from d → e (a) and its dynamic range from d → u (b). . . . . . . . . 6.15 Block diagram of standard ∆=M=K conguration: interconnection system M connecting uncertainty block ∆ and stabilizing controller K. ∆ and M can be concatenated by a upper linear fractional transformation to describe the perturbed plant P used for µ-synthesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 Achieved robustness margin for dierent uncertainty ranges (a) and resulting perturbation shaping functions (b), applying ∆-K iteration: peak values γ for dierent uid ow volumes to evaluate the maximal network subsection size and manipulating the inverse additive uncertainty to shape the perturbation signal. . . . . . 6.17 Singular values of the nominal generalized plant with quasi-optimal robust PI controller K * P I : evaluating for robust performance and stability with respect to the Euclidean vector norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.18 Closed-loop frequency responses (a) and step responses (b) from d → y: implementing extended PI controller KP I with low-pass lter and quasi-optimal robust PI controller K * P I , when optimizing for the reduced generalized plant (using joint ing (a) and closing (b) a cutting head: nominal plant, expected upper and lower bounds from control design compared to resulting step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi 6.25 ing (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.29 Step responses of extended quasi-optimal PI controller at 200 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.30 Step responses of extended quasi-optimal PI controller at 300 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.31 Step responses of extended quasi-optimal PI controller at 400 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.32 Step responses of optimized state feedback controller at 100 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.33 Step responses of optimized state feedback controller at 200 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using nonlinear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.35 Step responses of optimized state feedback controller at 400 MPa, when opening (a) and closing (b) a cutting head: nominal plant, upper and lower bounds from control design compared to step responses from numerical simulations, using non-linear model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxxvii 6.36 responses from experiment. Nozzle 1 closes while nozzle 2 remains open (a), nozzle 1 opens while nozzle 2 remains close (b), nozzle 2 closes while nozzle 1 remains close (c) and nozzle 2 opens while nozzle 1 remains open (d). . . . . . . . . . . . 6.40 Disturbance steps at 200 MPa operating point for symmetric cutting heads (E-9): resulting step responses from non-linear simulation model and measured step responses from experiment. Nozzle 1 closes while nozzle 2 remains open (a), nozzle 1 opens while nozzle 2 remains close (b), nozzle 2 closes while nozzle 1 remains close (c) and nozzle 2 opens while nozzle 1 remains open (d). . . . . . . . . . . . 6.41 Switching pattern considering 4 transitions for 200 MPa operating pressure of experiment E-4: desired switching states resulting in dierent pump rates for disturbance rejection by means of control. . . . . . . . . . . . . . . . . . . . . . . . . 6.42 Switching pattern considering 4 transitions for 200 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means of control. . . . . . . . . . . . . . . . . . . . . . . . . 6.43 Disturbance steps at 200 MPa operating point for asymmetric cutting heads (E-4): resulting step responses from non-linear simulation model. Nozzle 1 closes while nozzle 2 remains open (a), nozzle 1 opens while nozzle 2 remains close (b), nozzle 2 closes while nozzle 1 remains close (c) and nozzle 2 opens while nozzle 1 remains open (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.44 Disturbance steps at 200 MPa operating point for symmetric cutting heads (E-9): resulting step responses from non-linear simulation model. Nozzle 1 closes while nozzle 2 remains open (a), nozzle 1 opens while nozzle 2 remains close (b), nozzle 2 closes while nozzle 1 remains close (c) and nozzle 2 opens while nozzle 1 remains open (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66.47 Switching pattern for 200 MPa operating pressure of experiment D-2, considering an initial PI controller: desired switching states resulting in dierent pump rates for disturbance rejection by means of control to reference pressure. . . . . . . . . xxxviii 6.48 2), considering the quasi-optimal PI controller: resulting step responses from nonlinear simulation model. Nozzle 2 closes (a) and opens (b) while nozzle 1 remains open. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.54 Disturbance steps at 200 MPa operating point for asymmetric distributed pumps (D-5), considering the quasi-optimal PI controller: resulting step responses from nonlinear simulation model. Nozzle 2 closes (a) and opens (b) while nozzle 1 remains open. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.55 Applied concept for independent control design compared to decentralized control: while the independent control design considers other network subsections as exogenous disturbances, the distributed control design takes the dynamic coupling between neighbouring pumps into account. . . . . . . . . . . . . . . . . . . . . . .

7.1 Overview of managing approaches, aiming at hierarchical distributed control for high-pressure pumps: model predictive control, control allocation and cooperative control are distinguished with emphasis on distributed algorithms. . . . . . . . . 7.2 7.5 Application of dynamic average consensus to interconnected high-pressure pumps: local pump rate and information from neighbouring pump agents are used to estimate the average water consumption. . . . . . . . . . . . . . . . . . . . . . . . xxxix 7.6 pump: distributed pump agents estimating the desired reference pressure for different switching states and various rate of convergence. . . . . . . . . . . . . . . . 7.12 Distributed balancing causing each high-pressure pump to average pump rate: distributed pump agents estimating the desired reference pressure for dierent switching states and various rate of convergence. . . . . . . . . . . . . . . . . . . 7.13 Average consensus (left

  . . xl A.11 Overview of experiment B-5: fast switching disturbance for a 1 pump -1 cutting head setup at 200 MPa operating point. . . . . . . . . . . . . . . . . . . . . . . . A.12 Fast switching disturbance for a 1 pump -1 cutting head setup: zoom-in at dierent sample times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.13 Overview of experiment E-9 considering a 1 pump -2 cutting head setup at 350 MPa with dierent nozzle sizes: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles. . . . . . . . . . . . . . . . . . A.14 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 350 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b). . . . . . . . . . . . . . . . A.15 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 350 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b). . . . . . . . . . . . . . . . A.16 Overview of experiment E-13 considering the 1 pump -2 cutting head setup at 350 MPa with nozzles of the same size: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles. . . . . . . . . . . . . . A.17 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 350 MPa: opening of cutting head 1 (a) and closing of cutting head 1 (b). . . . . . . . . . . . . . . . A.18 Zoom-in pressure trends of the 1 pump -2 cutting head setup at 350 MPa: opening Disturbance steps at 200 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open. . . . . B.5 Disturbance steps at 200 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed. . . . . B.6 Disturbance steps at 350 MPa operating point, switching states (a): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains open. . . . . B.7 Disturbance steps at 350 MPa operating point, switching states (b): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains closed. . . . . B.8 Disturbance steps at 350 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open. . . . . B.9 Disturbance steps at 350 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed. . . . . B.10 Inverse switching pattern B for 200 MPa operating pressure of experiment E-11: .15 Disturbance steps at 350 MPa operating point, switching states (a): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains open. . . . . B.16 Disturbance steps at 350 MPa operating point, switching states (b): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains closed. . . . . B.17 Disturbance steps at 350 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open. . . . . B.18 Disturbance steps at 350 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed. . . . . Switching pattern A for 350 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . B.21 Inverse switching pattern B for 350 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . B.22 Disturbance steps at 200 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open. . . . . B.23 Disturbance steps at 200 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed. . . . . B.24 Disturbance steps at 350 MPa operating point, switching states (a): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains open. . . . . B.25 Disturbance steps at 350 MPa operating point, switching states (b): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains closed. . . . . .31 Disturbance steps at 200 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open. . . . . B.32 Disturbance steps at 200 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed. . . . . B.33 Disturbance steps at 350 MPa operating point, switching states (a): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains open. . . . . B.34 Disturbance steps at 350 MPa operating point, switching states (b): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 1 switches while nozzle 2 remains closed. . . . . Disturbance steps at 350 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed. . . . . 323 Properties of a hydraulically driven high-pressure pump (Type JCP 22 LC): with a hydraulic pumping unit driven by a 22 kW asynchronous motor. . . . . . . . . . 1.2 Properties of cutting head nozzles: nozzles inner diameter and operating pressures determining the output uid ow (l/min). . . . . . . . . . . . . . . . . . . . . . . 1.3 Work station setups: waterjet machining classied according to typical work station setups found in industry and assigned to waterjet applications. . . . . . . . . 1.4 Cutting head congurations: typical parametrisation for common waterjet appli-Conguration for use case 3: measurements derived on dierent test bench setups to validate high-pressure network models and low-level controllers as well as to verify the high-level managing design. . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Switching patterns: specication of switching patterns with respect to typical manufacturing processes used for experiments. . . . . . . . . . . . . . . . . . . . . . . 2.7 Properties of a electrically driven high-pressure pump: high pressure pump prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 High-pressure pump hardware conguration: hardware used to built an electrically driven pump that consists of two pumping units. . . . . . . . . . . . . . . . . . . 2.9 Parametrization of motion control: control loop bandwidth, forward gains, feedback lter and load observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Congurations for numerical simulations with boundary conditions A and B: initial conditions, resulting equilibrium points and relative delays per piping length. . . 3.3 Simulation error comparing distributed and lumped parameter model: modelling a piping of 20 m length with dierent numbers of homogeneous segments. . . . . 3.4 Excerpt of experiments for use case 1 and 2: measurements derived on dierent test bench setups to identify dierent network parameters. . . . . . . . . . . . . . 3.5 Design of experiment for parameter identication to determine the pressure-dependant bulk modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Identied coecients of the parametric model and its condence intervals describing the pressure-dependent bulk modulus. . . . . . . . . . . . . . . . . . . . . . . 3.7 Identied discharge coecient with linear regression and its condence interval. . 3.8 Identied friction loss coecient with linear regression and its condence intervals. 4.1 List of nodes and branches for topology (a): assignment of the dierent highpressure components to nodes k and branches kj. . . . . . . . . . . . . . . . . . . 4.2 Model parameters for topology (a): parameters of each segment, corresponding to (4.12), (4.13) and grouped in nodes and branches. . . . . . . . . . . . . . . . . . . 4.3 List of nodes and branches for topology (b): assignment of the dierent highpressure components to nodes .10 rmse (MPa) to compare the dierent experiments, derived for contour cutting with a single cutting head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 rmse (MPa) to compare the dierent experiments, derived for contour cutting with asymmetric and symmetric cutting heads. . . . . . . . . . . . . . . . . . . . 4.12 rmse (MPa) to compare the dierent experiments, derived for cavity cutting with a single cutting head. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 rmse (MPa) to compare the dierent experiments, derived for cavity cutting with asymmetric and symmetric cutting heads. . . . . . . . . . . . . . . . . . . . . . . 4.14 Excerpt of experiments for use case 3: measurements derived with dierent test bench setups and used for model validation. . . . . . . . . . . . . . . . . . . . . . 4.15 rmse (MPa) to compare the dierent experiments, derived for contour cutting Time domain control objectives for disturbance rejection: hard constraints for unit step d(t) → y(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Time domain control objectives for reference tracking: soft constraints for unit step r(t) → y(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Uncertainty range of varying parameters obtained from test bench setup and highpressure pump design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Uncertainty range of lumped parameters used for transfer function description. . 5.5 Signals scaling to bound the amplitudes of an interconnection system to unit gain. 5.6 Properties of the measurement signal derived from 2 pressure gauges for dierent measurements at 100, 200, 300 and 400 MPa operating points. . . . . . . . . . . . 5.7 System gain for approximated and extended non-linear model: varying a single parameter, while holding each other parameter on nominal value. . . . . . . . . . 5.8 System gain for approximated linearised model: varying a single parameter, while holding each other parameter on nominal value to compare u → y and d → y. . . 5.9 Time constants for approximated linearised model: varying a single parameter, while holding each other parameter on nominal value for u → y. . . . . . . . . . . Achieved performance by means of detailed numerical simulations for dierent control structures: comparing achieved closed-loop performance from simulation with results from control design, using simplied and linearised model. . . . . . . 6.9 Excerpt of experiments for use case 2: measurements derived with dierent test bench setups and used for controller validation. . . . . . . . . . . . . . . . . . . . 6.10 Achieved performance by means of experiments at the test bench for dierent network topologies: comparing closed-loop performance from measurements with results from simulations, using detailed simulation model. . . . . . . . . . . . . . 6.11 Achieved performance by means of simulations for dierent network topologies: providing closed-loop performance from simulations, using detailed simulation model with quasi-optimal robust PI controller. . . . . . . . . . . . . . . . . . . . Excerpt of experiments for use case 3: measurements derived on dierent test bench setups used for controller validation. . . . . . . . . . . . . . . . . . . . . . . 7.1 Excerpt of experiments for use case 3: measurements derived on dierent test bench setups used for high-level managing. . . . . . . . . . . . . . . . . . . . . . . 7.2 Considered conguration used to verify the high-level managing. . . . . . . . . .

	List of Tables Résumé étendu en français
	A.25 Zoom-in of fast switching disturbance for a 2 pump -2 cutting head setup with symmetric topology: cutting head 1 remains open while head 2 switches periodically.302 A.26 Zoom-in of fast switching disturbance for a 2 pump -2 cutting head setup with asymmetric topology: cutting head 1 remains open while head 2 switches periodically.302 B.1 Inverse switching pattern B for 200 MPa operating pressure of experiment E-6: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . B.2 Switching pattern A for 350 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. xliv 1.1 xlviii xlix

of cutting head 1 (a) and closing of cutting head 1 (b). . . . . . . . . . . . . . . . A.19 Overview of experiment E-7: fast switching disturbances for a 1 pump -2 cutting head setup at 200 MPa operating point. . . . . . . . . . . . . . . . . . . . . . . . A.20 Fast switching disturbance for a 1 pump -2 cutting head setup: zoom-in when switching of head 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.21 Fast switching disturbance for a 1 pump -2 cutting head setup: zoom-in when almost simultaneously switching of head 1 and 2. . . . . . . . . . . . . . . . . . . A.22 Fast switching disturbance for the 1 pump -2 cutting head setup: zoom-in when switching head 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.23 Overview of experiment D-

3 considering fast switching disturbance for a 2 pump -2 cutting head setup with symmetric topology: cutting head 1 remains open while head 2 switches periodically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.24 Overview of experiment D-6 considering fast switching disturbance for a 2 pump -2 cutting head setup with asymmetric topology: cutting head 1 remains open while head 2 switches periodically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Inverse switching pattern B for 350 MPa operating pressure of experiment E-9: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . xli B.4 desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . B.11 Switching pattern A for 350 MPa operating pressure of experiment E-13: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . B.12 Inverse switching pattern B for 350 MPa operating of experiment E-13: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . B.13 Disturbance steps at 200 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open. . . . . B.14 Disturbance steps at 200 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed. . . . . BB.19 Inverse switching pattern B for 200 MPa operating pressure of experiment E-6: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . xlii B.20 B.26 Disturbance steps at 350 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open. . . . . B.27 Disturbance steps at 350 MPa operating point, switching states (d): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains closed. . . . . B.28 Inverse switching pattern B for 200 MPa operating pressure of experiment E-11: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . B.29 Switching pattern A for 350 MPa operating pressure of experiment E-13: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . B.30 Inverse switching pattern B for 350 MPa operating of experiment E-13: desired switching states resulting in dierent pump rates for disturbance rejection by means control to reference pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . BB.35 Disturbance steps at 350 MPa operating point, switching states (c): resulting step responses from non-linear simulation model and measured step responses from experiment on test bench. Nozzle 2 switches while nozzle 1 remains open. . . . . xliii B.36 cations with respect to possible work station setups. . . . . . . . . . . . . . . . . 1.5 Manufacturing processes: waterjet machining classied according to typical manufacturing processes found in industry and assigned to waterjet applications. . . . 1.6 Properties of cutting water: mains water is subject to softening and ltering to full requirements for waterjet machining. . . . . . . . . . . . . . . . . . . . . . . 2.1 Test bench hardware conguration: hardware used for data acquisition and to control the switching heads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Test bench network topologies: setups applied to the test bench and modelling subject for simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Conguration for use case 1: measurements derived on dierent test bench setups to verify the robust low-level control design and the graph-based modelling methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conguration for use case 2: measurements derived on dierent test bench setups to validate high-pressure network models and low-level controllers as well as to identify network parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 3.1 Physical properties of pure water: characteristics of water (H 2 O) at 25°C and 0.1 MPa (1 bar) [Schmidt and Grigull, 1981]. . . . . . . . . . . . . . . . . . . . . xlv 3.2 k and branches j, k. . . . . . . . . . . . . . . . . . 4.4 Piping segmentation for topologies (a) and (b): parameters used to simulate the piping sections, considering the short and long network topology, respectively. . . 4.5 Piping segmentation for topologies (c) and (d): parameters used to simulate the piping sections, considering the symmetric and asymmetric network topology, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Piping segmentation for topologies (e), (f) and (g): parameters used to simulate the piping sections, considering the symmetric, asymmetric and decentralized distributed network topology, respectively. . . . . . . . . . . . . . . . . . . . . . . . 4.7 Excerpt of experiments for use case 1: measurements derived on dierent test bench setups to verify the graph-based modelling methodology. . . . . . . . . . . 4.8 Correlation coecients R and rmse to compare the model's accuracy for dierent operating pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Excerpt of experiments for use case 1 and 2: measurements derived with dierent test bench setups and used for model validation. . . . . . . . . . . . . . . . . . . 4with asymmetric and symmetric pumps. . . . . . . . . . . . . . . . . . . . . . . . 4.16 rmse (MPa) to compare the dierent experiments derived, for cavity cutting with asymmetric and symmetric pumps. . . . . . . . . . . . . . . . . . . . . . . . . . . xlvi 5.1 6.1 Achieved performance for state feedback controllers derived with full and reduced generalized plant: comparing achieved closed-loop performance with the control objectives for waterjet machining. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Achieved performance for initial PI controller without low-pass ltering and extended PI controller with low-pass ltering : comparing achieved closed-loop performance with control objectives for waterjet machining. . . . . . . . . . . . . . . 6.3 Optimization for uncertain parameter V δ : nding a maximal network subsection size for which γ-value remains below 1. . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Achieved performance for extended PI controller and quasi-optimal PI controller, both with low-pass ltering: comparing achieved closed-loop performance with control objectives for waterjet machining. . . . . . . . . . . . . . . . . . . . . . . 6.5 Uncertainty range of varying parameters, where the arithmetic mean value is considered for nominal parameter values. . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Model parameters used for numerical simulations, subject of control design verication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Considered use cases used for numerical simulations, subject of control design verication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 6.12 Comparing robust PI controller to quasi-optimal robust PI controller and robust PI gain scheduling in terms of closed loop step response and obtained uncertainty range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlvii 6.13

  Le débit d'eau fourni par chaque pompe peut par conséquent diérer. Par ailleurs, un phénomène d'interférence entre les pompes distribuées peut être occasionné par leur couplage dans le réseau. Les pompes voisines peuvent compenser ces uctuations additionnelles mais au prix d'une surconsommation d'énergie. Pour contrer ce phénomène d'interférence, la mise en ÷uvre d'un mécanisme de coopération entre les diérentes pompes permet d'équilibrer l'eort global de commande. Chaque pompe doit garantir des performances robustes et la stabilité autour de son voisinage. Cette thèse propose un mécanisme de gestion haut niveau des pompes qui assure leur coopération et permet d'augmenter le degré de exibilité et d'évolution des réseaux haute pression.

  illustre comment l'emploi d'une description non structurée des incertitudes permet de modéliser les variations des paramètres présentes dans les installations. Une réécriture du modèle incertain sous une forme standard ∆-M à l'aide d'une transformation fractionnelle linéaire supérieure permet d'obtenir un modèle perturbé. Ce modèle est le point de départ de la synthèse du régulateur H ∞ réalisée au cours du chapitre suivant.

Table 1 .

 1

			Pressure intensifiers	Attenuators
	Properties	Value	Unit
	Maximal pressure Input uid ow	400 2.8	(MPa) (l/min)	Hydraulic drive
	Drive power	22	(kW)
	Frame size	1.15x1.1x1.05	(m)
	Weight	800	(kg)
	Noise level	74	(dB)

1 Properties of a hydraulically driven high-pressure pump (Type JCP 22 LC): with a hydraulic pumping unit driven by a 22 kW asynchronous motor.

Table 1 .

 1 2 Properties of cutting head nozzles: nozzles inner diameter and operating pressures determining the output uid ow (l/min).

		Pressure (MPa)
		100 200 300 400
	Nozzles inner	0.10 0.13 0.19 0.23 0.26

diameters (mm) 0.15 0.30 0.42 0.51 0.59 0.20 0.53 0.74 0.91 1.05 0.25 0.82 1.16 1.42 1.64 0.30 1.18 1.67 2.05 2.37 0.35 1.61 2.28 2.79 3.22 0.40 2.10 2.98 3.64 4.21 On/off-valve Nozzle holder High-pressure inlet Pneumatics supply Waterjet Figure 1.2 Example of a cutting head used for pure water applications: pneumatically actuated on/o-valve and nozzle holder generating the waterjet. 2

Table 1 .

 1 4: Cutting head congurations: typical parametrisation for common waterjet applications with respect to possible work station setups.

		Operating Nozzles of Equivalent	Water
		pressure	diameter	diameter consumption
		(MPa)	(mm)	(mm)	(l/min)
	Surface cleaning	200	16 x 0.10	1 x 0.40	2.98
	Pure water cutting	300	4 x 0.15	1 x 0.30	2.04
	Abrasive cutting	400	1 x 0.25	1 x 0.25	1.55

Table 1 . 5 :

 15 Manufacturing processes: waterjet machining classied according to typical manufacturing processes found in industry and assigned to waterjet applications.

		Cycle time On/o-ratio Surface Pure water Abrasive
		(s)	(%)	cleaning	cutting	cutting
	Stripping 30 ... 600	100	x
	Contour	3 ... 60	80 ... 90		x	x
	Cavity	0.5 ... 2.0	50 ... 70		x	x

Table 1 .

 1 6: Properties of cutting water: mains water is subject to softening and ltering to full requirements for waterjet machining.

	Properties	Value	Unit
	Electric conductivity	< 20	(mS/cm)
	pH value	6.8 -7.5
	Hardness	< 0.1	(°fH)
	Particle quantity	< 100	(mg/l)
	Particle size	< 10	(µm)

  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Test bench for scalable high-pressure networks . . . . . . . . . . 2.2.1 Test bench setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Investigated network topologies . . . . . . . . . . . . . . . . . . . . 2.2.3 Use case denition . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Investigated switching patterns . . . . . . . . . . . . . . . . . . . .

	2
	System description
	Contents
	2.1

Table 2 .

 2 1: Test bench hardware conguration: hardware used for data acquisition and to control the switching heads.

	Components	Type	Properties
	Control unit (PLC)	1x Rockwell, CompactLogix ###	1 ms update time.
	Digital output module 1x Rockwell, Compact I/O ###
	Analogue input module 1x Rockwell, Compact I/O ###	0.2% accuracy, 16 bit resolution, ≥ 0.7 ms sampling rate.
	Pressure gauges	4x Gefran, ###	0.1% accuracy,

< 1 ms response time. modelled in MATLAB Simulink, applying the proposed graph-based modelling methodology, given in Chapter 3. This aims at validated high-pressure network models, which are useful to simulate future waterjet facilities.

Table 2 .

 2 2: Test bench network topologies: setups applied to the test bench and modelling subject for simulation.

	Topologies	Pumps Heads	Utilization	Use
					cases
	(a) Short piping (b) Long piping	1	1	Control design and modelling verication	1
	(c) Symmetric network (d) Asymmetric network	1	2	Parameter identication, model and controller validation	2
	(e) Symmetric distribution			Extended model and
	(f) Asymmetric distribution	2	2	extended controller validation,	3
	(g) Decentralized distribution			managing verication

Table 2 . 3 :

 23 Conguration for use case 1: measurements derived on dierent test bench setups to verify the robust low-level control design and the graph-based modelling methodology.

	Meas. Topology Pressure	Nozzle Switching	Utilization
			(MPa)	(mm) pattern
	R-1		100 . . . 400	0.25	Strip.	Mdl. verif.
	B-1	(a) Short	200	0.35	Contour	Ctr. verif.
	B-2		350	0.25	Contour	Ctr. verif.
	B-3				Contour Param. ident.
	B-4		200	0.35	Contour	Mdl. valid.
	B-5	(b) Long			Cavity	Mdl. valid.
	B-6 B-7		350	0.25	Contour	Param. ident. Mdl. valid.

Table 2 . 4 :

 24 Conguration for use case 2: measurements derived on dierent test bench setups to validate high-pressure network models and low-level controllers as well as to identify network parameters.

	Meas. Topology	Pressure Nozzle 1 Nozzle 2 Switching	Utilization
			(MPa)	(mm)	(mm)	pattern
	E-1		200	0.25	0.25	Contour	Param. ident.
	E-2		350	0.15	0.15	Contour	Param. ident.
	E-3 E-4	(c) Sym. net.	200	0.3	0.2	Contour Contour Mdl. & Ctr. valid. Param. ident.
	E-5					Cavity	Mdl. valid.
	E-6 E-7		350	0.2	0.1	Contour	Param. ident. Mdl. & Ctr. valid.
	E-8 E-9 E-10 E-11	(d) Asym. net.	200 350	0.25 0.15	0.25 0.15	Contour Contour	Param. ident. Mdl. & Ctr. valid. Param. ident. Mdl. & Ctr. valid.

  Table2.5. Various measurements have been realized with the same topology and conguration, but with respect to other switching patterns only applied at the cutting head 2, while cutting head 1 remains open for any time. This allows for investigating the eects of a single source of disturbance aecting two pumps at dierent network locations. Two experiments have been realized for each topology. A rst experiment investigated a symmetric cutting head conguration, where both cutting heads hold a 0.2 mm nozzle. The subsequent experiment considered an asymmetric conguration, where head 1 holds a 0.1 mm nozzle and head 2 a 0.3 mm nozzle. All three switching patterns have been applied with respect to each test bench setup, considering an application for surface stripping, contour cutting and cavity cutting. Basically, all of them are of interest when verifying the high-level managing.

	Meas. Topology	Pres. Noz. 1 Noz. 2 Switch.	Utilization
			(MPa)	(mm)	(mm) pattern
	D-1					Strip.	Mdl. val.
	D-2	(e) Sym. dis.	200	0.2	0.2	Contour Mdl. & Ctr. val.
	D-3					Cavity	Mdl. val.
	D-4					Strip.	Mdl. val.
	D-5	(f) Asym. dis.	200	0.2	0.2	Contour Mdl. & Ctr. val.
	D-6					Cavity	Mdl. val.
	D-7 D-8	(g) Dezen. dis.	350	0.1	0.3	Contour Cavity	Manag. Manag.
	2.2.4 Investigated switching patterns
	Dierent experiments have been conducted while opening and closing cutting heads with dened
	switching patterns. These experiments vary with respect to the cutting head conguration, net-
	work topology and switching pattern. Particularly, two switching patterns are dened considering

Measurements considering the stripping application are used for an extended model validation, while measurement considering cavity cutting are used for an extended controller validation. The measurements considering contour cutting are used for model validation as well as for controller validation.

Table

2

.5: Conguration for use case 3: measurements derived on dierent test bench setups to validate high-pressure network models and low-level controllers as well as to verify the high-level managing design. the investigated manufacturing processes: an application for contour cutting and an application for cavity cutting. The cycle times and on/o-ratios of Table

2

.6 have been exemplary chosen to realize the dierent switching patterns.

Table 2

 2

	.6: Switching patterns: specication of switching patterns with respect to typical manu-
	facturing processes used for experiments.
	Cycle time On/o-ratio
		(s)	(%)
	Contour	60	50
	Cavity	0.5	50

Table 2 .

 2 7 Properties of a electrically driven high-pressure pump: high pressure pump prototype.

			Linear actuators
			# # #
	Properties	Value	Unit
	Maximal pressure Input uid ow	480 3.2	(MPa) (l/min)	Piston pumps
	Drive power	27.5	(kW)
	Frame size	1.45x1.0x0.55	(m)
	Weight	ca. 600	(kg)
	Noise level	ca. 60	(dB)

Table 2 .

 2 8: High-pressure pump hardware conguration: hardware used to built an electrically driven pump that consists of two pumping units.

	Components	Type	Properties
	Control unit (PLC) 1x Rockwell, CompactLogix ###	1 ms update time.
	Power supply	1x Rockwell, Kinetix ###
	Frequency converter 2x Rockwell, Kinetix ###	25 µs cycle time.
	Synchronous motor 2x Wittenstein, ###	131.8 Hz dynamic bandwidth.
	Absolute encoder	2x Sick, ###	12 bit Resolution, 0.3°s accuracy, 200 kHz output frequency.

Table 3 .

 3 1: Physical properties of pure water: characteristics of water (H 2 O) at 25°C and 0.1 MPa (1 bar)[START_REF] Schmidt | Properties of water and steam in SI-units[END_REF].

	Properties	Symbol Value	Unit
	Fluid density		997	(kg/m 3 )
	Characteristic speed of sound	a	1484	(m/s)
	Bulk modulus	K	2.1739 × 10 9 (Pa)

Table 3 .

 3 2: Congurations for numerical simulations with boundary conditions A and B: initial conditions, resulting equilibrium points and relative delays per piping length.

			Velocity v(t)		Pressure p(t)	Rel. delay
	Sim. t = 0 (m/s) (m/s) lim t→∞	Rel. ratio t = 0 (%) (MPa) (MPa) lim t→∞	Rel. ratio (%)	t = 0 (ms/m) (ms/m) lim t→∞
	A-1	419	461	10	20	30	50	2.4	2.2
	A-2	663	729	10	50	74	48	1.5	1.4
	A-3	937	1031	10	100	145	45	1.1	1.0
	A-4 1325 1458	10	200	288	44	0.8	0.7
	B-1	2.76	3.04	10	20	29	44	362.3	328.9
	B-2	4.36	4.80	10	50	72	44	229.4	208.3
	B-3	6.17	6.79	10	100	144	44	162.1	147.3
	B-4	8.72	9.60	10	200	288	44	114.7	104.2

Table 3 .

 3 3: Simulation error comparing distributed and lumped parameter model: modelling a piping of 20 m length with dierent numbers of homogeneous segments. Dist. param Lump. param. Segments n length ∆x rmse Max. rel. error Rel. sim. time Rel. sim. time

		(m)	(MPa)	( % )	(s/s)	(s/s)
	4	5	5.24	10.1	0.0381	0.0388
	10	2	2.87	7.8	0.0826	0.0291
	20	1	0.24	0.9	0.187	0.0309
	40	0.5	0.19	0.7	0.522	0.0351
	100	0.2	0.09	0.4	3.74	0.0589

Table 3 .

 3 4: Excerpt of experiments for use case 1 and 2: measurements derived on dierent test bench setups to identify dierent network parameters.

	Meas. Topology	Pressure Nozzle 1 Nozzle 2 Switching
			(MPa)	(mm)	(mm)	pattern
	B-3 B-6	(b) Long	200 350	0.35 0.25	--	Contour
	E-1		200	0.25	0.25
	E-2 E-3	(c) Sym. net.	350 200	0.15 0.3	0.15 0.2	Contour
	E-6		350	0.2	0.1
	E-8 E-10	(d) Asym. net.	200 350	0.25 0.15	0.25 0.15	Contour

Table 3 .

 3 5: Design of experiment for parameter identication to determine the pressure-dependant bulk modulus.

	Experiments Pumping chamber Initial position Piston velocity
			(###)	(mm/s)
	1...8	1	{0, 40, 80, 120}	{1, 2}
	9...16	2	{20, 60, 100, 140}	{1, 2}

Table 3 .

 3 8: Identied friction loss coecient with linear regression and its condence intervals.

	Coecient Fitted value 95% condence bounds Unit

λ

Table 4 .

 4 1: List of nodes and branches for topology (a): assignment of the dierent high-pressure components to nodes k and branches kj.

	k	Nodes:	kj	Branches:
	1	Pumping chamber 1	[13]	Check-valve 1
	2	Pumping chamber 2	[23]	Check-valve 2
	3	Coupling	[34]	Connector
	4 . . . 7 High-pressure piping [45] . . . [78] High-pressure piping
	8	Cutting head	[80]	Nozzle

Table 4 . 3

 43

	kj	Branches:

: List of nodes and branches for topology (b): assignment of the dierent high-pressure components to nodes k and branches j, k. k Nodes:

Table 4 .

 4 These are assigned to the pressures p P (t), p N (t) and p H (t). Table4.4 summarize the assignments of piping sections to homogeneous segments for topology (a) and (b). 4: Piping segmentation for topologies (a) and (b): parameters used to simulate the piping sections, considering the short and long network topology, respectively.

	, . . . , [14, 15], . . . , [21, 22], [22, 0]}	(4.16)

k ∈ {1, . . . , 22}

(4.17)

nodes of homogeneous segments. This yields a lumped parameter model of order 22. The simulated pressure states are evaluated at the segments 4, 14 and 21 for model validation.

Table 4 .

 4 5: Piping segmentation for topologies (c) and (d): parameters used to simulate the piping sections, considering the symmetric and asymmetric network topology, respectively.

		Symmetric network model
		Section Number of Segment	Discharge
	Section length L segments n length ∆x coecient ζ
	1 2 3	7.17 m 5.94 m 6.31 m	7 6 6	1.0243 m 0.9900 m 1.0517 m	### ### ###
		Asymmetric network model
		Section Number of Segment	Discharge
	Section length L segments n length ∆x coecient ζ
	1 2 3	7.17 m 0.91 m 11.34 m	7 1 11	1.0243 m 0.9100 m 1.0309 m	### ### ###

Table 4 .

 4

	6: Piping segmentation for topologies (e), (f) and (g): parameters used to simulate the
	piping sections, considering the symmetric, asymmetric and decentralized distributed network
	topology, respectively.
		Symmetric distributed network model
		Section Number of Segment	Discharge
	Section length L segments n length ∆x coecient ζ
	1 2 3 4 5	6.38 m 6.48 m 4.94 m 0.84 m 0.94 m	6 6 5 1 1	1.1383 m 1.0808 m 0.9880 m 0.8400 m 0.9400 m	### ### ### ### ###
		Asymmetric distributed network model
		Section Number of Segment	Discharge
	Section length L segments n length ∆x coecient ζ
	1 2 3 4 5	10.82 m 2.03 m 5.86 m 0.63 m 1.16 m	11 2 6 1 1	0.9832 m 1.0150 m 0.9767 m 0.6300 m 1.1600 m	### ### ### ### ###
		Decentralized distributed network model
		Section Number of Segment	Discharge
	Section length L segments n length ∆x coecient ζ
	1 2 3 4	6.08 m 5.50 m 7.98 m 0.41 m	6 6 8 1	1.0125 m 0.9167 m 0.9975 m 0.4100 m	### ### ### ###
	5	1.38 m	1	1.3800 m

Table 4 .

 4 7: Excerpt of experiments for use case 1: measurements derived on dierent test bench setups to verify the graph-based modelling methodology.

	Meas. Topology Pressure	Nozzle Switching
		(MPa)	(mm) pattern
	R-1	(a) Short 100 . . . 400	0.25	Strip.

Table 4

 4

	Meas. Topology	Pressure Nozzle 1 Nozzle 2 Switching
			(MPa)	(mm)	(mm)	pattern
	B-4 B-5	(b) Long	200	0.35	-	Contour (Appendix A.1) Cavity
	B-7		350	0.25	-	Contour (Appendix A.1)
	E-4 E-5	(c) Sym. net.	200	0.3	0.2	Contour Cavity	(Appendix A.1)
	E-7		350	0.2	0.1	Contour (Appendix A.1)
	E-9 E-11	(d) Asym. net.	200 350	0.25 0.15	0.25 0.15	Contour Contour (Appendix A.1)

.9: Excerpt of experiments for use case 1 and 2: measurements derived with dierent test bench setups and used for model validation.

  4

		p P	2.3083	2.2394 1.2013 0.7246
	( (c) Sym. network )	p N	2.9671	2.5961 1.4402 0.6662
	( 200 MPa )	p H1	2.8849	2.5689 1.4331 0.5950
		p H2	2.2483	2.2453 1.2345 0.5925
	E-7	p P	2.8172	2.2289 4.1477 1.0991
	( (c) Sym. network ) ( 350 MPa )	p N p H1	1.4212 2.1638	2.5854 2.2301 0.3928 2.3143 3.0765 0.6746	(Appendix A.1)
		p H2	1.5375	2.3619 2.3801 0.5502
	E-9	p P	6.0825	0.4364 1.2065 5.0967
	( (d) Asym. network ) p N	6.7886	1.1651 1.3519 5.3109
	( 200 MPa )	p H1	6.7740	1.2433 1.3076 5.4195
		p H2	7.1273	0.7716 1.2476 5.2175
	E-11	p P	2.6541	1.0427 1.6741 10.4430
	( (d) Asym. network ) p N ( 350 MPa ) p H1	2.6247 2.7952	1.5006 1.7329 11.2671 1.6923 1.8193 10.9516	(Appendix A.1)
		p H2	2.3882	1.3909 1.6246 10.9348

  5

	p P	2.9488	4.8987	4.7719
	( (c) Sym. network ) p N ( 200 MPa ) p H1	3.7786 3.8152	5.7320 5.6843	5.1903 5.0483	(Appendix A.1)
	p H2	3.2608	4.4720	4.6131

Table 4 .

 4 14: Excerpt of experiments for use case 3: measurements derived with dierent test bench setups and used for model validation.

	Meas. Topology	Pressure Nozzzle 1 Nozzzle 2 Switching
			(MPa)	(mm)	(mm)	pattern
	D-1					Strip.
	D-2	(e) Sym. dis.	200	0.2	0.2	Contour
	D-3					Cavity
	D-4					Strip.
	D-5	(f) Asym. dis.	200	0.2	0.2	Contour
	D-6					Cavity

Table 4 .

 4 15: rmse (MPa) to compare the dierent experiments, derived for contour cutting with asymmetric and symmetric pumps.

Table 4 .

 4 16: rmse (MPa) to compare the dierent experiments derived, for cavity cutting with asymmetric and symmetric pumps.

	Measurement	Pos.	Fig. (a)
			d 1 = 1
			d 2 = {0, 1}
	D-3	p P 1	3.0554
	( (e) Sym. distributed ) ( 200 MPa )	p P 2 p N	3.3177 2.9057	(Appendix A.2)
		p H	3.1889
	D-6	p P 1	9.0982
	( (f) Asym. distributed ) p P 2 ( 200 MPa ) p N	7.6913 8.8256	(Appendix A.2)
		p H

  Figure 5.2 Block diagram of standard ∆=M conguration: interconnection system M and uncertainty block ∆, containing the parametric uncertainties. the input and output channels, such as M 11 ∈ G p×q , M 12 ∈ G p×m , M 21 ∈ G n×q and M 22 ∈ G n×m . The matrix M 11 holds transfer functions to assign the uncertainty block

					Δ
			d		e
			w		P M	z
	The interconnection transfer function matrix M is then partitioned as
	e z	= M	d w	=	M 11 M 12 M 21 M 22	d w	, d = ∆ e ,	(5.3)
	with respect to

The interconnection system holds the exogenous input w(t) ∈ R m and the exogenous output z(t) ∈ R n , where the uncertainty block is interconnected by d(t) ∈ R q and e(t) ∈ R p . ∆ from d(t) to e(t), providing an upper feedback loop. Evaluating

(5.3)

from w(t) to z(t) gives the transfer function for the nominal plant G n (s) = M 22 .

Table 5 .

 5 1: Time domain control objectives for disturbance rejection: hard constraints for unit step d(t) → y(t).

Table 5 .

 5 2: Time domain control objectives for reference tracking: soft constraints for unit step

	Requirements	Objectives Conditions
	Rise time (90% r(t)): |y(t)| ≥ 0.9 t ≥ 10 seconds
	Overshoot:	|y(t)| ≤ 1.0 t ≥ 0 seconds
	Steady-state error:

r(t) → y(t).

Table 5 .

 5 6: Properties of the measurement signal derived from 2 pressure gauges for dierent measurements at 100, 200, 300 and 400 MPa operating points.

	Operating point	σ 2	SNR Sensor Sample set Number of	σ 2	SNR
		×10 -3	(dB)			samples	×10 -3	(dB)
	100 MPa	3.99 61.381	1	A	539	5.40 60.251
					B	539	2.95 62.653
				2	A	539	4.33 60.998
					B	539	3.22 62.653
	200 MPa	3.18 70.699	1	A	339	3.13 70.835
					B	339	3.17 70.774
				2	A	339	3.09 70.764
					B	339	3.34 70.430
	300 MPa	3.60 73.557	1	A	259	3.71 73.534
					B	259	3.03 74.341
				2	A	259	3.50 73.685
					B	259	4.22 72.791
	400 MPa	3.93 75.371	1	A	219	4.27 75.088
					B	219	3.84 75.530
				2	A	219	4.04 75.234
					B	219	3.64 75.660

Table 5 .

 5 7: System gain for approximated and extended non-linear model: varying a single parameter, while holding each other parameter on nominal value.

	Param. y min	y max	Rel. range
	×10 6	×10 6

Table 5 .

 5 8: System gain for approximated linearised model: varying a single parameter, while holding each other parameter on nominal value to compare u → y and d → y.

		Param. y min	y max	Rel. range
			×10 6 ×10 6	(%)
	Control	H δ	### ∞ [-50, + ∞]
		x δ	74.72 1494 [-93, + 41]
	Disturbance	x δ	2.000 800.0 [-100, +100]

Table 5 .

 5 9: Time constants for approximated linearised model: varying a single parameter, while holding each other parameter on nominal value for u → y.

		Param. τ min	τ max	Rel. range
			(s)	(s)	(%)
	Control	H δ	###	∞	[-50, + ∞]
		x δ V δ	0.0179 0.3576 [-93, + 41] ### ### [-67, + 67]
		K δ	0.1932 0.3671 [-24, + 45]

Table 6 .

 6 1: Achieved performance for state feedback controllers derived with full and reduced generalized plant: comparing achieved closed-loop performance with the control objectives for waterjet machining.

		K SF		KSF	Objective
		nom.	max.	nom.	max. (Sect. 5.3)
	Peak gain γ-value:	0.6067	0.6167	≤ 1.0
	Rise time (90% r(t)):	3.9144 3.9144 3.9144 3.9144	< 10
	Settling time (2% e(t)): 0.1300 0.1368 0.1270 0.1340	< 0.5
	Overshoot:	0.0739 0.0927 0.0742 0.0936	< ±0.1
	6.2.3 Robust PI control design

Table 6 .

 6 2: Achieved performance for initial PI controller without low-pass ltering and extended PI controller with low-pass ltering : comparing achieved closed-loop performance with control objectives for waterjet machining.

		K P I		KP I	Objective
		nom.	max.	nom.	max. (Sect. 5.3)
	Peak gain γ-value:	1.5610	0.6354	≤ 1.0
	Rise time (90% r(t)):	3.9144 3.9144 3.9144 3.9144	< 10
	< 0.5 0.1182 0.1492 0.0753 0.0963 < ±0.1 Settling time (2% e(t)): 0.1017 0.1217 0.1236 0.1318 Overshoot:

Table 6 . 4 :

 64 Achieved performance for extended PI controller and quasi-optimal PI controller, both with low-pass ltering: comparing achieved closed-loop performance with control objectives for waterjet machining.

		nom.	KP I	max.	nom.	K * P I	Objective max. (Sect. 5.3)
	Peak gain γ-value:	0.6354		1.0	≤ 1.0
	Rise time (90% r(t)):	3.9144 3.9144 3.9144 3.9144	< 10
	Settling time (2% e(t)): 0.1236 0.1318 0.1445 0.1538	< 0.5
	Overshoot:	0.0753 0.0963 0.0700 0.0934	< ±0.1

Table 6 .

 6 5: Uncertainty range of varying parameters, where the arithmetic mean value is considered for nominal parameter values.

				Nominal value Min value Max value
	Flow resistance H δ ×10 ###	###	0	###
	Fluid volume	V δ	(cm 3 )	48.28	15.87	80.69
	Bulk modulus	K δ	(GPa)	###	###	###
	Operating point x δ	(MPa)	200.5	1	400
	methodology, see Section 3.3. It employs a high-pressure network, interconnecting a high-pressure
	pump to two cutting heads, as shown in the diagram of Figure 6.22. This results in a simulation
	model of order 35.

Table 6 .

 6 6: Model parameters used for numerical simulations, subject of control design verication.

		Section Number of Segment	Discharge
	Section length L segments n length ∆L coecient ζ
	1, 2, 3	0.91 m	10	0.091 m	###
	Table 6.7: Considered use cases used for numerical simulations, subject of control design veri-
	cation.
	Topology Pressure Control eort Nozzle 1 Nozzle 2 Switching
	(MPa)	( -)	(mm)	(mm)	pattern
		100	0.439
	Sym. net.	200 300	0.609 0.744	0.2821 0.2821 mm Contour
		400	0.839

Table 6 .

 6 8: Achieved performance by means of detailed numerical simulations for dierent control structures: comparing achieved closed-loop performance from simulation with results from control design, using simplied and linearised model.Suboptimal PI controller K PIAn excellent match is obtained by applying the initial robust PI controller(6.22) on the detailed simulation model. The plots of Figure6.24 -6.27 compare the expected nominal step responses from controller synthesis with the linearised plant and the set of quasi-nominal responses from numerical simulations at 100, 200, 300 and 400 MPa operating pressures. It shows the scaled pressure dierences ∆y around the operating point and the corresponding control signals ∆u. The observed variations stay within expected upper and lower bounds derived from control design. It is recognized that the disturbance is rejected with expected settling time and overshoot. For the 100 and 200 MPa case, the detailed simulations reveal slightly degradation of the settling time. This results, particularly for the 100 MPa case, into an increased overshoot. The nominal response remains within expected bounds.

			Ctr. design	Verication (Simulations)	Objective
		Pressure (MPa)	40 ... 400	100	200	300	400	(Sect. 5.3)
	K P I	max. settling time: max. overshoot:	0.1217 0.1492 0.1319 0.0989 0.0825 0.0773 < ±0.1 < 0.5 0.3960 0.2470 0.2200 0.1970
	K * P I	max. settling time: max. overshoot:	0.1538 0.0934	0.1370 0.0970 0.0760 0.0700 0.0603 0.0567 0.0493 0.0460	< 0.5 < ±0.1
	KSF	max. settling time: max. overshoot:	0.1368 0.0927	0.1160 0.0900 0.0780 0.0670 0.0710 0.0652 0.0568 0.0509	< 0.5 < ±0.1

  6.28 -6.31 compare the expected nominal step response from controller synthesis with the linearised plant and the set of quasi-nominal responses from numerical simulations at 100, 200, 300 and 400 MPa operating pressures. Again, it shows the scaled pressure dierences ∆y around the operating point and the corresponding control signals ∆u. The observed variations stay within expected upper and lower bounds derived from control design. It is recognized that any disturbance is rejected with expected settling time and overshoot. For the 100 MPa case, the detailed simulations reveal a slight degradation of the settling time, which results in an increased overshoot. The nominal response remains within expected bounds.

Table 6 .

 6 10: Achieved performance by means of experiments at the test bench for dierent network topologies: comparing closed-loop performance from measurements with results from simulations, using detailed simulation model.

			Measurement	Simulations	Objective
		Transitions	1-4	5-8	1-4	5-8	(Sect. 5.3)
	E-4 (200 MPa) max. overshoot: max. settling time: 1.3989 0.9429 0.9470 0.6310 0.1723 0.0937 0.1839 0.0850 < ±0.1 < 0.5
	E-7	max. settling time: 0.9656 0.7580 0.2750	-	< 0.5
	(350 MPa) max. overshoot:	0.0800 0.0507 0.0734 0.0185	< ±0.1
	E-9 (200 MPa) max. overshoot: max. settling time: 1.1820 1.1879 0.6992 0.7070 0.1487 0.1566 0.1311 0.1302 < ±0.1 < 0.5
	E-11	max. settling time: 0.3855 0.5676 0.3582 0.3882	< 0.5
	(350 MPa) max. overshoot:	0.0431 0.0446 0.0447 0.0448	< ±0.1

Table 6 .

 6 12: Comparing robust PI controller to quasi-optimal robust PI controller and robust PI gain scheduling in terms of closed loop step response and obtained uncertainty range.

		Robust KP I nom. max.	Optimal K * P I nom. max.	Gain-sched. K * PI min. max.
	Peak gain γ-value:	0.6354	1.0	1.0
	Settling time (2% e(t)): 0.1236 0.1318 0.1445 0.1538 0.0615 0.0846
	Overshoot: Uncertainty V δ × 10 -4 :	0.0753 0.0963 0.0700 0.0934 0.0414 0.0629 5.8996 9.5091 18.089

Table 6 .

 6 13: Excerpt of experiments for use case 3: measurements derived on dierent test bench setups used for controller validation. All experiments have been derived for a single operating pressure of 200 MPa, taking the switching pattern for contour cutting into account. Wile cutting head 1 remains open for any time, cutting head 2 will open and close for a time period of 60 seconds. Cutting heads open and close with respect to the predened switching pattern given by the disturbance signal d

	Meas. Topology	Pressure Nozzle 1 Nozzle 2 Switching
			(MPa)	(mm)	(mm)	pattern
	D-2	(e) Sym. dis.	200	0.2	0.2	Contour
	D-5	(f) Asym. dis.	200	0.2	0.2	Contour

Table 7 .

 7 1: Excerpt of experiments for use case 3: measurements derived on dierent test bench setups used for high-level managing.

	Meas. Topology	Presure Nozzle 1 Nozzle 2 Switch.
			(MPa)	(mm)	(mm) pattern
	D-7	(g) Dezen. dis.	350	0.1	0.3	Contour

Une contribution importante de ce chapitre porte sur la démarche globale de synthèse d'un régulateur pour une pompe installée dans un réseau haute pression de topologie inconnue. Le réseau doit être découpé en sous-sections dont la pression intérieure est régulée par une pompe. La longueur de la sous-section est alors considérée comme un paramètre incertain lors de la synthèse du régulateur. Une procédure itérative ∆-K est proposée an de déterminer la longueur maximale de la sous-section pour laquelle les performances souhaitées en boucle fermée sont garanties. Cette procédure permet d'eectuer une synthèse de type H ∞ tout en augmentant itérativement l'intervalle de l'incertitude ∆. Cette démarche aboutit à un régulateur K qui maximise la taille admissible de la sous-section. Comparativement à la synthèse initiale, une augmentation de 61% de la longueur de la sous-section a pu être obtenue.Enn, un premier régulateur de type PI a fait l'objet d'une validation expérimentale à l'aide du banc d'essais. Les mesures obtenues ont été comparées aux prédictions fournies par les modèles de simulation. Ces résultats montrent la bonne cohérence entre les simulations et la réalité. En raison d'un dysfonctionnement du banc test, cette phase de validation expérimentale n'a malheureusement pas pu être menée plus loin, par exemple, pour tester le régulateur par retard de phase. Néanmoins, grâce à la bonne qualité des prédictions fournies par le modèle, le régulateur par retard de phase a fait l'objet de plusieurs validations en simulation en considérant diverses topologies couplant deux pompes à deux têtes de coupe. Ces modèles en boucle fermée servent au cours du chapitre suivant à tester et à valider un mécanisme de gestion haut niveau.

Illustrations courtesy of Jet Cut Power GmbH, Schönenwerd.

Illustrations courtesy of Jet Cut Power GmbH, Schönenwerd.

Illustrations courtesy of Jet Cut Power GmbH, Schönenwerd.

1.3. Principles of high-pressure generation

2.2. Test bench for scalable high-pressure networks

This section investigates the simulation accuracy for high-pressure networks, when applying the graph-based modelling methodology for congurations with two cutting heads. For that, the pressure generation in a high-pressure network becomes disturbed, when switching the dierent cutting heads in open and close position. Two cutting heads are distributed in the network, while a single high-pressure pump operates at a dened position. This pump is in closed-loop control, aiming at a constant reference pressures of 200 and 350 MPa, respectively. For this reason, the pump holds by default a PI controller(6.46), as specied in Section 6.4. It will adjust its induced overall uid ow, such as to compensate any switching disturbances. Thus, a wide variety of uid ow rates from 0 to 2.45 l/min will be obtained, depending on the switching state of each cutting head. Here, the graph-based modelling methodology has been applied to model dierent network topologies, which interconnect a high-pressure pump with two cutting heads.

Conclusion and perspectives

Acknowledgments

for energy ecient waterjet machining.

Summary

It has been shown that a pressure-dependent bulk modulus is essential to track the transient behaviour within a desired pressure range. Hence, applying a constant bulk modulus results in an incorrect transient behaviour, that converges towards the desired steady-state. On the other hand, a constant uid density causes a transient behaviour similar to the measurement, but with an incorrect steady-state. Therefore, a pressure-dependent uid density is desired to meet the steady-state over a wide pressure range. Table 4.8 lists the correlation coecient R and rmse for all the discussed models. This evaluation is based on the experimental and simulation data over a time interval of 30 seconds at a constant operating point. All models achieve a correlation coecient of R > 0.9687. As the graph-based modelling methodology obtains a rmse < 2.42 for all observed operating points, an increase to rmse > 5.10 has been observed for both models, using constant parameter. Modelling approaches that either consider a variable uid density (e.g. [START_REF] Fabien | Dynamic modelling and identication of a waterjet cutting system[END_REF], [START_REF] Ferretti | Abrasive waterjet intensier model for machine diagnostics[END_REF]) or a variable bulk modulus (e.g. [START_REF] Hountalas | Development of a fast and simple simulation model for the fuel injection system of diesel engines[END_REF]], [START_REF] Lino | Nonlinear modelling and control of a common rail injection system for diesel engines[END_REF]) are sucient for simulations around a specic operating point. However, the proposed methodology improves the simulation accuracy, when a wide pressure range is desired. 4.3.2 High-pressure network model validation (1 pump conguration) Summary for 1 pump congurations Contour cutting: the above section investigates the contour cutting at 200 MPa operating point, applied on the topologies (c) Symmetric network and (d) Asymmetric network. It considers extended topologies, including the switching of two cutting heads. The topology (b) Long piping has been initially introduced, see Appendix A.1. This basic topology can be considered as a reference, when switching a single cutting head. Table 4.10 compares the rmse for the residuals between measured and simulated data for the experiments with a single cutting head. Table 4.11 provides the rmse, considering the experiments with two cutting heads. The residuals have been evaluated for a period of 2 seconds, when switching a cutting head. Thus, each column corresponds to a zoomed-in plot, where each row provides a measurement position. Supplementary measurements and simulations have been obtained at an operating point of 350 MPa pressure. Also these results are summarized in the and discussed in the Appendix A.1. 16.6050 6.2981 In general, the simulated pressure trends follow the measured dynamics, for the switching of dierent cutting heads at various network position, with reasonable accuracy. The amplitude of disturbance varies with respect to the cutting head conguration, causing the pump to run on its entire operating range. Whenever all cutting heads shut close, the system reaches a pure integral behaviour and becomes very sensitive of any change of input uid ow. Complementary, it is recognized that the cutting head is not able to entirely shot close. Thus, in real application, a small output ow remains that causes a steady pressure loss (see Figure 4.20 (a/d)). The simulations remain about their static pressure as the idealized cutting head model does not consider any eects of leakages. This results in an increased rmse. Referring to the basic topologies (Table 4.10), a relative error of about 1% is observed, when opening a cutting head (d → 1) at 200 MPa. The error increases to 2%, for the 350 MPa operating point, and it becomes further increased to above 4%, when closing a cutting head (d → 0). The relative error for the extended topologies (Table 4.11) range between 0.3 -0.7%, insofar as these unfavourable cases for closed cutting heads are not evaluated. In addition, the cutting head will close with a time delay of nominal 144 ms and open with 45 ms delay. These delays can vary within a standard deviation of about 8 and 2.5 ms, respectively, as investigated in the system description (see Chapter 2). This deviation can cause varying residuals, when a cutting head is switching far o the nominal delay time.

Upper linear fractional transformation (ULFT)

Matrix manipulations with respect to the augmented interconnection system (5.3) allows for verifying the upper linear fractional transformation (ULFT) [Safonov, 1981] (5.4) of M and ∆, which describes the perturbed plant G p (s) = F u (M, ∆) that provides the closed-loop system z = G p (s) w, taking the plant perturbations into account.

P=K conguration

A closed-loop control system can be rearranged in a standard P=K conguration [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. It will separate the controller block K and establishes the interconnection system P, as depicted in the block diagram of Figure 5.3. This new interconnection system holds the exogenous input channel w(t) ∈ R m and the exogenous output channel z(t) ∈ R n , where the controller block is interconnected by the control variable u(t) ∈ R q and measurement variable y(t) ∈ R p .

P

Lower linear fractional transformation (LLFT)

Matrix manipulations with respect to the augmented interconnection system (5.5) allow for verifying the lower liner fractional transformation (LLFT) [Safonov, 1981] F (P, K) = P 11 + P 12 K(I -P 22 K) -1 P 21 (5.6) of P and K, which describes the generalized plant T (s) = F (P, K) that provides the closed-loop system z = T (s) w, as desired for control design.

It follows that a nominal feedback system M is robustly stable, with respect to the uncertainty ∆, if and only if [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF] µ ∆ (M) ≤ 1 β , (5.19) whereby β denotes an uncertainty bound for guaranteed stability.

For the general case of structured uncertainties holds [START_REF] Gu | Robust Control Design with MATLAB[END_REF] Nominal stability ⇔ M internally stable , (5.20a) Nominal performance ⇔

(5.20d)

Due to diculties for calculating the structured singular value µ ∆ , it is estimated by lower and upper bounds, such as

(5.21)

The lower bound can be obtained by the spectral radius ρ and the upper bound can be given by the largest singular value σ.

Considering a complex unstructured uncertainty and if ∆ is a full matrix yields ρ(M) = µ ∆ (M) = σ(M). For this case, the above conditions for robustness are simplied

(5.22b)

An uncertain system is then guaranteed stable for all normalized structured uncertainties ∆, such as ∆(s) ∞ ≤ 1.

Internal stability

A system Σ is internally stable, if it contains no hidden unstable poles and when applying a bounded exogenous signal to any system input results in a bounded response at every system output (BIBO stability, see [START_REF] Gu | Robust Control Design with MATLAB[END_REF]). Hence, internal stability is given, if and only if all poles p i lie in the left hand plane (LHP), such as Re {λ i (A)} < 0 ∀ i, where λ i (A) denotes the eigenvalues of a system in state space description. The poles are obtained by the roots of the characteristic equation: det (sI -A) = 0.

Problem formulation

The initial control design problem considers the decentralized and robust control of decoupled high-pressure pumps, following the approach in [START_REF] Marquez | Decentralized robust PI controller design for an industrial utility boiler -an IMC method[END_REF] and [Iftar, 2013]. A robust low-level controller is thereby designed, attenuating the impact from the switching of cutting heads to a network subsection. This aims to reduce the coupling between neighbouring subsections by means of local disturbance rejection, using the available high-pressure pumps. Further investigations can expand this approach by considering the dynamics of overlapping network sections. This is promising to enhance the resulting closed-loop system performance, as suggested in [START_REF] Benlatreche | Robust decentralised control strategies for large-scale web handling systems[END_REF] and [START_REF] Swarnakar | A design framework for overlapping controllers and its industrial application[END_REF]. However, a distributed control approach that exchanges information between controllers, see for example To analyse the input to output gain at steady-state, the non-linear plant model (5.38) has been investigated around equilibrium. Dening dx/dt = 0 results in the equilibrium point

where it is recognized that lim d→0 y eq = ∞ .

(5.102)

On the other hand, the gains for the linearised model (5.47) have been obtained at equilibrium

( 5.103) where the time constants

dene the dynamics from u(t) → y(t) and d(t) → y(t), respectively. Applying u max → u 0 and d max → d 0 , varying a single parameter, while holding the other parameters on nominal value, give the sensitivity range. Table 5.7 and 5.8 summarize the results for the system gain considering the non-linear (5.38) and the linearised model (5.47), respectively. Where Table 5.9 evaluates the variations regarding the time constants.

The ow resistance and the nozzle cross section has the ability to increase the system gain to innity. Hence, the control channel must become zero, that the system output channel for H δ = 0 or S δ = 0 remains bounded. Considering the approximated model, which separates control and

Experimental studies

This section validates an initial PI controller, using measurements from the test bench, as introduced in Section 2.2 and comparing them with detailed simulations. For this, the robust PI controller (6.22) has been implemented on a high-pressure pump. Several high-pressure network topologies will be compared, as dened in Section 4.2, considering single pump setups with 2 cutting heads and multi pump setups of 2 pumps connected to 2 cutting heads.

All experiments have been derived for two operating points, each taking the switching pattern for contour cutting into account. A cutting head will open for a cutting period of 60 seconds, followed by a hold period in closed position for another 60 seconds. This switching pattern is periodically repeated for both cutting heads with a 90°phase shift, see Section 2.2 for details. The diagram of Figure 6.36 illustrates the experimental setup, as implemented for controller validation. The cutting heads open and close with respect to the predened switching pattern, as given by the disturbance signal

The pressure controller will then adjust the pump rate, using the control signal u(t), to obtain the desired reference pressure. The reference value r(t) and disturbance d(t) are measured to excite the extended high-pressure network models, for closed-loop simulation. This extended model considers the high-pressure pump, a network model and the low-level pressure controller in closed-loop. The pump output pressure p P (t) as well as the pump rate u(t) will be compared with simulated values (p P (t) and û(t)) to validate the PI controller. The measurement data have been captured with a sampling rate of 500 Hz. 6.36 Experimental setup for controller validation: pressure gauges measure the pressure p x (t) at dierent positions, the input uid ows Q(t) vary with respect to the switching disturbance d(t). Reference pressure r(t) and disturbance d(t) are used as the model inputs. The estimated pressures px (t) have been compared to the measured pressures.

The controller parametrization and test bench setup of this section, correspond to the model validation already used in Chapter 4. It is important to mention, that the controller implementation on the test bench does not correspond to the desired robust control design from Section 6.2. The closed-loop experiments on the test bench have been derived with the obtained controller from robust control design, but implementing a faulty controller scaling ψ f to the PLC. This provides controller parameters dierent from the desired robust control design.

The controller scaling results from the model-based trajectories for displacing the pistons such as to synchronize the pumping units. These trajectories are shifted to obtain a specic pressurization with respect to the desired operating pressure, see Section 2.3. The remaining cycle time is then available for hauling and lling with respect to the conditions (2.18 -2.22). Recalling the concept of camming, a varying operating pressure p(t) requires a change of the trajectory, represented in terms of a function F(p), and results in a dierent induced overall Dierent measurements have been derived for simulation and experimental validation, as listed in Table 6.9. The 2 extended measurements (E-4 and E-7) consider the symmetric network topology, where both cutting heads have been asymmetrically congured with nozzles of dierent inner diameter. The other 2 extended measurements (E-9 and E-11) consider the asymmetric network topology, while both cutting heads have been symmetrically congured with identical nozzles. This section provides an overview, plots extending these results are given in Appendix B.1. Applying the graph-based modelling methodology, according to the test bench conguration, results in dierent high-pressure network models. These models have been already introduced and experimentally validated in Chapter 4 by means of excessive measurements and simulations.

Contour cutting: asymmetric and symmetric cutting heads

Measurements have been obtained and compared to simulations for the 200 MPa and 350 MPa operating points. Disturbance steps have been applied on the test bench, when switching the cutting heads open and close. Considering the experiments E-4 and E-7, the alternately switching of two cutting heads results in 4 possible pump rates. That correspond to dierent uid ow consumptions, since both cutting heads hold nozzles of dierent inner diameters. Consequently, 8 transitions have been obtained with respect to the cutting head switching. Also considering the experiments E-9 and E-11, the switching results in 4 possible pump rates. This causes dierent uid ow consumption, as the piping sections to the cutting heads are of dierent length. For the subsequent investigations, these 8 transitions are realised by applying two alternative switching patterns, see also Section 2.2. The switching patterns on disturbance d = [d 1 , d 2 ], the measured pump rates u = [u 1 , u 2 ] and the measured pump output pressures p P = [p P 1 , p P 2 ] are shown in the plots of Figures 6.37 and 6.38. These plots overview the experiments E-4 and E-9, representing the asymmetric and symmetric cutting heads at 200 MPa. The plots for the inverse 4 transitions as well as the complementary experiments for controller validation at 350 MPa (E-7 and E-11) are provided in the Appendix B.1. The switching patterns are repeated 5 times to cover dierent parameter variations, that occur during continuous pump operation.

Contour cutting: symmetric and asymmetric distributed pumps

Measurements D-2 and D-5 have been obtained and compared to simulations, considering the 200 MPa operating point. Disturbance steps have been applied on the test bench, when cutting head 2 opens and closes. This results in 2 possible pump rates. The applied switching pattern on disturbance d = [d 1 , d 2 ], the measured pump rates u = [u 1 , u 2 ] and measured pump output pressures p P = [p P 1 , p P 2 ] are shown in the plots of Figure 6.47 for measurement D-2 and in the plots of Figure 6.48 for measurement D-5, respectively. It is observed that the pump rates for both pumps reach a dierent steady-state for symmetric distributed pumps, where asymmetric distributed pumps seem to reach a similar steady-state. Where the operating point of a high-pressure pumps depends on its position in the network, the asymmetry in the network conguration on the test bench seems to compensate the dierent pump rates of each prototype. The switching pattern is again repeated 5 times to cover dierent parameter variations that occurs during continuous pump operation. A high rate of convergence is only obtained, if the reference signal is slowly varying. As a consequence, to guarantee fast convergence, the dynamics of the reference signal needs to be taken into account. Nevertheless, this algorithm is scalable as the computation and communication eort does not grow with increased network size. It guarantees convergence to average, robustness for loss of agents and packet loss.

Managing objectives

Future waterjet facilities combine N equivalent high-pressure pumps to supply M work stations at a time. Each pump n generates an input uid ow Q n (t) to a network subsection. These subsections are interconnected to establish a high-pressure network with unknown topology ∆, see the high-pressure network in the diagram of Figure 7.2. Every work station m holds a cutting head of various conguration that causes an output uid ow Q m (t) with respect to an unknown switching state d m (t). The overall water consumption varies with respect to the opening and closing of the cutting heads. For steady-state, the sum of input uid ows, generated from all coupled pumps, corresponds to the sum of all output uid ows at the work stations. The maximally available input uid ows Q n of every pump n = {1, . . . , N } must satisfy the maximally possible water consumption Q m of every work station m = {1, . . . , M }, such as

Applying the distributed algorithms

Dynamic average consensus (7.4a -7.4b) allows interconnected agents i to agree to a global average value x i (t) → ū(t) without any centralized manager. In a subsequent step, it is required to update the local reference values r i to track the average for the subsystem states by means of control. Average consensus is used in [START_REF] Liu | Synchronization of multi-agent systems with delayed control input information from neighbors[END_REF] to generate the references for formation control, when positioning vehicles with dened deviation from the average position. It is also implemented in [START_REF] Mirali | First-order average consensus for cooperative control problems using novel weighting strategies[END_REF] for cooperative control, attaining the local subsystem states to reach the average values and eliminating the consensus error. In contrast to this, multiple distributed PI controllers manipulate the ow at an edge to balance the load for all vertices in a networked system [Wei and van der Schaft, 2013]. The states of a distributed parameter system converge to average value by means of distributed balancing [Demetriou, 2013], where the physical system dynamics of the high-pressure network has been taken into account to obtain consensus.

Average consensus for high-pressure pumps

Applying the extended dynamic average consensus algorithm (7.5a -7.5c) for the high-level managing of distributed pumps, aims to estimate the overall average pump rate x i (t) → ū(t) at every agent i, as illustrated in diagram of Figure 7.5.

It is then desired to manipulate each reference signal r i (t) by means of control, see [START_REF] Liu | Synchronization of multi-agent systems with delayed control input information from neighbors[END_REF] and [START_REF] Mirali | First-order average consensus for cooperative control problems using novel weighting strategies[END_REF], in order to allocate the overall water consumption of a high-pressure network to the available pumps, such as u i (t) → x i (t) ∀ t → ∞. Whereas the logical layer estimates the time-varying average, the physical layer will track this average value.

Distributed balancing for high-pressure pumps

Considering the high-pressure network as an additional communication channel, the pressure propagation between network subsections will be used to balance the high-pressure pumps. Adopting the approaches in [Wei and van der Schaft, 2013] and [Demetriou, 2013], each pump agent i is thereby assigned to derive an individual reference pressure r i (t) by exchanging its local pump rate u i On the other hand, distributed balancing is evaluated, using the same high-pressure network conguration of Table 7.1. The preliminary results of Figure 7.9 show again the pump output pressures, the corresponding pump rates, for both distributed pumps, and the applied switching patterns to the cutting heads. Every pump agent will adjust the local reference pressure with respect to a global reference of 350 MPa. The resulting pump output pressures will not necessary reach the global reference, but the pressure trends are expected to remain in close neighbourhood of the desired global reference. This compensates the dierent pressure losses along a uid ow path with respect to the pump positions. The low-level control still attenuates pressure uctuations due to the switching of cutting heads. However, the pump rates of each high-pressure pump will converge to the global time-varying average and satisfy the overall water consumption.

The distributed balancing (7.13a -7.13d) is found to balance the high-pressure pumps, even for a time-varying water consumption. The rate of convergence depends on the tuning variable α. The zoom-in plots of Figure 7.10 illustrate the local references for the pump agents, considering the dierent switching states (a) -(d) and various α-values. The rate of convergence is enhanced by increasing α, compare doted (α = 0.01), dashed (α = 0.1) and full lines (α = 1). The obtained local reference values are applied on the low-level controllers. This maintains to track the average pump rate, while compensating the dierent pressure losses along the uid ow path. The plots of Figure 7.11 show the resulting pressure trends at the pump outtakes, where the plots of Figure 7.12 present the convergence of the individual pump rates to a global average.

Application on a 3 pump setup

Finally, average consensus and the proposed distributed balancing are applied to the introduction example of a future waterjet facility (see Figure 7.4), taking 3 pumps and 4 cutting heads into account. The cutting heads are congured as given in Table 7.2. They will alternately switch open and close considering contour cutting. The plots (1a-1d) of Figure 7.13 show the oating operation and the estimated average pump rates, while the plots (2a-2d) realize a balanced pump operation by means of distributed balancing.

The average consensus performs as expected to estimate the global average pump rate with reasonable rate of convergence (1a). Pressure uctuations become attenuated by means of lowlevel control (1b), where every pump obtains an individual pump rate, contributing to the overall water consumption (1c). The water consumption varies with respect to the cutting head switching states. Also the distributed balancing performs as expected and causes each pump to converge to the the time-varying average water consumption. The pump agents provide individual reference values (1a), which are realized for every pump output pressure (2b). The pump rates converge towards average value (2c) with respect to the time-varying water consumption. The start up and shut down of the pump follows a reference value ramping. Both is convenient to observe the reference value tracking that will be used to verify the simulation model for a wide pressure range. The plots of Figure A.1 and A.2 shows the simulated and measured pressure trends during start up and shut down for a operating point of 200 MPa (measurement B-4) and 350 MPa (measurement B-7), respectively. The input and system pressure are given in the rst plot, both output pressures in the second plot and the corresponding pump rate in the third plot.

A.1.3 Cavity cutting: single cutting head

The cutting head will switch open and close with a period of 0.5 seconds whereas the pump tries to maintain an operating point of 200 MPa. The uid ow consumption for measurement B-5 will vary as shown in Figure A.11. The cutting head is again congured with a nozzle of 0.35 mm inner diameter. Thus, a maximal uid ow rate of 2.4 l/min results for a pump rate of 100%.

A.2 Multiple cutting heads experiments A.2.1 Contour cutting: asymmetric cutting heads

A continuous operating point of 350 MPa pressure is chosen. Two cutting heads will open and close alternately with a phase shift of 90°. Considering the measurement E-9, cutting head 1 holds a nozzle of 0.2 mm inner diameter while the cutting head 2 is congured with a 0.1 mm nozzle considering the 350 MPa setup. As a consequence four dierent uid ows will result Q(t) ∈ {0, 0.27, 1.01, 1.28} l/min. The plots of Figure A.13 shows input pressure trend, input uid ow and cutting head switching pattern for simulation and measurement over an entire switching cycle. Figure A.13 Overview of experiment E-9 considering a 1 pump -2 cutting head setup at 350 MPa with dierent nozzle sizes: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles.

Zoomed-in plots for every measurement positions are given in the subsequent plots, when rst opening cutting head 1 followed by opening cutting head 2 and then closing head 1 followed by closing head 2. The plots of Figure A.14 show the zoom-in for opening and closing cutting head 1, where the plots of Figure A.15 show the zoom-in for opening and closing cutting head 2. The input and system pressure are given in the rst plot, both output pressures in the second plot and the corresponding switching pattern in the third plot.

A.2.2 Contour cutting : symmetric cutting heads

A continuous operating point of 350 MPa pressure is chosen. Two cutting heads will open and close alternately with a phase shift of 90°. Considering the measurement E-13, both cutting heads hold a nozzle of 0.15 mm inner diameter. Four dierent uid ows will result Q(t) ∈ {0, 0.61, 0.71, 1.32} l/min. The uid ow diers slightly between each nozzle due to the asymmetric network conguration which causes dierent pressure losses across each piping section. The plots in Figure A.16 show input pressure trend, input uid ow and cutting head switching pattern for simulation and measurement over an entire switching cycle. Figure A.16 Overview of experiment E-13 considering the 1 pump -2 cutting head setup at 350 MPa with nozzles of the same size: pressure trend p P (t) and pump rate u(t) due to switching pattern d(t) for selected switching cycles.

Zoomed-in plots for every measurement positions are given in the subsequent plots, when rst opening cutting head 1 followed by opening cutting head 2 and the closing head 1 followed by closing head 2. The plots of Figure A.17 show the zoom-in for opening and closing cutting head 1, where the plots of Figure A.18 show the zoom-in for opening and closing cutting head 2. The input and system pressure are given in the rst plot, both output pressures in the second plot and the corresponding switching pattern in the third plot.

Cavity cutting: asymmetric cutting heads

Two cutting heads will switch open and close with a period of 0.5 seconds whereas the pump tries to maintain an operating point of 200 MPa. For a rst case, cutting head 1 is switching only while the cutting head 2 remains closed. For a second case, cutting head 2 will join causing both cutting heads to switch almost synchronously. For a third case, cutting head 1 closes entirely while the cutting head 2 continuous to switch. The resulting pressure trends and pump rate for measurement B-7 is given in Figure A.19. The cutting head 1 is again congured with a nozzle of 0.3 mm inner diameter where a 0.2 mm nozzle is installed in cutting head 2.

B.1.2 Contour cutting: symmetric cutting heads

Measurements E-11 and E-13 were obtained and also compared to simulations, considering the 200 MPa and 350 MPa operating points, respectively. Disturbance steps were applied on the test bench when switching the dierent cutting heads open and close. This alternately switching results in 4 possible pump rate of dierent uid ow rates, since the piping sections to the cutting heads are of dierent length. MPC evolved as an eective tool to handle complex systems. It nds wide application in process industry. Recent developments in MPC are overviewed in [Mayne, 2014]. MPC solves multivariable control problems in a centralized meaner when generating the optimal output variables while predicting the system states for a nite horizon to achieve stability. As a consequence, control is subject to online optimization and requires an accurate model. The application of centralized MPC to an air conditioning system which generates optimal set-points for decentralized controllers is presented in [START_REF] Elliott | Decentralized model predictive control of a multi-evaporator air conditioning system[END_REF].

Large-scale systems give constraints to centralized control approaches due to network and computational reasons. That again requires to decompose the initial control problem into a set of possibly decoupled subsystems. A controller is then independently designed for each subsystem. In the framework of decentralized model predictive control, each decentralized controller will generate its output variables subject to optimize a local optimization perspective. No information is shared between the controllers. Hence, this approach deteriorates closed-loop performance and stability for any insucient decoupling of neighbouring subsystems. Some MPC techniques are shortly discussed in this sequel.

Hierarchical MPC

Hierarchical methods enhancing decentralized MPC for large-scale systems are discussed in [START_REF] Marquez | Hierarchical control of large scale systems: A zone control approach[END_REF]. The control task is thereby allocated to dierent layers. The top-layer MPC solves a global optimization problem in a centralized meaner to derive optimal set points for the mid-layer controllers. These decentralized MPC's calculate the output trajectories for its subsystems, required to obtain the overall perspectives, where local controllers at low-level generate the control outputs. In addition, a zone control was introduced in [START_REF] Marquez | Hierarchical control of large scale systems: A zone control approach[END_REF] that allows the decentralized MPC's to realize a set point range. This strategy aims at enhancing the convergence to optimum.

Distributed MPC

Recent research work raises increasing attention to distributed model predictive control (DMPC). A detailed survey is given in [START_REF] Christodes | Distributed model predictive control: A tutorial review and future research directions[END_REF]. In the framework of DMPC, the distributed controllers share informations, e.g. local state estimations and set point trajectories, to improve closed-loop stability and performance. The application of an DMPC scheme to a smart grid is presented in [START_REF] Xu | Distributed predictive control for energy management of multi-microgrids systems[END_REF], which is subject to maximizing the overall energy consumption. The smart grid consists of multiple interconnected micro grids. This distributed network is dicult to handle with a centralized optimization. Thus, a controller is designed for each micro grid and the power ow information is shared among these DMPC's.

Non-cooperative and cooperative algorithms are distinguished. Considering non-cooperative DMPC, each controller follows its local optimization criteria, e.g. [START_REF] Farina | Distributed non-cooperative MPC with neighbor-to-neighbor communication[END_REF]. On the other hand, cooperative DMPC allows distributed controllers to retrieve global optimization objectives, e.g. [START_REF] Trodden | Cooperative distributed MPC of linear systems with coupled constraints[END_REF]. It is discussed in [START_REF] Rawlings | Coordinating multiple optimization-based controllers: New opportunities and challenges[END_REF] that DMPC does not guarantee performance and stability if a global objective is missing, even though information between neighbours is exchanged. However, a global objective regarding all DMPC's increases the amount of data that is exchanged. Solving the global optimization problem in a decentralized meaner involves to locally estimate the state trajectory of the other subsystems and the subsystem's states to be updated periodically by means of communication. A cooperative DMPC scheme is shown in [START_REF] Ferramosca | Cooperative distributed MPC for tracking[END_REF] that guarantees convergence to the centralized optimum of the overall system. It is essential to calculate suitable initial conditions for the rst iteration to ensure convergence to optimum.

C.1.3 Cooperative control and synchronization

Consensus in terms of network synchronization of agents and cooperative control of distributed systems are closely related. The dierent distributed protocols aims at consensus between multiple autonomous agents in a networked system while tracking a desired set point reference. Each agent is basically a local optimizer, working at subsystem level. Consensus is achieved when all agents reach the desired set point. The reference set point is thereby not available to all agents. Typically, information is only exchanged between neighbouring agents.

Discrete time dynamic agents and continuous time agents are distinguished. Continuous time agents often show integral action where distributed Kalman lter, distributed least square and other algorithms are used to obtain synchronization. Considering robustly controlled highpressure pumps, discrete dynamics may adequately describe a possible pump agents.

Literature further considers output synchronization and state synchronization. Output synchronization aims at regulating the output of each agent to a desired value, while state synchronization regulates the state of each agent to a desired value.

Solely searching databases about average consensus, an approach discussed later, almost 5'000 papers were published in 1997, already over 10'000 in the year 2007 and over 20'000 in the year 2017. Cumulatively almost 300'000 papers are known until today dealing with dierent subjects of the average consensus problem. Consequently, only some major results are reviewed in this sequel.

Finite-time consensus

Convergence to a consensus value depends on the network topology. Dierent algorithms are found to guarantee consensus in nite time. A synchronization protocol taking into account the control inputs of neighbouring agents is considered in [START_REF] Liu | Synchronization of multi-agent systems with delayed control input information from neighbors[END_REF]. This approach improves the control performance and guarantees convergence, independent of the network topology. A decentralized synchronization protocol sharing state values of neighbouring agents is discussed in [START_REF] Yuan | Decentralised minimum-time consensus[END_REF]. Even since the amount of network members is not known to each agent, the convergence value is computed in minimum time before consensus is eectively reached.

Average consensus

Most approaches for average consensus deploy an iterative scheme where every agent updates its estimated average value using its value and the estimation from neighbouring agents. This procedure manages all agents to converge to the average value of the initial system state. Some optimal strategies to solve the average consensus problem are discussed in [START_REF] Delvenne | Optimal strategies in the average consensus problem[END_REF]. It compares the convergence behaviour in terms of the network topology and gives strategies for which consensus is guaranteed in nite-time.

Protocols that guarantee nite-time convergence typically aim at synchronizing the nodes on an arbitrary value that is not necessarily specied. However, the network topology has to be a priori known to choose appropriate weighting parameters. To obtain this design requirement, a self-conguration procedure deriving appropriate weighting parameters is discussed in [START_REF] Dung | Distributed design of nitetime average consensus protocols[END_REF]. Applying a distributed matrix factorization allows for retrieving appropriate weighting parameters. An optimal weighting strategies is essential to obtain fast convergence time. A rst-order average consensus protocol that guarantees consensus in nite-time was used in [START_REF] Mirali | First-order average consensus for cooperative control problems using novel weighting strategies[END_REF]. The convergence time was enhanced for cooperative control when choosing adequate weights.

Dynamic consensus

The consensus value can correspond to the average control eort of interconnected high-pressure pumps. In this particular use case, the nal average value will vary from the initial average since system state and network topology changes. Dynamic consensus provides a powerful framework to track these time-varying systems.

Dynamic average consensus is studied in [START_REF] Zhu | Discrete-time dynamic average consensus[END_REF]. In contrast to static average consensus, all agents have to agree to the average of time varying signals. This work further introduces a higher-order average consensus protocol capable to reduce steady-state error.

A distributed continuous time consensus protocol is considered in [START_REF] Wang | Distributed dynamic average consensus for nonlinear multi-agent systems in the presence of external disturbances over a directed graph[END_REF] where the system states are subject to converge towards time-varying input signals while attenuating disturbances which acts on the state variables. Here, the convergence relays on the initial condition of the average consensus algorithm. A discrete method for dynamic averaging considering robust initial conditions is given in [START_REF] Montijano | Robust discrete time dynamic average consensus[END_REF].

Disturbance attenuation

Disturbance attenuation is an additional property regarding the high-pressure network. It is shown in [Lunze, 2014] that a disturbance acting on a subsystem may aect other subsystems due to the cooperative control mechanism. It is proposed that a disturbed agent is dismissed from communication as long as excessive disturbance occurs. On the other hand, an observerbased synchronization is presented in [START_REF] Peymani | H ∞ almost regulated synchronization and H ∞ almost formation for heterogeneous networks under external disturbances[END_REF]. Almost consensus between disturbed agents is obtained by attenuating the disturbance impact on the state error.

Event-triggered communication

Along with discrete time consensus it rises the questions of which minimal information set and update rate is required for consensus in desired time. That further allows for improving communication eciency in a data network. For that, event-triggered communication for the dynamic average consensus problem is discussed in [START_REF] Kia | Distributed event-triggered communication for dynamic average consensus in networked systems[END_REF], reducing the communication eort when restricting the update to the agents with largest remaining estimation error. In addition to that, conditions on connectivity are investigated in [START_REF] Nowzari | Distributed event-triggered coordination for average consensus on weight-balanced digraphs[END_REF] to derive convergence when a network topology is recongured.

Distributed parameter system

A special case of state consensus is its application on distributed parameter systems as examined in [Demetriou, 2013]. It aims at ensuring agreement between the states of a dynamic system, described with partial dierential equations, while tracking a control objective.

In that meaner, uid ow networks controlled by decentralized PI controllers are proposed in [Wei and van der Schaft, 2013] where pressure states are subject to load balancing. It is shown that consensus is not guaranteed for every equilibrium point, especially with strongly coupled systems.

Abstract

This thesis provides a design framework that aims to improve the energy eciency of future waterjet facilities in compliance with industry 4.0. This framework involves the modelling of high-pressure networks and the robust control design for a new generation of electrically driven high-pressure pumps. The Part I of this manuscript classies typical waterjet applications and denes use cases to validate the proposed design framework. It presents common hydraulically driven pumps and introduces the functional principle of novel electrically driven pumps. A test bench employs these use cases to investigate the decentralized operation of two pump prototypes with electrical drive systems. The work of Part II develops an approach for the exible modelling of high-pressure networks. This modelling assigns the equations from uid dynamics to the graph theory. The simulation accuracy is thereby improved by introducing a pressure-dependent bulk modulus and relating that to a pressure-dependent uid density. This approach is implemented to provide a generic simulation toolbox, which is eventually applied to model specic high-pressure networks with respect to the dened use cases. The resulting high-pressure network models are validated by means of measurement data obtained from the test bench. Finally, Part III proposes a decentralized approach of the robust control design for electrically driven high-pressure pumps, each capable to stabilize a network subsection. Dierent uncertainty descriptions with various varying parameter are evaluated to realize a robust H ∞ synthesis of structured controllers. The complexity of the resulting optimization problem due to controller syntheses is then reduced by introducing the concept of joint shaping functions. Moreover, introducing the procedure of ∆-K iteration employs guaranteed robustness while maximizing the permissible uncertainty range. The validated high-pressure network models are used to verify stability and performance for the decentralized control design. Simulations reveals that multiple pumps will operate at arbitrary pump rates, if any managing is missing. Hence, algorithms for distributed balancing by means of average are investigated to overcome this downside.

Keywords: Waterjet machining, high-pressure network, graph-based modelling, uid dynamics, robust control, decentralized control, dynamic average consensus, distributed balancing.