Mr. Yoann Mr Youness Lami

Mr Hervagault

Vincent Benjamin

El Mehdi

Keywords: routing protocol, 2, 4, 1 Metric selection

It is a great milestone in one's life to engage in the doctoral study and success the Ph.D. degree. This doctoral thesis, like most research works, is the result of a curious and inquisitive spirit, coupled with plenty of hard work and persistence. Naturally, it was difficult at times, but overall, the fulfilling moments far exceeded the hardship -and I owe that in a world of people, to whom I will always be grateful. Without their generous supports throughout, this thesis would not become possible.

First of all, I would like to express my sincere gratitude to my supervisors Prof. Laurent LEFÈVRE and Assoc. Prof. Denis GENON-CATALOT for their continuous support of my Ph.D. study and related research, for their patience, motivation, and immense knowledge. Their guidance helped me in all the time of research period and writing of this thesis. They are knowledgeable and helpful mentors who have not only given me advice on the academic side, but also shared a lot of experience in life. Their patience and high availability when I needed assistance in research benefited me greatly. It has been an honor to be their Ph.D. student. They have taught me, both consciously and unconsciously, how good automatic control and computer science can be combined. I appreciate all their contributions of time, ideas, and funding to make my Ph.D. experience productive and stimulating. The joy and enthusiasm that they have for their research were contagious and motivational for me, even during tough times in the Ph.D. pursuit

Table of contents

Nomenclature

A multi-layer approach to distributed control problems for irrigation channels with communication constraints

The irrigation always is a major concern for human societies and it is more serious with rapid urbanization and climate change [START_REF] Ahmad | Water and Energy Scarcity for Agriculture: Is Irrigation Modernization the Answer?[END_REF]. Currently, the performance of irrigation channels manually operated can no longer meet new challenges of irrigated areas in both water conservation and operating aspects [START_REF] Zeleke | Hydraulic and Operational Performance of Irrigation System in View of Interventions for Water Saving and Sustainability[END_REF][START_REF] Malaterre | La modernisation des canaux d'irrigation : apports de l'automatisation pour la gestion opérationnelle[END_REF][START_REF] Li | Performance analysis of irrigation channels with distributed control[END_REF][START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF][START_REF] Plusquellec | Modern Water Control in Irrigation: Concepts, Issues, and Applications[END_REF]. The automation of engineering works is a possible way of improving the operational management [START_REF] Ahmad | Water and Energy Scarcity for Agriculture: Is Irrigation Modernization the Answer?[END_REF][START_REF] Malaterre | La modernisation des canaux d'irrigation : apports de l'automatisation pour la gestion opérationnelle[END_REF]. Automatic control systems can be designed to achieve desired objectives of water resource management while reducing investment and operating costs [START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF][START_REF] Malaterre | La modernisation des canaux d'irrigation : apports de l'automatisation pour la gestion opérationnelle[END_REF][START_REF] Puig | Model predictive control of combined irrigation and water supply systems: Application to the Guadiana river[END_REF][START_REF] Imam Sutrisno | Distributed model predictive control and application to irrigation canal[END_REF][START_REF] Charbonnaud | A supervised robust predictive multi-controller for large operating conditions of an open-channel system[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF][START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF][START_REF] Mareels | Systems engineering for irrigation systems: Successes and challenges[END_REF]. From these perspectives, this thesis presents the control problems of an irrigation network with limited communication and a multi-layer approach to solving these problems in a distributed manner. As an interdisciplinary subject, it uses the concepts from automatic control, hydraulic engineering, and computer networks in an attempt to bring together these advantages. The detailed discussions for each layer and the demonstration with analytical and simulation results are described throughout several chapters. The first layer to be considered is the hydraulic network composed of free-surface channels, hydraulic structures and mesh subnetwork of pressurized pipes [START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF]. By adopting a model-based control strategy [START_REF] Rawlings | Model Predictive Control : Theory and Design[END_REF], the control of this large-scale complex system requires a tractable model, which adequately captures system dynamics and the interactions among subsystems (different irrigation channel models are surveyed in [START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF][START_REF] Zhuan | Models and Control Methodologies in Open Water Flow Dynamics: A Survey[END_REF]). Based on the top-down approach to deal with the system complexity, the first chapter presents a "modular modeling" that involves partitioning the system into different scalable and reusable subsystems (i.e., modules) and making use of well-defined modular interfaces (the approach is detailed in [START_REF] Steven | GSFLOW-Coupled Ground-Water and Surface-Water Flow Model Based on the Integration of the Precipitation-Runoff Modeling System (PRMS) and the Modular Ground-Water Flow Model (MODFLOW-2005)[END_REF]). Modules are independently modeled and contained in a Model Library as model components, which can be re-used for other water systems. For instance, the reach component in the Model Library is a discrete-time nonlinear model for a reach obtained by coupling the Saint-Venant (SV) equations for describing the physics of free-surface fluid [START_REF] Dos | A multi-model approach to Saint-Venant Equations: A stability study by LMIs[END_REF][START_REF] Lozano | Simulation of automatic control of an irrigation canal[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF][START_REF] Belaud | Hydraulic Modeling of an Automatic Upstream Water Level Control Gate for Submerged Flow Conditions[END_REF][START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF] and the Lattice Boltzmann (LB) method for fluid simulation [START_REF] Kruger | The Lattice Boltzmann Method[END_REF][START_REF] Liu | Lattice Boltzmann method for the Saint-Venant equations[END_REF][START_REF] Zhang | Lattice Boltzmann method for overland flow studies and its experimental validation[END_REF][START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF]. As demonstrated in [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF][START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF][START_REF] Diemer | Discrete controllability of distributed parameters systems using lattice Boltzmann models: An application to the shallow water equations[END_REF], the chosen LB method is an efficient and powerful numerical tool (in terms of accuracy, numerical stability and computational time) to simulate the free-surface flows respecting the conservation laws of macroscopic variables. Other components of the Model Library are hydraulic structure models. The hydraulic structures are usually treated as internal boundaries of reaches and modeled by using algebraic relationships between the flow and pressure variables [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF]. Eventually, the complete model of a considered system is built by selectively assembling appropriate components from the Model Library. The channel model can be used as a forecasting model to predict some water quantities (e.g., water heights, flow rates) at different points along the channels at a time, corresponding to specified initial conditions [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF]. The advantages of the "modular modeling" approach are the flexibility and the effectiveness of the modeling process in terms of model construction, testing, and reuse-ability of system components [START_REF] Steven | GSFLOW-Coupled Ground-Water and Surface-Water Flow Model Based on the Integration of the Precipitation-Runoff Modeling System (PRMS) and the Modular Ground-Water Flow Model (MODFLOW-2005)[END_REF].

To enable the exchange of the information among the control system's components, a communication network is considered in the second layer. In networked control systems [START_REF] Bemporad | Networked Control Systems[END_REF][START_REF] Zhang | Survey on Recent Advances in Networked Control Systems[END_REF][START_REF] Rachana | Networked Control System: Overview and Research Trends[END_REF], an efficient communication between their components plays an important role in achieving global objectives [START_REF] Zhang | A brief survey of recent results on control and filtering for networked systems[END_REF][START_REF] Tan | Necessary and Sufficient Stabilizing Conditions for Networked Control Systems with Simultaneous Transmission Delay and Packet Dropout[END_REF][START_REF] Rachana | Hydrology and Hydraulic Systems: Fourth Edition[END_REF]. However, the study of the network architecture and the interactions between networked components is challenging due to the presence of heterogeneous devices and communication issues such as time delay, data packet loss, etc. [START_REF] Kansara | Sensor based Automated Irrigation System with IOT : A Technical Review[END_REF][START_REF] Zhang | Survey on Recent Advances in Networked Control Systems[END_REF]. Solving these problems is investigated in the second chapter by introducing a hybrid network architecture [START_REF] Li | A survey on topology control in wireless sensor networks: Taxonomy, comparative study, and open issues[END_REF][START_REF] Yu | A survey on wireless sensor network infrastructure for agriculture[END_REF]132] and a dynamic routing design ( [132], Section 5.2) based on Quality of Service (QoS) requirements of control applications. A hybrid network refers to any network that contains two or more different communications standards or multiple topological structures [START_REF] Aleksandar | Novel Architecture for Networked Control Systems[END_REF][START_REF] Kim | Remote Sensing and Control of an Irrigation System Using a Distributed Wireless Sensor Network[END_REF][START_REF] Biao Ren | The hybrid mobile wireless sensor networks for data gathering[END_REF]. In terms of QoS, some applications (and users) demand just the performance, which is guaranteed from the network "as the best that could be done under the circumstances", others may require the satisfaction of stronger performance criteria (such as dropped packet rate, out-of-order delivery, error rates, bit rate, throughput, transmission delay, availability, jitter, etc.) (see more details in [132], Section 5.4). Eventually, the second chapter focuses on the ways to provide QoS that meet the requirements of control applications when integrating all control system components in a hybrid network [START_REF] Karkazis | Evaluating routing metric composition approaches for QoS differentiation in low power and lossy networks[END_REF][START_REF] Xia | QoS Challenges and Opportunities in Wireless Sensor/Actuator Networks[END_REF]. The dynamic routing is a process determining the optimal path that data packets should follow through a network to arrive at a specific destination [132] and it uses routing algorithms and protocols to monitor and respond to changes in network topology [START_REF] Kumar | Routing in IPv6 over low-power wireless personal area networks (6LoWPAN): A survey[END_REF][START_REF] Fischione | Design principles of wireless sensor networks protocols for control applications[END_REF]. The purpose of dynamic routing is to help prevent packet delivery failure, improving network performance and relieving network congestion [132]. For network routing design in order to satisfy the QoS requirements of control applications, a weighted composition of some standard metrics is proposed so that the routing protocol using the composite metric achieves convergence, loop-freeness and path-optimality properties [START_REF] Capone | An Energy Efficient and Reliable Composite Metric for RPL Organized Networks[END_REF][START_REF] Chang | Energy-Efficient Oriented Routing Algorithm in Wireless Sensor Networks[END_REF][START_REF] Karkazis | Evaluating routing metric composition approaches for QoS differentiation in low power and lossy networks[END_REF][START_REF] Olivier | Energy-based routing metric for RPL[END_REF][START_REF] Liu | Joint routing and sleep scheduling for lifetime maximization of wireless sensor networks[END_REF]. This composition method shows the correlation between combined routing metrics and required network performance criteria. In this chapter, the composition approach is analytically demonstrated on the application of composite metrics with the Routing Protocol for Low-power and lossy network (RPL) [START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF][START_REF] Zahariadis | Design Guidelines for Routing Metrics Composition in LLN[END_REF], specified by Internet Engineering Task Force (IETF -www.ietf.org). Simulations of the dynamic routing in the hybrid network are also performed and different network performance criteria are evaluated through extensive simulation scenarios. The comparison of simulation results can validate the benefits of this composition approach for dynamic routing in networked control systems.

As the irrigation network is a large-scale complex system, the control of this system is chosen in the distributed way in order to increase the scalability and reliability [START_REF] Van Overloop | Human-in-the-Loop Model Predictive Control of an Irrigation Canal[END_REF][START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF][START_REF] Puig | Model predictive control of combined irrigation and water supply systems: Application to the Guadiana river[END_REF][START_REF] Imam Sutrisno | Distributed model predictive control and application to irrigation canal[END_REF][START_REF] Igreja | Application of distributed model predictive control to a water delivery canal[END_REF][START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF][START_REF] Van Overloop | Model Predictive Control on Open Water Systems[END_REF]. In distributed control, accounting for the interactions of subsystems, the coordination and/or the cooperation of controllers requires exchanging the information among controllers [START_REF] Elmahdi | Decentralized Control Framework and Stability Analysis for Networked Control Systems[END_REF][START_REF] Alur | Compositional modeling and analysis of multi-hop control networks[END_REF][START_REF] Maurice | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF][START_REF] Tipsuwan | Control methodologies in networked control systems[END_REF]. However, challenges arise when controllers may lose the synchronization and when the shared information may encounter the divergence due to the switching topology and the imperfection in communication [START_REF] Li | Robust Tracking Control of Networked Control Systems With Communication Constraints and External Disturbance[END_REF][START_REF] Deroo | Distributed control design with local model information and guaranteed stability[END_REF]. Managing the divergent (or outdated) information is considered in the third chapter as an information consensus problem and solved by using an asynchronous consensus protocol [START_REF] Qin | Recent Advances in Consensus of Multi-Agent Systems: A Brief Survey[END_REF][START_REF] Lin | Distributed Continuous-time Optimization: Nonuniform Gradient Gains, Finite-time Convergence, and Convex Constraint Set[END_REF][START_REF] Zhou | Constrained consensus of asynchronous discrete-time multi-agent systems with time-varying topology[END_REF][START_REF] Li | Designing Fully Distributed Consensus Protocols for Linear Multi-Agent Systems with Directed Graphs[END_REF][START_REF] Ren | Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues[END_REF][START_REF] Gao | Asynchronous consensus of continuous-time multi-agent systems with intermittent measurements[END_REF][START_REF] Xiao | Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. Based on multi-agent system (MAS) paradigm, the consensus approach provides a solution guaranteeing that all controllers have a consistent view of some data values needed for action computation [START_REF] Ren | Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues[END_REF][START_REF] Olfati-Saber | Consensus and Cooperation in Multi-Agent Networked Systems[END_REF][START_REF] Ren | Information Consensus in Multivehicle Cooperative Control[END_REF]. By exploring information exchange topologies under proposed communication assumptions (e.g., directed graphs, connectivity, and asynchronism) [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF], we find the necessary and sufficient conditions for the information consensus to be reached asymptotically using the defined asynchronous protocol. The designed consensus protocol can be integrated into the distributed scheme so that the shared information among controllers asymptotically converges to the newest values after applying an action. As a result, the performance of the control system is improved in comparison with the decentralized or distributed implementation strategies where interaction information is not up to date. Some simulation scenarios are given to illustrate the efficiency of this approach for the distributed control of irrigation channels modeled by the Lattice Boltzmann method.

Finally, the third layer introduces the optimal reactive control system developed for the regulatory control of large-scale irrigation channels under a Distributed Cooperative Model Predictive Control (DCMPC) framework [START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF][START_REF] Malaterre | La modernisation des canaux d'irrigation : apports de l'automatisation pour la gestion opérationnelle[END_REF][START_REF] Puig | Model predictive control of combined irrigation and water supply systems: Application to the Guadiana river[END_REF][START_REF] Imam Sutrisno | Distributed model predictive control and application to irrigation canal[END_REF][START_REF] Charbonnaud | A supervised robust predictive multi-controller for large operating conditions of an open-channel system[END_REF][START_REF] Igreja | Application of distributed model predictive control to a water delivery canal[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF][START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF][START_REF] Camponogara | Distributed model predictive control[END_REF]. Whereas different approaches to the control of irrigation systems have been developed and applied to real channels all over the world (as surveyed in [START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF][START_REF] Mareels | Systems engineering for irrigation systems: Successes and challenges[END_REF][START_REF] Charbonnaud | A supervised robust predictive multi-controller for large operating conditions of an open-channel system[END_REF]), the DCMPC is chosen because the channel control requires taking into account system inputs/outputs, the interactions among subsystems, and the structural/operational constraints and processing the delay compensation [START_REF] Borrelli | Predictive Control for linear and hybrid systems[END_REF][START_REF] Rawlings | Model Predictive Control : Theory and Design[END_REF]. The fourth chapter discusses different control implementation strategies (i.e., centralized, decentralized and distributed strategies) and how the cooperative communication among local controllers can be included to improve the performance of the overall system. In the proposed DCMPC scheme, each controller shares the interaction variable (e.g., the flow rate through a controlled gate) for action computation of the upstream neighbor and a coordination variable (e.g., its set-points) for the cooperation with the downstream neighbor. For simulations, a particular benchmark of an irrigation channel modeled by the Lattice Boltzmann method is considered. The comparisons of simulation results among the proposed control approaches validate the benefits of the distributed cooperative control approach. The numerical results show up the improvement of some global performance criteria such as response time, overshoot limit, but larger computation delay in distributed cooperative control with regard to decentralized control. By these results, we have extended the state-of-the-art in the control of irrigation channels with the coherent combination of the modeling by the Lattice Boltzmann method and the centralized/decentralized/distributed model predictive control approaches.

Outline and contribution of the thesis

Motivated by challenging control and communication problems in distributed control of largescale irrigation channels, this thesis presents a multi-layer approach to solving these problems through several chapters organized as follows:

• Chapter 1 contributes the component-based modeling of irrigation channels using the Lattice Boltzmann method. Modularity in modeling involves the modularized and parameterized modeling technique and the creation of a Model Library. The proposed modular approach reduces the system design and review cycle that can significantly improve the design consistency of a large-scale system. In simulations, it takes less time and enables a more efficient co-simulation. The model has a decisive role in the understanding and diagnosis of irrigation channels.

• Chapter 2 investigates in designing a hybrid network to cope with the heterogeneity of wide-area networked control systems and the communication issues. In the hybrid network, a composite metric is proposed for dynamic routing in order to satisfy different QoS requirements of control applications. The resulting network model can be used to dynamically compute the QoS-related costs (or probabilities) of different paths between the components of networked control systems (NCSs). These are the important concerns when considering the information exchange in NCSs.

• Chapter 3 aims to solve updating problems of the information shared among controllers in a networked control system with communication constraints. These problems are considered as an information consensus problem and solved by using an asynchronous consensus protocol. The advantages of this approach are that, even when the neighboring controllers lose the direct communication, the shared information is still able to be updated with the benefit of consensus convergence. The integration of the consensus protocol into a control scheme is demonstrated that it still guarantees the control performance, but with the robustness of the changes in communication topology due to link/node failures and time delays.

• Chapter 4 aims to implement and compare different control schemes in which costs, constraints, profiles are taken into account in decentralized and distributed constrained optimization problems (i.e., via the MPC design); and the cooperation among local MPC controllers can be included in order to improve the performance of the overall system. In brief, the original contributions of this thesis are as follows:

• combining the LB method for fluid simulation and "modular modelling" approach, we provide a parameterizable Model Library of a large-scale irrigation system suitable for adequately capturing its dynamics and the interactions among the subsystems. This Model Library can be reused for other water systems;

• to cope with the heterogeneity and the communication issues when integrating all components of a wide-area networked control system into a unified network infrastructure, we propose a hybrid network architecture and the dynamic routing with a composite metric over this network;

• to address challenging QoS requirements of control applications in network design, we show that several performance criteria related to the communication needs for control applications can be optimized by using a metric composition approach;

• for the convergence of the information shared among controllers, we design an asynchronous consensus protocol and establish conditions for a class of "leader-followers" systems to asymptotically reach the consensus under fixed or time-varying directed exchange topology;

• we propose a global topology for the "leader-switching" system to satisfy multiple information consensus problems;

• we integrate the consensus protocol into the DCMPC algorithm

• we propose a decentralized implementation of the control system where a local MPC controller for each subsystem is separately designed. They compute the action using only the local information;

• the distributed cooperative controllers are designed by taking into account the information shared by neighboring controllers (e.g., system output and set-points) to compute the action. They cooperate with their neighbors by minimizing the influence of an applied action with the regularization objective of their neighbors;

• extensive simulation results are provided through different scenarios, which validate the proposed predictive control schemes and highlights their strengths and weakness in terms of global performances. These chapters emphasize the potential interest of the multi-layer approach, more precisely its efficiency and reliability for supervision, multi-objective optimization and distributed control of complex water transport systems.

Chapter 1

Modeling of irrigation channels using the Lattice Boltzmann method

The use of a hydraulic computing model allows making some simulations in order to deeply know about system operations, predict hydraulic variables, evaluate management rules, and diagnose malfunctions. This chapter provides an overview of the methods used for the modeling of free-surface hydraulic as found in natural rivers or artificial channels. The emphasis is on the application of the Lattice Boltzmann (LB) method to model free-surface flow characterized by fluid hydrodynamics. Organizationally, Section 1.1 introduces a general irrigation channel with various geometric and characteristic variables that are useful for hydraulic calculation. In the channel, the flow can be treated in different patterns (e.g., uniform or non-uniform) or in different functional regimes (i.e., steady or varied) corresponding to spatial and temporal variations. The Saint-Venant (SV) equations presented in Section 1.2 constitute a distributed-parameter model, which is commonly used to provide a detailed description of one-dimension (1D) free-surface flow. The detailed model makes it possible to understand the phenomena associated with the physical system. Using the SV model, the analysis of a steady-state regime of non-uniform flow is useful for considering the evolution of water heights (or flow rates) in the channel as a function of the boundary conditions. Section 1.3 is dedicated to the modeling of an irrigation channel using the LB method. The LB model allows generating the dynamics of a complex water system from various boundary conditions. In particular, the one-dimension three-velocities (D1Q3) LB model can be effectively used for the modeling of a reach (or a segment). For coupling different reaches, the hydraulic structures (such as gates, weirs, or mixed gates and spillway) are used as internal boundaries and modeled by mathematical relations between mesoscopic variables and hydraulic variables. In Section 1.4, the discrete-time LB equations are solved to simulate the flow of irrigation channel using Bhatnagar-Gross-Krook (BGK) collision model. We are particularly interested in the modularized and parameterized modeling technique, which is used to create a Model Library of system components (e.g., reach, gate, weir or mixed). As an example, a section of the Bourne channel in France is considered for the analysis and simulation schemes. The simulation results presented in Section 1.4.2 show the coherence of flow behavior corresponding to the specified initial conditions and when introducing perturbations. Finally, Section 1.5 summarizes the aspects presented in this chapter.

Irrigation channels

The operational management of irrigation channels often is unsatisfactory due to the complexity of considered systems [START_REF] Ahmad | Water and Energy Scarcity for Agriculture: Is Irrigation Modernization the Answer?[END_REF][START_REF] Malaterre | La modernisation des canaux d'irrigation : apports de l'automatisation pour la gestion opérationnelle[END_REF][START_REF] Li | Performance analysis of irrigation channels with distributed control[END_REF][START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF][START_REF] Burt | Modern water control and management practices in irrigation: Impact on performance[END_REF]. In traditional irrigation channels, water is conducted, from a source of supply (e.g., lakes, rivers, dams), along the main channels (possibly diverted to secondary channels) and then to distribution networks or to the natural drainage systems. An irrigation channel (or free-surface channel) generally is an open-channel hydraulic system providing a desired flow of water [START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF]. When the channel is directly connected to a water source such as a lake or a river, the water supply is fairly reliable in normal weather [START_REF] Rachana | Hydrology and Hydraulic Systems: Fourth Edition[END_REF]. Even though, water must be regulated to avoid using so much water in one area while other areas suffer. For this purpose, some hydraulic structures are required to regulate the flow and deliver the correct amount of water to different branches of the system and onward to the irrigated fields. There are four main types of hydraulic structures [START_REF] Rachana | Hydrology and Hydraulic Systems: Fourth Edition[END_REF][START_REF] Vasquez | Hydraulique a surface libre[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF]: erosion control structures, distribution control structures, crossing structures and water measurement devices. Erosion control structures (e.g., drop structures or chutes) are required to reduce the bottom slope of channels lying on steeply sloping land in order to avoid high velocity of the flow and risk of erosion. Distribution control structures (e.g., gates, weirs, or mixed gates and spillway) are installed at appropriate locations in order to control the water heights, flow rates or volumes at some points along the channel. Crossing structures (e.g., flumes, culverts, and inverted siphons) are necessary to carry irrigation water across roads, hillsides, and natural depressions. The most commonly used water measurement devices are water level and flow-rate meters. From the main or secondary channels, pumping stations are used to withdraw water upward into the distribution network (e.g., mesh pressurized network or field ditches). Occasionally, when an irrigation channel traverses a great distance or must navigate through the changes in elevation, temporary reservoirs are commonly built to store water and then refill irrigation channels through dams. Popularly, the drainage system at channel downstream removes the excess of water from the channels (e.g., caused by oversupply, rainfall, high water level in the river).

The channel managers usually face different operational management problems. Their main objectives are: (1) improving irrigation water management in order to increase productivity;

(2) save water, minimize energy consumption and exploration costs; while (3) guaranteeing the safety of the infrastructure among others. The automation of hydraulic structures is a possible way to achieve the desired objectives of operational management while reducing or sometimes eliminating human intervention. In principle, automatic control of such a system requires the description of the system dynamics represented by a model. However, modeling the complete system is a difficult task due to the complexity (e.g., dimensionality, uncertainty, and information structure constraints) and because unknown external perturbations may affect its behavior.

Characterization of irrigation channels

Generally, the irrigation channel is a very large and complex system characterized by important time delays between inputs and outputs (due to water transportation), strong non-linear dynamics (mainly around the hydraulic structures), unknown or unmeasurable perturbations (e.g., inflows created by rainfall, high river level and outflows created by stealthy withdrawal, infiltration and evaporation losses), and interactions between subsystems. We recall briefly in the next sections the definition of some common parameters and variables, which can be used for system modeling.

Geometric parameters

We limit our study to a free-surface channel (e.g., single bed river or irrigation channel) characterized by fixed banks and bottom bed, a surface subjected to atmospheric pressure, and the watercourse can be reasonably considered as rectilinear (see Fig. 1.1). The geometry of the channel can then be perfectly defined by a succession of cross-sections, perpendicular to the direction of flow (i.e., flow axis) [START_REF] Vasquez | Hydraulique a surface libre[END_REF]. Assuming that the free (or open-air) surface is horizontal from one channel bank to the other and velocities are homogeneous in a cross-section, the bank-to-bank and vertical components of the flow can be neglected. As a result, this leads to one-dimensional modeling (or 1D model) that allows all geometric parameters and hydraulic variables to be considered as functions of the abscissa l (m) measured on the flow axis and the time t (s) [START_REF] Vasquez | Hydraulique a surface libre[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF]. For simplicity, we denote a function, for example, f (l,t) by f . Definition 1.1.1. Geometric parameters [START_REF] Vasquez | Hydraulique a surface libre[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF].

As shown in Fig. 1.1,

• the surface width or top width, denoted by B (m), is the width of the channel at the level of the free surface,

• the water height, denoted by h (m), is the maximum water height measured between the bottom bed and the free surface,

• the wetted perimeter, denoted by P (m), is the perimeter of a cross-section excluding the length of the free surface,

• the wetted section area, denoted by S (m 2 ), is the portion of the section wetted by the water in the cross-section of the channel,

• the hydraulic height, denoted by D h = S/B (m), corresponds to the ratio between the wetted section area and the top width of the channel,

• the hydraulic radius, denoted by R h = S/P (m), is the ratio between the wetted section area and the wetted perimeter,

• the bed slope, denoted by I (m/m), is commonly expressed as the ratio in of the difference between upstream and downstream heights to the length of the channel,

• the free-board of the channel is the anticipated height of the bank above the highest water level, it is required to guard against over-topping by waves or unexpected increase in water level.

The parameters B, h, P, S, and R h depend on the flow rate (therefore, they are not geometric constants) whereas I is a geometric constant. According to the shape of their cross-section, the channel is called rectangular, trapezoidal, triangular, circular, parabolic or irregular (as shown in Fig. 1.1.b). The most commonly used cross-sections in irrigation and drainage systems are rectangular or trapezoidal forms with small side slope (e.g., usually less than 5%¸ [START_REF] Vasquez | Hydraulique a surface libre[END_REF]). When the cross-section, bed slope and roughness do not vary according to the flow axis, the channel is prismatic.

Hydraulic variables

Some hydraulic variables needed to quantify the motion of water (or fluid) are: Definition 1.1.2. Hydraulic variables [START_REF] Vasquez | Hydraulique a surface libre[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF].

• the velocity at a point of the flow, denoted by v = ∆l ∆t (m/s) (where ∆l is 1D lattice spacing and ∆t is time step), is the velocity of the particle passing at this point in the mean time,

• the average velocity, denoted by u (m/s), is the average of all velocities in a cross-section,

• the flow rate, denoted by Q = Bhu (m 3 /s), is the volume of water flowing through a cross-section perpendicular to the flow axis per unit time.

In addition, the study of fluid dynamics involves other parameters characterizing the environment such as the acceleration of gravity, denoted by g (m/s 2 ); the pressure, denoted by ρ (N/m 2 ); the viscosity, denoted by µ (kg/m * s); the various force terms, denoted by F (N/kg), etc. The density of the water, denoted ρ w , is 1000 (kg/m 3 ) in the case of clear water without suspended matter. The water in motion also exerts a perpendicular thrust and a tangential frictional force on the banks of the channel. The solution of a dynamic fluid problem typically involves the calculation of various characteristic variables of the fluid (e.g., flow depth, flow rate, velocity, pressure, density, etc.) as functions of space and time.

Classifications of the flows

The free-surface flows can be classified and described in various ways, according to the variability of hydraulic characteristics (e.g., flow heights and velocities) with respect to time and space.

Definition 1.1.3. Classifications of the flows [START_REF] Vasquez | Hydraulique a surface libre[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF].

The fundamental types of flows related to free-surface hydraulics are:

• Variability over time:

-Steady (or permanent) flow -the flow heights and velocities do not change over time or are assumed to be constant during the considered time interval. Note that the flow in the channels is rarely permanent. Nevertheless, the temporal variations are, in some cases, slow enough so that the flow can be considered as a succession of steady state (i.e., defined as a quasi-permanent regime).

-Unsteady (or non-permanent) flow -the flow heights and velocities do change with time.

• Variability in space:

-Uniform flow -the flow heights and velocities remain invariable in various sections of the channel. A truly uniform flow is rarely found in rivers, but rather in channels of great length, with constant cross-section and slope.

-Varied (or nonuniform) flow -the flow heights and velocities change along the length of the channel. A varied flow may be accelerated or decelerated depending on whether the velocities increase or decrease in the direction of movement. Varied flow can be further classified as either rapidly or gradually varied.

* Rapidly varied flow -the flow heights and velocities change abruptly (sometimes with discontinuities) over a comparatively short distance. This is usually manifested in the vicinity of a singularity, such as a weirs, shrinkage, hydraulic jump or sudden drop.

* Gradually varied flow -the flow heights and velocities only change over a long distance.

Besides, we also use the definition of continuous flow when the flow rate is constant throughout the channel. This is the case for a steady flow. Therefore, the continuous flow can be described using the continuity equation for the steady flow. Whereas the spatially varied or discontinuous flow is considered when the flow rate is nonuniform along the channel. This happens when water enters and/or leaves the channel along the course of the flow. The flow can be described using the continuity equation for continuous unsteady flow.

The classifications of flows are also based on the states of flow. The behavior of a free-surface flow is governed by the effects of viscosity and gravity. Surface tension has a minor contribution and does not play a significant role to be a governing factor in most circumstances. Depending on the effect of viscosity, relatively to inertial forces (as represented by the Reynolds number), the flow can be either laminar, turbulent, or transitional (see more details in [START_REF] Sajjadi | Lattice Boltzmann method and RANS approach for simulation of turbulent flows and particle transport and deposition[END_REF][START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF]).

Example of an irrigation channel

We introduce the Bourne channel (BC) in France as an example for the system analysis and modeling. The block diagram of the BC irrigation network presented in Fig. 1.2 shows multiple sections (i.e., reaches) of a primary channel interconnected through the Gates (G), Mixed gates and spillways (M) or Pumping stations (P). The main trunk, BC, connects further two secondary channels (S2, S3). For system modeling, the irrigation network is decomposed into three types of system components: (1) the first category includes the reaches of primary channel and secondary channels; (2) the second category contains the hydraulic structures (such as G, M, and P) interconnecting the reaches; (3) the third category is a mesh subnetwork of pressurized pipes (the modeling of pipe network is ignored in this thesis). The Bourne channel has several problems, whether technical, economic or compliance. Although endowed with a historical seniority since the 17th century, the construction of numerous engineering structure was carried out for many years and the services of agricultural crops are popularly started from the 18th century. The BC main channel of 46 km in length (excluding subnets of secondary and tertiary channels) is used for the irrigation of large cultivation areas (about 25, 000 hectares) with gravitational and aspersion techniques. The water in the channel is taken from the Bourne river at the dam of Auberives. The upstream flow is regulated and limited to 7 (m 3 /s); the complement of the water level, if necessary, is taken by pumping in the 1.1 Irrigation channels | 13 Isère river. Along the channel, the pumping stations withdraw the water in order to supply a mesh sub-network of pressurized pipes. The total length of pressurized pipes can attain nearly 1000 (km). Finally, at the top of the plots, the water is distributed to irrigated lands on demand. Moreover, when the irrigation is no longer necessary in the favorable seasons, the water is re-conducted to turbines for the electricity production and the electricity is sold to the "Réseau de Transport d'Électricité" (RTE). The control structures attached to the free-surface channel are gates or mixed. Actually, the measuring instruments are probes or meters of old technology for the evaluation of levels and flow-rates at some locations. Two downstream reservoirs conserve the unused water at the end of the main channel. The socio-economic context of the BC have been the subject of a more precise description, thus making it possible to formulate the problematic of the distributed control of an irrigation network under communication constraints (see more details in [START_REF] Nguyen | Optimal reactive control of hybrid architectures : A case study on complex water transportation systems[END_REF]).

Necessity of a model for the irrigation channel

Technical improvement of channel management requires the knowledge of the hydrodynamic behavior of the flow and the characteristics of installed hydraulic structures. Hydrodynamic models sufficiently detailed enable a good understanding of the overall hydraulic characteristics of large-scale irrigation schemes [START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF][START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF]. They are useful for the evaluation of the impacts of different operational options on hydraulic performance and main problems faced by the managers of irrigation channels. In particular, automatic control techniques such as Model Predictive Control (MPC) make use of the models for the controller design [START_REF] Borrelli | Predictive Control for linear and hybrid systems[END_REF][START_REF] Rawlings | Model Predictive Control : Theory and Design[END_REF][START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF][START_REF] Imam Sutrisno | Distributed model predictive control and application to irrigation canal[END_REF][START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF][START_REF] Van Overloop | Model Predictive Control on Open Water Systems[END_REF][START_REF] Camponogara | Distributed model predictive control[END_REF]. The benefits of a hydrodynamic model offered to channel managers can be to:

• avoid resorting to experimentation field and thus disrupting the functional operation,

• test and evaluate channel management rules through simulations on the model and to develop control methods,

• reproduce existing management rules and diagnose malfunctions. Numerous models exist to represent the hydrodynamic behavior of a free-surface flow. These models can principally be grouped into two families.

• Empirical or conceptual approach (Experience ⇒ Model). This is a macroscopic approach which considers the system as a whole and focuses only on global behavior.

• Deterministic or theoretical approach (Model ⇒ Experience). This approach consists of decomposing a system and its operations into sub-systems and micro-phenomena, modeling them and then constructing a recomposition model. These models are based on the principles and equations of mechanics in order to represent some hydraulic phenomena of the system. For the irrigation channels, we adopt the deterministic approach for system modeling because the resulting mechanical models can study the real motion of the water. In addition, this approach is motivated by many works (as surveyed in [START_REF] Zhuan | Models and Control Methodologies in Open Water Flow Dynamics: A Survey[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF][START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF]) and the validation theoretically or through experimentation [START_REF] Lozano | Simulation of automatic control of an irrigation canal[END_REF][START_REF] Puig | Optimal predictive control of water transport systems: Arrêt-Darré/Arros case study[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF]. This makes it possible to increase the performance of the control and supervision systems.

Modeling of an irrigation channel using classical Saint-

Venant equations

We consider a non-permanent and non-uniform flow of an irrigation channel supposed to be rectangular. The equations of Barré de Saint-Venant established in 1871, also called 1D Shallow Water equations (1DSW), are mostly used to model unsteady flow gradually varied with free-surface channels (see [START_REF] Gejadze | Discharge estimation under uncertainty using variational methods with application to the full Saint-Venant hydraulic network model[END_REF], [START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF], Section 2.1 and Appendix A). These hyperbolic partial differential equations are in fact derived from the depth-integrating of the Navier-Stokes equations with some assumptions and simplifications (presented in [START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF], Section 2.1). They consist in two relations, the first being the mass conservation (also called, continuity equation) and the second, the momentum balance (also called, dynamic equation) as follows:

∂ t (h) + ∂ l (hu) = 0, (1.1)
∂ t (hu) + ∂ l ( 1 2 gh 2 + hu 2 ) = F, (1.2)
where: h is the water height, u is the depth-average velocity of the flow, g is the acceleration of gravity, and F is force term. Different external forces (e.g., forces of gravity, pressure or friction) representing the interactions of the system with the environment can be integrated into SV equations. Especially used for simplicity, a simple force term, F, is calculated by using the bed slope I and the friction J at the bottom of the channel [START_REF] Jian | A lattice Boltzmann model for the shallow water equations[END_REF], that is:

F = gh(I -J). (1.3)
Commonly, the friction, J, is deduced by the Manning-Strickler empirical formula [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF]:

J = n 2 u 2 ( Bh (B+2h) ) 4 3
,

with n (sm -1 3 ) is the Manning-Strickler coefficient. Following Eqs. (1.1) and (1.2), we can initialize the water height profile along the channel using the boundary conditions. Assuming that the upstream flow rate and the downstream water height are fixed at the constant values Q 0 and h 0 , the water height profile of the steady flow (i.e., ∂ t h = 0 and ∂ t u = 0) in the channel of length L is expressed by:

∂ l h = gh(I -J) gh -u 2 , (1.5)
1.2 Modeling of an irrigation channel using classical Saint-Venant equations | 15

where, the boundary conditions:

h(L) = h 0 , u = Q 0 Bh and J = n 2 Q 2 0 B 2 h 2 ( Bh B+2h ) 4 3
. The solution can be obtained by integrating the ordinary differential equation for h(l).

As shown in Eq(s). (1.5), the uniform flow (i.e., ∂ l h = 0 and ∂ l u = 0) in a channel is attained when the condition I = J holds (i.e., the friction forces equilibrate the gravitational forces). The relationship between uniform water height, h e , and the steady-state water flow, Q e , is deduced by using the Eq(s). (1.4) and the definition Q e = Bh e u as follows: The fundamental assumptions necessary for the SV equations to be valid are the followings:

Q e = √ I
• the flow is considered as one-dimensional, rectilinear and the variation of channel width along the flow axis is small,

• the vertical pressure distribution vertically is hydrostatic and the vertical acceleration is negligible,

• the velocity is uniform over the cross-section, the average velocity is used in the calculation,

• the average bed slope is small (so that the cosine of elevation angle may be replaced by unity),

• the effect of friction can be taken into account through resistance laws used for steady flow or the viscosity is negligible,

• finally, the channel is supposed to transport clear water.

The resolution of the Eqs. (1.1) and (1.2) allows to define the temporal variations of the water heights h or flow rates q along the flow axis l. The analytical resolution of the SV equations is impossible in most real cases, but the numerical resolution is now quite common on microcomputers using finite difference methods (such as Preissmann Implicit Scheme) or finite element methods (introduced in [67], Section 2.2). For these methods, the initial and boundary conditions are needed for the computation. The choice of boundary conditions depends on the flow characteristics since a change in the boundary conditions may change the flow characteristics. We can choose either the water height or the flow rate at the channel upstream or downstream as boundary conditions.

The modeling method using SV equations has been validated on an experimental channel located at the University of Evora in Portugal in the Gignac experimental platform project (2000)(2001)(2002)(2003)(2004)(2005)(2006) financed by Cemagref ( [START_REF] Litrico | Experimental validation of a methodology to control irrigation canals based on Saint-Venant equations[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF], Section 11), on Arrêt-Darré/Arros dam-river system located in the Southwest region of France [START_REF] Puig | Optimal predictive control of water transport systems: Arrêt-Darré/Arros case study[END_REF], on the "Sector B-XII del Bajo Guadalquivir", Lebrija, Spain [START_REF] Lozano | Simulation of automatic control of an irrigation canal[END_REF], and otherwise, on lab experimentations [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF][START_REF] Belaud | Hydraulic Modeling of an Automatic Upstream Water Level Control Gate for Submerged Flow Conditions[END_REF].

Modeling of an irrigation channel using Lattice Boltzmann method 1.3.1 Hydrodynamic model of free-surface flow

As demonstrated in [START_REF] Sajjadi | Lattice Boltzmann method and RANS approach for simulation of turbulent flows and particle transport and deposition[END_REF][START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF][START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF][START_REF] Diemer | Discrete controllability of distributed parameters systems using lattice Boltzmann models: An application to the shallow water equations[END_REF], the LB method is an efficient and powerful numerical tool (in terms of accuracy, numerical stability and computational time) to simulate the freesurface flow while respecting the conservation laws of macroscopic variables (as described by the Eqs. (1.1) and (1.2)). The variables used in the LB model are the density distributions Fig. 1.3 Modeling a reach of length L using Lattice Boltzmann (LB) method illustrates the density distributions, f i , the velocities of particles, v i , in a spatial direction i, i = {1, 2, 3}.

(denoted by f i for a spatial direction i), representing the density of particles which enter into a site l at a time t with a velocity (denoted by v i for the particles from the direction i) [START_REF] Kruger | The Lattice Boltzmann Method[END_REF]. According to [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF], Section 2.2, the one-dimension three-velocities (D1Q3) LB method is effectively used to model the irrigation channel in one dimension l and with three velocities [v 1 v 2 v 3 ] corresponding to three directions i = {1, 2, 3} (as illustrated in Fig. 1.3), where:

[v 1 v 2 v 3 ] = [0 v (-v)], v = ∆l ∆t
(∆l is lattice spacing, ∆t is time step). Based on the theory of cellular automata [START_REF] Chopard | Cellular automata modeling of physical systems[END_REF], the dynamics of the flow are described by the intrinsic interactions among particles. These interactions can be represented by two consecutive phases [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF]: (1) the collision phase -at the time t, a density distribution f in i of particles from the direction i entering into a site with velocity v i collide with other particles from other directions entering into the same site; (2) the streaming phaseduring the next time step (t + ∆t), a new density distribution f out i of particles emerging from the collision phase move to a new lattice site with a new velocity. These phases can be mathematically 1.3 Modeling of an irrigation channel using Lattice Boltzmann method | 17 expressed by: Collision:

f out i (l,t) = f in i (l,t) + Ω i ( f in (l,t)), Streaming: f in i (l + v i ∆t,t + ∆t) = f out i (l,t), (1.7)
where: i = {1, 2, 3}, Ω i represents the collision term depending on the vector of all distributions f in i (l,t), denoted by f in (l,t). Among the models used in the study of collision, the Bhatnagar-Gross-Krook (BGK) collision model is known for its stability and flexibility in describing free-surface flow properties (see the details in [START_REF] Kruger | The Lattice Boltzmann Method[END_REF], Section 10.7). Using this model, the collision term, Ω i , is expressed by a relaxation of density distributions, f i , with respect to the local equilibrium distributions, f e i , that is:

Ω i ( f in (l,t)) = 1 τ ( f e i (l,t) -f in i (l,t)), (1.8)
where: τ is the relaxation time corresponding to the viscosity in a fluid model. As in SV modeling, the LB density distributions must verify for each site and at each time step, the conservation of mass (related to the water height, h), momentum (related to the flow rate per unit of section width, given by hu), and momentum tensor, that is:

∑ 3 i=1 f in i = ∑ 3 i=1 f out i = h ∑ 3 i=1 v i f in i = ∑ 3 i=1 v i f out i = hu ∑ 3 i=1 v 2 i f in i = ∑ 3 i=1 v 2 i f out i = 1 2 gh 2 + hu 2 (1.9)
The equilibrium density distributions must also satisfy these conservations as follows:

3 ∑ i=1 f e i = h 3 ∑ i=1 v i f e i = hu 3 ∑ i=1 v 2 i f e i = 1 2 gh 2 + hu 2 (1.10)
Solving these equations Eq(s). (1.10) allows determining the equilibrium density distributions, f e i , as functions of the macroscopic variables h and u, as follows:

f e 1 = h - 1 2v 2 gh 2 - 1 v 2 hu 2 f e 2 = 1 4v 2 gh 2 + 1 2v hu + 1 2v 2 hu 2 f e 3 = 1 4v 2 gh 2 - 1 2v hu + 1 2v 2 hu 2 (1.11)
In order to increase the accuracy of the resulting simulation scheme, different external forces representing the interactions of the system with the environment, can be integrated into the LB models. For instance, force models described by Zou [START_REF] Jian | A lattice Boltzmann model for the shallow water equations[END_REF], or by Gou [START_REF] Guo | Discrete lattice effects on the forcing term in the lattice Boltzmann method[END_REF], or by Exact Different Method [START_REF] Kupershtokh | A lattice Boltzmann equation method for real fluids with the equation of state known in tabular form only in regions of liquid and vapor phases[END_REF] can be added to the free-surface LB models (see more details in [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF] and [START_REF] Kruger | The Lattice Boltzmann Method[END_REF], Section 6). For a channel model designed for supervision and control purposes, a simple force is appropriate [START_REF] Jian | A lattice Boltzmann model for the shallow water equations[END_REF]. This simple force term is evaluated at the current site by a discretization of the classical force terms in the water flow equations (see the Eq(s). (1.3)).

With the choice of D1Q3 for lattice topology, modeling a free-surface flow using the LB method in the presence of the simple external force results in the local dynamics [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF]:

f i (l + v i ∆t,t + ∆t) = f i (l,t) + 1 τ ( f e i (l,t) -f i (l,t)) + ω i ∆t c 2 s v i F (1.12)
where: ω i and c s are the parameters determined by the geometry of the lattice so that the isotropy of the model is preserved [START_REF] Jian | A lattice Boltzmann model for the shallow water equations[END_REF]. The following values are chosen for ω i and c s :

ω 1 = 2 3 , ω 2 = ω 3 = 1 6 , c 2 s = ∑ 3 i=1 ω i v 2 = v 2 3
. The Eq(s). (1.12) can be put in the form of a representation by directions as follows:

f 1 (l,t + ∆t) = f 1 (l,t) + 1 τ ( f e 1 -f 1 (l,t)) f 2 (l,t + ∆t) = f 2 (l -v∆t,t) + 1 τ ( f e 2 -f 2 (l,t)) -∆t 2v F f 3 (l,t + ∆t) = f 3 (l + v∆t,t) + 1 τ ( f e 3 -f 3 (l,t)) + ∆t 2v F (1.13)
The global dynamics of the LB model (i.e., the dynamics of the density distributions at all points of the lattice) at a time t + ∆t are defined as the functions of density distributions at preceding instants and neighboring points. By applying the values of equilibrium density distributions calculated by the Eq(s). (1.11) to the Eq(s). (1.13), the global dynamics can be expressed as follows:

f 1 (l,t + ∆t) = f 1 (l,t) + 1 τ ( f e 1 -f 1 (l,t)) = 1 τ [(τ -1 2φ 2 ) (1 -1 2φ 2 -F r φ ) (1 -1 2φ 2 + F r φ )]    f 1 (l,t) f 2 (l,t) f 3 (l,t)    = A 1    f 1 (l,t) f 2 (l,t) f 3 (l,t)    f 2 (l,t + ∆t) = f 2 (l -v∆t,t) + 1 τ ( f e 2 -f 2 (l -v∆t,t)) + ∆t 2v F = ∆t 2v F + 1 τ [( 1 4φ 2 ) ( 1 4φ 2 + F r 2φ + τ -1 2 ) ( 1 4φ 2 -F r 2φ -1 2 )]    f 1 (l -v∆t,t) f 2 (l -v∆t,t) f 3 (l -v∆t,t)    = ∆t 2v F + A 2    f 1 (l -v∆t,t) f 2 (l -v∆t,t) f 3 (l -v∆t,t)    1.
3 Modeling of an irrigation channel using Lattice Boltzmann method | 19

f 3 (l,t + ∆t) = f 3 (l + v∆t,t) + 1 τ ( f e 3 -f 3 (l + v∆t,t)) -∆t 2v F = -∆t 2v F + 1 τ [( 1 4φ 2 ) ( 1 4φ 2 + F r 2φ -1 2 ) ( 1 4φ 2 -F r 2φ + τ -1 2 )]    f (l + v∆t,t) f (l + v∆t,t) f (l + v∆t,t)    = -∆t 2v F + A 3    f 1 (l + v∆t,t) f 2 (l + v∆t,t) f 3 (l + v∆t,t)    , (1.14)
where: φ = v √ gh is the ratio of the wave speed to the lattice speed, and F r = u √ gh is the Froude number.

By normalizing the time step ∆t, t -∆t = k -1, t = k, t + ∆t = k + 1 and denoting lattice positions l j+1 = l j + v∆t, l j-1 = l j -v∆t, the dynamics of free-surface flow can be represented as a discrete-time systems using state-space representation as follows:

x(k + 1) = A(x(k))x(k) + F, (1.15)
where: the discrete-time local states, x(k) ∈ R 3N×1 , of the channel of length L (corresponding to N discretized points l j , j = 1, . . . , N) are defined by:

x(k) = [ f 1 (l 1 , k) f 2 (l 1 , k) f 3 (l 1 , k) . . . f 1 (l j , k) f 2 (l j , k) f 3 (l j , k) . . . f 1 (l N , k) f 2 (l N , k) f 3 (l N , k)] T , (1.16)
the state matrix, A ∈ R 3N×3N , and the matrix related to forces, F, are determined by:

A =                                    A 1 0 0 • • • 0 0 0 0 0 • • • 0 A 2 0 A 3 0 • • • 0 0 0 A 1 0 • • • 0 0 A 2 0 0 • • • 0 0 0 0 A 3 • • • 0 0 0 0 A 1 • • • 0 0 0 A 2 0 • • • 0 0 0 0 0 • • • 0 0 . . . 0 0 0 • • • A 1 0 0 0 0 • • • 0 0 0 0 0 • • • 0 A 3 0 0 0 • • • 0 A 1 0 0 0 • • • A 2 0 A 3 0 0 • • • 0 0                                    and F =    0 ∆t 2v F -∆t 2v F    (1.17)
Note that it is a nonlinear system represented by: x(k + 1) = A(x(k))x(k), since the elementary matrices A i depend on the Froude number and φ , which themselves depend on the macroscopic variables h and u determined by:

h = 3 ∑ i=1 f i = f 1 + f 2 + f 3 q = hu = 3 ∑ i=1 v i f i = v( f 2 -f 3 ) (1.18)
The Chapman-Enskog expansion on the LB model, Eq(s). (1.15), allows recovering the SV continuity and dynamic equations, Eqs. (1.1) and (1.2), as demonstrated in [START_REF] Liu | Lattice Boltzmann method for the Saint-Venant equations[END_REF]. Their results show the good applicability of the LB method in solving the SV equations.

Boundary conditions used to generate the dynamics of the irrigation channel

The LB method allows integrating a wide variety of boundary conditions. For instance, these conditions can be imposed by either defining macroscopic conditions (e.g., water height or flow rate at the boundaries) or using lattice-type boundary conditions (e.g., bounce-back, periodic or reflexive conditions) (see more details in [START_REF] Kruger | The Lattice Boltzmann Method[END_REF], Section 5). Due to the non-respect of conservation laws for some lattice-type boundary conditions (reported in [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF][START_REF] He | Analysis and boundary condition of the lattice Boltzmann BGK model with two velocity components[END_REF]), we have chosen in this study to directly impose the macroscopic variables at the boundaries. We consider a reach of length L (corresponding to N discretized points l j , j = 1, . . . , N) as shown in Fig. 1.4. Unknown density distribution, f 2 (l 1 ), at the site l 1 and unknown density distribution, f 3 (l N ), at the site l N , are calculated differently in the case where either the water height, h, or the flow rate, Q, is imposed at upstream or downstream ends of the reach.

l 1 l 2 l 3 l N-1 l N ✲✛ ✲✛ f 2 (l 1 ) f 3 (l 1 ) f 1 (l 1 ) f 1 (l N ) f 2 (l N ) f 3 (l N )
Fig. 1.4 Unknown density distribution, f 2 (l 1 ), at the site l 1 and unknown density distribution, f 3 (l N ), at the site l N , are calculated differently according to the boundary conditions. The Eq(s). (1.18) can be applied in these cases to determine the unknown density distributions:

• Water flow rate, Q 1 , imposed at upstream l 1 vB( f 2 (l 1 ) -f 3 (l 1 )) = Q 1 ⇒ f 2 (l 1 ) = Q 1 vB + f 3 (l 1 ) (1.19) • Water flow rate, Q N , imposed at downstream l N vB( f 2 (l N ) -f 3 (l N )) = Q N ⇒ f 3 (l N ) = f 2 (l N ) - Q N vB (1.20)
1.3 Modeling of an irrigation channel using Lattice Boltzmann method | 21

• Water height, h 1 , imposed at upstream l 1

f 1 (l 1 ) + f 2 (l 1 ) + f 3 (l 1 ) = h 1 ⇒ f 2 (l 1 ) = h 1 -f 1 (l 1 ) -f 3 (l 1 ) (1.21) • Water height, h N , imposed at downstream l N f 1 (l N ) + f 2 (l N ) + f 3 (l N ) = h N ⇒ f 3 (l N ) = h N -f 1 (l N ) -f 2 (l N ) (1.22)
However, the oscillation of water heights or flow rates causes a fast change in density distributions and vice-versa. A relaxation algorithm as described in [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF], Section 5.4 may be necessary in some situations to prevent numerical instabilities.

Coupling D1Q3 LB models of multiple reaches

The operational management of irrigation channels is done through controlled hydraulic structures and well-defined control policies. These hydraulic structures are used as inter-connectors between the reaches, but also to measure or control the hydraulic variables. Among the interconnectors, we have classical distribution control structures (e.g., submerged gates, weirs, mixed gates and spillway), but also pumping stations and branching structures such as T or Y junctions. These various inter-connectors can be modeled by mathematical relations, which can integrate the state of inter-connectors and the variation of hydraulic variables.

Fig. 1.5 Lattice schemes are presented for some interconnections between two reaches. The flow rate through a gate, Q g , (or a weir, Q w , or mixed, Q m ), are expressed by the relations with water heights at upstream, h, and downstream, h ′ , of the gate (or weir or mixed). For the pumping station, the flow rates before, Q, and after, Q ′ , the pump are normally deduced by the pumping flow rate such that:

Q = Q ′ + Q p .
• Submerged gate -In a submerged gate as presented in Fig. 1.5, the flow rate through the gate, Q g , is calculated by the well-known gate equation ( [START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF], Section 6.2), which is based on the difference between the water heights at upstream, h, and downstream, h ′ , of the gate, that is:

Q g = B g α g θ 2g(h -h ′ ), (1.23)
where: B g is the gate width, α g is the gate coefficient, θ is the gate opening, and g is the acceleration of gravity.

• Overshot weir (also, immovable spillway) -it can be considered as level thresholds to be reached before having a flow to the downstream and thus guarantee a minimum height at upstream. The flow rate through the overshot weir, Q w , can be written ( [START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF], Section 6.2) as follows:

Q w =    B w α w 2g(h -h w ) 3 , i f h ≥ h w 0, otherwise (1.24)
where: B w is the weir width, α w is the weir coefficient, h is water height at upstream of weir and h w is the weir level.

• Mixed of gates and spillway -it is common to use a mixed structure consisting of different elements (e.g., a gate and a spillway or two gates) placed in parallel to interconnect two reaches. The flow rates, Q M , through a mixed of two gates, g 1 , g 2 , and one spillway, w, for example, are described by following equations:

Q M = Q g 1 + Q g 2 + Q w , Q g 1 = B g 1 αg 1 θ g 1 2g(h 1 -h ′ 1 ), Q g 2 = B g 2 αg 2 θ g 2 2g(h 2 -h ′ 2 ), Q w =    B w α w 2g(h -h w ) 3 , i f h ≥ h w 0, otherwise (1.25)
• Pumping station -the flow rate and water height just before, Q and h, and after the pump, Q ′ and h ′ , are described by the following equations:

Q = Q ′ + Q p , h = h ′ , (1.26)
where: Q p is the flow rate withdrawn by the pump.

• Branching junction -in the case of the main channel that splits into two branches or diverts to secondary channels (see Fig. 1.6), the water heights are the same for three branches 1.3 Modeling of an irrigation channel using Lattice Boltzmann method | 23 whereas the upstream flow rate is divided into two parts as follows:

h = h ′ = h ′′ , Q = Q ′ + Q ′′ .
(1.27)

Fig. 1. [START_REF] Belaud | Hydraulic Modeling of an Automatic Upstream Water Level Control Gate for Submerged Flow Conditions[END_REF] The main channel diverting to a secondary channel is represented by a T branching junction. The upstream flow rate is divided into two parts:

Q = Q ′ + Q ′′ .
When coupling two reaches modeled by the D1Q3 LB method, some of the distribution functions before, ( f i ), and after, ( f ′ i ), the interconnection are unknown. The unknown LB variables can be determined by solving the equations describing the physical properties of the coupling (as presented in Eqs. (1.23) to (1.27)) and the definition of macroscopic variables (see Eq(s). (1.18)). For example, the coupling of two reaches by a submerged gate requires that the unknown distributions, f 3 and f ′ 2 for two reaches, must be computed using the gate equation, Eq(s). (1.23), and the definition of water height, h, and flow rate through the gate, Q g as Eq(s). (1.18). As a result, we obtain f 3 and f ′ 2 by solving following equations:

vB( f 2 -f 3 ) = Q g ⇒ f 3 = f 2 - Q g vB (1.28) vB( f ′ 2 -f ′ 3 ) = Q g ⇒ f ′ 2 = f ′ 3 + Q g vB (1.29)
In the same manner for other "inter-connectors" (such as weirs, pumping stations or branching junctions), a system of (non)linear equations can be solved by numerical methods to obtain the unknown variables.

Validation of D1Q3 LB method

Some laboratory validations (as reported in [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF][START_REF] Diemer | Discrete controllability of distributed parameters systems using lattice Boltzmann models: An application to the shallow water equations[END_REF]) have been experimentally realized on the micro-channel at LCIS (at Valence, France). Geometrical parameters of the micro-channel are described in [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF] such as the length 8(m), the width 0.1(m), the rated water height 0.1(m) and some other network parameters (e.g.,). A first test for the force term is performed on a steady flow in a micro-channel with deformed bottom [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF]. A second test verifies the speed of wave propagation for the model [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF]. The last test compares the steady case of the LB model to the exact solution of the SV equations [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF]. The modeling methodology is validated through comparison to the traditional practices and further correlated to the lab measurements. A comparative study between the LB method and some conventional methods (e.g., Finite Volume and Preissmann methods) shows some advantages of the LB method in terms of performance and precision. For example, it is 30 times faster than the Finite Volume method and 90 times faster than the Preissmann method (see the details in [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF]). By integrating appropriate forces (e.g., Zhou's force and Guo-Chopard's force), the resulting model gives better precision. All these results show the potential of LB approach to simulate a realistic complex irrigation network.

Fluid simulations

The LB modeling requires a lot of data from the channel (e.g., geometry of the channel, longitudinal profile, roughness coefficient) that are not always available in large-scale channels. In this work, some terms have been neglected in the equations in order to create a simplified model of free-surface flow [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF]. The successive simulations of the obtained model are necessary for calibration and validation using data gathered on a practical channel. The resulting simulator can be used as a management and diagnostic tool for the efficient water management of the considered channel. For example, the LB simulation results of the hydrodynamic model can help for the comprehension of local behaviors with regards to global states. On this basis, the control engineer investigates the new features for the designed controllers, making the model compatible with predefined mathematical criteria or the refinement of initial conditions.

Modularity in modeling of free-surface channels using a library

In the literature, the works related to LB modeling of irrigation channels usually consider the system of one, two or multiple reaches as a whole [START_REF] Diemer | Discrete controllability of distributed parameters systems using lattice Boltzmann models: An application to the shallow water equations[END_REF][START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF][START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF][START_REF] Litrico | Modeling and Control of Hydrosystems[END_REF]. Therefore, the resulting models are difficult to apply to other systems, due to the complex interconnection considered in these systems. This section proposes a modular approach using parameterized modeling blocks gathered in a Model Library. The proposed modular approach reduces the system design and review cycle. This can significantly improve the design consistency for large-scale systems.

In simulations, it takes less time and enables a more efficient co-simulation. The concepts allow the decomposition of complex systems into tractable subsystems and a set of interactions. Subsystems are the modeled using the LB method or mathematical equations and the sub-models are added to a Model Library as components. These components are re-usable and scalable for generating a parameterized complete model. For the irrigation channel, the modular modeling methodology is used to generate the Model Library of reaches and control structures. As an example, a Simulink/MatLab model library created for an irrigation channel is shown in Fig. 1.7.

Simulation schemes and results

As a demonstration of the modeling methodology, we perform Simulink/MatLab simulations for a section of the Bourne channel, from the Mondy gate (a mixed gate and spillway structure) to the branching junction with secondary channel S2 (see Fig. For numerical time-interpretation of the resulting equations, the system firstly needs to be initialized at steady state of flow by the following procedure:

h S θ 1-Orme θ 2-Orme Q P θ 1-Mondy Q M 1.64m 0.20m 0.20m 0m 3 /s 0.60m 2m 3 /
• Choose the initial water height of steady flow at S2, h S (l S , 0), the steady flow rate is calculated by Q S (l S , 0) = B 3 * h S (l S , 0) * u (in the case of a rectangular section, otherwise Q S (l S , 0) = S(h S (l S , 0)) * u).

• Integrate the steady-flow Eq(s). (1.5) with the boundary conditions at S2 to obtain the water height h S (l), the steady-state profile for l ∈ [l O , l S ].

• Using the gate relation Eq(s). (1.23), calculate the flow rate through the mixed at Orme, Qg O (l O , 0), by selecting gate openings, θ 1-Orme and θ 2-Orme .

• Integrate the Eq(s). (1.5) with the boundary conditions at Orme to obtain the water height h(l), the steady-state profile for l ∈ [l MdM , l O ].

• The flow rate at "Monts du Matin", Q MdM , is calculated by the pump relation, Eq(s). (1.26) where the pumping flow rate is Q P (l P , 0) .

• Integrate the Eq(s). (1.5) with the boundary conditions at "Monts du Matin" to obtain the water height h(l), the steady-state profile for l ∈ [l M , l MdM ].

• By selecting the initial flow rate at the Mondy gate, Q M (l M , 0), and the gate opening, θ 1-Mondy , Eq(s). (1.23) can determine the water height at upstream of the Mondy gate.

• The initial water height profile for the considered section is initialized. The initial values used in simulations are presented in Table 1.3 and an overview of the initialized section is shown in Fig. 1.11. Starting with this equilibrium profile for the considered section, At time t = 80 (s), the opening of the Mondy gate, θ 1-Mondy , is changed in a way to linearly increase the flow rate at the upstream by 50% from its initial value Q M (l M , 0) at time 1000 (s). At time t = 1200 (s), the gate openings at Orme, θ 1-Orme and θ 2-Orme , increase gradually to 0.3 (m) at t = 2000 (s). At "Monts du Matin", the pumping flow rate is increased from 0 (m 3 /s) to 0.2 (m 3 /s) between t = 5000 (s) and t = 5500 (s) and stabilized. The simulation scenario is illustrated in Figs. 1.17 and 1.18.

The simulations are done with Matlab R2016b®on a computer Intel®Core™i5-4310U CPU 2.0GHz. The simulation results presented in Fig. 1.12 (for water height profile along the section) and Fig. 1.13 (for flow rate profile) show a coherent behavior of the considered section. Indeed, we see that (1) there is a wave propagation phenomenon at a constant speed from the upstream to the downstream; (2) the water heights change when the flow passes through the mixed structures at "Mondy" and at "Orme"; and (3) the discontinuity of flow rates corresponds to the withdrawal of the pumping station at "Monts du Matin". The first coherent results motivate a model of a more complex section or the whole channel, which can potentially be constructed in the same way (see Section 1.4.3). Finally, let us remark that the model still needs to be calibrated and validated during the field campaigns, with surveys and orders from channel operators and precise corresponding experimental measurements.

Considering a real irrigation channel and data requirements for model setup

Another more complex section of the Bourne channel (shown in Fig. 1.2) from the Auberives barrage to the branching T junction with the secondary channel S2, is considered for the simulation to illustrate hydraulic aspects. This section (with a length of approximately 37 (km) long) comprises twelve reach segments, three mixed control structures, and eight pumping stations withdrawing the water along the main channel. For reliable hydraulic simulations, some adequate data on water conveyance and distribution infrastructure are needed to put into the model. The collected field data can be classified into three main categories: (1) channel geometry, longitudinal profile, and environment parameters; (2) the details of control structures and their operations; (3) flow characteristics such as water heights, flow rates through gates and pumping flow rates. All of these data are not available in Bourne channel. We have used the maximum information provided by the "Syndicat d'Irrigation Drômois" (SID) operator on the irrigation network topology, channel geometry, details of control structures, and operation rules. The canal geometry and longitudinal profiles were determined for the whole channel length and cross-section data were measured at fixed intervals by a campaign of channel surveying (files supplied by SID operator). Control structures such as submerged gates, mixed structures were carefully located and their dimensions were measured. Moreover, their detailed features such as gate levels, opening heights and their conditions were noted for each structure. The global parameters necessary for the modeling of the considered section are summarized in Tables 1.1 to 1.5.

The modeling of the considered section using the LB method is described in Section 1.3. Similarly, the model of the second considered section is created as shown in Fig. 1.14 by coupling different components of the Model Library (shown in Fig. 1.7) such as reach, pump and mixed components through their interactions. As mentioned in Section 1.4.2, for numerical time-interpretation of the resulting equations, the considered section firstly needs to be initialized at steady state of flow by the similar procedure applied on the overall section using the boundary conditions and initial values given in Fig. 1.11. An overview of the initialized section is shown in Fig. 1.16.

Based on the section model, a simulation scenario is proposed for hydraulic presentation. For example, assuming that the gate openings of three mixed controlled structures change as described in Fig. 1.17, the flow rate at Auberives evolves from 0 to 2.2 (m 3 /s) in time between 0 and 1000 (s) and pumping flow rates at different pumping stations evolve as shown in According to simulation results, we see that there is a wave propagation phenomenon at constant speed from the upstream to the downstream of the considered section; the water heights change when the flow passes through the mixed structures at "Ecanciere", "Mondy" and at "Orme"; and the discontinuity of flow rates corresponds to the withdrawal of the pumping stations at "Monts du Matin", "Mondy", "Tendillon", "Martinet","Papelissier", "Meymans", "Eymeux", "Baume".

In consequence, based on LB method and the parameters of the flow (e.g., channel geometry, environment parameters, the details of control structures and their operations, and flow characteristics), it is possible to determine the longitudinal profile of the water line at various sections in the channel and thus, to localize the phenomena which generate and perhaps quantify the flow and perturbations themselves. Therefore, the natural animation of the flow in the channel can be supported by modeling the evolution in time and in space of the flow and disturbances generated by the different situations.

From a complexity point of view, with the code written in Simulink/Matlab and the specified computer used to perform these simulations, the computation time is 7 (s). The computation time estimated for the flow in future 24 (h) is 84 (s). A simulation over one year for the above network described with 374 sites requires about 8.5 (h) of CPU times. In addition, a C++ implementation typically is 100 times faster without any code optimization (as discussed in [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF]). It can reduce the time needed for the 1-year simulation to the order of a few minutes of CPU times with the same spatial and temporal resolutions. Even for real-scale complex irrigation networks (e.g., with more secondary channels, hydraulic structures, pumping stations, reservoirs, etc.), the simulations can respect real-time requirements. This computing efficiency may be useful for scale-reduced experimental micro-channels typically characterized by fast dynamics associated with low frictions and short reaches (e.g., only a few meters long) or for other application examples of fluid flow (see more details in [START_REF] Marcou | Modélisation et contrôle d'écoulements à surface libre par la méthode de Boltzmann sur réseau[END_REF]). Moreover, the D1Q3 LB method is analyzed in detail and compared with other numerical schemes (such as an implicit finite difference scheme and a finite volume approach) in [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF]. This work has shown that the LB method is 10 times faster than the Finite Volume method and 100 times faster than the Preissmann method (see [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF], Section 5). With appropriate forces (Zhou force and Guo-Chopard strength), the LB model gives the better precision, but the numerical stability region of the LB model is limited by the Courant condition and sub-critical flow conditions.

Summary

In this chapter, we have presented the LB method with its application to the modeling and the simulation of the free-surface flow in irrigation channels. Geometric parameters and hydraulic variables characterizing an irrigation system are first introduced to provide a synthesis of physical quantities in the modeling framework. The SV (or 1DSW) equations are briefly represented as a classical distributed-parameter model for transient free-surface flow. The LB approach to solving the SV equations is introduced in details. It is a mesoscopic technique bridging the molecular movements and macroscopic physical variables. The aspects concerning boundary conditions were discussed in 1D for network-type conditions, but also for conditions that allow directly imposing macroscopic variables. A coupling methodology to interconnect several 1DSW models is developed by introducing mathematical relations of LB variables and hydraulic variables. Taking into account the boundary conditions, internal interconnections and the interactions with the environment, fluid simulations show the potential of the LB approach to simulate a realistic complex network of irrigation channels.

Chapter 2

Composite metric design for the dynamic routing in networked control systems using a hybrid architecture Networked control systems (NCSs) allow integrating components (e.g., controllers, sensors, actuators) and algorithms of control systems into a global network infrastructure. Based on this infrastructure, the control information (e.g., reference inputs, controlled variables, system inputs/outputs, feedback values, and collected data) can be exchanged among the components through wired and/or wireless communication subnetworks. A hybrid network architecture involving different types of subnetworks to be interconnected is needed so that the components can share, collaborate and cooperate efficiently to accomplish different tasks. Therefore, an efficient communication among the system components plays an important role in achieving global objectives. This chapter aims to design a hybrid network to cope with the heterogeneity of wide-area NCS and a composite metric for the dynamic routing over the hybrid network in order to satisfy different Quality of Service (QoS) requirements of control applications. The resulting network model can be used to dynamically compute the QoS-related costs (or probabilities) of different paths between NCS components. These are the important concerns while considering the information exchange in NCSs.

To describe the motivation and deal with the routing problem of complex communication network, the following sections will present: (1) the challenging communication constraints possibly imposed on a wide-area NCS, an example is given by the Bourne channel (BC) case (Section 2.2.1); (2) the benefits of a hybrid network using IPv6 over Low power Wireless Personal Area Network (6LoWPAN) technology to unify the communication among NCS components (Section 2.2.2, and Section 2.2.3); (3) different QoS requirements of control applications can be imposed to the network services (Section 2.2.4); (4) the design of a composite metric used by a dynamic routing protocol for satisfying these requirements (Section 2.3); (5) a case study for the design of a composite metric used by Routing Protocol for Low power and lossy network (RPL) over a network with mesh topology (Section 2.4); and finally [START_REF] Belaud | Hydraulic Modeling of an Automatic Upstream Water Level Control Gate for Submerged Flow Conditions[END_REF] the simulations of this case study in some extensive scenarios.

Introduction

Networked control systems (NCSs) are evolving into distributed NCSs, which integrate distributed components (e.g., controllers, sensors, actuators), and distributed algorithms of control systems into a global network infrastructure [START_REF] Zhang | Survey on Recent Advances in Networked Control Systems[END_REF][START_REF] Aleksandar | Novel Architecture for Networked Control Systems[END_REF][START_REF] Rachana | Hydrology and Hydraulic Systems: Fourth Edition[END_REF]. This vision is even more realistic with the proliferation of the ubiquitous Internet of Things (IoT) [START_REF] Sethi | Internet of Things : Architecture, Protocols, and Applications[END_REF]. Indeed, the sensing/actuating devices in NCSs can greatly benefit from the recent IoT developments. Many works have been done in order to emphasize the complementary contributions in both fields of automatic control and wireless sensor/actuator networks (WSANs). In particular, two lines of research have significant practical implications [START_REF] Zhang | Survey on Recent Advances in Networked Control Systems[END_REF][START_REF] Rachana | Hydrology and Hydraulic Systems: Fourth Edition[END_REF]: (1) network-aware control that aims at modifying control algorithms to cope with the imperfection in communication; and (2) control-aware networking and communication that attempts to modify network protocols and radio links for better real-time control performance. As an example of the former category, research works have discussed the control methodologies in NCSs [START_REF] Tan | Necessary and Sufficient Stabilizing Conditions for Networked Control Systems with Simultaneous Transmission Delay and Packet Dropout[END_REF], the modeling, analysis, and design of NCSs (see [START_REF] Elmahdi | Decentralized Control Framework and Stability Analysis for Networked Control Systems[END_REF][START_REF] Alur | Compositional modeling and analysis of multi-hop control networks[END_REF] and reference therein). Especially, many works were interested in system stabilization and control in the presence of communication constraints [START_REF] Li | Event-based network consensus with communication delays[END_REF][START_REF] Liang | Optimal Control and Stabilization for Networked Control Systems with Packet Dropout and Input Delay[END_REF][START_REF] Rojas | On the infimal Signal-to-Noise Ratio limitations for measurable disturbance rejection[END_REF][START_REF] Nguyen | Signal-to-noise ratio for irrigation canal networked control system (MSC'2015)[END_REF][START_REF] Braslavsky | Feedback Stabilization over Signal-to-Noise Ratio Constrained Channels[END_REF]. Whereas the works in the latter category have investigated in a joint design of control algorithm and network protocol [START_REF] Aleksandar | Novel Architecture for Networked Control Systems[END_REF][START_REF] Al-Dabbagh | Design Considerations forWireless Networked Control Systems[END_REF][START_REF] Fischione | Design principles of wireless sensor networks protocols for control applications[END_REF][START_REF] Park | Wireless Networked Control System Co-Design[END_REF] or introduced network configurations adapted to control systems [START_REF] Elmahdi | Decentralized Control Framework and Stability Analysis for Networked Control Systems[END_REF][START_REF] Di Girolamo | Co-design of controller and routing redundancy over a wireless network[END_REF]. However, these works have considered control algorithms without [START_REF] Park | Wireless Networked Control System Co-Design[END_REF] or with simple routing strategies such as multi-path static routing [START_REF] Alur | Compositional modeling and analysis of multi-hop control networks[END_REF][START_REF] Di Girolamo | Co-design of controller and routing redundancy over a wireless network[END_REF]. Moreover, the components of wide-area NCSs are commonly heterogeneous devices integrated into different types of networks. They demand to pay much more attention in network design, especially routing design to enable the interoperability and Quality-of-Service (QoS) satisfaction. The co-design of NCS and network often requires making some assumptions on network models with probabilities of delays or packet loss for different paths [START_REF] Di Girolamo | Co-design of controller and routing redundancy over a wireless network[END_REF]. There is little information available on the routing optimization process combining multiple criteria (i.e., metrics) in NCSs. More concrete methods also need to be investigated on the computation of delay (or packet loss) probability in network models.

In certain wide-area NCSs, communication constraints might require a unified communication among components within a global network. For the design of this network, the present chapter aims at presenting the approaches to two problems:

• multiple applications on the same network infrastructure: by introducing a hybrid network to cope with the heterogeneity of NCSs and the 6LoWPAN technology suitable for a resource-limited network (e.g., including devices with low-power, limited memory and processing capabilities);

• QoS satisfaction: by designing a composite metric for dynamic routing over the hybrid network in order to satisfy different QoS requirements of control applications. As an example, one ArrowHead project demonstrator focusing on distributed control of Bourne Channel (BC) irrigation network in Drôme -France, is used to introduce an NCS with specific communication constraints (see detail in [START_REF] Nguyen | Optimal reactive control of hybrid architectures : A case study on complex water transportation systems[END_REF]). The BC communication setting is shown in Fig. 2.1.

Motivation: A hybrid network for NCSs

Communication constraints in NCSs

A study of networking aspects is always necessary for the design of wide-area NCSs. With regards to the needs of data exchange, each controller designed for a distributed NCS may have three basic actions [START_REF] Al-Dabbagh | Design Considerations forWireless Networked Control Systems[END_REF]: (1) gather the information recorded by sensors; (2) interact with other controllers for coordination and cooperation; and (3) send control signals to the actuators. To achieve these actions, controllers commonly make use of efficient transmission techniques for point-to-point communication, sometimes over a long range and under challenging limitations on data rate, operational energy consumption, etc. [START_REF] Al-Dabbagh | Design Considerations forWireless Networked Control Systems[END_REF][START_REF] Tan | Necessary and Sufficient Stabilizing Conditions for Networked Control Systems with Simultaneous Transmission Delay and Packet Dropout[END_REF]. To give an example, some communication constraints are encountered in large-scale BC control system such as long communication range and large network coverage, wireless-imposed network type, heterogeneous and resource-limited devices, dynamic topology, and lack of regular maintenance. The serious difficulties in the current state of the BC control system are frustrating some technological proposals in terms of interoperability, scalability, network design, and operations. Particularly for network routing, some standard protocols used in Internet stack (e.g., Routing Information Protocol -RIP, Open Shortest Path First -OSPF) or the protocols, especially designed for WSANs (e.g., Adhoc On-Demand Distance Vector -AODV, Dynamic MANET On-demand routing protocol -DYMO, Dynamic Source Routing -DSR) perform much worse for this task as discussed in [START_REF] Sheng | A survey on the IETF protocol suite for the internet of things: Standards, challenges, and opportunities[END_REF] and some routing protocols are compared on the basis of different criteria in [START_REF] Kumar | Routing in IPv6 over low-power wireless personal area networks (6LoWPAN): A survey[END_REF]. Dealing with these challenges requires the network design to adopt constraint-based networking technologies (such as LoRa® technology from Semtech providing long-range communications) or to optimize the dynamic routing process.

A hybrid network for heterogeneous components of the NCS

An NCS with communication constraints described above will take advantage of the unified communication among components within a hybrid network. This type of network refers to any network that contains two or more different communications standards or multiple topological structures [START_REF] Li | A survey on topology control in wireless sensor networks: Taxonomy, comparative study, and open issues[END_REF][START_REF] Yu | A survey on wireless sensor network infrastructure for agriculture[END_REF]132]. It enables different types of subnetworks to be interconnected so that the components can share, coordinate and cooperate efficiently to accomplish different tasks. A hybrid network for NCS (as shown in Fig. 2.2) is commonly composed of traditional WSNs [START_REF] Ojha | Wireless sensor networks for agriculture: The state-of-the-art in practice and future challenges[END_REF][START_REF] Yu | A survey on wireless sensor network infrastructure for agriculture[END_REF], embedded networks (as defined in [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF], Section 1.1) and Internet Protocol (IP) networks (e.g., local network infrastructure connected to the Internet) [132]. Although each subnetwork is different in nature, but the network services are required to be homologous through the hybrid network.

To benefit from available technologies and the large-scale deployment of the Internet, it may be expected that NCS components should be integrated into Internet-connected networks [START_REF] Sheng | A survey on the IETF protocol suite for the internet of things: Standards, challenges, and opportunities[END_REF]. In this integration, each component playing the role of a node in the global network, is able to dynamically join the Internet. In addition, with recent technological progress materializing IoT, the future Internet is foreseen to be extended as a worldwide network of interconnected devices or objects (as mentioned in [START_REF] Sheng | A survey on the IETF protocol suite for the internet of things: Standards, challenges, and opportunities[END_REF]). Wireless network technology continues to play an ever-increasing role in data network architecture throughout the large range of electronic component markets. However, deploying an NCS configured to access the Internet (see Fig. 2.3) raises new challenges, which need to be tackled before such an integration. [START_REF] Sheng | A survey on the IETF protocol suite for the internet of things: Standards, challenges, and opportunities[END_REF] highlighted and discussed the challenges newly emerging on the capabilities of WSAN devices such as the mobility management, quality of service (QoS), and network configuration in addition to their usual sensing/actuating functionality. Many open challenges remain, mostly due to the complex Fig. 2.3 A hybrid network architecture composed of multiple Simple LoWPANs is proposed for NCS due to low mobility between subnetworks [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF]. deployment characteristics and the stringent requirements imposed by various desired services of such hybrid network. For example, the control applications cannot directly profit by IP technology because of limited processing and transmission capabilities of nodes, non-permanent power source, small packet size or the overhead of IP header, etc. The promising 6LoWPAN technology allows IP networking for resource-limited devices.

6LoWPAN technology suitable for the NCS hybrid network

As surveyed in [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF], some possible technologies can be considered while addressing the Internet integration of WSANs such as ZigBee (ZigBee Alliance), Machine-to-Machine (ETSI M2M), Future Internet (e.g., EU 4WARD project) and 6loWPAN (IETF). Nevertheless, non standard or especially non-IP networks tailored to specific applications seem inappropriate for building a global infrastructure. If all of the system components are connected through a single open standard protocol (e.g., IP), they might benefit without any extra effort, by seamless connectivity, unique addressability, and rich applicability. This shows a strong trend of convergence on standardization, both in industry and in academia, that is clearly steering towards an Internet-based approach. The 6LoWPAN and ROLL WGs of IETF are actively performing this standardization by specifying the 6LoWPAN technology and RPL protocol.

The design of a hybrid network for NCSs can take advantage of 6LoWPAN compared to other technologies (e.g., WiFi, ZigBee) such as scalability, self-organization, and low-power consumption [START_REF] Kumar | Routing in IPv6 over low-power wireless personal area networks (6LoWPAN): A survey[END_REF][START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF]. More advantages are presented in [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF], Section 1.1.1. The 6LoW-PAN is expected to become a standard networking technology supported on any constrained network [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF]. By this motivation, it shows up a wide range of interesting research proposals (e.g., RFC4919, RFC6568, RFC6606) and applications using 6LoWPAN (as surveyed in [START_REF] Kumar | Routing in IPv6 over low-power wireless personal area networks (6LoWPAN): A survey[END_REF], and [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF], Section 1.1.4). They offer solutions for making networks more reliable, adaptable, durable and manageable.

Different QoS requirements of control applications

A hybrid network is technologically realizable. However, one of the important concerns for the hybrid network design is the selection of network performance criteria. Whereas "QoS is the description and/or measurement of the performance of a network service as seen by network users" [START_REF] Karkazis | Evaluating routing metric composition approaches for QoS differentiation in low power and lossy networks[END_REF]132], selecting network performance criteria for the hybrid network can be based on QoS requirements of control applications. For instance, the level meters or pressure meters in mesh-pressurized subnetworks require minimum power consumption and minimum aggregation delay to reach their controllers. The control signals from controllers are expected to reach actuators (e.g., gate control motors) with the minimum packet loss rate, minimum aggregation delay, and in certain cases (e.g., with pressure regulating valves), minimum power consumption. Whereas the controllers are implemented on devices usually attached to a power source, they only require minimum aggregation delay and packet loss rate.

Eventually, the QoS-based performance criteria considered in the NCS hybrid network are packet loss rate, network latency, and network lifetime. The challenging factors influencing on network performance in the hybrid network are the varying channel characteristics, bandwidth allocation, fault tolerance levels and hand-off support for heterogeneous wireless networks. Network software may help to address those challenges by using dynamic routing.

Dynamic routing design for the hybrid network

There is a state-of-the-art introduction to various dynamic routing mechanisms (e.g., flooding, shortest path, distance vector routing, link state routing, etc.) in [132], Section 5.2. Especially for 6LoWPAN networks, the work [START_REF] Kumar | Routing in IPv6 over low-power wireless personal area networks (6LoWPAN): A survey[END_REF] presented a survey of existing routing protocols, a taxonomy of routing requirements and parameters for evaluating routing algorithms. The RPL presented in Section 2.4 is one of the common protocols used for the routing in 6LoWPAN networks [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF].

Network model and formal representation of metrics

It is crucial for routing design to address the relationship between routing metrics, path calculation algorithms, and packet forwarding schemes. For this goal, they should be properly represented in a network model. According to [START_REF] Luis | Algebra and algorithms for QoS path computation and hop-byhop routing in the Internet[END_REF], refined thereafter by [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], a network can be modeled by a strongly connected and directed graph G(H, L), also called digraph, where:

• H is a set of nodes;

• L is a set of edges representing links between nodes.

• Let H = {X 1 , X 2 , . . . , X n } be a node set of the graph G of n nodes. For i, j, s, d ∈ {1, . . . , n}, a link from node X i to node X j , denoted by l i j or (X i , X j ) ∈ L, has link cost, c(l i j ) or c(X i , X j );

2.3 Dynamic routing design for the hybrid network | 41 a path p from node X s to node X d , denoted by p(X s , X d ), has path cost, c(p);

a set of all paths from X s to X d discovered by a routing protocol, R, is denoted by R(X s , X d ) From the network model, the routing using a metric m can be mathematically represented as an algebra on top of a quadruplet: (P, ⊕, c, ≼), called Path Weight Structure (PWS), where P is a set of all possible paths, ⊕ is the path concatenation operation, c is a function that maps a path to a cost based on metric m and ≼ is an order relation.

A metric is specified in a domain, for example, integer, real, [1,+∞], etc. It can be either recorded or aggregated along a path [START_REF] Phillippe | Routing Metrics Used for Path Calculation in Low-Power and Lossy Networks[END_REF]. For a same aggregated metric, the path cost may be different depending on the aggregation rule such as additive, multiplicative, or concave. For a path p of n p nodes, denoted by p(P 1 , P 2 , . . . ,

P n p ), a metric is additive if: c(p) = c(P 1 , P 2 ) + • • • + c(P i p , P i p +1 ) + • • • + c(P n p -1 , P n p ), multiplicative if: c(p) = c(P 1 , P 2 ) * • • • * c(P i p , P i p +1 ) * • • • * c(P n p -1 , P n p )
, and concave if: c(p) = Max{c(P 1 , P 2 ), . . . , c(P i p , P i p +1 ), . . . , c(P n p -1 , P n p )} or Min{c(P 1 , P 2 ), . . . , c(P i p , P i p +1 ), . . . , c(P n p -1 , P n p )}. Metric order relation defines weighted comparison of paths leading to the next hop selection.

The following section presents the requirements for a routing protocol using a designed metric to operate correctly. These requirements can be expressed by the mean of PWS properties.

Requirements for the routing protocol using a designed metric

Applying a designed metric to a routing protocol cannot always guarantee the correct network operations. According to [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], in order to guarantee correct network operations, the routing protocol must satisfy three basic requirements: consistency, loop-freeness, and path-optimality. From the network model (see Definition 2.3.1), the routing requirements for a network can be formulated using PWS as follows: Definition 2.3.2. Routing requirements: consistency, loop-freeness, and path-optimality [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF].

• A routing protocol, R, is consistent if all paths discovered by R are consistent. For a path p discovered by R, assumed to contain n p nodes, denoted by p(P 1 , P 2 , . . . , P n p ), p is consistent if for any P i p , i p ∈ {1, . . . , n p -1} on path p, P i p must forward packets (sent from P 1 to P i p ) to the next hop P i p +1 on path p;

• A routing protocol R is optimal if R chooses the optimal path among all paths discovered by R, to route packets. Following PWS formulation, (P, ⊕, c, ≼), the optimal path p min from nodes X s to X d discovered by R, p min ∈ R(X s , X d ) ⊆ P, is defined by: for any

p k ∈ R(X s , X d ) : c(p min ) ≼ c(p k );
• A loop-free path p is defined by: for any X i , X j ∈ H, i, j ∈ {1, . . . , n} on path p, if i ̸ = j then X i is different from X j . Loop-freeness is the most important requirement for a routing protocol.

Formally, these routing requirements are verified by considering isotonicity and monotonicity properties of PWS defined as follows:

Definition 2.3.3. Isotonicity and monotonicity properties [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF].

According to [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], isotonicity is defined as: ∀p, q, r ∈ P, quadruplet PWS (P, ⊕, c, ≼) is left-isotonic: if c(p) ≼ c(q) then c(r ⊕ p) ≼ c(r ⊕ q), or right-isotonic: if c(p) ≼ c(q) then c(p ⊕ r) ≼ c(q ⊕ r), or strict-isotonic: if it is left-isotonic and right-isotonic. Monotonicity is defined as: ∀p, r ∈ P, quadruplet PWS (P, ⊕, c, ≼) is left-monotonic: if c(p) ≼ c(r ⊕ p), or right-monotonic: if c(p) ≼ c(p ⊕ r), or strict-monotonic: if it is left-monotonic and rightmonotonic.

Important following lemma related to isotonicity and monotonicity, is presented from Sobrinho [START_REF] Luis | Network routing with path vector protocols[END_REF], Proposition 1 and reported by [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], Section III.B, for a hop-by-hop routing: Lemma 2.3.1. Sobrinho's Theorems [START_REF] Luis | Network routing with path vector protocols[END_REF][START_REF] Luis | Algebra and algorithms for QoS path computation and hop-byhop routing in the Internet[END_REF][START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF] The strict isotonicity and monotonicity are sufficient for correct hop-by-hop routing based on optimal paths. Proof. See [START_REF] Luis | Network routing with path vector protocols[END_REF], Proposition 1 and [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], Section III.B.

Then, the necessary and sufficient conditions are relaxed by Yaling [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF] for different protocols ( [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], Table I). Since the chapter focuses on the specific RPL routing within a 6LoWPAN network, the Yaling's theorem [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF] presented in Lemma 2.4.1 are used to establish the requirements related to PWS properties for RPL routing protocol. The following section recalls different methods involved in the metric composition approach.

Metric composition methods

To cover QoS requirements imposed by control applications while taking into account the characteristics and constraints of the underlying network, the routing protocol must adopt an appropriate routing metric [START_REF] Phillippe | Routing Metrics Used for Path Calculation in Low-Power and Lossy Networks[END_REF]. Whereas primary standardized metrics could achieve a specific performance criterion, the combination of multiple primary metrics might lead to the optimization of various performance aspects [START_REF] Trakadas | Routing metric selection and design for multi-purpose WSN[END_REF][START_REF] Zahariadis | Design Guidelines for Routing Metrics Composition in LLN[END_REF][START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF]. The Gouda methods [START_REF] Gouda | Maximizable routing metrics[END_REF] addressed this combination question by defining two distinct approaches, namely lexical metric composition and additive metric composition. Definition 2.3.4. Metric composition methods proposed by Gouda [START_REF] Gouda | Maximizable routing metrics[END_REF].

Following the formulation of quadruplet (M , ⊕, c, ≺) defined in [START_REF] Gouda | Maximizable routing metrics[END_REF], two routing metric sets are considered:

(M 1 , ⊕ 1 , c 1 , ≺ 1 ) and (M 2 , ⊕ 2 , c 2 , ≺ 2 )
, where: ≺ i , i ∈ {1, 2} is an order relation over a set of metric values M i , ⊕ i is the path (or link) cost aggregation operation and c i is the function that maps a path (or a link) to a cost. A relation ≺ lex over the set M 1 × M 2 is called a lexical composition for the ordered pair (≺ 1 , ≺ 2 ) if and only if, for every (m 1 , m 2 ) and

(m ′ 1 , m ′ 2 ) in M 1 × M 2 , the following relation: (m 1 , m 2 ) ≺ lex (m ′ 1 , m ′ 2 ) ⇔ (m 1 ≺ 1 m ′ 1 ) ∨ [(m 1 = m ′ 1 ) ∧ (m 2 ≺ 2 m ′ 2 )
] is obtained. The additive composition relation ≺ add over the set M 1 × M 2 is defined as by:

(m 1 , m 2 ) ≺ add (m ′ 1 , m ′ 2 ) ⇔ (m 1 + m 2 ) ≺ (m ′ 1 + m ′ 2 ).
The lexical metric combination intentionally provides a strict priority between inspected metrics and it is not necessary for the primary metrics to hold the same order relation. However, in the case where the most prioritized metric is not equal (e.g., it is not associated with a limited set of integer values), the second-order metric will likely never be considered. The second approach, "additive metric combination", is more interesting when combining multiple metrics with the same order relation (either ≺ or ≻). This additive manner when combining multiple metrics allows tuning some performance criteria by using weight factors (e.g., α i , i ∈ {1, 2} for the metric m i ). Nevertheless, it imposes the same properties for all considered metrics such as domain, aggregation rule, and order relation. Therefore, the primary routing metrics with different properties are usually transformed into derived metrics holding the same properties before combining them into an additive composite metric.

A composite routing metric

To design a composite metric based on a combination of multiple primary metrics, the works [START_REF] Zahariadis | Design Guidelines for Routing Metrics Composition in LLN[END_REF][START_REF] Karkazis | Evaluating routing metric composition approaches for QoS differentiation in low power and lossy networks[END_REF][START_REF] Trakadas | Routing metric selection and design for multi-purpose WSN[END_REF] have mentioned some issues to be considered for general applications. In particular, the composite metric must hold monotonicity and isotonicity properties (see Definition 2.3.3) so that the protocol achieves the consistency, loop-freeness and optimality properties (see Definition 2.3.2 and Lemma 2.3.1). Other concerns are about scalability, path stability, continuity (i.e., small variations in metric values must result in small variations in the composite metric value) [START_REF] Trakadas | Routing metric selection and design for multi-purpose WSN[END_REF][START_REF] Zahariadis | Design Guidelines for Routing Metrics Composition in LLN[END_REF].

In the considered hybrid network, assuming that three routing metrics are used in combination to deal with some network QoS requirements of control applications aforementioned. All of these metrics are strict monotonic and strict isotonic. Then, they will be used in an additive composition. Given that the strict priority between these metrics does not seem absolutely necessary, the additive metric combination (presented in Section 2.3.3) is more appropriate. Proposition 2.3.1. A composite metric for the hybrid network.

Based on the additive composition of primary metrics, the proposed composite metric is defined as follows:

m h = α 1 * m 1 + α 2 * m 2 + α 3 * m 3 , (2.1)
where:

α i ∈ R, i ∈ {1, 2, 3} such that: α i ∈ [0, 1] and ∑ 3 i=1 α i = 1
, and m 1 , m 2 , m 3 are the primary isotonic and monotonic metrics. Based on the composite metric definition, the cost of a path p is calculated as follows:

c h (p) = α 1 * c 1 (p) + α 2 * c 2 (p) + α 3 * c 3 (p), (2.2)
The metric m h , defined by Eq(s). (2.1), is isotonic and monotonic. By Lemma 2.3.1, the hopby-hop routing protocol using the metric m h satisfies three routing requirements: consistency, loop-freeness and path optimality (see Definition 2.3.2).

Proof. According to the formulation of PWS quadruplet (P, ⊕, c, ≼) (see Definition 2.3.1), three derived metrics can be represented as (P, ⊕, c i , ≼), i = {1, 2, 3}. The function c h (by Eq(s). (2.2))

that maps a path to a cost based on additive composite metric m h , is defined as the addition of weighted terms c i . From monotonicity and isotonicity properties of each metric m i , i ∈ {1, 2, 3}, those are (by Definition 2.3.3): ∀p, q, r ∈ P,

α 1 * c 1 (p) < α 1 * c 1 (r ⊕ p) α 2 * c 2 (p) < α 2 * c 2 (r ⊕ p) α 3 * c 3 (p) < α 3 * c 3 (r ⊕ p) c h (p) = 3 ∑ i=1 α i * c i (p) < 3 ∑ i=1 α i * c i (r ⊕ p) = c h (r ⊕ p)
where: 0 ≤ α i ≤ 1, and ∑ 3 i=1 α i = 1. Therefore, m h is left-monotonic. Similarly, the rightmonotonicity property is proved for the metric m h . In the same manner, the costs of paths p, q, r are computed by Eq(s). (2.2) and the left-isotonicity property of each metric m i , i ∈ {1, 2, 3} deduce following inequalities:

α 1 * c 1 (p) < α 1 * c 1 (q) ⇒ α 1 * c 1 (r ⊕ p) < α 1 * c 1 (r ⊕ q) α 2 * c 2 (p) < α 2 * c 2 (q) ⇒ α 2 * c 2 (r ⊕ p) < α 2 * c 2 (r ⊕ q) α 3 * c 3 (p) < α 3 * c 3 (q) ⇒ α 3 * c 3 (r ⊕ p) < α 3 * c 3 (r ⊕ q) c h (p) < c h (q) ⇒ c h (r ⊕ p) < c h (r ⊕ q).
Therefore, m h is left-isotonic. Similarly, right-isotonicity property: c h (p) < c h (q) ⇒ c h (p ⊕ r) < c h (q ⊕ r) can be obtained for m h .

Challenges in the design of a composite metric

The diversity of network QoS requirements motivates the design of a composite metric to cope with routing problems in the hybrid network. However, the non-trivial composition method (e.g., linear composition) is challenging because of the strong difference of metric definitions, the complex computation of dynamic paths for wide-area network and fast adaptation in the routing decision [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF]. It could lead to routing instability, non-optimal paths, increased latency, and packet loss due to small temporary loops [START_REF] Phillippe | Routing Metrics Used for Path Calculation in Low-Power and Lossy Networks[END_REF][START_REF] Trakadas | Routing metric selection and design for multi-purpose WSN[END_REF]. As discussed in [START_REF] Shelby | 6LoWPAN: the wireless embedded internet[END_REF], Section 4.2, the design of routing metrics plays an important role in (1) capturing specific characteristics of the target network; (2) optimizing some performance aspects; (3) having an impact on different routing protocol (causing the problems of instability, sub-optimality, loop creation); and (4) ensuring the use of consistent path calculation mechanisms.

Therefore, not all metrics can be used on any network. A particular case study is considered in Section 2.4 to demonstrate the metric composition approach to RPL routing adapted for NCSs. The process of composite metric design usually includes: (1) the selection of the primary metrics in order to preserve their routing properties; (2) the metric (re)quantification and derivation; (3) the composition of derived metrics; and finally (4) the demonstration with a routing protocol for consistency, loop-freeness and path optimality (see Definition 2.3.2).

Case study: Networked control systems using the RPL routing protocol

This chapter, especially focuses on the dynamic routing design for the hybrid network using 6LoWPAN technology. Routing problems of the hybrid network (e.g., the limited resources of nodes, how to route information in a multi-hop network, mesh topology frequently changed due to mobility, unreliable radio links, and specific application requirements) concern with both Internet and 6LoWPAN topologies [START_REF] Sheng | A survey on the IETF protocol suite for the internet of things: Standards, challenges, and opportunities[END_REF]. Consequently, the routing design for hybrid network involves considering properties of nodes and wireless links to provide the QoS as expected by control applications. The hybrid network using 6LoWPAN technology can take advantage of RPL protocol for dynamic routing because they are specified by two closely linked IETF Working Groups (i.e., 6LoWPAN and ROLL) and RPL is specified for a low-power and lossy network [START_REF] Kumar | Routing in IPv6 over low-power wireless personal area networks (6LoWPAN): A survey[END_REF][START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF].

Background. RPL routing: requirements, architecture, algorithms, and protocols [START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF][START_REF] Kumar | Routing in IPv6 over low-power wireless personal area networks (6LoWPAN): A survey[END_REF].

From the IETF point of view, requirements of control applications as discussed above can belong to two categories: "industry" [RFC5673] (low power, high reliability, ease of installation and maintenance); and "building" [RFC5867] (auto-configuration, manageability). Terminologically, the constrained network in RPL context is also called "Low power and Lossy Networks" (LLN) [RFC7102]. In hybrid network, the dynamic routing is chosen to operate at the network layer (called "route-over" in [RFC6606], pp. 4-6) in order to make the link-layer transmission techniques transparent to IP and to profit by hop-by-hop forwarding decisions. The IETF ROLL WG is developing a potential solution for routing problem in LLN by defining a routing protocol, called "IPv6 Routing Protocol for LLN" (RPL) [RFC6550]. RPL is a proactive, distance-vector routing protocol (based on distributed Bellman-Ford algorithm and hop-by-hop forwarding), especially optimized for multi-point to point data flows. It cannot only operate on 6LoWPAN, but also on a large number of physical networks (refer to [RFC3819]). Functionally, it specifies how to build a "Directed Acyclic Graph" (DAG) containing all devices and routers using an objective function (OF), a metric set and some constraints. The DAG is itself composed of one or many sub-acyclic graphs, called "Destination-Oriented Direct Acyclic Graph" (DODAG). Each DODAG has a root (or sink node) representing a single point for external destinations. The RPL paths are usually optimized for packets to be transmitted through this sink node. A routing metric is typically based on the network information such as distance, bandwidth, network load, hop count, delay, maximum transmission unit (MTU), reliability and communications costs (see more in [START_REF] Phillippe | Routing Metrics Used for Path Calculation in Low-Power and Lossy Networks[END_REF]).

The NCS hybrid network is considered as a class of network in which both the routers (i.e., intermediate nodes) and devices are constrained (i.e., LLNs). In hybrid network, the RPL protocol is appropriate to provide a unified routing mechanism whereby multipoint-to-point traffic (e.g., from sensors towards a controller) and the point-to-multipoint traffic (e.g., from a controller to the actuators or other controllers) are supported. Following Lemma considers the necessary and sufficient conditions for the RPL protocol to satisfy three routing requirements: consistency, loop-freeness, and path-optimality (see Definition 2.3.2).

As an example, some communication constraints (presented in Section 2.2.1) and operational criteria for BC control and supervision are concerned such as the reliability of the paths (e.g., when sending control signal to actuators), low energy consumption (as always with batterypowered devices) and short average delay (e.g., for information gathering or control sending) in data transmission. These criteria can infer the selection of three primary metrics from Table 2.1 such as link Expected Transmission count (ETX), Node remaining Energy (NE) and Hop count of a Path (HP) (as illustrated in Fig. 2.4 and summarized in Table 2.2). None of these Fig. 2.4 An example of the data communication from level sensor S i to controller C i in the hybrid network: a path p(S i , . . . , P i , . . . , P 1 ,C i ) is taken as an example. For a node P i on the path p, the path cost from P i to C i is calculated by: c(P i ,C i ) = c(P i , P i-1 ) + c(P i-1 ,C i ). Packet loss rate (also, transmission rate)

primary metrics by itself is suitable for use in the hybrid network to satisfy QoS requirements of control applications. Therefore, to achieve specific objectives of NCS using one homologous RPL instance, a composite metric (i.e., a weighted combination of primary metrics) should be considered. Besides, other approaches refer to the operations of RPL routing with multiple instances separately, each uses one primary metric for a specific data flow. Nevertheless, that needs the coordination of root nodes globally to do the unilateral allocation of instance identification (see more detail of RPL Instance IDs in [START_REF] Phillippe | Routing Metrics Used for Path Calculation in Low-Power and Lossy Networks[END_REF]).

Metric quantification

For a routing problem using shortest path, the strong difference of metric domains, aggregation rules and order relations require the re-definition and derivation of the primary metrics in order to be used in composition method. The following Lemma is helpful for how to redefine a metric.

Lemma 2.4.2. PWS properties of a metric [START_REF] Karkazis | Evaluating routing metric composition approaches for QoS differentiation in low power and lossy networks[END_REF] Any metric, (a) which is additive over the path; (b) provides a positive cost for any link, and (c) whose order relation is the "less than or equal" defined for real numbers, is strictly monotonic and strictly isotonic.

Proof. Following the PWS formulation (P, ⊕, c, ≼) of an additive metric m, assuming that the path costs c(p) > 0, c(q) > 0 for any paths p, q ∈ P, then c(p ⊕ q) = c(p) + c(q) > c(p) and c(q ⊕ p) = c(q) + c(p) > c(p). Therefore, the metric m is strictly monotonic by Definition 2.3.3. In the same manner, for any paths p, q, r ∈ P such that: c(p) > 0, c(q) > 0, c(r) > 0, we obtain: c(p) < c(q) ⇔ c(p)+ c(r) < c(q)+c(r) ⇔ c(p ⊕r) < c(q ⊕r) and c(p) < c(q) ⇔ c(r)+ c(p) < c(r) + c(q) ⇔ c(r ⊕ p) < c(r ⊕ q), By Definition 2.3.3, the metric m is strictly isotonic.

Based on Lemma 2.4.2, the redefinition of selected primary metrics in Table 2.1 is necessary for obtaining (1) additive metrics over a path, (2) positive cost of any link, and (3) "less than or equal" order relations.

Link Expected Transmission Count (ETX)

In a control system, it is required that control signals must successfully reach the system actuators and the information exchanged between controllers must be accurate and stable. These requirements may be satisfied by imposing certain link reliability on the paths. As shown in Table 2.1, the primary metrics related to link reliability include Link Throughput, Latency, Quality Level and Expected Transmission Count (ETX). The easily measurable and adequate metric used in control application routing is Link ETX. It is commonly defined as the average number of transmissions (and re-transmissions) to successfully deliver a packet over a link. To avoid the complicated measurement and probability calculation, [START_REF] Karkazis | Evaluating routing metric composition approaches for QoS differentiation in low power and lossy networks[END_REF] proposed a Link ETX computation based on the number of packets delivered with or absence of acknowledgments (ACKs) (here, the control application is assumed to use the acknowledgment mechanism for link reliability). For a link l, the mathematical formula of Link ETX is given by: ET X(l) = (transmitted packets) (packets with ACKs)

= (N s + N f ) N s , (2.3)
where N s (and N f ) is the number of packets successfully delivered with acknowledgments (correspondingly, the absence of acknowledgments). A path of m links l i , i = {1, . . . , m} has Path ETX: ET X(p) = ∑ m i=1 ET X(l i ). Regarding the metric formulation (see Definition 2.3.1), the quadruplet PWS (P, ⊕, c, ≼) represents a set of paths P, capable of concatenation ⊕, the additive cost function c that maps the link reliability properties to Link ETX value, the order relation ≼ is "less than or equal". As a result, the Objective Function related to Path ETX can be described as min{ET X(p)}, i.e., the best path is the one with a minimum value of Path ETX. Based on Lemma 2.4.2, this metric ET X is monotonic and isotonic.

Node remaining Energy (NE)

The energy issue is one of the most important concerns in control system due to the fact that a lot of devices are battery-powered and lack of regular maintenance (see Section 2.2.1). The nodes whose residual energy is lower than a threshold value, are disabled and create black-holes in the network unless certain redundancy is added to the network topology. In addition, the Link ETX metric discussed above usually provides for all data flow few similar paths which rapidly become congested. Concentrating data flow on mostly same paths results in energy exhaustion of nodes along the paths. The RPL protocol takes this issue into account by proposing another primary metric Node remaining Energy (NE) for path calculation algorithm. Many research works have considered different schemes such as node deployment, topology control, routing protocols, data reduction, duty/sleep schedule, etc. to provide a lowest energy-oriented path in routing selection (see [START_REF] Zhao | An Energy-Efficient Region-Based RPL Routing Protocol for Low-Power and Lossy Networks[END_REF][START_REF] El-Aaasser | Energy Aware Classification for Wireless Sensor Networks Routing[END_REF]). The energy-aware challenge is the complexity of a network that requires a dynamic constraint-based routing. Furthermore, there is no simple abstraction that adequately covers the broad range of power sources and energy storage devices used in existing LLN nodes such as mains-powered, primary batteries, energy scavengers, etc. The dynamic energy-aware metric is proposed by [START_REF] Phillippe | Routing Metrics Used for Path Calculation in Low-Power and Lossy Networks[END_REF] as a useful percentage of remaining energy in similar nodes. The mathematical definition of this metric can be given in terms of measurable elements, either the energy or the power. In NCS example, assuming the current energy in battery E now can be estimated, the remaining energy ratio of node X can be calculated by:

NE(X) = (initial energy) (stored remaining energy) = E 0 E now , (2.4)
where E 0 is the initial energy of node battery. This metric is additive along the path, that is: for any path p of m nodes X i , i = {1, . . . , m}, NE(p) = ∑ m i=1 NE(X i ). Regarding the metric formulation (see Definition 2.3.1), the quadruplet PWS (P, ⊕, c, ≼) represents a set of paths P, capable of concatenation ⊕, the additive cost function c that maps the node energy properties to NE value, the order relation ≼ is "less than or equal". As a result, the Objective Function can be described as min{NE(p)}, i.e., the best path is the one with minimum Path remaining Energy. Based on Lemma 2.4.2, this metric NE is monotonic and isotonic.

Hop count of a Path (HP)

Many routing protocols use HP as a metric to report the number of traversed nodes along the path. It is also used as a constraint to indicate the maximum number of hops (i.e., hop limit) that a path may follow. HP(X) = number of traversed nodes along a path p from X to root node.

(2.5)

The control system considers this metric as a mean of time delay when a node desires to reach destination node rapidly. By minimizing the number of traversed nodes, the number of transmissions (but not re-transmission) is also reduced with equally expected link quality. As a result, the total energy consumption or aggregation delay may be minimized. To formally represent HP metric, the quadruplet PWS (P, ⊕, c, ≼) (see Definition 2.3.1) is used in the mapping of link cost c to 1 for any link. The path HP is equal to the number of links along the path. For example, the Path HP of a path p comprising m nodes (excluding source node) and also m links l i , i = {1, . . . , m} is:

HP(p) = ∑ m i=1 c(l i ) = m.
The order relation ≼ is "less than or equal", which is applied on the integer. The Objective Function can be described as min{HP(p)}, i.e., the best path is the one with minimum Path HP. Based on Lemma 2.4.2, this metric HP is monotonic and isotonic.

Remark 2.4.1. Three primary metrics (as quantified in Section 2.4.2) are strict monotonic and strict isotonic. So the Lemma 2.3.1 also holds.

Since the PWS left-isotonicity and left-monotonicity properties (here, strict isotonicity and monotonicity) of three primary metrics, above quantified, are guaranteed, the RPL routing protocol using separately one of these primary metrics will achieve the consistency, loop-freeness and path-optimality (by Lemma 2.4.1). These primary metrics with different properties (i.e., domains, aggregation rules, order relation) need to be transformed into derived metrics holding the same properties before combining them into an additive composite metric. The Table 2.3 shows the derived metrics suitable to be used for dynamic routing in the NCS hybrid network. Since monotonicity and isotonicity properties have been demonstrated with the use of primary metrics (as mentioned in Section 2.4.2), it is easy to deduce the same properties for their derived metrics in Table 2.3. Indeed, they are all additive, corresponding costs are positive, and order relations are "less than or equal", so they are all monotonic and isotonic by Lemma 2.4.2. Then, these derived metrics are used for the design of a composite metric. Considering the NCS hybrid network as shown in Fig. 2.4, the composite metric m h (see Eq(s). (2.1)) can be defined by combining three routing metrics in Table 2.3 as follows:

Metric composition

m h = α 1 * (1 - 1 ET X ) + α 2 * (1 - 1 NE ) + α 3 * (1 - 1 HP ), (2.6)
where:

α i ∈ R, i ∈ {1, 2, 3} such that: α i ∈ [0, 1] and ∑ 3 i=1 α i = 1.
For any path p of m links (and m nodes except the source node), the path cost c h (p) (see Eq(s). (2.2)) is calculated as follows:

c h (p) = α 1 * c 1 (p) + α 2 * c 2 (p) + α 3 * c 3 (p), (2.7)
where:

c 1 (p) = m ∑ i=1 c 1 (l i ) = m ∑ i=1 (1 - 1 ET X(l i ) ), c 2 (p) = m ∑ i=1 c 2 (X i ) = m ∑ i=1 (1 - 1 NE(X i ) ), c 3 (p) = m ∑ i=1 c 3 (X i ) = m ∑ i=1 (1 - 1 HP(X i )
).

(2.8) By Lemma 2.4.1, the RPL protocol using the composite metric m h can achieve consistency, loop-freeness, and path-optimality (as defined in Definition 2.3.2).

The weight parameters (i.e., α i ) in Eq(s). (2.6) can be chosen to enable the relative importance of three metrics or to tune for the best performance criteria corresponding to the target network. For simulation, the following section presents a particular network benchmark using different (primary and composite) metrics with RPL protocol, which implements the distributed Bellman-Ford algorithm and hop-by-hop forwarding scheme.

Simulation results

Considered network topology

For the control of irrigation channels (see Fig. 2.1), the network topology is assumed to contain 144 nodes covering all possible communicating devices (e.g., water level or pressure meters, actuators, controllers) for a considered section of the BC irrigated field. These nodes are assumed homogeneous with the same power consumption scheme and transmission capacity (except unconstrained root nodes). They are scattered in the field of 2D-grid 1200mX1200m along channel section (including a part of mesh pressurized network). For simplicity, the topology is set up so that all nodes are horizontally and vertically aligned and equally distributed with a hop distance of 100m. Their data exchange using wireless links operates in the range of 125m without obstruction. The channel capacity is set to maximum 255 bi-directional flows. Each node, assumed stationary in this example, is assigned the same integer for its identifier (ID) and network address.

A modeling and simulation framework, called J-Sim simulator (presented in [START_REF] Sobeih | J-Sim: A simulation and emulation environment for wireless sensor networks[END_REF]) is used in network simulations. An extension for RPL support in J-Sim (introduced in [START_REF] Karkazis | Evaluating routing metric composition approaches for QoS differentiation in low power and lossy networks[END_REF]) is based on the IETF specifications [RFC6550, version 17] and it operates on top of IEEE 802.11 standard as a link layer (MAC). Moreover, the integrated simulator also offers the capability of evaluating some performance criteria while using a composite metric. The basic parameters for network simulation are summarized in Table 2

Performance evaluation

The percentage of packet loss (%), average latency (ms), and network lifetime (s) are considered as the performance criteria in this example. The packet loss ratio is computed from the number of packets that were dropped by lossy links (or buffer overflow) and a total number of emitted packets. The average latency is computed by the average of delays for a number of packets transmitted from the reference flow. Concerning power consumption, a node which is heavily traversed by data traffic (e.g., n16) is monitored. It is also the first node depleted from its energy. Network lifetime is considered as network duration until 5% of nodes (i.e., about 8 nodes in this network topology) which get their energy depleted and therefore the network cannot fully guarantee its functionality. During the simulation of all scenarios above, three performance criteria have been evaluated and recorded (see Table 2.6). These results are then compared as illustrated in Fig. 2.6. They show that a composite metric can improve several performance criteria while primary metrics when used separately can only improve, but better, the corresponding performance criterion. For example, routing with only HP metric (first points 1.0HP in Fig. 2.6) gives the smallest latency (21.79ms), a consequent network lifetime (2320.68s), but increases in the packet loss ratio (about 60%) compared to other scenarios. The use of ETX metric in different compositions (e.g., 0.5HP + 0.5ET X,0.2HP + 0.3ET X + 0.5NE) can reduce the packet loss ratio to about 0% with a short latency and network lifetime. The presence of NE metric in composite metrics (e.g., 0.2HP + 0.8NE, 0.2HP + 0.3ET X + 0.5NE) of different scenarios has distinctly improved the network lifetime to about 50% (6000s). As a result, by tuning around the weight parameters, the desired performance can be achieved for the designed hybrid network.

Results of other approaches

This section introduces some results of research works considering similar primary metrics (i.e., NE, HP, and ETX) or composite metrics in the RPL routing protocol. The work [START_REF] Capone | An Energy Efficient and Reliable Composite Metric for RPL Organized Networks[END_REF] defined a novel metric, namely "Lifetime and Latency Aggregateable Metric" (L2AM) for RPL routing protocol. It allows the RPL protocol to compute minimum path cost, based on a combination of both data reliability (defined by ETX) and node residual energy, thus to prolong the network lifetime. In the simulation scenario shown in [START_REF] Capone | An Energy Efficient and Reliable Composite Metric for RPL Organized Networks[END_REF], Fig. 3, RPL routing with L2AM metric reaches a network lifetime gain of 21% over RPL with ETX metric. Given the same overall degree of network reliability, the results from a wide variety of simulation scenarios show that L2AM clearly outperforms the standard ETX metric in terms of network lifetime. Whereas the work [START_REF] Chang | Energy-Efficient Oriented Routing Algorithm in Wireless Sensor Networks[END_REF] proposed an energy-oriented routing mechanism to improve RPL routing protocol by incorporating ETX and residual energy metrics for path selection in LLN networks. The simulation result of the given scenario (see [START_REF] Chang | Energy-Efficient Oriented Routing Algorithm in Wireless Sensor Networks[END_REF], Fig. 11) showed that the overall network lifetime for proposed routing algorithm increases about 12% compared to the standard RPL mechanism.

The work [START_REF] Karkazis | Evaluating routing metric composition approaches for QoS differentiation in low power and lossy networks[END_REF] showed that by combining the Hop count metric and the node remaining energy with the additive composition approach, a significant energy savings (up to 16% for nodes participating in forwarding data from multiple data sections) can be reached without heavily compromising other performance criteria such as latency or packet loss. Interestingly, the work [START_REF] Olivier | Fuzzy-based routing metrics combination for RPL To cite this version[END_REF] designed and implemented a new objective function for RPL to optimize more than one network performance aspect using fuzzy-based routing metric combination. The example scenario used fuzzy inference system to combine ETX, delay, and node remaining power into one unique value. Experimental results of this work show that by using the composite metric with objective function, RPL routing obtains better results. These are for both network energy distribution and packet reception rates (6.06% at throughput 12 packets/min), compared to the ETX scenario (9.40%). Another work [START_REF] Liu | Joint routing and sleep scheduling for lifetime maximization of wireless sensor networks[END_REF] studied network lifetime maximization of wireless sensor networks through joint routing and sleep schedule. The proposed Iterative Geometric Programming (IGP) algorithm drastically outperforms the performance of optimal iterative separate routing and sleep scheduling method by an average of 29% over a large range of traffic rates. Compared to the design with optimal routing, but fixed sleep scheduling, the proposed IGP algorithm prolongs the lifetime by an average of 284%.

Conclusion

Through analytical demonstration and simulation results, this chapter has shown that several performance criteria related to QoS requirements of control applications, can be optimized for dynamic routing by using a metric composition approach. The additive combination method uses and derives some standard metrics characterizing the LLNs in order to preserve their routing properties. The designed composite metric also verifies the PWS properties guaranteeing the consistency, loop-freeness, and path optimality of the considered routing protocol. As a result, the design of a hybrid network for wide-area NCSs while taking into account the QoS requirements of control applications can be helped. Besides, it is necessary to emphasize that the QoS requirements in this chapter seem mild due to the context of control applications and the results are only shown by simulations. The practical networks with more complex characteristics, possibly produce different results. Nevertheless, the approach outlined in this chapter can be extended or replicated in other applications. The perspectives of this work aim to show the implementation of the designed composite metric in a practical network and obtain experimental results in order to validate the benefits of composition approach.

Chapter 3

Asynchronous information consensus in the distributed control of irrigation channels

In distributed model predictive control (DMPC), the global performance of the control system is improved by taking into account the interactions of subsystems, the coordination and/or the cooperation of controllers. For these tasks, the controllers need to exchange the information. However, the shared information may encounter some divergence due to time-varying topologies and communication issues. This chapter presents an approach to the challenges due to loss of the synchronization or imperfection in communication among controllers, by introducing an information consensus problem and an asynchronous consensus protocol. By exploring the information exchange topologies under proposed communication assumptions (e.g., directed graphs, connectivity, and asynchronism), we find the necessary and sufficient conditions for the information consensus to be reached asymptotically using the defined asynchronous protocol. The asynchronous consensus can then be integrated into the DMPC scheme so that the shared information among controllers asymptotically converges to the newest values after applying an action. The aim of this chapter is to (1) establish conditions for a class of "leader-followers" systems using an asynchronous linear protocol to asymptotically reach the consensus under fixed or time-varying directed communication topologies; (2) propose a global topology for "leaderswitching" systems to satisfy the generalized consensus problems. For this purpose, the following sections present: (1) the definition of interaction variables incorporated into subsystem models and the information shared by controllers for coordination and cooperation (Section 3.2.2); (2) the formulation of a consensus problem with communication assumptions for the distributed control of irrigation channels (Section 3.3); (3) the design of an asynchronous protocol solving the consensus problem (Section 3.3.1); (4) the convergence analysis of the consensus processes (Section 3.3.3); (5) generalized consensus problems for a leader-switching system (Section 3.3.4). Through extensive simulation results, Section 3.4 demonstrates the benefits of the information consensus approach for the convergence problems of information states of all controllers once a controller applies an action to the subsystem.

Introduction

A fundamental concern in distributed model predictive control (DMPC) of large-scale systems is the coordination and/or the cooperation of groups of controllers to achieve better global performance [START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF][START_REF] Van Overloop | Human-in-the-Loop Model Predictive Control of an Irrigation Canal[END_REF][START_REF] Puig | Model predictive control of combined irrigation and water supply systems: Application to the Guadiana river[END_REF][START_REF] Negenborn | Coordinated model predictive reach control for irrigation canals[END_REF]. Some challenges of DMPC stem from the fact that the MPC method continuously uses the dynamic models of systems for the prediction of system outputs [START_REF] Borrelli | Predictive Control for linear and hybrid systems[END_REF][START_REF] Rawlings | Model Predictive Control : Theory and Design[END_REF]. An observation in such distributed control is that the interactions among subsystems influence not only local control decisions, but also the overall system optimality [START_REF] Imam Sutrisno | Distributed model predictive control and application to irrigation canal[END_REF][START_REF] Igreja | Application of distributed model predictive control to a water delivery canal[END_REF][START_REF] Negenborn | Coordinated model predictive reach control for irrigation canals[END_REF]. Therefore, in order to get better results for the global optimization, interaction variables involved in the prediction models need to be updated. The values of these variables are deduced from the neighbor states that are known only by exchanging information among controllers. For improving the control performance by neighboring cooperation, controllers might also share their objectives and/or constraints [START_REF] Nguyen | Distributed Model Predictive Control of Irrigation Systems using Cooperative Controllers[END_REF][START_REF] Negenborn | Coordinated model predictive reach control for irrigation canals[END_REF]. In addition, controllers need to use some shared knowledge about state variables in order to coordinate their behaviors [START_REF] Negenborn | Coordinated model predictive reach control for irrigation canals[END_REF]. As a result, the efficient distributed control of a large-scale system requires that all controllers have a consistent view of the shared information for the coordination, the cooperation or the consideration of subsystems' interactions [START_REF] Ren | Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues[END_REF][START_REF] Olfati-Saber | Consensus and Cooperation in Multi-Agent Networked Systems[END_REF]. The instantaneous values of shared information recorded by a controller are defined as its information state. Due to time-varying topology and communication issues (e.g., time delays, data packet dropouts, non-uniform sampling and quantization errors, missing and/or noisy measurements), controllers may lose the synchronization or may encounter updating problems for the shared information [START_REF] Christopher | Information Propagation Algorithms for Consensus Formation in Decentralized Multi-Agent Systems[END_REF][START_REF] Fang | On Communication Requirements for Multi-agent Consensus Seeking[END_REF]. Our approach to these problems is the so-called information consensus problem [START_REF] Qin | Recent Advances in Consensus of Multi-Agent Systems: A Brief Survey[END_REF][START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF][START_REF] Ren | Overview of Consensus Algorithms in Cooperative Control[END_REF]. By integrating a consensus process into a control scheme, all controllers try to asymptotically reach a common knowledge on certain values of interest after executing an action that changes these variables. Even when neighboring controllers loss the direct communication, the interaction/coordination variables are still able to be updated with the benefit of consensus convergence. A consensus protocol comprises interaction rules specifying the information exchange among controllers over communication networks [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF]. Moreover, it is impossible to fully synchronize all controllers in a large-scale control system. They cannot update their information states at the same time. Therefore, we focus on the information consensus in asynchronous mode [START_REF] Zhou | Constrained consensus of asynchronous discrete-time multi-agent systems with time-varying topology[END_REF][START_REF] Lee | Stability Analysis of Large-Scale Distributed Networked Control Systems with Random Communication Delays : A Switched System Approach[END_REF][START_REF] Qin | Stationary consensus of asynchronous discrete-time second-order multi-agent systems under switching topology[END_REF][START_REF] Gao | Asynchronous consensus of continuous-time multi-agent systems with intermittent measurements[END_REF][START_REF] Xiao | Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays[END_REF][START_REF] Mehyar | Distributed Averaging on Asynchronous Communication Networks[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. First investigated in the field of computer science, especially in Multi-Agent Systems (MASs), consensus problems have recently attracted much attention in a large range of application areas in automatic control. Motivated by some pioneering works such as the simplified model of flocking [START_REF] Vicsek | Novel Type of Phase Transition in a System of Self-Driven Particles[END_REF], the coordination of mobile autonomous agents using nearest neighbor rules [START_REF] Jadbabaie | Coordination of mobile autonomous agents using nearest neighbor rules[END_REF], and the theoretical framework for the analysis of consensus algorithms in multi-agent networked systems [START_REF] Olfati-Saber | Consensus and Cooperation in Multi-Agent Networked Systems[END_REF], many interesting works have been done for control applications (as surveyed in [START_REF] Wang | An overview of consensus problems in constrained multi-agent coordination[END_REF][START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF][START_REF] Ren | Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues[END_REF]). For instance, consensus problems can be found in two categories (classified by [START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF]): (1) formation control with application to multi-vehicle systems [START_REF] Biao Ren | The hybrid mobile wireless sensor networks for data gathering[END_REF] or networked multiple mobile robot systems [START_REF] Sheng | Consensus Formation Control for a Class of Networked Multiple Mobile Robot Systems[END_REF], or multi-unmanned aerial vehicle (UAV) systems [START_REF] Kuriki | Consensus-based Cooperative Formation Control with Collision Avoidance for a Multi-UAV System[END_REF]; and (2) non-formation cooperative control with application to role/task assignment, cooperative timing, cooperative search, payload transport and air traffic control [START_REF] Wollkind | Automated conflict resolution for air traffic management using cooperative multiagent negotiation[END_REF][START_REF] Tomlin | Conflict resolution for air traffic management: a study in multiagent hybrid systems[END_REF]. Whereas this approach brings along numerous advantages over traditional control schemes, especially in networked control systems (NCSs) [START_REF] Guinaldo Losada | Asynchronous Control for Networked Systems[END_REF], challenges arise in the information exchange due to the imperfection in communication. For dealing with communication issues in a distributed NCS, many works have been investigated in system stabilization that sets some limitations in control performance when introducing communication constraints (see [START_REF] Li | Robust Tracking Control of Networked Control Systems With Communication Constraints and External Disturbance[END_REF][START_REF] Rojas | On the infimal Signal-to-Noise Ratio limitations for measurable disturbance rejection[END_REF][START_REF] Rojas | Fundamental Limitations in Control over a Communication channel[END_REF][START_REF] Braslavsky | Feedback Stabilization over Signal-to-Noise Ratio Constrained Channels[END_REF] and references therein). The survey of recent progress in NCS, [START_REF] You | Survey of Recent Progress in Networked Control Systems[END_REF] have discussed various network conditions required for different control purposes (such as the minimum rate coding for stabilizability of linear systems in the presence of time-varying channel capacity, the critical packet loss condition for stability of the Kalman filter with intermittent observations, network topology for coordination of networked MASs, as well as event-based sampling for energy and communication efficiency). Another interesting overview on the theoretical development of NCSs under some challenging issues induced from communication networks is discussed in [START_REF] Zhang | Survey on Recent Advances in Networked Control Systems[END_REF]. In recent years, there has been a renewed interest in consensus problems introduced in NCSs with communication constraints (see [START_REF] Ma | Leader-following consensus of multi-agent systems with limited data rate[END_REF][START_REF] Xu | Event-Triggered Schemes on Leader-Following Consensus of General Linear Multiagent Systems Under Different Topologies[END_REF][START_REF] Wang | Cooperative Containment Control of Multiagent Systems Based on Follower Observers With Time Delay[END_REF][START_REF] Li | Event-based network consensus with communication delays[END_REF][START_REF] Lu | Distributed optimization of first-order discrete-time multi-agent systems with event-triggered communication[END_REF][START_REF] Wang | Pinning consensus for multi-agent systems with non-linear dynamics and time-varying delay under directed switching topology[END_REF][START_REF] Ren | Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues[END_REF]). The consensus protocols presented in these works usually operate in a synchronous way. Although being relatively sparse, the asynchronous consensus of MASs has been discussed in some related works (e.g., [START_REF] Gao | Asynchronous consensus of multiple second-order agents with partial state information[END_REF][START_REF] Qin | Stationary consensus of asynchronous discrete-time second-order multi-agent systems under switching topology[END_REF][START_REF] Xiao | Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF][START_REF] Mehyar | Distributed Averaging on Asynchronous Communication Networks[END_REF][START_REF] Beard | Synchronization of Information in Distributed Multiple Vehicle Coordinated Control[END_REF]). However, the updating problem of shared information (e.g., interaction among subsystems, coordination variables) among controllers considered as a consensus problem has been rarely discussed in the literature. As an example, the following sections will introduce a case study presenting DMPC scheme for irrigation channels and details of the associated consensus problem.

Distributed model predictive control of irrigation channels

We consider now an irrigation channel with multiple reaches interconnected through hydraulic control structures such as submerged gates, mixed gates and spillways (see Fig. 1.2). The irrigation channel is provided some local controllers implemented for the control of water flow at some points by adjusting the settings of control structures. In decentralized control schemes, these local controllers operate without being conscious of the presence of other controllers or subsystems. They compute the control action based only on local information without taking into account the information exchanged with other controllers. An observable drawback of such a decentralized control scheme is the reduced performance at a system-wide level due to unexpected and unanticipated interactions among subsystems by local controllers. For improving the global system performance, a distributed control is adequate in which controllers consider a negotiation, a coordination or a cooperation among them by exchanging the necessary information. An example of the DMPC scheme for the control of irrigation channels is given in Fig. 3.1. For the distributed control scheme, we observe that the computational complexity and theoretical properties (e.g., feasibility, stability and robustness) depend closely on the choice of the prediction model, control objectives, and constraints.

Prediction model

The modeling of an irrigation channel using the LB method is introduced in Section 1.3.1. The dynamics of LB model are represented by Eq(s). (1.13) and rewritten as follows:

f 1 (l,t + ∆t) = f 1 (l,t) + 1 τ ( f e 1 -f 1 (l,t)) f 2 (l,t + ∆t) = f 2 (l -v∆t,t) + 1 τ ( f e 2 -f 2 (l,t)) -∆t 2v F f 3 (l,t + ∆t) = f 3 (l + v∆t,t) + 1 τ ( f e 3 -f 3 (l,t)) + ∆t 2v F (3.1)
Accounting for the interactions of subsystems, the coordination or the cooperation of controllers requires the consideration of interaction variables or coordination variables from subsystem models. The following sections consider these variables suitable to be used for the coordination of controllers.

Definition and management of shared information

We consider here an irrigation channel comprising n reaches interconnected by the submerged gates (as shown in Fig. 3.2). The DMPC control of this system is divided over n controllers, {C 1 ,C 2 , . . . ,C i , . . . ,C n }. The information shared by controllers is chosen depending on the control setting and system modeling. For example, as [START_REF] Negenborn | Coordinated model predictive reach control for irrigation canals[END_REF] adopted deterministic discrete-time linear integral delay models for channel control, the exchange of the flow rate at the downstream of each reach is sufficient for neighbor local controllers to compute actions because the inflow of downstream reach is equal to the outflow of upstream one. For our case study, we assume to use the downstream configuration (see Fig. 3.2) in which the control actions apply on the upstream gate of each reach i , i ∈ {1, 2, . . . , n} to adjust the gate opening, θ i (manipulated input), in order to regulate the downstream flow rate, Q i (controlled variables). Some measurements are assumed to be available such as the water heights at upstream, (h i us ), and downstream, (h i ds ), of each gate, the flow rate of water through a gate, Q i g . To take into account the interactions among subsystems and the coordination of controllers, they need to share interaction variables. Using the prediction model Eq(s). (3.1) and the DMPC scheme shown in Fig. 3.3, we can easily identify that the necessary information exchanged by the neighbors of the controller i at control step k,

denoted by χ i in (τ), τ ∈ [k, k + 1], contains: { f i-1 1 (l L i-1 , k), f i-1 2 (l L i-1 , k), f i+1 3 (l us , k)}.
Thus, the management of the shared information needs to separate neighbor set M i for each controller, C i : M i = {C i 1 ,C i 2 , . . . ,C i mi } into upstream neighbor sub-set, U i , and downstream neighbor subset, D i . For simplicity, the information shared by controller C i to its neighbors at time τ is chosen as:

χ i out (τ) = [ f i 3 (l us , τ) f i 1 (l L i , τ) f i 2 (l L i , τ)] T .
In inputs, the controller C i receives the information shared by its neighbors at time τ as follows:

χ i in (τ) = [χ i in|C i 1 (τ) χ i in|C i 2 (τ) . . . χ i in|C i mi (τ)] T .
We observe that the gate equations (Eq(s). (4.4)) permit the computation of these interaction variables, once the neighbor controllers share the flow rate through the gate at their position. Therefore, the flow rate through each gate (Q i g ) is an appropriate variable to be used for coordination among controllers. The coordination variables, Q i g , are considered in this chapter as the shared information for the consensus problem (see Fig. 3.4). In a control system of n controllers, {C 1 ,C 2 , . . . ,C i , . . . ,C n }, the information vector of the controller C i is defined by:

χ i = [χ i 1 χ i 2 . . . χ i n ] T
, where:

χ i j = Q i g j ∈ R
is the flow rate through gate j recorded by the controller C i (see the example in Fig. 3.6). Other variables can appropriately be used as coordination or cooperation variables depending on distributed control schemes (an example of the cooperation of controllers is given in Chapter 4), but the corresponding consensus problems can be solved in the same manner. Timing for sharing information depends on the synchronization mechanism and the coordination method. The following section investigates the consensus while controllers exchange the information for coordination in an asynchronous way.

Asynchronous information consensus in distributed control of irrigation channels

The efficient distributed control of irrigation channels requires that all controllers have to exchange, coordinate and cooperate towards optimizing global objectives. In our distributed control scheme, each controller proceeds with (1) collecting the information from its neighbors;

(2) updating the information state; (3) computing control actions according to control laws; (4) applying the control actions; [START_REF] Beard | Synchronization of Information in Distributed Multiple Vehicle Coordinated Control[END_REF] sharing and performing the consensus process for the convergence of the shared information among all controllers (as illustrated in Fig. 3.5). The controllers designed in this manner, are very close to the concept of agents [START_REF] Wooldridge | An Introduction to MultiAgent Systems[END_REF]. Indeed, they possess limited capabilities of information gathering, processing skills, and actions. They also have communication capability and perform the coordination and/or the cooperation based on the information exchanged among them [START_REF] Van-Thang | A multiagent approach using model-based predictive control for an irrigation canal[END_REF][START_REF] Negenborn | Coordinated model predictive reach control for irrigation canals[END_REF]. Due to communication issues, controllers may encounter a divergence in the shared information, which can result in unexpected actions. We approach the problem of divergent information shared by controllers (i.e., agents) by an information consensus problem [START_REF] Christopher | Information Propagation Algorithms for Consensus Formation in Decentralized Multi-Agent Systems[END_REF][START_REF] Ren | Information Consensus in Multivehicle Cooperative Control[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. The information consensus guarantees that controllers sharing information over an unreliable and dynamically changing network topology, have a consistent view of critical information for coordination and cooperation tasks [START_REF] Ren | Information Consensus in Multivehicle Cooperative Control[END_REF]. This problem is then solved by using a consensus protocol designed to achieve the convergence of information states of controllers. In DMPC scheme, the consensus process can be integrated into the control algorithms in a receding fashion in order to provide most updated values needed for adequate action computation. In addition, our MAS-like control system is characterized by some asynchronism features (as listed in [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]) such as:

1. each controller operates according to its own clock;

2. no guarantee on delivery time or successful delivery;

3. the exchange topology is time-dependent. Therefore, the consensus problem is investigated in this chapter using an asynchronous framework. An overview of basic concepts of consensus problems in MASs and asynchronous consensus protocols for maintaining system stability property is presented in [START_REF] Lee | Convergence Analysis of Asynchronous Consensus in Discrete-time Multi-agent Systems with Fixed Topology[END_REF][START_REF] Qin | Stationary consensus of asynchronous discrete-time second-order multi-agent systems under switching topology[END_REF][START_REF] Gao | Asynchronous consensus of continuous-time multi-agent systems with intermittent measurements[END_REF][START_REF] Xiao | Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. Under proposed asynchronism assumptions, [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF] introduced a theoretical mapping in which models of synchronous and asynchronous MASs with various communication patterns (such as directed and delayed communication) can be addressed in the same framework. The work [START_REF] Qin | Stationary consensus of asynchronous discrete-time second-order multi-agent systems under switching topology[END_REF] is concerned with the asynchronous consensus problem of discrete-time second-order MAS under dynamically changing communication topology, in which the asynchrony means that each controller detects the neighbors' state information to update its information state by its own clock. Under a fixed topology, [START_REF] Lee | Convergence Analysis of Asynchronous Consensus in Discrete-time Multi-agent Systems with Fixed Topology[END_REF] studied the convergence conditions for asynchronous consensus problems in multi-agent systems such that the asynchronous consensus value converges to the synchronous consensus value. In recent years, there has been a tremendous amount of renewed interest in leader-follower configurations [START_REF] Franzè | A distributed model predictive control scheme for leader-follower multi-agent systems[END_REF][START_REF] Liu | Leader-following consensus for discrete-time multi-agent systems with parameter uncertainties based on the event-triggered strategy[END_REF][START_REF] Shang | Leader-Follower Fixed-Time Group Consensus Control of Multiagent Systems under Directed Topology[END_REF][START_REF] Abdessameud | Leader-Follower Synchronization of Euler-Lagrange Systems With Time-Varying Leader Trajectory and Constrained Discrete-Time Communication[END_REF][START_REF] Xu | Event-Triggered Schemes on Leader-Following Consensus of General Linear Multiagent Systems Under Different Topologies[END_REF][START_REF] Zhao | Leader-follower optimal coordination tracking control for multi-agent systems with unknown internal states[END_REF][START_REF] Gao | Leader-following consensus for discrete-time descriptor multi-agent systems with observer-based protocols[END_REF]. Based on the standard topologies (graphs) of information flow, these systems can be viewed as single-Fig. 3.5 Approach to the problem of divergent information shared among controllers by an information consensus problem. This problem is then solved by an asynchronous consensus protocol designed to achieve the convergence of information states of controllers. leader-multiple-followers systems [START_REF] Xiao | Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays[END_REF]. This chapter follows a rather different approach, called the asynchronous information consensus for a leader-switching system. In this system, the leader role of a controller is not fixed and chosen depending on the instant of an event in the system provoking a change in information states of the controllers. Solving this problem requires the design of an asynchronous consensus protocol such that the information states of all controllers eventually converge to the newest values once a controller (which becomes a leader) applies an action that changes the values of these variables. For example, the control system of irrigation channels switches the leader when a controller applies an action on a gate that changes the system inputs and consequently, the output, Q g , is also changed. In consequence, the information states of all controllers need to be updated (an example is shown in Fig. 3.6). Henceforth, we use an asynchronous framework for the design and the performance analysis of consensus protocol possibly implemented in coordinated controllers.

Problem statements for asynchronous discrete-time system

From MAS point of view, our control system is characterized as a discrete-time system, in which the information flow among controllers is directed and the communication topology may be time-dependent (also called switching topology) [START_REF] Olfati | Consensus problems in networks of agents with switching topology and time-delays[END_REF]. As the subsystem states change over time, each controller attempts to reach an agreement with others on some values deduced from their states. We formulate a consensus problem for the agreement of all controllers by introducing the information states of controllers. The information state vector of a controller records the instantaneous values of the information shared by all controllers and itself. The asynchronous consensus protocol is eventually investigated to solve the convergence of information states to the newest values as shared by its neighbors (an example is presented in Fig. 3.6). By Fig. 3.6 An example of consensus problem in distributed control of irrigation channels is used to illustrate the asynchronous information consensus of the leader-switching system. In this example, the controller, C 2 , applies a computed action, u 2 , to the subsystem, which changes the value of its information state, χ 2 . C 2 actually becomes the leader. The consensus process is started by C 2 for providing the convergence of information element χ 2 of all controllers. accomplishing the consensus processes, the information state vectors of all controllers must contain the same values asymptotically. In this chapter, we are interested in the consensus problem of an asynchronous discrete-time MAS-like system under fixed or time-varying topology in the presence of communication issues such as time delays [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF][START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF].

Assumptions 3.3.1. Regularity assumptions [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF].

The convergence analysis of asynchronous consensus process requires making some communication assumptions (introduced in [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF], Section II as regularity assumptions) such as:

1. any delay is bounded (see [START_REF] Olfati | Consensus problems in networks of agents with switching topology and time-delays[END_REF], section X.A for quantitatively bounded delay), only a finite number of updating instants can occur within any finite time interval;

2. a controller does not fail to be updated or is updated infinitely often;

3. all controllers can access their own states and at least one of these controllers can achieve it without delay.

The basic idea for a consensus protocol design is to impose similar dynamics for updating information states of all controllers [START_REF] Ren | Information Consensus in Multivehicle Cooperative Control[END_REF]. Solving a consensus problem consists in designing an update law so that the information states of all controllers converge to common values.

We consider a control system of n controllers, C = {C 1 ,C 2 , . . . ,C i , . . . ,C n }, indexed using an index set N = {1, 2, . . . , n}. The information state of controller C i , i ∈ N, is defined as:

χ i = [χ i 1 χ i 2 . . . χ i n ] T . All controllers share a common state space χ i ∈ R n×1 . Let t 0 < t 1 < • • • < t a < .
. . be event-based discrete-time instants [START_REF] Xu | Event-Triggered Schemes on Leader-Following Consensus of General Linear Multiagent Systems Under Different Topologies[END_REF][START_REF] Li | Event-based network consensus with communication delays[END_REF] when an event (e.g., applying an action) occurs in the control system provoking a change in information states of controllers. We assume that, at the time instant t a , the controller C i applies the control action which changes the information state: χ i (t a ). The controller C i becomes then the actual leader and starts the consensus processes executed on other controllers (i.e., followers) so that the change is updated in their information states. The updating instants for asynchronous information exchanged among controllers are denoted by τ 0 < τ 1 < • • • < τ u < . . . . By accomplishing the consensus process, the consensus vector defined as:

χ(τ) = [χ 1 (τ) . . . χ i (τ) . . . χ n (τ)] T = [χ 1 i (τ) . . . χ i i (τ) . . . χ n i (τ)] T (3.2)
converges to the same value χ i (τ 0 ) = χ i i (t a ) for all elements. An example of the asynchronous framework used for the convergence of the information exchanged among controllers is illustrated in Fig. 4.5, which shows different time instants and intervals.

For the control system, we consider a localized consensus protocol [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF], in which each controller updates its information state using latest-known state values of its neighbors and itself such that [START_REF] Olfati | Consensus problems in networks of agents with switching topology and time-delays[END_REF]:

χ i (τ u+1 ) = χ i (τ u ) + λ u i (τ u ) (3.3)
where: λ > 0 is step-size and u i (τ u ) is a linear consensus protocol defined by:

u i (τ u ) = n ∑ j=1 A i j (χ j (τ u ) -χ i (τ u )) (3.4)
in which A i j , i, j ∈ N represents neighboring information flow (i.e., the magnitude of A i j (τ) can represent time-varying relative confidence of the controller C i in the information shared by the controller C j at time τ or the relative reliability of information exchange links between them). Using the protocol Eq(s). (3.4), the dynamics of discrete-time consensus process Eq(s). (3.3) while λ = 1 can be described by:

χ i (τ u+1 ) = χ i (τ u ) + n ∑ j=1 A i j (χ j (τ u ) -χ i (τ u )) = (1 - n ∑ j=1, j̸ =i A i j )χ i (τ u ) + n ∑ j=1, j̸ =i A i j χ j (τ u ) = A ii χ i (τ u ) + n ∑ j=1, j̸ =i A i j χ j (τ u ) = n ∑ j=1 A i j χ j (τ u ) (3.5)
where, we set A ii = (1 -∑ n j=1, j̸ =i A i j ). Whereas numerous asynchronous models have been proposed and successfully applied to some practical problems [START_REF] Victor | A short introduction to asynchronous systems[END_REF][START_REF] Frommer | On asynchronous iterations[END_REF][START_REF] Bertsekas | Parallel and Distributed Computation: Numerical Methods[END_REF], the asynchronous model proposed in [START_REF] Su | Further results on convergence of asynchronous linear iterations[END_REF] is one of the prevalent models in asynchronous theory [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. According to [START_REF] Su | Further results on convergence of asynchronous linear iterations[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF], the time-varying topology and communication delays can be integrated into the asynchronous consensus protocol 3.4, for updating information states of a controller C i , as follows:

χ i (τ u+1 ) =    ∑ n j=1 A i j (τ u )χ j (τ u -d i j (τ u )), i ∈ S(τ u ) χ i (τ u ), otherwise (3.6)
where: for i, j ∈ N, χ i (τ 0 ) is initial information state of controller C i ; A i j (τ) is neighboring information flow between controllers C i and C j and A ii = (1 -∑ n j=1, j̸ =i A i j ); d i j (τ) are timevarying communication delays; S(τ) is the set of already updated controllers at the instant τ. The Eq(s). (3.6) is also called the asynchronous consensus protocol in the literature [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF].

The matrices A(τ) = [A i j (τ)] are non-negative stochastic matrices, which have an eigenvalue equal to 1 and a corresponding eigenvector equal to

1 n = [1 1 . . . 1] T , that is: A(τ)1 n = 1 n , ∀τ.
The matrices A(τ) represent the time-varying communication topologies of the control system and are principally explored in the convergence analysis of the asynchronous consensus problem.

The global information consensus is asymptotically achieved for all controllers when the information states of all controllers tend to the initial value of the actual leader. Assuming that the leader is C i : χ i (τ 0 ) = χ i i (t a ) and for any initial values of followers: χ j (τ 0 ), ( j ̸ = i), the asymptotic achievement of the consensus means:

∀ j ∈ N, ||χ j (τ) -χ i (τ 0 )|| → 0, as τ → τ +∞ . (3.7)
Formally, the Assumptions 3.3.1 used in the convergence analysis of asynchronous consensus can be formulated as follows: Under the Assumptions 3.3.2, we investigate the reachability of a consensus while the information exchange among controllers is in asynchronous mode. Concretely, we search the conditions for information states of controllers to converge to the consensus point, depending on the initial state of the root node. Normally, asynchronous consensus points are determined depending on the set of updated controllers, delays, and initial states of controllers [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. The convergence analysis is based on the properties of the matrix A representing the communication topologies of the control system.

Preliminaries

Communication topology of MAS-like systems can be represented by a directed graph (called digraph). In a digraph, each node corresponds to an agent (i.e., a controller) and each edge represents an available communication channel. Based on algebraic graph theory, matrix theory, and control theory, we discuss the convergence of asynchronous consensus protocol in networks of dynamic agents with fixed and time-varying topology. Following definitions and results are useful for solving the asynchronous consensus problem. Definition 3.3.1. Background on graph theory (see more detail in [START_REF] Ren | Information Consensus in Multivehicle Cooperative Control[END_REF][START_REF] Olfati | Consensus problems in networks of agents with switching topology and time-delays[END_REF])

The information exchange among agents in MAS is modeled by a weighted directed graph, G(C, E, A), of order n, where:

• C = {C 1 ,C 2 , . . . ,C n } is a finite non-empty set of n nodes (i.e., agents), indexed using an index set N = {1, 2, . . . , n};

• E ⊆ CxC is a set of edges (e.g., ordered pairs of nodes e i j = (C i ,C j ), i, j ∈ N, representing a unidirectional information exchange link from C i to C j );

• A = [A i j
] is an adjacency matrix defined with non-negative elements A i j ≥ 0, ∀i, j ∈ N and A i j > 0 if and only if e ji = (C j ,C i ) ∈ E;

• the neighbor set M i of a node C i ∈ C is defined by M i = {C j ∈ C | e ji = (C j ,C i ) ∈ E}.
Due to dynamic information flow among agents, the time-varying digraph, G(τ), represents the communication pattern at a time τ. All possible graphs defined for a group of n agents can be denoted by: G = {G 1 , G 2 , . . . , G m }, where: m, the cardinality of G, is a finite number. The union of a set of p graphs, {G g 1 , G g 2 , . . . , G g p }, each graph has a node set C of n elements, is a graph G comprising the same node set C and the edge set equal to the union of edge sets of p graphs. Some graph properties involved in the convergence analysis of asynchronous consensus, are defined in [START_REF] Horn | Matrix analysis[END_REF][START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF][START_REF] Olfati-Saber | Consensus and Cooperation in Multi-Agent Networked Systems[END_REF][START_REF] Ren | Information Consensus in Multivehicle Cooperative Control[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF] as follows:

• A digraph, G, is strongly connected if there is a directed path from C i to C j and also a directed path from C j to C i for any pair of distinct nodes C i ,C j ∈ C;

• A spanning tree of a digraph is a tree formed by graph edges connecting all nodes of the graph. The digraph G has a rooted directed spanning tree if and only if G has at least one node (called root node) with a directed path to all of the other nodes;

• The graph associated with a matrix

A ∈ R n×n , denoted by G A , is a digraph on n nodes C i , i ∈ N, such that there is a directed edge in G A from C j to C i if and only if A i j > 0 for any pair of nodes C i ,C j ∈ C. Definition 3.3.2.
Background on non-negative matrix theory (see more detail in [START_REF] Horn | Matrix analysis[END_REF][START_REF] Meyer | Chapter 8: Perron-Frobenius Theory of Nonnegative Matrices[END_REF]) Let R n×n (n > 1) be a set of all (n × n) real matrices and N = {1, 2, . . . , n} be an index set. A matrix A = [A i j ] ∈ R n×n is:

• non-negative, denoted by A ≥ 0, if all of its components are non-negative real numbers, that is:

A i j ∈ R, A i j ≥ 0, ∀i, j ∈ N ([ 44
], Definition 6.2.17);

• (row) stochastic if sum of elements of each row is unity, that is: ∑ n j=1 A i j = 1, ∀i ∈ N ( [START_REF] Horn | Matrix analysis[END_REF], Definition 8.7);

• irreducible or indecomposable if A cannot be conjugated into block upper triangular form by a permutation matrix ([44], Definition 6.2.22). By [START_REF] Horn | Matrix analysis[END_REF], Theorem 6.2.24, the matrix A is irreducible if and only if (I n + |A|) n-1 > 0 or its associated graph G A is strongly connected;

• primitive if A is irreducible and A has a unique nonzero eigenvalue of maximum modulus ( [START_REF] Horn | Matrix analysis[END_REF], Definition 8.5.0). By [START_REF] Horn | Matrix analysis[END_REF], Theorem 8.5.2, the matrix A is primitive if and only if A k > 0, for some natural number k ∈ N + . By [START_REF] Horn | Matrix analysis[END_REF], Corollary 8.5.8 (Wielandt), the non-negative matrix A is primitive if and only if A n 2 -2n+2 > 0. The eigenvalues of a real square matrix A are complex numbers that make up the spectrum of the matrix. The exponential growth rate of the matrix powers A k as k → +∞ is controlled by the eigenvalue of A with the largest absolute value (spectral radius ρ(A)).

The Perron-Frobenius (P-F) theorem describes the properties of the leading eigenvalue and the corresponding eigenvectors when A is a non-negative irreducible matrix as follows: Let A ∈ R n×n be a non-negative irreducible matrix with spectral radius ρ(A) = f , also called P-F eigenvalue. The following statements hold:

• f is a positive, real number, algebraically simple eigenvalue of A.

• There is a unique positive real vector r = [r i ] ∈ R n×1 such that: Ar = f r (i.e., r is right P-F eigenvector corresponding to P-F eigenvalue f ).

• There is a unique positive real vector ℓ = [l i ] ∈ R n×1 such that: ℓ T A = ℓ T f (i.e., ℓ is left P-F eigenvector corresponding to P-F eigenvalue f ).

• Cesàro averages ( [START_REF] Horn | Matrix analysis[END_REF], Theorem 8.6.1)

lim k→+∞ 1 k k ∑ i=0 A i f i = rℓ T (3.8)
where: ℓ and r are the left and right P-F eigenvectors of A corresponding to the P-F eigenvalue f (i.e., Ar = f r and ℓ T A = ℓ T f ) and are normalized so that ℓ T r = 1.

• If A has exactly p distinct eigenvalues of maximum modulus, then A is similar to e 2πik/p A for each k = 0, 1, . . . , (p -1) and the maximum-modulus eigenvalues of A are e 2πik/p f , each has algebraic multiplicity 1 ([44], Theorem 8.4.6).

Based on Theorem 3.3.1, the following results state the convergence of a primitive matrix. • A k is non-negative and primitive for every integer k ≥ 1

• lim k→+∞ ( A f ) k = rℓ T (3.9)
where the matrix rℓ T is spectral projection, called P-F projection, corresponding to P-F eigenvalue f .

lim k→+∞ A k = 1 n ℓ T (3.10)
Based on these fundamental results of the graph and matrix theories, we discuss in the next section the convergence conditions for asynchronous systems presented in Section 3.3.1.

Convergence analysis of the asynchronous information consensus

The communication topologies play a key role in consensus problems [START_REF] Liu | Leader-following consensus for discrete-time multi-agent systems with parameter uncertainties based on the event-triggered strategy[END_REF]. This section explores the communication topologies to derive appropriate conditions for the convergence of an asynchronous consensus process. Considering an asynchronous consensus protocol in the information exchange among controllers may break down the convergence properties obtained with the synchronous communication [START_REF] Lee | Convergence Analysis of Asynchronous Consensus in Discrete-time Multi-agent Systems with Fixed Topology[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. By introducing the Assumptions 3.3.2, we demonstrate that the asynchronous consensus protocol guarantees the stability property (i.e., the convergence of information states). The results about the consensus problems of synchronous and asynchronous systems may be found in the literature (e.g., [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF][START_REF] Ren | Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues[END_REF][START_REF] Olfati-Saber | Consensus and Cooperation in Multi-Agent Networked Systems[END_REF][START_REF] Ren | Consensus seeking in multiagent systems under dynamically changing interaction topologies[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF][START_REF] Ren | Consensus of information under dynamically changing interaction topologies[END_REF][START_REF] Strikwerda | A convergence theorem for chaotic asynchronous relaxation[END_REF]). These results are related to the distributed coordination, distributed optimization, formation control, cooperative control and consensus problems of MAS-like systems (e.g., multiple mobile robotic systems, unmanned aerial vehicles, automated highway systems, satellite clusters) in which all agents reach a consensus asymptotically when their states converge to a common value. As stated in Section 3.3.1, we are henceforth interested in the communication topologies represented by non-negative stochastic matrices.

For the stability of asynchronous systems, the following results are fundamental to give a direction for the convergence analysis of asynchronous consensus process. Lemma 3.3.3. Fixed point of asynchronous system ([28], Lemma 1, [START_REF] Su | Further results on convergence of asynchronous linear iterations[END_REF], Theorem 1 and [START_REF] Lubachevsky | A Chaotic Asynchronous Algorithm for Computing the Fixed Point of a Nonnegative Matrix of Unit Spectral Radius[END_REF]).

Let the communication topology is fixed for the asynchronous system represented by Eq(s). (3.6), that is: A(τ) = [A i j (τ)] = A, ∀τ. The fixed point of asynchronous system Eq(s). (3.6) is asymptotically stable under the class of regular asynchronism (Assumptions 3.3.2) if the spectral radius, ρ(A), of non-negative matrix A = [A i j ], is less than unity.

• The graph G A associated with A, has a spanning tree, if and only if 1 is the unique eigenvalue of A with maximum modulus.

The results in Corollary 3.3.2 summarize the conditions for the convergence of asynchronous systems based on the communication topology represented by an adjacency matrix. We distinguish two cases of the communication topology.

The consensus of asynchronous one-leader-multiple-followers systems under fixed communication topology

We consider now the first case in which the topology is fixed, and consequently, the matrix

A(τ) = [A i j (τ)] = A, ∀τ.
Based on the convergence results of synchronous systems [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF][START_REF] Ren | Information Consensus in Multivehicle Cooperative Control[END_REF][START_REF] Olfati-Saber | Consensus and Cooperation in Multi-Agent Networked Systems[END_REF][START_REF] Ren | Consensus of information under dynamically changing interaction topologies[END_REF], and Corollary 3.3.2, we derive the conditions for the convergence of asynchronous consensus process as follows:

Theorem 3.3.2. The consensus of asynchronous systems under fixed communication topology ([28], Theorem 2, [START_REF] Su | Further results on convergence of asynchronous linear iterations[END_REF], Theorem 11,[START_REF] Pott | On the convergence of asynchronous iteration methods for nonlinear paracontractions and consistent linear systems[END_REF], Section 4). For the asynchronous system Eq(s). (3.6) with fixed communication topology represented by a non-negative stochastic matrix A and under the Assumptions 3.3.2, the global consensus Eq(s). (3.7) is asymptotically reachable if and only if the graph G A associated to A, has a unique spanning tree rooted by the specified root node.

Proof. (⇐) Based on Lemma 3.3.4 and the Assumptions 3.3.2, demonstrating that the asynchronous system Eq(s). (3.6) can achieve global consensus asymptotically, is equivalent to show that:

•

χ(τ) → C1 n , as τ → τ +∞ • C = χ i i (τ 0 ) = χ i i (t a ) Indeed,
A is a non-negative stochastic matrix with positive diagonal elements, the Corollary 3.3.2 shows that the matrix A is NSP and lim p→+∞ A p = 1 n l T , where: 1 n and ℓ are the right and left eigenvectors corresponding to the eigenvalue 1. By Lemma 3.3.4 and the Assumptions 3.3.2, the information states converge χ(τ) → C1 n , as τ → τ +∞ . In other words, the asynchronous system Eq(s). (3.6) asymptotically achieves the global consensus. The graph G A associated with A, has a unique spanning tree rooted by the specified root node (i.e., C i in this case), so the information states of all nodes asymptotically converge to the initial state of the root node, that is:

C = χ i i (τ 0 ) = χ i i (t a
). (⇒) When the asynchronous system Eq(s). (3.6) asymptotically achieves the global consensus, we suppose that the graph associated with the matrix A does not have a spanning tree. By Corollary 3.3.2, the algebraic multiplicity of eigenvalue 1 of A is m 1 > 1. The Perron-Frobenius theorem (see Theorem 3.3.1) ensures that A p is similar to e 2πik/m 1 A and trA p = e 2πik/m 1 trA p . Since e 2πik/m 1 is real and positive only if k is an integer multiple of m 1 , this is impossible if A has any positive main diagonal entry and k is not divisible by m 1 . Therefore, the graph associated with the matrix A has a spanning tree. If the spanning tree is not unique and rooted by the specified root node, the information state χ(τ) converges to a consensus point, depending on the computation (i.e., depending on initial states of controllers, the delays, and update sets). A counterexample is given in [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF], Example 1.

With regards to the synchronous case, the consensus points of the asynchronous system are normally different depending on the computation (i.e., depending on the set of updated controllers, delays, and initial states) [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. Especially, information states of all the nodes asymptotically converge to the initial state of the root node when the interaction graph has a unique rooted spanning tree.

The consensus of asynchronous one-leader-multiple-followers system under timevarying communication topologies

The following theorem provides the convergence results in the second case where the communication topologies are time-varying. The topologies change in a control system while communication links between controllers may be unreliable due to disturbances, loss and/or subject to communication range limitations. Consider a dynamic graph G(τ) ∈ G, representing the information exchange among n controllers at the time τ. The time-varying weight A i j (τ) of each edge belongs to a finite set of arbitrary positive numbers b. The asynchronous discrete-time consensus protocol defined in Eq(s). (3.6) achieves the convergence (Eq(s). (3.7)) asymptotically if and only if there exists an infinite sequence of uniformly bounded, non-overlapping time intervals [τ u , τ u+1 ), u ≥ 0, starting at τ 0 = 0, and across each such interval, the union of interaction graphs has a spanning tree rooted at the specified root node.

Proof. The Theorem 3.3.3 is a particular case of [START_REF] Fang | Cooperative Control of Multi-Agent Systems with Information Flow Contraints[END_REF], Theorem 2.3, [START_REF] Moreau | Stability of multiagent systems with time-dependent communication links[END_REF], Proposition 1, or [START_REF] Ren | Consensus of information under dynamically changing interaction topologies[END_REF], Theorem 3.3 with one spanning tree routed by the specified root node. Different proof techniques are proposed in these works. See more detail in [START_REF] Fang | Cooperative Control of Multi-Agent Systems with Information Flow Contraints[END_REF], Theorem 2.3

In brief, the Theorem 3.3.3 shows that asynchronous information consensus under timevarying communication topologies can be asymptotically achieved if the union of directed interaction graphs has a spanning tree frequently enough in the evolution of consensus process. From the asynchronous point of view, the same synchronous system under the time-varying topologies may be mapped to different asynchronous systems with different spanning trees [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. Therefore, the consensus point depends on the sequence of the spanning trees in the course of the computations (e.g., as stated in Lemma 3.3.4, the constant C is computationally dependent).

Generalized consensus problems in a leader-switching system

In this section, we study the consensus problems of discrete-time leader-switching systems with fixed topologies representing a distributed control system. The notions of "leader" and "follower" roles are extensively used in broad applications related to MAS, such as leader-follower control, leader-following consensus, leader-follower synchronization and leaderless coordination [START_REF] Zhao | Leader-follower optimal coordination tracking control for multi-agent systems with unknown internal states[END_REF][START_REF] Liu | Leader-following consensus for discrete-time multi-agent systems with parameter uncertainties based on the event-triggered strategy[END_REF][START_REF] Shang | Leader-Follower Fixed-Time Group Consensus Control of Multiagent Systems under Directed Topology[END_REF][START_REF] Abdessameud | Leader-Follower Synchronization of Euler-Lagrange Systems With Time-Varying Leader Trajectory and Constrained Discrete-Time Communication[END_REF]. In the asynchronous control system, a controller can play two roles alternatively, that is a leader or a follower, or both simultaneously in different consensus processes. The actual leaders are the controllers that determine the ultimate information state of the control system. For the structural decomposition of MAS-like systems, [START_REF] Xiao | Consensus problems in discrete-time multiagent systems with fixed topology[END_REF] proposed leaders-followers configurations and established the necessary and sufficient conditions for an agent to be a leader and for the convergence of consensus problems. The global system performance is commonly related to the interactions among controllers ( [START_REF] Olfati | Consensus problems in networks of agents with switching topology and time-delays[END_REF], Theorem 8).

In previous sections, we have proposed an asynchronous consensus protocol Eq(s). (3.4) and established the necessary and sufficient conditions for this protocol to solve a consensus problem (Theorem 3.3.2) in the single-leader-multiple-follower configuration [START_REF] Xiao | Consensus problems in discrete-time multiagent systems with fixed topology[END_REF]. The convergence results are based on the communication topologies considered as standard topologies. To accomplish the updating of the information vector of a controller defined in Section 3.2.2, different consensus processes are involved in the control system characterized as a leaderswitching system. Therefore, we need the global topology so that different consensus processes of the whole control system converge (evidently, to different consensus points). Based on common characteristics of these consensus processes and the combination of standard topologies, the global topology can be designed for the convergence of all consensus processes. Consider a control system of n controllers, C = {C 1 ,C 2 , . . . ,C i , . . . ,C n }, indexed using an index set N = {1, 2, . . . , n}. The information state of a controller C i , i ∈ N, is defined as:

χ i = [χ i 1 χ i 2 . . . χ i n ]
T . All controllers share a common state space χ i ∈ R n×1 . An elementary consensus problem is described as follows:

Definition 3.3.3. Elementary consensus problem P i Let t 0 < t 1 < • • • < t a < .
. . be event-based discrete-time instants. Assuming that, at time instant t a , a controller C i applies a control action which changes its output: χ i i (t a ), and so the information state: χ i (τ). The controller C i becomes the actual leader and starts the consensus process applied on other controllers (i.e., followers) so that the change is updated in their information states. The updating instants for asynchronous information exchange among controllers are denoted by τ 0 < τ 1 < • • • < τ u < . . . . By accomplishing the consensus process, the consensus vector defined as:

χ(τ) = [χ 1 (τ) χ 2 (τ) . . . χ n (τ)] T (3.11)
converges to the same value χ i (τ 0 ) = χ i i (t a ) for all elements. The elementary consensus problem (denoted by P i ), is solved by the asynchronous protocol (denoted by F i ), defined by Eq(s). (3.4). The conditions for the convergence of asynchronous consensus protocol, F i , are related to the standard topology (denoted by T i ) or the corresponding adjacency matrix (denoted by A i ). As stated in Theorem 3.3.2, the convergence conditions are one of following statements:

• Graph associated with A i (denoted by G i ) has a unique spanning tree rooted by the root node C i .

• Or, the non-negative stochastic matrix A i is primitive.

3.3 Asynchronous information consensus in distributed control of irrigation channels | 75

• Or, by Corollary 3.3.2, lim k→+∞ A k i = 1 n ℓ T i , where: k ∈ N + , 1 n and ℓ i are right and left positive P-F eigenvectors corresponding to P-F eigenvalue 1, that is: ℓ i > 0, ℓ T i A i = ℓ T i , and are normalized such that: ℓ T i 1 n = 1. We introduce the concept of consensus function for the consensus problems of discrete-time leader-switching systems with fixed topologies. Definition 3.3.4. Consensus function F [START_REF] Xiao | Consensus problems in discrete-time multiagent systems with fixed topology[END_REF] and F-consensus problem [START_REF] Xiao | Consensus problems in discrete-time multiagent systems with fixed topology[END_REF][START_REF] Olfati-Saber | Consensus and Cooperation in Multi-Agent Networked Systems[END_REF].

A consensus function, F, corresponding to the consensus protocol F, that solves a consensus problem P (also called F-consensus problem), can be expressed by:

F : R n×1 -→ R F(χ(τ)) -→ C, (3.12)
where:

C = χ i (τ 0 ) is the initial state of controller C.
For simplicity, in the case of an asynchronous system with the complete updating set S(τ) = {1, . . . , n}, ∀τ and zero delays (see the mapping between asynchronous and synchronous systems in [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]), the Eq(s). (3.6) can be presented by:

χ(τ u+1 ) = Aχ(τ u ) χ(τ u+1 ) = A k χ(τ 0 ) (3.13)
Based on Corollary 3.3.2 and Lemma 3.3.4, lim k→+∞ A k = 1 n ℓ T and χ(τ u ) → C1 n , as τ → τ +∞ , the consensus function, F, is defined for this specific asynchronous system as:

F(χ(τ)) = ℓ T χ(τ) (3.14)
where: 1 n and ℓ are right and left positive P-F eigenvectors corresponding to P-F eigenvalue 1, that is: ℓ T A = ℓ T , and are normalized such that: ℓ T 1 n = 1. For an asynchronous system mapped into a synchronous system, the consensus process can be illustrated using the consensus function as follows:

F(χ(τ 1 )) = F(Aχ(τ 0 )) = F(χ(τ 0 )) F(χ(τ 2 )) = F(Aχ(τ 1 )) = F(χ(τ 0 )) F(χ(τ 3 )) = F(Aχ(τ 2 )) = F(χ(τ 0 )) . . . . . . F(χ(τ u+1 )) = F(Aχ(τ u )) = F(χ(τ 0 )) (3.15)
For the complete control system, n standard topologies (or single-leader-multi-follower configuration), {T 1 , . . . , T i , . . . , T n }, represented by adjacency matrices {A 1 , . . . , A i , . . . , A n }, are involved.

Assumptions 3.3.3. Elementary F i -consensus problem

Assuming that all standard topologies, T i , i ∈ N, represented by NSP matrices,

A i , i ∈ N, consensus functions F i , , i ∈ N solve F i -consensus problem. That is: ∀i ∈ N, lim k→+∞ A k i = 1 n ℓ T i (3.16)
where: k ∈ N + , 1 n and ℓ i are right and left positive P-F eigenvectors corresponding to P-F eigenvalue 1, that is:

ℓ i > 0, ℓ T i A i = ℓ T i ,
and are normalized such that:

ℓ T i 1 n = 1.
A priori, the following convex combination of these standard topologies [START_REF] Xiao | Consensus problems in discrete-time multiagent systems with fixed topology[END_REF] can create a solution that solves the consensus problems for a class of "leader-switching" systems:

χ 1 (τ u+1 ) = A 1 χ 1 (τ u ) χ 2 (τ u+1 ) = A 2 χ 2 (τ u )
. . .

χ n (τ u+1 ) = A n χ n (τ u )            =⇒ χ(τ u+1 ) = Aχ(τ u ) (3.17)
Let

e i = [0 . . . 1 (row i) . . . 0] T ∈ R n×1 and α i ∈ [0, 1], such that: ∑ n i=1 α i = 1. Proposition 3.3.1. Consensus function F for a F-consensus problem.
Based on the matrix A defined as follows:

A = n ∑ i=1 α i A i (3.18)
the consensus protocol Eq(s). (3.17) can solve the global consensus problem of leader-switching system (called F-consensus problem) using the following consensus function:

F(χ(τ)) = ℓ T χ(τ) (3.19)
where: ℓ = ∑ n i=1 α i ℓ i .

Proof. Obviously, the matrix A defined in Proposition 3.3.1, has following properties:

• non-negative: ∀i ∈ N, A i ≥ 0, and

α i ∈ [0, 1] ⇒ A = ∑ n i=1 α i A i ≥ 0 • stochastic: ∀i ∈ N, A i 1 n = 1 n and ∑ n i=1 α i = 1, α i ∈ [0, 1] ⇒ A1 n = (∑ n i=1 α i A i )1 n = 1 n • lim k→+∞ A k = 1 n ℓ T : By Eq(s). (3.16): ∀i ∈ N, lim k→+∞ A k i = 1 n ℓ T i , ℓ T i 1 n = 1 and α i ∈ [0, 1]
, we obtain:

lim k→+∞ A k = lim k→+∞ n ∑ i=1 (α i A i ) k = n ∑ i=1 α i 1 n ℓ T i = n ∑ i=1 1 n α i ℓ T i = 1 n ℓ
T and:

ℓ T A = ( n ∑ i=1 α i ℓ i ) T ( n ∑ i=1 α i A i ) = n ∑ i=1, j=1 α i α j ℓ T i A i 3.3 Asynchronous information consensus in distributed control of irrigation channels | 77 = n ∑ i=1 α 2 i ℓ T i + n ∑ i=1, j=1,i̸ = j α i α j ℓ T i A j = n ∑ i=1 (α 2 i ℓ T i + α i ℓ T i n ∑ i̸ = j α j ) = n ∑ i=1 (α 2 i ℓ T i + (α i -α 2 i )ℓ T i ) = n ∑ i=1 α i ℓ i = ℓ
T Therefore, the consensus protocol defined by Eq(s). (3.17) solves the F-consensus problem.

Remark 3.3.2. By selecting α i , i ∈ N appropriately, the proposed protocol Eq(s). (3.17) can solve any F-consensus problem. However, due to the complexity of a considered system, not all protocols that solve a consensus problem can be expressed by the form Eq(s). (3.17).

Example 3.3.1. An example of a simple leader-switching system.

• A standard topology represented by the following matrix A i :

A i =            (1 -α i1 ) 0 . . . α i1 . . . 0 0 (1 -α i2
. . . (1 -α in )            ∈ R n×n (3.20)
where: α i j ∈ (0, 1], ∀i, j ∈ N = {1, 2, . . . , n}, such that: α i j = α ji .

• Consensus protocol: χ i (τ u+1 ) = A i χ i (τ u ), i ∈ N solves F i -consensus problem.

• Consensus function: F i (χ) = ℓ T i χ, where: ℓ i = [0 0 . . . 1 row i . . . 0] T and ℓ T i A i = ℓ T i . The global topology for a leader-switching system is given in summary form as follows:

A =       (1 -∑ i̸ =1 α i α i1 ) α 2 α 21 . . . α n α n1 α 1 α 12 (1 -∑ i̸ =2 α i α i2 ) . . . α n α n2 . . . . . . . . . . . . α 1 α 1n α 2 α 2n . . . (1 -∑ i̸ =n α i α in )       (3.21)
where: α i ∈ [0, 1], α i j ∈ (0, 1], ∀i, j ∈ N, such that: ∑ n i=1 α i = 1. By Proposition 3.3.1, the protocol Eq(s). (3.17) solves the F-consensus problem with the consensus function Eq(s). (3.19). An example of the global topology for the system with five controllers is given in Fig. 3.7.

In this section, we have discussed the feasibility and the stability of asynchronous consensus protocol in achieving the convergence of information states of controllers. Based on the stability theory of asynchronous system and proposed communication assumptions, we have established the necessary and sufficient conditions for the information consensus to be reached asymptotically using the asynchronous protocol defined in Section 3.3.1. The results are obtained for an asynchronous system under the fixed and time-varying communication topologies and bounded time-varying communication delays. The consensus points of information states are different depending on the initial state of the actual root node. Furthermore, the consensus protocol added to control algorithm will be demonstrated to still guarantee the control performance, but with robustness to changes in exchange topology due to link/node failures and time delays (see Section 4.2.3).

Simulation results

We have performed simulations of the consensus problems for one controller (when it becomes a leader) of one-leader DMPC control system to show the convergence of designed asynchronous consensus protocol such that the information states of all controllers update the newest value from the leader. Simulations of the leader-switching system, when different consensus processes are executed in the control system, are also performed to illustrate the possibility of switching the leader using the parameterizable global topologies.

Based on the example shown in Fig. 3.6, we consider a control system of n = 4 controllers, C = {C 1 ,C 2 ,C 3 ,C 4 }, indexed using an index set N = {1, 2, 3, 4}. The communication topologies are constructed from the Eq(s). (3.21), where: if C i is a root, α i = α r = 0.9, otherwise,

α i = (1 -α r )/n, α i j = 1/n, ∀i, j ∈ N, such that: ∑ n i=1 α i = 1. The information state of controller C i , i ∈ N, is defined as: χ i = [χ i 1 χ i 2 χ i 3 χ i 4 ]
T . All controllers share a common state space: χ i ∈ R 4×1 , ∀i ∈ N. The initial information states of all controllers (except the leader) are assumed to be random values between 0 and 5 (m 3 /s). Let t 0 < t 1 < • • • < t a < . . . be event-based discrete-time instants when a controller applies an action provoking a change in information states of controllers. Assuming that, at time instant t i a , the controller C i applies the control action which changes the information state: χ i (t i a ). The controller C i becomes the actual leader and starts the consensus process executed on other controllers (i.e., followers) so that the change is updated in their information states. The sampling time in simulations is chosen as: dt = 0.05 (s). The updating instants of the controller C i are assumed to be every τ = 2 * i * dt (s) in order to illustrate the asynchronism. By Proposition 3.3.1, the protocol Eq(s). (3.17) solves the consensus problem of the system characterized by the matrix Eq(s). (3.21), in other words, the consensus vector defined as: χ(τ) = [χ 1 (τ) χ 2 (τ) χ 3 (τ) χ 4 (τ)] T converges to the same value χ i (τ 0 ) = χ i i (t i a ) for all elements. The simulation parameters are summarized in Table 3.1. The simulations were done in Matlab R2016b on a computer Intel®Core™i5-4310U CPU 2.0GHz. Convergence analysis of asynchronous consensus Fig. 3.9 Simulation results of asynchronous consensus protocol used in the one-leader control system with 10 controllers.

The first simulation scenario uses the example presented in Fig. 3.6. In this scenario, the controller C 2 applies a computed action u 2 to the subsystem, which changes the value of its information state χ 2 . C 2 actually becomes the leader. The consensus process is started by C 2 for providing the convergence of information states of all controllers. The simulation results are shown in Fig. 3.8. We see that the information states of four controllers, which start from different initial values, can reach asymptotically a common value after 6 (s). The updating instants are chosen to be different for every controller. For more complex control system with 10 controllers using the designed asynchronous consensus protocol, the simulation results presented in Fig. 3.9, show the convergence of their information states, which is achieved after 40 (s).

The second scenario explores the first scenario for testing the losses of direct communication. Assuming that the controller C 4 loses the communication in two directions with the controller C 1 at τ = 2 (s). Through simulation, the Fig. 3.10 shows a divergence of the information states over the simulation time T sim = 10 (s). In the other hand, in the case the controller C 4 loses Fig. 3.12 Simulation results of the asynchronous consensus protocol used in the leader-switching system. Four controllers C i are involved. As an example, the initial state of the leader C i is χ i = i.

The controller C i is assumed to apply a control action at time instant (i -1) * T sim/4. For τ between (i -1) * T sim/4 and i * T sim/4, the controller C i becomes the leader which updates all elements χ i of all controllers.

the communication in two directions with the controller C 3 at τ = 2 (s), the convergence of the information states still is achieved. Indeed, the loss of direct communication between C 4 and C 3 infers the elements α 34 = α 43 = 0 in the matrix Eq(s). (3.21). However, the adjacency matrix still has a unique eigenvalue with maximum modulus ρ(A) = 1. By Corollary 3.3.2, the controllers can reach the consensus asymptotically. These examples demonstrate that the change in topology of control system can cause a divergence or keep the convergence property.

In terms of communication aspects, these results help the control system design for considering the robustness of link/node failures.

The third simulation scenario analyzes the convergence of different consensus processes with different actual leaders in a control system. In this scheme, four controllers are involved in four different consensus processes. Assuming that at time instant t i a = (i -1) * T sim/4, the controller C i applies the control action which changes the information state: χ i (t i a ). The simulation results shown in Fig. 3.12 demonstrate that the system using the global communication topology can switch the leader by changing the values of α i , i ∈ N. The consensus is asymptotically reached with the random initial values of the information states and the controllers use the asynchronous consensus protocol.

Conclusion

The coordinated controllers in distributed control encounter some challenging problems in the presence of time-varying topology and communication issues such as loss of synchronization or the updating problem of interaction variables and coordination variables. We have approached these problems by formulating them as an asynchronous information consensus problem. In this chapter, we have determined the characteristics and properties of information exchange topologies for solving the asynchronous consensus problem with communication constraints such as communication delays. Under certain communication assumptions, the analytical demonstration shows that information consensus of asynchronous systems under fixed communication topologies can be achieved asymptotically if the associated graph has a spanning tree. For the systems with time-varying communication topologies, the asynchronous information consensus is asymptotically achieved if the union of directed interaction graphs across some intervals has a spanning tree frequently enough during the evolution of consensus processes. Based on these conditions, the asynchronous consensus protocol is then designed to achieve the convergence of information states of controllers. Integrating the consensus protocol in DMPC scheme results in the most updated values of interaction variables and coordination variables that the controllers can use for the action computation. Therefore, the computed actions are more appropriate to apply to subsystems than other cases with outdated values of coordination variables. The implementation of a consensus-based control algorithm in distributed controllers improves the control performance when compared to decentralized control. In addition, when the synchronization among controllers in distributed control strategy is not guaranteed, this asynchronous framework can be used to provide the control performance close to those of the synchronous case. The consensus algorithm has been applied to the example of a distributed control of irrigation channels modeled by using the Lattice Boltzmann method. The simulation results demonstrate the effectiveness and usefulness of the asynchronous consensus approach to distributed control.

Chapter 4 Distributed Model Predictive Control of irrigation channels using cooperative controllers

This chapter investigates the regulatory control of large-scale irrigation channels under a Distributed Cooperative Model Predictive Control (DCMPC) framework. Operationally, the regulatory control involves the regulation of water levels and/or flow rates at some points in the channel by manipulating available hydraulic structures (e.g., submerged gates, movable weirs, etc.) in order to satisfy real-time performance requirements. For this purpose, we firstly provide a discrete-time linearized model of the large-scale system suitable for adequately capturing its dynamics and the interactions among subsystems using the Lattice Boltzmann (LB) method. Next, we discuss different control implementation strategies (i.e., centralized, decentralized and distributed strategies) and how the cooperative communication among local controllers can be included to improve the performance of the overall system. For simulations, a particular benchmark of an irrigation channel is considered. Comparisons through simulations among the proposed control approaches validate the benefits of the distributed cooperative control approach. The rest of this chapter is organized as follows. Section 4.1 explores the methods and tools needed to address this problem. Section 4.2 introduces the distributed cooperative control approach for irrigation channels. Section 4.3 provides the simulation results under different scenarios, details comparison results and discussions. Finally, Section 4.4 draws with some conclusions and presents future perspectives.

Model predictive control of irrigation channels

Our approach to control irrigation channels is based on the optimization of an objective function to compute the best actions and the dynamic models to predict the system behaviors. The DMPC method has its roots in such optimal control while explicitly taking into account the constraints and the predictions [START_REF] Borrelli | Predictive Control for linear and hybrid systems[END_REF][START_REF] Rawlings | Model Predictive Control : Theory and Design[END_REF]. MPC is a well-known technique for handling control and state constraints while offering good performance specifications (the basic notions about MPC are presented in [START_REF] Rawlings | Model Predictive Control : Theory and Design[END_REF]). In principle, MPC allows a uniformity, that is, the same technique can be successfully applied for a wide range of control problems [START_REF] Borrelli | Predictive Control for linear and hybrid systems[END_REF]. The delay compensation is also one interesting characteristic that makes MPC particularly suitable for delay-varying systems such as irrigation channels. There are many works which employ different control strategies within an MPC framework for the control of irrigation channels (as surveyed in [START_REF] Horvath | New offset-free method for model predictive control of open channels[END_REF]). For example, [START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF] presents a methodology for the optimal management of combined water supply and navigability/sustainability in river systems based on the MPC. [START_REF] Van Overloop | Human-in-the-Loop Model Predictive Control of an Irrigation Canal[END_REF] presents a decentralized MPC scheme for large-scale systems whose components can exchange information through a network. [START_REF] Imam Sutrisno | Distributed model predictive control and application to irrigation canal[END_REF] proposes a scheme to determine the weighting values that are proportional to the control load for each subsystem. This weighting scheme is then applied to control the irrigation channel using Feasible-Cooperation MPC and Nash-bargaining MPC. [START_REF] Igreja | Application of distributed model predictive control to a water delivery canal[END_REF] proposes a distributed MPC algorithm for a water delivery channel using linearized and discretized Saint-Venant model. A methodology for the optimal management of a combined irrigation and water supply system based on MPC is proposed in [START_REF] Puig | Model predictive control for combined water supply and navigability / sustainability in river systems[END_REF][START_REF] Puig | Model predictive control of combined irrigation and water supply systems: Application to the Guadiana river[END_REF]. [START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF] focuses on distributed MPC strategy in which a discrete-time linear integral delay model is used for prediction. Their distributed control algorithm is based on the augmented Lagrangian duality method. [START_REF] Van Overloop | Human-in-the-Loop Model Predictive Control of an Irrigation Canal[END_REF][START_REF] Van Overloop | Model Predictive Control on Open Water Systems[END_REF] address a MPC methodology configured for water quantity control on open water systems modeled by the classical Integrator Delay model and the Saint-Venant model. [START_REF] Camponogara | Distributed model predictive control[END_REF] proposed a scheme to achieve some degree of coordination among agents (i.e., controllers) that solve MPC problems with locally relevant variables, costs, and constraints, but without solving a centralized MPC problem. In this scheme [START_REF] Camponogara | Distributed model predictive control[END_REF], agents within each neighborhood work sequentially. Each agent uses the same interior-point method (i.e., barrier method [START_REF] Borrelli | Predictive Control for linear and hybrid systems[END_REF]) with the same Lagrange multipliers to generate its iterations. The predictive control theory is also used in [START_REF] Sawadogo | Decentralized Predictive Controller for Delivery Canals[END_REF] to formulate a set of local controllers for the automatic operation of gates of each reach. The distributed control scheme presented in the work [START_REF] Sawadogo | Decentralized Predictive Controller for Delivery Canals[END_REF] takes into account explicitly the interactions among subsystems using Saint-Venant models. In MPC, the computational complexity and theoretical properties (e.g., feasibility, stability and robustness) depend closely on the choice of the prediction model, the control objectives and the constraints [START_REF] Rawlings | Model Predictive Control : Theory and Design[END_REF].

One of the Arrowhead project demonstrators (focusing on the automation of Bourne channel (BC) irrigation network in Drôme -France) introduces the distributed control of a large-scale irrigation system as a case study (for more details, see [START_REF] Nguyen | Optimal reactive control of hybrid architectures : A case study on complex water transportation systems[END_REF]). The structural diagram of BC irrigation system is shown in Fig. 1.2. This system is equipped with only local controllers developed to guarantee the water level at some points within a specified range by adjusting control structures such as submerged gates or mixed gates and spillway. Their action decisions are only based on local information (e.g., measured water level at gate location), without taking into account the potential coordination/cooperation with other controllers. To determine the acceptable ranges, a centralized coordinator is involved and proceeds with a course of actions: collecting measurements from the whole irrigation channel, optimizing, choosing values in a pre-defined way and pushing reference values to local controllers. Controlling a such system in a centralized way [START_REF] Montazar | Centralized Downstream PI controllers for the west canal of Aghili irrigation district[END_REF] would give rise to a large computational charge and set up a single point of failure. The important delays may occur in information gathering, from level sensors to the centralized coordinator, due to communication issues. The decentralized approach [START_REF] Elmahdi | Decentralized Control Framework and Stability Analysis for Networked Control Systems[END_REF][START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF] will scale better for the large-scale system. In the decentralized scheme, the objective of each controller is to locally determine the actions that optimize the behavior of the subsystem (i.e., by minimizing the specified local cost function). The drawback of decentralized control is that these local control actions are computed regardless of the effects on the overall system performance. The controllers do not consider a cooperation, a coordination or a negotiation among them. As a consequence, the global performance may be degraded due to the unexpected and unanticipated interactions among the subsystems. The distributed approach [START_REF] Panagiotis | Distributed model predictive control: A tutorial review and future research directions[END_REF][START_REF] Tian | Distributed model predictive control for multi-objective water system management[END_REF][START_REF] Filipe | Distributed LQG control of a water delivery canal[END_REF][START_REF] Sanchez | Distributed Control of Irrigation Canals[END_REF][START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF] combines the optimality of centralized approach and the less complexity of decentralized approach to provide a good compromise.

Motivated by the above-mentioned observations, the aims of this chapter is to implement and compare realistic and flexible control schemes in which:

• costs, constraints, profiles are taken into account in decentralized and distributed constrained optimization problems via an MPC design;

• the cooperation among local MPC controllers can be included in order to improve the performance of the overall system. Next, we employ the LB method for providing a prediction model of the irrigation channel suitable for capturing its dynamics and the interactions among the subsystems.

Prediction model

As already presented in Section 1.3.1, the modeling of a dynamic flow with D1Q3 LB method in the presence of a simple external force F, results in the Eq(s). (1.13), which are rewritten as follows:

f 1 (l,t + ∆t) = f 1 (l,t) + 1 τ ( f e 1 -f 1 (l,t)) f 2 (l,t + ∆t) = f 2 (l -v∆t,t) + 1 τ ( f e 2 -f 2 (l,t)) -∆t 2v F f 3 (l,t + ∆t) = f 3 (l + v∆t,t) + 1 τ ( f e 3 -f 3 (l,t)) + ∆t 2v F (4.1)
The macroscopic variables h and u are determined by Eq(s). (1.18) and rewritten as follows:

h = 3 ∑ i=1 f i = f 1 + f 2 + f 3 q = hu = 3 ∑ i=1 v i f i = v( f 2 -f 3 ) (4.2)
In distributed scheme, we also consider the coupling of two reaches modeled by D1Q3 LB method with a gate in a submerged regime. The flow rate through gate Q g depends on the difference between the water heights at upstream h and downstream h ′ around the gate, as expressed in Eq(s). (1.23):

Q g = B g αθ 2g(h -h ′ ) (4.3)
where: B g is the gate width, α is the gate coefficient and θ is the gate opening. In addition, the outflow Q of upstream reach is equal to the inflow Q ′ of downstream reach, that is:

Q = Q ′ = Q g .
The manipulated variable θ is deduced from following gate equations:

vB( f 2 -f 3 ) = vB ′ ( f ′ 2 -f ′ 3 ) = B g αθ 2g(h -h ′ ) (4.4) 4.1.1.

Linearization of a nonlinear LB model

For use in MPC method, we need a good model that is both descriptive enough to capture the most significant dynamics of the real irrigation channel and simple enough for solving the optimization problem [START_REF] Puig | Model predictive control for combined water supply and navigability / sustainability in river systems[END_REF][START_REF] Van Overloop | Human-in-the-Loop Model Predictive Control of an Irrigation Canal[END_REF][START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF][START_REF] Rawlings | Model Predictive Control : Theory and Design[END_REF]. The linearized D1Q3 LB model offers a good compromise. For further use, we consider the linearization of the LB dynamics (Eq(s). (4.1)) around an equilibrium height, h = h 0 , and an equilibrium velocity, u = u 0 , which can deduce an equilibrium flow rate: Q 0 = Bh 0 u 0 . The small derivations ε i , i = {1, 2, 3}, around the equilibrium points are expressed by: ε i = f if e i (h 0 , u 0 ), where: f i , i = {1, 2, 3}, are the solution of the Eq(s). (4.1).

l 1 l 2 l 3 l N-1 l N ✲✛ ✲✛ f 2 (l 1 ) f 3 (l 1 ) f 1 (l 1 ) f 1 (l N ) f 2 (l N ) f 3 (l N ) Fig. 4.1
The discretized points of a reach in D1Q3 LB model.

For a reach of length L (corresponding to N discretized points l j , j = 1, . . . , N), the state-space representation of discrete-time system described by Eq(s). (1.14) is rewritten as follows:

   f 2 (l j ,t + ∆t) f 1 (l j ,t + ∆t) f 3 (l j ,t + ∆t)    =    f 2 (l j-1 ,t) f 1 (l,t) f 3 (l j+1 ,t)    + 1 τ    f e 2 (l j-1 ,t) -f 2 (l j-1 ,t) f e 1 (l j ,t) -f 1 (l j ,t) f e 3 (l j+1 ,t) -f 3 (l j+1 ,t)    + ∆t 2v    F 0 -F    (4.5)
The simple force term, F, is calculated by Eq(s). (1.3):

F = gh   I - n 2 u 2 Bh B+2h 4/3   (4.6)
The equilibrium density distributions, f e i , follow the conditions of mass and momentum conservation such that:

∑ i f i = h = h 0 + ∑ i ε i , ∑ i v i f i = hu = h 0 u 0 + (ε 2 -ε 3 )v, ∑ i v 2 i f i = 1 2 gh 2 + hu 2 = 1 2 g(h 0 + ∑ i ε i ) 2 + (h 0 u 2 0 -u 2 0 ∑ i ε i + 2(ε 2 -ε 3 )vu 0 ). (4.7)
4.1 Model predictive control of irrigation channels | 87

These condition gives the following results for f e i (h, u):

f e 1 (h, u) = f e 1 (h 0 , u 0 ) + 1 - gh 0 v 2 + u 2 0 v 2 ∑ i ε i - 2u 0 v (ε 2 -ε 3 ), f e 2 (h, u) = f e 2 (h 0 , u 0 ) + 1 2 ( gh 0 v 2 - u 2 0 v 2 ∑ i ε i + 1 2 + u 0 v (ε 2 -ε 3 ) f e 3 (h, u) = f e 3 (h 0 , u 0 ) + 1 2 gh 0 v 2 - u 2 0 v 2 ∑ i ε i - 1 2 - u 0 v (ε 2 -ε 3 ) (4.8)
By combining Eq(s). (4.5) and Eq(s). (4.8), the linearization of the Eq(s). (4.5) provides the following equations (as detailed in [START_REF] Van Thang Pham | Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network[END_REF]):

   ε 2 (l j+1 ,t + ∆t) ε 1 (l j ,t + ∆t) ε 3 (l j-1 ,t + ∆t)    = M    ε 2 (l j ,t) ε 1 (l j ,t) ε 3 (l j ,t)    + N    ε 2 (l j ,t) ε 1 (l j ,t) ε 3 (l j ,t)    (4.9)
where: By normalizing the time step ∆t, t -∆t = (k -1), (t = k), t + ∆t = (k + 1), the dynamics of free-surface flow can be represented as a linearized discrete-time systems using state-space representation as follows:

M = 1 τ     τ -1 Ψ 2 e (1 -Fr 2 e ) 1 -1 Ψ 2 e (1 -Fr 2 e ) -2 Fr e Ψ e 1 -1 Ψ 2 e (1 -Fr 2 e ) + 2 Fr e Ψ e 1 2Ψ 2 e (1 -Fr 2 e ) τ -1 2 + 1 2Ψ 2 e (1 -Fr 2 e ) + Fr e Ψ e 1 2Ψ 2 e (1 -Fr 2 e ) -1 2 -Fr e Ψ e 1 2Ψ 2 e (1 -Fr 2 e ) 1 2Ψ 2 e (1 -Fr 2 e ) -1 2 + Fr e Ψ e τ -1 2 + 1 2Ψ 2 e (1 -Fr 2 e ) -Fr e Ψ e     N = ∆t 2v    1 1 1 0 0 0 -1 -1 -1       g(I + J e + 4J e
x(k + 1) = Ax(k) + Bu(k) (4.13)
where:

x(k) = ε 1 (l 1 , k) ε 3 (l 1 , k) . . . ε 2 (l j , k) ε 1 (l j , k) ε 3 (l j , k) . . . ε 2 (l N , k) ε 1 (l N , k) T , (4.14) u(t) = ε 1 (l 1 , k) ε 2 (l N , k) (4.15) A =                   a 22 a 23 0 0 0 • • • 0 0 0 0 0 0 0 a 31 a 32 a 33 • • • 0 0 0 0 0 a 12 a 13 0 0 0 • • • 0 0 0 0 0 0 0 a 21 a 22 a 23 • • • 0 0 0 0 0 . . . 0 0 0 0 0 • • • a 21 a 22 a 23 0 0 0 0 0 0 0 • • • 0 0 0 a 31 a 32 0 0 0 0 0 • • • a 11 a 12 a 13 0 0 0 0 0 0 0 • • • 0 0 0 a 21 a 22                   , B =                   a 21 0 0 0 a 11 0 0 0 . . . 0 0 0 a 33 0 0 0 a 23                   , (4.16)

Dynamics of a subsystem

We consider here an irrigation channel comprising n reaches interconnected by the submerged gates. Each reach (i.e., a subsystem), i , i ∈ {1, 2, . . . , n} has the length L i . The DMPC control of this system is divided over n controllers. Assuming that we use the downstream configuration (as shown in Fig. 4.2) in which the control actions apply on the upstream gate of each reach, i , to adjust the gate opening, θ i (manipulated input), in order to regulate the downstream flow rate, Q i (controlled variables). Each controller, C i , obtains the measurement available from sensors such as the water height at upstream, h i us , and at downstream, h i ds , of each gate and the flow rate, Q i , at the downstream of each reach. We also consider a lateral pumping discharge, Q i p , at point L p i of each reach as a perturbation. The dynamics of a subsystem are modeled by a linearized discrete-time model as presented above (Eq(s). (4.13)). For a reach i , i ∈ {1, . . . , n} of length L i (corresponding to N discretized points l j , j = 1, . . . , N), when integrating the interaction variables satisfy constraints on water height, gate opening, and on variation of gate opening as follows [START_REF] Negenborn | Coordinated model predictive reach control for irrigation canals[END_REF]:

h min ≤ h i ≤ h max θ min ≤ θ i ≤ θ max ∆θ min ≤ ∆θ i ≤ ∆θ max (4.19)
where the water height is given by: h

i (l, k) = h i (l, k) + h 0 with h i (l, k) = ε i 1 (l, k) + ε i 2 (l, k) + ε i
3 (l, k) and the gate opening is deduced from gate equations Eq(s). (4.4) with ε i = f if e i (h 0 , u 0 ), i = {1, 2, 3}, that is:

θ i (k) = vB(ε i 2 (l 1 , k) -ε i 3 (l 1 , k) + f e 2 -f e 3 ) B g α 2g(h i us (k) -h i ds (k)) (4.20)
where: h i us (k), h i ds (k) are the water heights at upstream and downstream ends of the considered gate (given by water level sensors).

These structural constraints, Eqs. (4.19) and (4.20), imply the constraints on the states (described by Eq(s). (4.17)) as follows:

x min ≤ x i ≤ x max ∆x min ≤ ∆x i ≤ ∆x max

Cost function

The regulatory control of irrigation channels expects an automatic adjustment of controlled structures with reliability and durability. For example, a gate will move up and down permanently to regulate the flow rate at the end of a reach close to reference values (set-points). In other words, a local controller seeks to achieve the objectives as follows (see [START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF][START_REF] Puig | Model predictive control of combined irrigation and water supply systems: Application to the Guadiana river[END_REF]):

• Minimize the deviation of controlled variable from the set-points in order to perform the regularization (first term of Eq(s). (4.22)),

• Minimize the sudden change in deviation of controlled variable (e.g., flow rate) from one control step to the next in order to encourage smooth change of this variable (second term of Eq(s). (4.22)),

• Minimize the change in sequence of computed actions in order to avoid the oscillation (third term of Eq(s). (4.22)). Therefore, we adopt the following cost function for each controller i :

J i (k) = (N p -1) ∑ n=0 (||z i (k + n|k) -r i (k)|| 2 Z 1 + ||∆z i (k + n|k)|| 2 Z 2 + ||∆u i (k + n|k)|| 2 U 1 ) (4.22)
where: z i is the controlled variable over the prediction horizon N p , r i (k) is the set-points, u i is system input over prediction horizon N p , and Z 1 , Z 2 ,U 1 are weighting matrices of appropriate dimensions.

In the forthcoming sections, we make use of the optimization problem formulated by Eq(s). (4.22) to discuss different implementation strategies (i.e., decentralized and distributed) and how the cooperative communication among local MPC controllers can be included to improve the performance of the overall system.

Decentralized control algorithm

For a centralized control scheme (as surveyed in [START_REF] Malaterre | Control of irrigation canals: why and how?[END_REF][START_REF] Zhuan | Models and Control Methodologies in Open Water Flow Dynamics: A Survey[END_REF]), the centralized MPC controller can obtain the measurements from available sensors (e.g., flow meters at downstream of each reach, water level meters around gates), makes the prediction based on the complete system model and determines actions for all controlled structures by optimizing a global cost function [START_REF] Van Overloop | Model Predictive Control on Open Water Systems[END_REF].

Because the controller has a global view of the system, this implementation can provide good overall performances, but with a high computational complexity and computation delay due to communication issues [START_REF] Weyer | Control of irrigation channels[END_REF][START_REF] Michael | Control of Large-Scale Irrigation Networks[END_REF][START_REF] Van Overloop | Model Predictive Control on Open Water Systems[END_REF]. To overcome the computational burden, a first choice would be to implement a decentralized scheme [START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF][START_REF] Elmahdi | Decentralized Control Framework and Stability Analysis for Networked Control Systems[END_REF][START_REF] Sawadogo | Decentralized Predictive Controller for Delivery Canals[END_REF]. A decentralized MPC approach can divide the control of the global system over multiple controllers. These local controllers operate without being conscious of the presence of other controllers or subsystems [START_REF] Sawadogo | Decentralized Predictive Controller for Delivery Canals[END_REF]. They compute the control action based only on local information without taking into account the information exchanged with other controllers. For instance, [START_REF] Sawadogo | Decentralized Predictive Controller for Delivery Canals[END_REF] presented the design of a predictive controller for decentralized control of delivery channel and the control performance is tested on an open-canal flow nonlinear model in realistic conditions. In the decentralized predictive control scheme presented in [START_REF] Sawadogo | Decentralized Predictive Controller for Delivery Canals[END_REF], a set of local controllers based on predictive control theory, has been designed for manipulating each gate with the objective of ensuring the water level at the downstream end of each reach close to the set-points under external perturbation conditions, such as on-demand deliveries. [START_REF] Elmahdi | Decentralized Control Framework and Stability Analysis for Networked Control Systems[END_REF] introduced a combination of decentralized control and networked control where control loops are closed through a network. In [START_REF] Elmahdi | Decentralized Control Framework and Stability Analysis for Networked Control Systems[END_REF], two design methods of decentralized control for non-networked systems were chosen as a base for the design of a controller for the networked systems, the first being an observer-based decentralized control, while the second is the well-known Luenberger combined observer-controller design. For another example, a multi-layer decentralized MPC approach is proposed and designed in [START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF] for its application to large-scale networked systems. In this work [START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF], the topology of the controller is structured in two layers and a system partitioning allows to establish a hierarchical flow of information between a set of controllers designed based on the MPC. Therefore, the strategy proposed by [START_REF] Ocampo-Martinez | Multi-layer Decentralized MPC of Large-scale Networked Systems[END_REF] results in a centralized optimization problem for considering the global control objectives, followed of a decentralized scheme for reaching the local control objectives. For our case study, the details on the decentralized algorithm construction can be synthesized in Algorithm 4.1. estimate the interaction variable d i (k) from gate equations (Eq(s). ( 4 determine the state x i (k + 1) and outputs z i (k), y i (k) at time instant k from (Eq(s). (4.17))

9:

go to the next step (k + 1) 10: end for The local MPC controller for each subsystem can be separately designed using the Algorithm 4.1. However, a distributed strategy can improve the global performance by taking into account the interactions among subsystems and the cooperation among controllers.

Distributed model predictive control using cooperative controllers

We consider the channel comprising a cascade of n reaches interconnected by submerged gates (as shown in Fig. 4.2). To account for the interactions among subsystems, the information exchange among controllers is required (see more details in Section 4.2.1). Assuming that the reliable data exchange among system components are guaranteed (e.g., by an asynchronous consensus process presented in Chapter 3). The cooperation is considered in our DCMPC scheme in such a way that all local objectives of the controllers have to be coordinated towards optimizing global objective (e.g., all controlled variables rapidly reach their set-points). Therefore, a controller needs to know the interaction variables for action computation and the coordination variables (e.g., set-points) for the neighboring cooperation. In consequence, the definition and management of shared information are of great importance for DCMPC scheme.

Definition and management of shared information

The selection of the information shared among controllers depends on the channel setting and system modeling. We consider the downstream configuration of our case study as presented in Section 4.1.1.2. When we couple two reaches i-1 and i with a gate controlled by controller C i , the upstream point l i us of the gate also is the downstream point of the reach i-1 , and the downstream point l i ds of the gate is the upstream point of the reach i . The flow rate through the gate, Q i g , is computed from the gate equations (Eq(s). (4.4)). In order to determine interaction variable d i and to compute the gate opening θ i , we may identify from the linearized LB model (Eq(s). (4.17)) and the gate equations (Eq(s). (4.4)) that the necessary information for the exchange among the controllers contains:

{ε i-1 1 (l L i-1 , k), ε i-1 2 (l L i-1 , k), ε i+1 3 (l us , k)}.
The information shared by controller C i with its neighbors can be chosen as:

[ε i 3 (l us , τ) ε i 1 (l L i , τ) ε i 2 (l L i , τ)] T .
In addition, the gate equations (Eq(s). (4.4)) permit the computation of these interaction variables, once the neighbor controllers share the flow rate Q g through the gate at their position. For improving the global performance by neighboring cooperation, a controller, C i , also shares its set-points, r i . Other variables (e.g., water heights) can appropriately be used as interaction/coordination variables depending on distributed control schemes. Timing for sharing information depends on asynchronization mechanism and also the coordination method for controllers. For example, the exchange of interaction/coordination variables among controllers is considered in Chapter 3 as an information consensus problem and solved by using an asynchronous consensus protocol.

Design of cooperative controllers

The cooperation among controllers can improve the global control performance. In DMPC scheme, accounting for the cooperation requires that all controllers have to compute actions towards optimizing local and global objectives. As an example, the designed controllers in [START_REF] Negenborn | Distributed model predictive control of irrigation canals[END_REF][START_REF] Negenborn | Coordinated model predictive reach control for irrigation canals[END_REF] are already cooperative in the way that they are subjected to common constraints. By using an augmented Lagrangian duality approach, the interconnecting constraints are removed from the local constraint set. They are then added to the cost function under additional linear cost terms and additional quadratic terms using Lagrange multipliers. For our case study, we consider again the channel comprising a cascade of n reaches interconnected by submerged gates (as shown in Fig. 4.2). In order to improve the global performance (e.g., response time of the overall system) of our control application, we aim to minimize the time needed for the controlled variables of all controllers to reach their set-points. The global regularization objective can be expressed by:

J global (k) = n ∑ i=1 |z i (k) -r i (k)| (4.23)
In the downstream configuration (as shown in Fig. 4.2) and from the gate equations (Eq(s). (4.4)), we observe that the flow rate through the controlled gate, estimated by a controller may be involved in neighbor regularization objective (i.e., the first term of Eq(s). (4.22)). Thus, in the proposed DCMPC scheme, each controller, i , shares the interaction variable, Q i g , for action computation of upstream neighbor and a coordination variable, r i-1 , for the cooperation with downstream neighbor (see Fig. 4.4). As a result, each controller C i has to optimize local Fig. 4.4 An example of DCMPC scheme in which each controller, i , shares the interaction variable, Q i g , for action computation of upstream neighbor and a coordination variable, r i-1 , for the cooperation with downstream neighbor. objectives (Eq(s). (4.22)) and also minimize deviations of the estimated flow rate through gate Q i g and the upstream neighbor set-points r i-1 for helping the upstream controller to faster reach the regularization objective. Based on Eq(s). C i can be reformulated as follows:

J i coop (k) = (N p -1) ∑ n=0 (||z i (k + n|k) -r i (k)|| 2 Z 1 + ||∆z i (k + n|k)|| 2 Z 2 + ||∆u i (k + n|k)|| 2 U 1 + ||q i (k + n|k) -r i-1 (k)|| 2 Q1 ) (4.24)
where: r i-1 (k) are the set-points of the reach i-1 , q i (k) is the estimated flow rate through the controlled gate over the prediction horizon N p , and Q 1 are weighting matrices of appropriate dimensions. The receding horizon control algorithm can be implemented in a distributed (and cooperative) control fashion as summarized in Algorithm 4.2.

Algorithm 4.2 Distributed MPC scheme for n controllers.

1: for i = 1, . . . , n do 2:

Inputs: Initial state x i (1), initial input u i (1)

3:

for k = 1 : k max do 4:

Inputs: predicted perturbation p i (k) and reference trajectory r i (k)

5:

receive the flow rate Q i+1 g through downstream gate from downstream controller C i+1 and the set-points r i-1 of upstream controller C i-1 6: determine the interaction variable d i (k) from gate equations (Eq(s). (4.4)) determine the state x i (k + 1) and outputs z i (k), y i (k) at time instant k from (Eq(s). (4.17))

10:

send the estimated flow rate through gate Q i g (k) = q i (k) + Q 0 and set-points r i (k + 1)

11:

go to the next step (k + 1)

12:

end for 13: end for

Distributed control scheme integrated consensus protocol for cooperative controllers

The asynchronous framework used for the convergence of the information exchanged among controllers is illustrated in Fig. 4.5. In this framework, each controller exchanges information asynchronously and updates its information states with the latest-known (possibly outdated) information from its local neighbors. The information exchange among controllers using the consensus protocol defined by Eq(s). (3.6) allows taking into account the dynamically changing communication topologies, bounded time-varying time delays, and asynchronous mode in the same framework. A controller starts the asynchronous consensus process after applying the Fig. 4.5 Asynchronous framework for consensus problem in distributed control of irrigation channels. Four controllers C A ,C E ,C M ,C O are involved in DCMPC scheme. After applying the computed optimal action to the subsystem, the corresponding controller becomes the actual leader and starts a consensus process. control action to the subsystem, estimating the flow rate through the controlled gate and sending this estimated flow rate. The details of the DCMPC scheme with consensus process can be summarized in Algorithm 4.3. In this scheme, after computed action is applied to the subsystem, the controller becomes the actual leader, which starts a consensus process (see an example in Fig. 3.6). We define a new time scale τ ∈ [τ 0 , τ max ) for solving consensus problem. As presented in Chapter 3, the asynchronous consensus protocol is designed in order to obtain the convergence of information states of controllers. When the information consensus among controllers is achieved, the controller can wait for the next control step. The speed of reaching a consensus as well as the performance analysis of the consensus protocol are the key in the selection of different time sampling.

The next section provides the simulation of different scenarios and a particular benchmark for the considered irrigation channel.

Simulation results

This section presents some simulation results of various control schemes for the lab-scale microchannel presented in Fig. 4.2 with n = 3 reaches. For simulations, we first consider a centralized control scheme (using a single controller controlling all of considered gates), then a decentralized determine the state x i (k + 1) and outputs z i (k), y i (k) at time instant k from (Eq(s). (4.17))

10:

send estimated flow rate through gate Q i g (k) = q i (k) + Q 0 and set-points r i (k + 1)

11:

start consensus process guaranteeing the estimated flow rate through gate received by other controllers converge to the same value Q i g (k)

12:

for τ = τ 0 : τ max do 13:

χ i (τ 0 ) = Q i g (k)

14:

apply the asynchronous consensus protocol Eq(s). (3.6) 15:

stop when the convergence conditions Eq(s). (3.7) are satisfied.

16:

end for 17:

go to the next step (k + 1)

18:

end for 19: end for control scheme (using several separate local controllers as presented in Section 4.1.4), a distributed control scheme (using several controllers exchanging the flow rates through controlled gate as presented in Section 4.2) and finally a distributed cooperative control scheme (using several cooperative controllers exchanging the flow rates through controlled gate and the set-points as detailed in Section 4.2.2). The simulation results of different control approaches are eventually compared according to performance criteria such as the response time, overshoot limit, and computational time. The response time for global performance is defined as the time needed for all reaches to attain per i = 95%, i = {1, 2, 3} of their set-points. The overshoot limit is defined as the maximum overshot of three reaches with regards to their set-points. For all simulations, we have used the same parameters (see Table 4.1, Table 4.2) and constraints (see Table 4.3). The simulations were done in Matlab R2016b on a computer Intel®Core™i5-4310U CPU 2.0GHz. In order to properly analyze the entire evolution of our system, we have chosen the sampling period: ∆t = 0.05 (s), prediction horizon: N p = 0.5 (s), communication delay among the For the chosen prediction horizon and weighting matrices, we see that:

• In terms of response time, the controlled variable has reached the set-points faster with the centralized and distributed/distributed cooperative schemes compared to the decentralized scheme. In the distributed cooperative scheme, each controller has to optimize local objectives (as in the decentralize scheme) and also minimize the deviations of the estimated flow rate through controlled gate and the set-points of upstream neighbor, so the response time of the overall system is faster than in the decentralized scheme.

• In terms of overshoot limits, the simulations of these schemes have shown that the overshot is more important for distributed and distributed cooperative schemes with regard to the centralized and decentralized schemes.

• For the comparison of the computational time, we have performed the simulations of four schemes in the same operational conditions. However, the distributed cooperative scheme does not provide a decisive advantage in terms of computational time due to the computation of cooperative terms during the prediction horizons (see Eq(s). (4.24) and Algorithm 4.2). In terms of computational time, we concluded that the computational time can be reduced by 16.72% using the decentralized approach, 12.68% using the distributed approach while using a distributed cooperative approach as a reference. The centralized control presents interesting performances in terms of response time, and overshoot. The decentralized control needs more time for all reaches to attain their objectives. We can deduce that the distributed cooperative control is a good compromise between global control performance and computational speed. In addition, the cooperative strategy gives a shorter response time and smaller overshot with regards to the distributed non-cooperative control. It is also noted that four approaches have rejected external perturbations due to the common use of the MPC method.

Afterward, we have implemented the asynchronous consensus protocol for the distributed control using the DCMPC scheme. The details are presented in Section 4.2.3. For simulations, we re-use the same parameters and constraints as in previous simulations (see Table 4.1, Table 4.2, Table 4.3). To demonstrate the benefits of the consensus process added to DCMPC scheme, we consider two simulation scenarios: a DCMPC scheme with the loss of direct communication and a DCMPC scheme integrated an asynchronous consensus protocol. The first scenario assumes that at t = 10 (s), controller C 2 does not receive the information shared by controller C 3 during 25 (s) and similarly, at t = 20 (s), controller C 1 does not receive the information shared by controller C 2 during 25 (s). The second scenario implements the DCMPC scheme as presented in Algorithm 4.3. The simulations of the first scenario are illustrated in Fig. 4.12. For the second scenario, the simulation results are the same as in Fig. 4.7 but with longer computational time. The control performance obtained by simulations of these scenarios have been summarized in Table 4.5. Through the simulation results presented in Fig. 4.13, we see that the consensus process can help the coordination/interaction variables to be updated to the newest values and the controlled variables have reached the set-points. As a result, the control objectives are still achieved in the distributed control scheme integrated the consensus protocol, but with the robustness of loss of communication. In addition, we have tested the change of the prediction horizons N p = 5 (s) in simulations in order to observe the system behaviors. The simulation results presented in Fig. 4.14 show the same behavior as with prediction horizon N p = 0.5 (s) presented in Fig. 4.7.

Briefly, the proposed distributed cooperative control combines the optimality of centralized approach and the reduction in terms of computation charge of decentralized approach to provide a good compromise as shown in simulation results.

Conclusions

This chapter investigated the control of irrigation channels using a distributed cooperative MPC framework. Using the LB method, we have firstly provided a discrete-time linearized model of the irrigation channel that captures its dynamics and the interactions among the subsystems. Different control implementation strategies (e.g., decentralized and distributed strategies) were then discussed and compared. Several simulations over a particular benchmark of the irrigation channel have validated the proposed approaches. The MPC framework used within the four control strategies has demonstrated that: (1) it respects the constraints on the controlled and manipulated variables; (2) it avoids excessive variations on the manipulated variables allowing better use of actuators; and (3) it rejects external perturbations. Future work will concentrate on more comparisons with other linear/nonlinear optimization-based control approaches over different benchmarks available in the literature. Also, different cost criteria and economic MPC implementations will be thoroughly investigated.

Chapter 5

Conclusions and future research

Control applications for water systems are more and more complex due to the inherently distributed nature of these physical systems or due to the spatially integration of heterogeneous devices and communicators with their own control laws and autonomous behaviors. This thesis presents the distributed control of water transport systems such as irrigation channels or water drainage systems. The dynamics of the water transportation in these systems can be represented by shallow water free-surface models. The Lattice Boltzmann (LB) method is used for fluid simulation and mesoscopic description of dynamics. The distributed control of the system involves the exchange of information among the components of the control system (e.g., controllers, sensors, actuators, etc.) and creates certain communication problems. Solving communication problems requires the design of communication network including heterogeneous devices, thus based on a hybrid architecture. Our approach to different Quality of Service requirements of control applications (i.e., the information gathering from sensors, the cooperation of controllers, the transmission of the computed control decision to actuators) is based on the design of a composite metric used by a dynamic routing protocol. An asynchronous information consensus process has been added to the distributed control algorithm to reinforce the robustness to changes in communication topology due to link/node failures and time delays. For the distributed control, an optimization based control has been developed with a cost function accounting for the performance and the cooperation of controllers, while encouraging smooth change in the controlled variables and avoiding oscillations. The details of these points described throughout several chapters are briefly outlined as follows:

• Discrete-time nonlinear LBM model of irrigation channels: Chapter 1 was dedicated to the component-based modeling of the irrigation channels using the Lattice Boltzmann method and mathematical relations. Modularity in modeling involves the modularized and parameterized modeling technique and the creation of Model Library. The proposed modular approach can reduce the system design and review cycle and significantly improve the design consistency of a large-scale system. In simulation, it takes less time and enables a more efficient co-simulation. The model had a decisive role in the understanding, and diagnosis of irrigation channels.

• Dynamic routing design for networked control systems using hybrid architecture: Chapter 2 has investigated the design of a hybrid network to cope with the heterogeneity of wide-area networked control systems and a composite metric for the dynamic routing over this hybrid network in order to satisfy some QoS requirements. The resulting network model can be used to dynamically compute the QoS-related costs (or probabilities) of different paths among the components of networked control systems (NCSs). These are important concerns while considering the information exchange in NCSs.

• Asynchronous information consensus for networked control systems: Chapter 3 has focused on solving updating problems of the information shared by controllers in a networked control system with communication constraints. These problems are considered as a consensus problem. The design of a distributed cooperative MPC scheme and an appropriate asynchronous consensus protocol is presented such that the shared information among controllers asymptotically converges to the newest values.

• Distributed cooperative MPC framework: Chapter 4 has implemented and compared different control schemes in which costs, constraints, profiles are taken into account in decentralized and distributed constrained optimization problems (i.e., via an MPC design). The cooperation among local MPC controllers has been included in order to improve the performance of the overall system. At some points, this thesis is considered to have following contributions:

• using the LB method for fluid simulation, we provide a Model Library for the hydraulic components in large-scale irrigation systems, suitably designed to capture the dynamics and interactions among the subsystems;

• to address challenging QoS requirements of control applications in network design, we show that several performance criteria related to the communication needs for control applications can be optimized by using the metric composition approach;

• for the convergence of shared information among controllers, we design an asynchronous consensus protocol and establish conditions for a class of "leader-followers" systems to asymptotically reach the consensus under fixed or time-varying directed communication topology;

• we propose a global topology for the "leader-switching" system (representing a more practical control system) to satisfy multiple consensus problems;

• we integrate the consensus process into the DCMPC algorithm

• we propose a decentralized implementation of control system where the local MPC controller for each subsystem is separately designed. They compute the action using only the local information;

• the distributed cooperative controllers are designed by taking into account the information of neighboring controllers (e.g., the output and reference value) to compute the action.

| 107

They cooperate with their neighbors by minimizing the influence of the applied action with regularization objective of their neighbors;

• extensive simulation results are provided through different scenarios which validate the proposed predictive control schemes and highlights their strengths and weakness in terms of global performances. This thesis provides a first analysis for some problems related to the distributed control of cooperative systems based on the concepts from control, computer science, telecommunication and multi-agent theories. This joint approach to networked control systems (NCSs) will make possible the realization of robust and efficient control/supervision with strong constraints, as well as the emergence of new and attractive applications. However, although the layered approach adopted in this thesis holds the potential to solve complex problems using collective intelligence, there still exist many research issues to study and experimental validations to carry on. Some perspectives of this work for future research are to:

• in the short term,

analyze quantitatively and qualitatively the characteristic of the distributed cooperative approach,

study the behavior of the control system in degraded mode; particularly, the contribution of the distributed cooperative approach with regards to the classical approach;

use more complex models of the irrigation systems and integrate the interaction with the environment (e.g., weather forecasting, water losses, etc.) to better represent the phenomena related to hydraulics of the water flow;

study the effects of uncertainties on the proposed control algorithms and the robustness against these uncertainties;

determine the convergence time of the consensus approach depending to the characteristics of the networks;

study the effects of the losses in communication networks and propose the possibility of network reconfiguration;

study the influence of the consensus on the performance of distributed cooperative control.

• for future projects, perform the operational management of Bourne irrigation channel (extended over 46 km) for the optimal distribution of water resources; the optimal balance of electricity production and consumption (pumping versus turbine), which has represented a differential of 80 000 e in 2016 by SID -Syndicat d'Irrigation Drômois;

manage the architecture of networked control systems for the valves and pumping stations of the Bourne canal by a proprietary LoRa® network (actually, the communication equipped with redundant GSM links, and thus, depended on an operator has represented a communication budget of 60 000 e/year in 2015);

modernize the irrigation and drainage systems in the Mekong Delta (Vietnam) with the scalability and complexity issues (e.g., CARE project). The Mekong Delta is a vast maze of rivers, swamps, islands, and villages surrounded by rice paddies. This region encompasses a large area of over 40, 500 km 2 and 2.6 million hectares of irrigated land for agriculture.

  irrigation channels using the Lattice Boltzmann method 1.1 Irrigation channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Characterization of irrigation channels . . . . . . . . . . . . . . . . . . 1.1.2 Geometric parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Hydraulic variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Classifications of the flows . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Example of an irrigation channel . . . . . . . . . . . . . . . . . . . . . 1.1.6 Necessity of a model for the irrigation channel . . . . . . . . . . . . . 1.2 Modeling of an irrigation channel using classical Saint-Venant equations . . . . 1.3 Modeling of an irrigation channel using Lattice Boltzmann method . . . . . . . 1.3.1 Hydrodynamic model of free-surface flow . . . . . . . . . . . . . . . . 1.3.2 Boundary conditions used to generate the dynamics of the irrigation channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Coupling D1Q3 LB models of multiple reaches . . . . . . . . . . . . . 1.3.4 Validation of D1Q3 LB method . . . . . . . . . . . . . . . . . . . . . 1.4 Fluid simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Modularity in modeling of free-surface channels using a library . . . . 1.4.2 Simulation schemes and results . . . . . . . . . . . . . . . . . . . . . 1.4.3 Considering a real irrigation channel and data requirements for model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Composite metric design for the dynamic routing in networked control systems 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Motivation: A hybrid network for NCSs . . . . . . . . . . . . . . . . . . . . . 2.2.1 Communication constraints in NCSs . . . . . . . . . . . . . . . . . . . 2.2.2 A hybrid network for heterogeneous components of the NCS . . . . . . 3.6 An example of consensus problem in distributed control of irrigation channels . 3.7 An example of the global topology for the system with four controllers . . . . . 3.8 Simulation results -Asynchronous consensus protocol is used in the one-leader control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Simulation results -Asynchronous consensus protocol is used in the system with 10 controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Simulation results -Asynchronous consensus protocol is used in the first scenario with the loss of direct communication . . . . . . . . . . . . . . . . . . . . . . 3.11 Simulation results -Asynchronous consensus protocol is used in the second scenario with the loss of direct communication . . . . . . . . . . . . . . . . . 3.12 Simulation results -Asynchronous consensus protocol is used in the leaderswitching system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Discretized points of a reach in D1Q3 LB model . . . . . . . . . . . . . . . . . 4.2 Schematic of the flow rate control in downstream configuration. . . . . . . . . 4.3 Variables are involved in the flow rate control of a reach . . . . . . . . . . . . . 4.4 An example of DCMPC scheme in which each controller shares the interaction and coordination variables . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5 Asynchronous framework for consensus problem in the distributed control of irrigation channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Performance comparison of four MPC settings . . . . . . . . . . . . . . . . . . 4.7 Simulation of distributed cooperative control with perturbation . . . . . . . . . 4.8 Simulation of distributed control with perturbation . . . . . . . . . . . . . . . 4.9 Simulation of decentralized control with perturbation . . . . . . . . . . . . . . 4.10 Simulation of centralized control with perturbation . . . . . . . . . . . . . . . 4.11 Evolution of gate openings in four control settings . . . . . . . . . . . . . . . . 4.12 Simulation results -the distributed control with the loss of direct communication among controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Simulation results -the distributed control with integrated consensus . . . . . . 4.14 Simulation results -the distributed control with another prediction horizon . . . List of tables 1.1 Geometric parameters of the irrigation channel. . . . . . . . . . . . . . . . . . 1.2 Structural parameters of mixed hydraulic structures. . . . . . . . . . . . . . . . 1.3 Boundary conditions and initial values are used in simulation scenarios. . . . . 1.4 Simulation parameters of the irrigation channel. . . . . . . . . . . . . . . . . . 1.5 Global simulation parameters for the considered section of BC. . . . . . . . . . 2.1 Some primary routing metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 An example of routing metrics used in NCS hybrid network. . . . . . . . . . . 2.3 Derived routing metrics used in NCS hybrid network. . . . . . . . . . . . . . . 2.4 Basic network simulation parameters of the hybrid network. . . . . . . . . . . 2.5 Some simulation scenarios for the dynamic routing with different metrics. . . . 2.6 Simulation results of a hybrid network using a composite metric. . . . . . . . . 3.1 Simulation parameters for the asynchronous consensus. . . . . . . . . . . . . . 4.1 Structural parameters of lab-scale micro-channel used in simulations. . . . . . 4.2 Simulation parameters of lab-scale micro-channel. . . . . . . . . . . . . . . . . 4.3 Constraints in the control of lab-scale micro-channel. . . . . . . . . . . . . . . 4.4 Simulation results of different implementation strategies. . . . . . . . . . . . . 4.5 Simulation results -different scenarios with integrated consensus . . . . . . . .

Fig. 1 . 1

 11 Fig. 1.1 The common parameters and variables are used to characterize a free-surface channel.

Fig. 1 . 2

 12 Fig. 1.2 Synoptic view of the Bourne channel (BC), which is used as an example of the freesurface channel.

Fig. 1 . 7 A

 17 Fig. 1.7 A Simulink/MatLab Model Library is created for the modeling and the simulation of an irrigation channel.

Fig. 1 .

 1 Fig. 1.8 A section of Bourne channel is considered for modeling and simulation.

Fig. 1 . 9 A

 19 Fig. 1.9 A Simulink/MatLab model is created for the simulation of the first considered section.

Fig. 1 .

 1 Fig. 1.10 Boundary conditions impose the water flow rate at Mondy gate, Q M , and water height at S2 junction, h S for the first considered section.

Fig. 1 .

 1 Fig. 1.11 The first considered section is initialized. The specified flow axis (i.e., 3.25e4 m to 3.62e4 m) is relative to global flow axis beginning at Auberives barrage.

Fig. 1 .Fig. 1 .

 11 Fig. 1.12 Simulation results -Water height profile in the first considered section. The specified flow axis (i.e., 3.25e4 m to 3.62e4 m) is relative to global flow axis beginning at Auberives barrage.

Fig. 1 .Fig. 1 .

 11 Fig. 1.14 A Simulink/MatLab model is created for the simulation of the second considered section using the Model Library.

Fig. 1 .

 1 Fig. 1.15 Boundary conditions are imposed for the second considered section.

Fig. 1 .Fig. 1 .Fig. 1 .

 111 Fig.1.[START_REF] Capone | An Energy Efficient and Reliable Composite Metric for RPL Organized Networks[END_REF] The second considered section is initialized.

Fig. 1 .

 1 Fig. 1.19 Simulation results -Water height profile along the second considered section corresponds to the specified simulation scenario.

Fig. 2 . 1

 21 Fig. 2.1 Communication setting of primary Bourne Channel (BC) in which the large network coverage and the wide communication range require the implementation of a large number of intermediate nodes that motivate a unified communication of control system components within a hybrid network.

Fig. 2 . 2

 22 Fig. 2.2 An example of a hybrid network for NCS. It can be composed of traditional WSNs, embedded networks, and Internet Protocol (IP) networks. Intermediate nodes can route any data packets.

Definition 2 . 3 . 1 .

 231 Network model and Path Weight Structure [152].

Proposition 2 . 4 . 1 .

 241 A composite metric for the considered NCS hybrid network.

Fig. 2 . 5

 25 Fig. 2.5 Simulations using JSIM framework -Three RPL instances are used in network routing with mesh network topology.

Fig. 2 . 6

 26 Fig. 2.6 Comparison of network performance criteria such as Packet Loss (%), Average Latency (ms), Network Lifetime (s) when using different metrics in RPL. Composite metrics are considered in different variations of weight parameters.

Fig. 3 . 1 A

 31 Fig. 3.1 A DMPC scheme for the control of irrigation channels.

Fig. 3 . 2 A

 32 Fig. 3.2 A downstream configuration for the distributed control of an irrigation channel. The controller C i is implemented for the control of water flow Q i at the downstream end of a reach i by adjusting the opening θ i of the gate at the upstream end.

Fig. 3 . 3

 33 Fig. 3.3 An example of DMPC scheme in which the controllers share interaction variables ({ f 1 (l L i-1 , k), f 2 (l L i-1 , k), f 3 (l us , k)}) at time k for action computation.

Fig. 3 . 4

 34 Fig. 3.4 An example of DMPC scheme in which the controllers share flow rates through their gates (Q g ) for coordination.

Assumptions 3 . 3 . 2 .

 332 Communication assumptions[START_REF] Su | Further results on convergence of asynchronous linear iterations[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. 1. 0 ≤ d i j (τ) ≤ D max ≤ +∞, ∀i, j ∈ N, ∀τ, where: D max is a constant (regulated delays); 2. +∞ τ=τ c S(τ) = C, ∀τ c (admissible updating sets); 3. A ii > 0, ∀i ∈ N, and ∃i ∈ N : d ii (τ) = 0, ∀τ; 4. χ i (τ 0 ) > 0, ∀i ∈ N to avoid considering the trivial consensus point [0 0 . . . 0].

Theorem 3 . 3 . 1 .

 331 Perron-Frobenius theorem for non-negative irreducible matrices ([44], Theorem 8.4.4 and [83], Chapter 8)

Lemma 3 . 3 . 1 .

 331 ([44], Lemma 8.5.4) If A ∈ R n×n is a non-negative irreducible matrix and all its main diagonal elements are positive, then A n-1 > 0, and by definition, A is primitive. Lemma 3.3.2. ([44], Theorem 8.5.1 and Lemma 8.5.5) If A ∈ R n×n is a non-negative and primitive matrix with spectral radius ρ(A) = f , the vectors r and ℓ are respectively right and left P-F eigenvectors corresponding to P-F eigenvalue f , then:

Corollary 3 . 3 . 1 .

 331 ([44], Theorem 8.5.1) If A ∈ R n×n is a non-negative, stochastic and primitive matrix (NSP) with spectral radius ρ(A) = f = 1, the vectors r = 1 n and ℓ are respectively right and left P-F eigenvectors corresponding to P-F eigenvalue f = 1, then:

Theorem 3 . 3 . 3 .

 333 The consensus of asynchronous systems under time-varying communication topologies ([27], Theorem 2.3, [85], Proposition 1, and [110], Theorem 3.3).

Fig. 3 . 7

 37 Fig. 3.7 An example of the global topology for the system with four controllers illustrates the parameterizable adjacency matrix A.

Fig. 3 . 8

 38 Fig. 3.8 Simulation results of the asynchronous consensus protocol used in the one-leader control system (see Fig. 3.6).

Fig. 3 .

 3 Fig.3.10 Simulation results of asynchronous consensus protocol used in oneleader control system in a scenario in which the controller 4 loses the direct communication with the controller 1 at τ = 2 (s).

Fig. 3 .

 3 Fig.3.11 Simulation results of asynchronous consensus protocol used in oneleader control system in a scenario in which the controller 4 loses the direct communication with the controller 3 at τ = 2 (s).

Fig. 4 . 2

 42 Fig. 4.2 Schematic of the flow rate control in downstream configuration.

( 4 . 21 )

 421 The computation of control action has to take into account of these constraints, Eqs. (4.[START_REF] Chopard | Cellular automata modeling of physical systems[END_REF]) and (4.21).

Fig. 4 . 3

 43 Fig. 4.3 Variables are involved in the flow rate control of a reach using the MPC method.

Algorithm 4 . 1 1 : 3 : 4 :

 41134 Decentralized MPC scheme for a controller C i . Inputs: Initial state x i (1), initial input u i (1), predicted perturbation p i (k) and reference trajectory r i (k) 2: for k = 1 : k max do Inputs: predicted perturbation p i (k) and reference trajectory r i (k) measure the flow rate Q i at downstream of each reach 5:

  (4.22), the cost function of cooperative controller 4.2 Distributed model predictive control using cooperative controllers | 95

4. 3 97 Algorithm 4 . 3 2 : 3 : 4 : 5 : 6 : 8 :

 39743234568 Simulation results | Distributed MPC scheme for n controllers with consensus process.1: for i = 1, . . . , n do Inputs: Initial state x i (1), initial input u i (1) for k = 1 : k max do Inputs: predicted perturbation p i (k) and reference trajectory r i (k) receive the flow rate Q i+1 g through downstream gate from downstream controller C i+1 and the set-points r i-1 of upstream controller C i-1 determine the interaction variable d i (k) from gate equations (Eq(s). (4.4))7:obtain U i * k (x i (k))by solving optimization problems (Eq(s). (4.24)) for prediction horizon N p under constraints (Eqs. (4.19) and (4.21)) apply the first element u i * k of U i * k to the subsystem (assumed at t i a ) 9:

Fig. 4 . 6

 46 Fig. 4.6 Performance comparison of four MPC settings: centralized, decentralized, distributed and distributed cooperative control.

Fig. 4 . 7

 47 Fig. 4.7 Distributed cooperative control with perturbation Q p -Variation of downstream flow rates (z ri = Q ri -Q 0 ) with regards to equilibrium points Q 0 of three reaches (ri = {r1, r2, r3}).

Fig. 4 . 8

 48 Fig. 4.8 Distributed control with perturbation Q p -Variation of downstream flow rates (z ri = Q ri -Q 0 ) with regards to equilibrium points Q 0 of three reaches (ri = {r1, r2, r3}).

Fig. 4 . 9

 49 Fig. 4.9 Decentralized control with perturbation Q p -Variation of downstream flow rates (z ri = Q ri -Q 0 ) with regards to equilibrium points Q 0 of three reaches (ri = {r1, r2, r3}).

Fig. 4 .

 4 Fig. 4.10 Centralized control with perturbation Q p -Variation of downstream flow rates (z ri = Q ri -Q 0 ) with regards to equilibrium points Q 0 of three reaches (ri = {r1, r2, r3}).

Fig. 4 .

 4 Fig. 4.11 Four control settings with perturbations -Evolution of gate openings (θ ri ) of three reaches (ri = {r1, r2, r3}) respecting the constraints specified in Table4.3

Fig. 4 .

 4 Fig. 4.12 Simulation results -the distributed control of micro-channel with loss of direct communication among controllers.

Fig. 4 .

 4 Fig. 4.13 Simulation results -the distributed control of micro-channel with the integrated consensus.

Fig. 4 .

 4 Fig. 4.14 Simulation results -the distributed control of micro-channel with the prediction horizon N p = 5s.

Table 1 .

 1 1 Geometric parameters of the irrigation channel.

	Segment	Length (L) Width (B) Slope (I)	Manning coef. (n) Space st. (∆x)
	1	100 m	5.12 m	2.400e-4	0.033	100 m
	2	2600 m	5.20 m	2.441e-4	0.033	100 m
	3	1000 m	5.40 m	2.440e-4	0.033	100 m

Table 1 .

 1 2 Structural parameters of mixed hydraulic structures.

	Mixed	Gate width	Gate	Spillway	Spillway	Spillway
	structure	(Bg)	coeff. (α)	width (Bsw)	height (Hsw)	coeff. (α s w)
	at Mondy	3.97 m	0.66	0.80 m	1.3 m	0.35
	at Orme	2.8 m & 0.9 m	0.66	1.91 m	1.7 m	0.35

Table 1 .

 1

3 Boundary conditions and initial values are used in simulation scenarios.

Table 1 .

 1 [START_REF] Beard | Synchronization of Information in Distributed Multiple Vehicle Coordinated Control[END_REF] Global simulation parameters for the considered section of BC.

	Parameter	Value Comment
	Total length	36200 m
	Number of reaches	12
	Number of controlled	3 Orme, Mondy, Ecanciere
	structures
	Number of pumping stations	8 Baume d'Hostun, Eymeux, Meymans,
		Papelissier, Martinet, Tendillon, Mondy,
		Monts du Matin
	Number of discretized points	374 Space lattice dx = 100m
	Simulation time	8000 s Time step dt = 10s

Table 2 .

 2 2 An example of routing metrics used in NCS hybrid network.

	Metric	Expected network performance
	Node Energy (NE)	Network lifetime
	Hop count (HP)	Network latency (also, energy consumption)
	Link ETX (ETX)

Table 2 .

 2 3 Derived routing metrics used in NCS hybrid network.

	Metric	Derived metric Domain Aggr. rule Order rel.
	Link ETX	(1 -1/ETX)	[0,1]	additive	min.
	Node energy	(1 -1/NE)	[0,1]	additive	min.
	Hop count	(1 -1/HP)	[0,1]	additive	min.

Table 2 .

 2 .4. 4 Basic network simulation parameters of the hybrid network.

	Parameters	Setting value
	Nodes	144 nodes on grid 1200m X 1200m
	Communication range 125m in Tx/Rx, interference excluded
	Layered protocols	FSP/UDP/RPL/802.11/PHY
	Data packets/Size	620p/30B, 620p/50B, 1600p/30B
	Energy model	1.6W Tx, 1.2W Rx, 1.15W Idle
	Simulation time	6000s
	2.5.2 Simulation scenarios

Simulations are performed with three root nodes (e.g., n10, n5, n0) representing the controller nodes equipped with Internet connection and power line. They receive data files comprising of 620 packets and 1600 packets (30 bytes/packet, and 50 bytes/packet) from 6 sensor nodes (e.g., n122, n112, n104, n84, n75, n70) by using File Service Protocol (FSP) in the application layer. This simple application protocol uses User Datagram Protocol (UDP) as the transport layer. The same network topology is used through all three scenarios in which the RPL protocol uses three instances with only one metric among ETX, NE, and HP at a time. Other scenarios are simulated with the designed composite metrics in different variations of weight parameters. In addition, another scenario with three different RPL instances, each uses a metric among ETX, NE, and HP, is investigated for performance comparison. The simulation lasts 6000s totally. The data flow from n122 to n10 were mainly observed as the reference flow in all scenarios to see what has been improved in terms of achieved performance while comparing different metrics (see Table

2

.5).

Table 2 .

 2 5 Some simulation scenarios for the dynamic routing with different metrics.

	Controllers	n0	n5	n5	n5 n5 n10
	Sensors/Actuators	n122 n112 n104 n84 n75 n70
	Reference information flow			n122-n10

Table 2 .

 2

	Metric	Packet loss (%)	Latency (ms)	Net. lifetime (s)
	1.0HP	1060/1600=66.25 21.793497981029844 2320.6778962369626
	1.0ETX	24/1600=1.50	21.719895608480915 2025.1918250257256
	0.2HP+0.8NE	116/1600=7.25	24.184583240256114	6000
	0.5HP+0.5ETX	24/1600=1.50	21.719895608480915 2025.1918250257256
	0.2HP+0.3ETX+0.5NE	26/1600=1.63	22.392553597808604	6000
	3 separate instances	1076/1600=67.25 21.88401692824656	2254.934818305747

[START_REF] Belaud | Hydraulic Modeling of an Automatic Upstream Water Level Control Gate for Submerged Flow Conditions[END_REF]

Simulation results of a hybrid network using a composite metric.

  ) . . .

				α i2	. . .	0
	. . .	. . .	. . .	. . .	. . .	. . .
	0	0	. . . 1 (row i) . . .	0
	. . .	. . .	. . .	. . .	. . .	. . .
	0	0	. . .	α in

Table 3 .

 3 1 Simulation parameters for the asynchronous consensus.

	Parameters	Setting value
	Nodes (n)	4
	Time sampling (dt)	0.05s
	Updating instants of controllers i	2*i*dt
	Time delay (d i j )	2*i*j*0.01s
	Simulation time (T sim)	6s, 10s, and 40s

Table 4 .

 4 1 Structural parameters of lab-scale micro-channel used in simulations.

	Length of the reach (L i )	7m
	Length until the pumping station (Lp i )	4m
	Width of the reach (B i )	0.1m
	Bed slope (I i )	2.6e-3
	Manning coefficient (n i )	1/97
	Performance coefficient of the gate (α i )	0.8

Table 4 .

 4 4 Simulation results of different implementation strategies. Perf. criterion Resp. time Overshoot Comput. time

		(s)	(m 3 /s)	(s)
	Centralized	8.4000 5.3577e-05	31.644207
	Decentralized	23.5500 1.2516e-04	26.846662
	Distributed	16.6000 2.3128e-04	28.149515
	Cooperative	11.6000 1.6854e-04	32.235586

Table 4 .

 4 [START_REF] Beard | Synchronization of Information in Distributed Multiple Vehicle Coordinated Control[END_REF] Simulation results of different scenarios.

	Perf. criterion	Resp. time Overshoot Comput. time
		(s)	(m 3 /s)	(s)
	With loss of communication	23.3000 1.6854e-04	28.480434
	With integrated consensus	11.6000 1.6854e-04	52.989587
	With N p = 5s	11.6500 1.6856e-04	183.914195

Acknowledgements

support of the Artemis Arrowhead European project under grant agreement number 332987 is acknowledged.

Lemma 2.4.1. Yaling's Theorems [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF] refines the Lemma 2.3.1.

The class of protocols based on distributed Bellman-Ford algorithm and hop-by-hop forwarding can find the shortest path if and only if the corresponding PWS is left-isotonic (see [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], Theorem 7). For these protocols to achieve consistent and loop-free routing, the necessary and sufficient conditions are that the PWS is left-monotonic (see [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], Theorem 8 and 9).

Proof. See [START_REF] Yaling | Design guidelines for routing metrics in multihop wireless networks[END_REF], Section VI (and refer back to [START_REF] Luis | Network routing with path vector protocols[END_REF]).

Functionally, the specification of RPL protocol can be interpreted using network model and PWS (Definition 2.3.1). Within an RPL instance, the rank (i.e., the best path cost) is defined as the approximate distance value of a node to the DODAG root. The Objective Function (OF) defines how to map one or more metrics and constraints into the path cost (i.e., path cost definition). Based on distributed Bellman-Ford algorithm solving the shortest path routing problem, a node will select its parent node with the smallest distance value to DODAG root (i.e., the order relation is minimum). For example, considering the nodes X,Y, Z ∈ H, and Y and Z are potential parent nodes of X, the distance values of the paths through Y and Z are calculated such that: rank X (viaY ) = rank Y + c(X,Y ) and rank X (viaZ) = rank Z + c(X, Z), where c(X,Y ) is a link cost from X to Y (i.e., the aggregation rule of path cost in this case is additive). If rank X (viaY ) < rank X (viaZ) then rank X = rank X (viaY ) and Y is the parent node of X. The redefinition of link cost can generate the inconsistent or non-optimal routing. Consequently, when the metric is redefined in order to satisfy QoS requirements, it is required to demonstrate that the RPL routing protocol continue to guarantee the consistency, loop-freeness, and path-optimality properties (see Definition 2.3.2). The process of metric re-definition encompasses the selection, re-quantification, derivation and composition of several primary metrics.

Metric selection

Based on static and/or dynamic metrics, paths can be calculated by routing algorithms using qualitative or quantitative methods [START_REF] Phillippe | Routing Metrics Used for Path Calculation in Low-Power and Lossy Networks[END_REF][START_REF] Trakadas | Routing metric selection and design for multi-purpose WSN[END_REF]. Many primary metrics have been proposed in [START_REF] Phillippe | Routing Metrics Used for Path Calculation in Low-Power and Lossy Networks[END_REF] for LLNs to capture different network characteristics and to meet the application requirements as shown in Table 2.1. These metrics should be redefined or differently re-quantified for specific applications as the way presented in this chapter. Therefore, the necessary conditions for the convergence of asynchronous iterations are ρ(A) < 1 [START_REF] Su | Further results on convergence of asynchronous linear iterations[END_REF] and ρ(A) = 1 for specific cases [START_REF] Pott | On the convergence of asynchronous iteration methods for nonlinear paracontractions and consistent linear systems[END_REF][START_REF] Lubachevsky | A Chaotic Asynchronous Algorithm for Computing the Fixed Point of a Nonnegative Matrix of Unit Spectral Radius[END_REF][START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF]. The following lemmas provide further results.

Lemma 3.3.4. The convergence for a non-negative irreducible matrix ([28], Lemma 2 and [START_REF] Su | Further results on convergence of asynchronous linear iterations[END_REF]).

Let consider the asynchronous system Eq(s). (3.6) characterized by the non-negative irreducible matrix A, which has the spectral radius, ρ(A) = 1 and the corresponding right eigenvector, r, that is: Ar = r. Under the Assumptions 3.3.2, there exists a bounded positive constant C ∈ R such that: χ(τ) → Cr, as τ → τ +∞ , where the value of C depends on A, χ(τ 0 ), S(τ), d i j (τ).

Following the Lemma 3.3.4, the information states χ(τ) converge to C1 n when r is similar to 1 n . Tight lower and upper bounds for the constant C are given in [START_REF] Lubachevsky | A Chaotic Asynchronous Algorithm for Computing the Fixed Point of a Nonnegative Matrix of Unit Spectral Radius[END_REF]. Assuming that there is a link from each node to itself (as mentioned in Assumptions 3.3.2), the diagonal elements of A are positive, the following lemma approaches the convergence based on the properties of the matrix A with positive diagonal elements, the associated graph G A and Perron-Frobenius theorem (see Theorem 3.3.1). Lemma 3.3.5. Unique eigenvalue with a maximum modulus of a matrix and the associated graph has a spanning tree ([110], Lemma 3.4).

• If a non-negative matrix A = [A i j ] ∈ R n×n has the sum of each row equal to the same positive constant f > 0, then f is an eigenvalue of A with a corresponding right eigenvector 1 n , such that: A1 n = f 1 n and the spectral radius ρ(A) = f .

• The eigenvalue f of A has algebraic multiplicity equal to 1, if and only if the graph associated with A, G A , has a spanning tree.

• If the graph G A associated with A, has a spanning tree and A ii > 0, then f is the unique eigenvalue with maximum modulus.

The Lemma 3.3.5 can derive the following results for an NSP matrix.

Corollary 3.3.2. Unique eigenvalue equal to 1 of a non-negative stochastic matrix with positive diagonal elements (see Lemma 3.3.1, Lemma 3.3.2, and [START_REF] Fang | Information consensus of asynchronous discrete-time multi-agent systems[END_REF], Lemma 3). Let A = [A i j ] ∈ R n×n be a non-negative stochastic matrix with positive diagonal elements. The following statements hold:

• A is primitive, so A is NSP.

• lim k→+∞ A k = 1 n ℓ T , where: k ∈ N + , 1 n and ℓ are right and left P-F eigenvectors corresponding to P-F eigenvalue 1, that is: ℓ T A = ℓ T , and are normalized such that:

among subsystems into this model, we obtain:

where, as illustrated in Fig. 4.3, we define:

• the discrete-time local states of each reach as:

• the inputs as:

• the perturbations as:

• the interaction variables among subsystems as:

• the controlled variable as:

• the measured output as:

• the estimated flow rate through controlled gate as:

• the water heights along the channel as:

• the corresponding matrices as:

For distributed control scheme, we also consider the interaction/coordination variables, χ i , in the information exchange among controllers (see Section 4.2.1).

Structural and operational constraints

The operational management of irrigation channels is subject to constraints on the channel structure (e.g., limits of the gate opening, reliability, and robustness of equipment, limits of reservoir capacity) as well as on channel operations (e.g., overflow risks, bank stability, minimum off-take levels). Assuming that the solutions for the control of the considered irrigation channel illustrated in Figs. [START_REF] Alur | Compositional modeling and analysis of multi-hop control networks[END_REF].2 and 4.3 and modeled by the discrete-time systems in Eq(s). (4.17) must

3 Constraints in the control of lab-scale micro-channel.

Gate opening 0m ≤ θ ≤ 0.5m Variation of gate opening -0.02m ≤ ∆θ ≤ 0.02m Constraints on water heights 0.10m ≤ h ≤ 0.25m Constraints on state variables -0.5 ≤ x ≤ 0.5 -0.5 ≤ ∆x ≤ 0.5 Constraints on the inputs -0.002 ≤ u ≤ 0.002 -0.004 ≤ ∆u ≤ 0.004 controllers: T c = 0.01 (s) and simulation time: T = 45 (s). For each reach, we assume having constraints on the values of manipulated variable: θ min = 0 (m); θ max = 0.5 (m); ∆θ min = -0.02 (m); ∆θ max = 0.02 (m). These constraints infer the limits on the inputs and the local states as follows: -0.002 ≤ u ≤ 0.002; -0.004 ≤ ∆u ≤ 0.004; -0.5 ≤ x ≤ 0.5; -0.5 ≤ ∆x ≤ 0.5.

The control objective for the reach i , (denoted by the subscript ri

by acting on the opening θ ri of the upstream gate. In addition, we also tested the ability of MPC to reject a perturbation. We have considered, for example, a lateral withdrawing at the location L p = 4 (m) of each reach with the constant discharge Q p = 0.8e -3 (m 3 /s), beginning at time instant k = 5 (s) to the end of the simulation.

As shown in Figs. 4.7 to 4.10, the controlled variables z ri , starting from 0 (Q ri = Q 0 ), have reached the set-points r ri through the use of all four control settings and the computed values of the manipulated variables θ ri , starting from θ max , fully respect the specified constraints (see Fig. 4.11). The performance of each control approach given by the simulation results have been summarized in Table 4.4 and Fig. 4.6.