Various kinds of manufacturing systems can be modeled and analyzed by Timed Event Graphs (TEGs). These TEGs are a particular class of timed Discrete Event Systems (DESs), whose dynamic behavior is characterized only by synchronization and saturation phenomena. A major advantage of TEGs over many other timed DES models is that their earliest behavior can be described by linear equations in some tropical algebra structures called dioids. This has led to a broad theory for linear systems over dioids where many concepts of standard systems theory were introduced for TEGs. For instance, with the (max,+)-algebra linear state-space models for TEGs were established. These linear models provide an elegant way to do performance evaluation for TEGs. Moreover, based on transfer functions in dioids several control problems for TEGs were addressed. However, the properties of TEGs, and thus the systems which can be described by TEGs, are limited. To enrich these properties, two main extensions for TEGs were introduced. First, Weighted Timed Event Graphs (WTEGs) which, in contrast to ordinary TEGs, exhibit event-variant behaviors. In WTEGs integer weights are considered on the arcs whereas TEGs are restricted to unitary weights. For instance, these integer weights make it straightforward to model a cutting process in a production line. Second, a new kind of synchronization called partial synchronization (PS) was introduced for TEGs. PS is useful to model systems where specific events can only occur in a particular time window. For example, consider a crossroad controlled by a traffic light: the green phase of the traffic light provides a time window in which a vehicle is allowed to cross. Clearly, PS leads to time-variant behavior. As a consequence, WTEGs and TEGs under PS are not (max,+)-linear anymore.

In this thesis, WTEGs and TEGs under PS are studied in a dioid structure. Based on these dioid models for WTEGs a decomposition of the dynamic behavior into an event-variant and an event-invariant part is proposed. Under some assumptions, it is shown that the eventvariant part is "invertible". Hence, based on this model, optimal control and model reference control, which are well known for ordinary TEGs, are generalized to WTEGs. Similarly, a decomposition model is introduced for TEGs under PS in which the dynamic behavior is decomposed into a time-variant and time-invariant part. Again, under some assumptions, it is shown that the time-variant part is invertible. Subsequently, optimal control, as well as model reference control for TEGs under PS is addressed. vii At this point I would like to say thank you to all the people, who supported my during my Ph.D. In particular, I am deeply grateful to my French supervisors Prof. Bertrand Cottenceau and Prof. Laurent Hardouin, for many discussions of my work, for helping me with all the administration stuff at Université d'Angers, for supporting me in some many different situations, especially during my stays in Angers.

In the same manner, I would like to thank my German supervisor Prof. Jörg Raisch for giving me the opportunity to work in this interesting field of research and for many discussions of my work. I am greatly

Zusammenfassung

Viele Produktions-und Fertigungsanlagen können mit Hilfe von Synchronisationsgraphen modelliert und analysiert werden. Diese Synchronisationsgraphen sind eine spezielle Klasse der zeitbehafteten Ereignisdiskreten Systemen, deren dynamisches Verhalten nur durch Synchronisations-und Sättigungsphänomene gekennzeichnet ist. Ein Vorteil dieser Synchronisationsgraphen gegenüber vielen anderen Modellen besteht darin, dass ihr schnellstes Verhalten durch lineare Gleichungen in einigen "tropischen" Algebren, den sogenannten Dioiden, beschrieben werden kann. Dies hat zu der Entwicklung einer umfangreichen Theorie für lineare Systeme in Dioiden geführt, wobei viele Konzepte aus der Standard Systemtheorie auf Synchronisationsgraphen übertragen wurden. Zum Beispiel die (max,+) Algebra biete elegante Analyseverfahren und Reglerentwurfsverfahren für Synchronisationsgraphen. Allerdings ist die Systemklasse, die mit Hilfe von Synchronisationsgraphen beschrieben werden kann, eingeschränkt. Zum Beispiel lassen sich Fertigungsanlagen mit Gruppierungs-oder Vereinzelungsschritten nicht mit Synchronisationsgraphen modellieren. Daher wurden einige Erweiterungen für Synchronisationsgraphen eingeführt. Zum einen wurden die Kanten von Synchronisationsgraphen mit ganzzahligen Gewichten erweitert. Diese gewichteten Synchronisationsgraphen weisen im Gegensatz zu gewöhnlichen Synchronisationsgraphen ereignisvariantes Verhalten auf und ermöglichen es nun Gruppierungsoder Vereinzelungsschritte zu beschreiben. Des Weiteren wurde eine neue Art der Synchronisation namens partieller Synchronisation (PS) eingeführt. Diese PS ist nützlich für die Modellierung von zeitvarianten Systemen, bei denen bestimmte Ereignisse nur in einem bestimmten Zeitfenster auftreten können. Ein solches Verhalten tritt zum Beispiel an einer Kreuzung mit Ampelsteuerung auf, die Grünphase der Ampeln beschreibt das Zeitfenster, in dem ein Fahrzeug die Kreuzung überqueren darf.

Aufgrund ihres ereignisvarianten bzw. zeitvarianten Verhalten können gewichteten Synchronisationsgraphen sowie Synchronisationsgraphen unter PS nicht mehr mit linearen Gleichungen in der (max,+) Algebra beschrieben werden. In dieser Arbeit werden gewichteten Synchronisationsgraphen und Synchronisationsgraphen unter PS in Dioiden modelliert. Basierend auf dieser Modellierung wird eine Zerlegung des dynamischen Verhaltens von gewichteten Synchronisationsgraphen in einen ereignisvarianten und einen ereignisinvarianten Teil vorgestellt. Analog wird für Synchronisationsgraphen unter PS gezeigt, dass ihr dynamisches Verhalten in einem zeitvarianten und zeitinvarianten Teil zerlegt werden kann. Unter speziellen Voraussetzungen wird gezeigt, dass dieser ereignisvarianten bzw. zeitvarianten Teile "invertierbar" ist. Dies ermöglicht die Übertragung von etablierten Analyseund Regelungsentwurfsverfahren von gewöhnlichen Synchronisationsgraphen auf die allgemeineren Klassen der gewichteten Synchronisationsgraphen und Synchronisationsgraphen unter PS. viii

Résumé

De nombreux systèmes de production peuvent être modélisés et analysés à l'aide de graphes d'événements temporisés (GET). Les GET forment une classe de systèmes à événements discrets temporisés (SEDT), dont la dynamique est définie uniquement par des phénomènes de synchronisation et de saturation. Un avantage majeur des GET par rapport à d'autres classes de SEDT est qu'ils admettent, sous certaines conditions, un modèle linéaire dans des espaces algébriques particuliers : les dioïdes. Ceci a conduit au développement d'une théorie des systèmes linéaires dans les dioïdes, grâce à laquelle de nombreux concepts de l'automatique classique ont été adaptés aux GET. Par exemple, l'algèbre (max,+) (i.e., le dioïde basé sur les opérations (max,+)) offre des techniques élégantes pour l'analyse et le contrôle de GET. Cependant, les conditions nécessaires pour modéliser un système à événements discrets par un GET sont très restrictives. Pour élargir la classe de systèmes concernés, deux extensions principales ont été développées. D'une part, les GET valués ont été introduits pour décrire des phénomènes d'assemblage et de séparation dans les systèmes de production. Cette extension se traduit par l'association de coefficients entiers aux arrêtes d'un graphe d'événements. Contrairement aux GET, ces systèmes ne sont pas invariants par rapport aux événements et ne peuvent donc pas être décrits par des équations linéaires dans l'algèbre (max,+). D'autre part, la synchronisation partielle (PS) a été introduite pour modéliser des systèmes dans lesquels certains événements ne peuvent se produire que pendant des intervalles prédéfinis. Par exemple, dans une intersection réglée par un feu tricolore, une voiture peut traverser l'intersection lorsque le feu est vert. Contrairement aux GET, ces systèmes ne sont pas invariants dans le domaine temporel et ne peuvent donc pas être décrits par des équations linéaires dans l'algèbre (max,+). Dans cette thèse, une modélisation des GET valués et des GET avec PS dans des dioïdes adaptés est présentée. A l'aide de ces dioïdes, une décompostion pour les GET valués (resp. GET avec PS) en un GET et une partie non-invariante dans le domaine des événements (resp. dans le domaine temporel) est introduite. Sous certaines conditions, la partie invariante est "invertible". Dans ce cas, les modèles et contrôleurs pour le GET valué ou le GET sous PS peuvent être directement dérivés des modèles et contrôleurs obtenus pour le GET associé. ix Publications Some ideas, results and figures have appeared previously in the following publications. The novel ideas, results and writing of publications [START_REF] Trunk | Model decomposition of weight-balanced timed event graphs in dioids: Application to control synthesis[END_REF][START_REF] Trunk | Output reference control for weightbalanced timed event graphs[END_REF][START_REF] Trunk | Model decomposition of timed event graphs under partial synchronization in dioids[END_REF][START_REF] Trunk | Modelling and control of periodic time-variant event graphs in dioids. Discrete Event Dynamic Systems[END_REF][START_REF] Trunk | Output reference control of timed event graphs under partial synchronization[END_REF] came from the first author. The second, third and fourth author provided a critical review to these publications and gave guidance on research direction.

Introduction

Discrete Event Systems (DESs), e.g. [START_REF] Cassandras | Introduction to Discrete Event Systems[END_REF], are systems where the dynamic behaviors are described by the occurrence of asynchronous discrete events. This class of systems is useful to model man-made systems -such as complex manufacturing lines, computer networks, and transportation networks -on a high level of abstraction. Typically, signals of such systems take discrete values that mostly belong to countable sets; for instance, the state of a machine could be busy, idle or broken. Furthermore, state changes are given by asynchronous events. For example, an operator can start a working process when the machine state is idle. At this particular time, the state of the machine changes from idle to busy. Many different modeling approaches have been introduced for DESs, among which are Petri nets and finite-state automata. These models give a formal way to describe how events are related to each other. Besides the logical order in which the events occur, in many applications, the time which elapses between consecutive events is important. In this case, the dynamic behavior of the system is described by timed DESs, e.g. by timed Petri nets or timed finite-state automata.

This thesis focuses on a particular class of timed DESs, where the dynamic behaviors are only governed by synchronization phenomena. Synchronization is essential in many systems; for instance, in public transportation networks, at a train station, the departure of trains may be synchronized with the arrival of other trains. In manufacturing systems, in order to start a task, the raw material is needed, and the required production machines must be ready. In a computer system, to perform a computation, data and the processing unit must be available. The time behavior of those systems can be naturally described by a subclass of timed Petri nets called Timed Event Graphs (TEGs). More precisely, TEGs are timed Petri nets where each place has exactly one upstream and one downstream transition and all arcs have the weight 1. An advantage of TEGs over the more general class of timed Petri nets is that the evolution of events can be described by recursive linear equations in a tropical algebra called (max,+)-algebra [START_REF] Heidergott | Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications[END_REF], or more generally in dioids [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF]. Within the last decades, this has led to the development of a broad theory of linear systems in dioids, including many methods for performance evaluation and controller synthesis. E.g. throughput analysis for TEGs can be stated as an eigenvalue problem in the (max,+)-algebra. The transfer function of a TEG is described by an ultimately cyclic series in a specific dioid called pM ax in vγ, δw , ', bq [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF]. Moreover, many control methods for linear systems in dioids have been studied, among which are: optimal feedforward control [START_REF] Cohen | Algebraic tools for the performance evaluation of discrete event systems[END_REF][START_REF] Menguy | Just-in-time control of timed event graphs: update of reference input, presence of uncontrollable input[END_REF], state and output feedback control [START_REF] Gaubert | Resource optimization and (min,+) spectral theory[END_REF][START_REF] Cottenceau | Model reference control for timed event graphs in dioids[END_REF][START_REF] Maia | Optimal closed-loop control of timed event graphs in dioids[END_REF][START_REF] Maia | On the model reference control for max-plus linear systems[END_REF][START_REF] Hardouin | Discrete-Event Systems in a Dioid Framework: Control Theory[END_REF] as well as observer based control [START_REF] Hardouin | Observer design for (max,+) linear systems[END_REF][START_REF] Hardouin | Observer-based controllers for max-plus linear systems[END_REF]. Moreover, in [START_REF] Schutter | Model predictive control for max-plus-linear discrete event systems[END_REF][START_REF] Schutter | Mpc for discrete-event systems with soft and hard synchronization constraints[END_REF], model predictive control for (max,+)-linear systems was introduced. It was also shown that the obtained results are suitable to handle scheduling problems in complex realworld systems. For instance, in [START_REF] Brunsch | Modeling and control of complex systems in a dioid framework[END_REF] dioid theory was applied to the modeling and the control of high throughput screening systems. These systems are used in the field of drug discovery of chemical and biological industries.

However, TEGs are quite restrictive in terms of their modeling capabilities. To enrich the model properties it is reasonable to consider weights (values in N " t1, 2, ̈̈̈u) on the arcs of TEGs. This leads to Weighted Timed Event Graphs (WTEGs), which have clearly more expressiveness and allow us to describe a wider class of systems. The weights are suitable to express batch (resp. split) processes; for instance, when several occurrences of events are needed to induce a following event or when one event can result in several following events. Clearly, such batch and split processes are quite common in many manufacturing systems; for instance, when a workpiece is cut into several parts. Another example in the field of computer science is provided by data streams in multirate digital signal processing. The weights are suitable to model data flow caused by up-and down-sampling. Unlike TEGs, WTEGs have an event-variant behavior and cannot be described by (min,+)-linear or (max,+)-linear systems anymore [START_REF] Cohen | Timed-event graphs with multipliers and homogeneous min-plus systems[END_REF]. Another restrictive property of TEGs is that they can only represent time-invariant systems. In order to describe time-variant behavior, in [START_REF] David-Henriet | Modeling and control for max-plus systems with partial synchronization[END_REF], a new form of synchronization, called partial synchronization (PS), has been introduced for TEGs. Such a partial synchronization is useful to describe systems where particular events can only occur in a specific time window. To motivate the practical relevance, let us consider an intersection controlled by a traffic light. A vehicle which arrives at the traffic light can only cross when the traffic light is green. If the vehicle arrives in the red phase, it has to wait for the next green phase. Therefore, the vehicle is delayed by a time that depends on its time of arrival at the intersection. The traffic light control causes a time-variant behavior which cannot be modeled by an ordinary TEG. In this thesis, dioid theory is applied to study the behavior of WTEGs as well as the behavior of TEGs under PS. Moreover, results for control synthesis of TEGs are generalized to the more general classes of WTEGs and TEGs under PS.

Motivation

A TEG can be conveniently modeled as a linear system over some dioids. For this, a counter function x : Z Ñ Z min , with Z min " ZYt̆8u, is associated with each transition giving the accumulated number of firings up to a time t. Using the particular dioid pM ax in vγ, δw , ', bq, it is straightforward to obtain transfer functions for TEGs. E.g., the earliest firing relation between an input transition and an output transition of a TEG is modeled by an ultimately cyclic series h P M ax in vγ, δw. This transfer function h maps an input counter function into an output counter function, which are respectively associated with the input transition and output transition of the TEG. The dioid pM ax in vγ, δw , ', bq was formally introduced in [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Cohen | Algebraic tools for the performance evaluation of discrete event systems[END_REF] and is based on the event-shift operator γ ν and time-shift operator δ τ with τ, ν P Z.

These operators map counter functions to counter functions in the following way: pγ ν xq ptq " xptq ̀ν and pδ τ xq ptq " xpt ́τq.

(1.1)

Time-shift operators model holding times associated with places and event-shift operators model initial markings of the places. For instance, see Figure 1.1, where x 1 and x 2 are associated with transition t 1 and t 2 . operators model the earliest behavior between input transition t 1 and output transition t 2 in the TEGs above. The holding time of two time units is modeled by the δ 2 operator and the three initial tokens by the γ 3 operator.

t 1 t 2 p 1 2 t 1 t 2 p 1 x 1 ptq
Moreover, by considering sums and compositions of these operators it is possible to describe the complete dynamic behavior of an ordinary TEG. As in conventional systems theory, transfer functions are convenient to solve some control problems. For instance, model reference control introduced for TEGs in [START_REF] Libeaut | Model matching for timed event graphs[END_REF][START_REF] Cottenceau | Model reference control for timed event graphs in dioids[END_REF][START_REF] Maia | Optimal closed-loop control of timed event graphs in dioids[END_REF] and [START_REF] Hardouin | Discrete-Event Systems in a Dioid Framework: Control Theory[END_REF] needs such an input-output representation in the dioid pM ax in vγ, δw , ', bq. Usually, the reference model describes the desired behavior and is as well specified in the dioid pM ax in vγ, δw , ', bq. To enforce this behavior, a controller is computed such that the closed-loop behavior follows the behavior of the reference model as close as possible, but is not slower than the reference. Therefore, it is also known as a model matching control problem. This control method is of practical interest for manufacturing systems. For instance, we can specify the desired throughput behavior of a production line in a reference model. The controller obtained from this reference optimizes the production process under the "just-in-time" criterion while guaranteeing the specified throughput. Thus, materials spend the minimum required time in the production line, which leads to a reduction of internal stocks.

The aim of this thesis is to describe the transfer behavior of extended TEGs, namely WTEGs and TEGs under PS, with a similar set of operators. This is necessary to extend the result for model reference control to the more general classes of WTEGs and TEGs under PS. In order to model the weights on the arcs in WTEGs, two new operators are considered, ] .

See Figure 1.2, for an example of how these operators can be used to manipulate a counter function. The dynamic behavior of a WTEG can then be described by sums and compositions of the operators tγ ν , δ τ , µ m , β b u in a dioid called pErrδss, ', bq.

To model the behavior of TEGs under PS, it is more convenient to associate dater functions instead of counter functions, with transitions. A dater function is a function x : Z Ñ Z max , with Z max " ZYt̆8u, where xpkq gives the time when the transition fires for the pk̀1q st time. To model periodic time-variant phenomena with dater functions, a new operator is introduced, i.e., for ω P N p∆ ω|ω xqpkq " rxpkq{ωsω.

Observe that this operator models a synchronization of the dater function with times t P tωk |k P Z max u. For instance, see Figure 1.3 where the operator ∆ 3|3 is applied to a dater function x 1 , thus the values ∆ 3|3 px 1 qpkq P t3k |k P Z max u. Therefore, with the ∆ 3|3 operator, we can model the earliest functioning of the TEG under PS given in Figure 1. [START_REF] Blyth | Residuation theory. International series of monographs in pure and applied mathematics[END_REF], where the PS of transition t 2 is given by a signal S 2 : Z Ñ t0, 1u where S 2 ptq " 1 for t P t3k |k P Zu and 0 otherwise. This signal enables the firing of transition t 2 at time t P Z where S 2 ptq " 1.

The dynamic behavior of a subclass of TEGs under PS, i.e. the class where PS of transitions are given by periodic signals, can be modeled by sums and compositions of the operators tγ ν , δ τ , ∆ ω|ω u in a dioid called pT rrγss, ', bq.

Related Work Weighted Timed Event Graph

For manufacturing systems and embedded applications, buffer size, throughput, and latency times are key features which can be analyzed and optimized. In general, we want to maximize the production rate (or data throughput) while keeping buffer size as small as possible. This kind of optimization problems have been widely studied in the context of WTEGs. Note that WTEGs are also referred to as Timed Weighted Marked Graphs and Timed 1. Introduction sition with one place and one token in the place is implicitly assumed. As a consequence, the considered models are a subclass of WTEGs, where it is assumed that a transition can potentially fire infinitely often concurrently.

Marking optimization for WTEGs are studied in [START_REF] Sauer | Marking optimization of weighted marked graphs[END_REF][START_REF] Toursi | Branch and bound approach for marking optimization problem of weighted marked graphs[END_REF][START_REF] He | Optimization of deterministic timed weighted marked graphs[END_REF][START_REF] He | Performance optimization for timed weighted marked graphs under infinite server semantics[END_REF][START_REF] He | An improved approach for marking optimization of timed weighted marked graphs[END_REF]. One main problem is to determine a minimal admissible marking for a given WTEG such that a given throughput is guaranteed. In the context of manufacturing systems, for instance, this yields a minimization of internal buffer sizes in an assembly line. In [START_REF] Toursi | Branch and bound approach for marking optimization problem of weighted marked graphs[END_REF], the problem is addressed based on a branch and bound algorithm. In [START_REF] Sauer | Marking optimization of weighted marked graphs[END_REF] and [START_REF] He | Optimization of deterministic timed weighted marked graphs[END_REF], heuristic methods are presented. In [START_REF] He | Performance optimization for timed weighted marked graphs under infinite server semantics[END_REF], the heuristic methods are compared to the optimal approach which is based on the transformation given in [START_REF] Nakamura | Cycle time computation in deterministically timed weighted marked graphs[END_REF] and has high complexity.

Dioid models of WTEG

For ordinary TEGs, it is known that their behavior can be described by linear equations over some dioids (or idempotent semirings) [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Heidergott | Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications[END_REF]. In [START_REF] Cohen | Timed-event graphs with multipliers and homogeneous min-plus systems[END_REF] and [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF], dioids based on a specific set of operators are introduced to describe the dynamic behavior of WTEGs. In [START_REF] Cohen | Timed-event graphs with multipliers and homogeneous min-plus systems[END_REF], a fluid version of WTEGs is investigated for which recurrent equations are obtained. Fluid WTEGs can be seen as continuous approximations of the WTEGs discussed in this thesis. A linearization is introduced for fluid WTEGs. Therefore, the behavior of a fluid WTEG can be analyzed by a (min,+)-linear system and approximate results can be obtained for the original WTEG. However, in some cases, the results obtained for the fluid WTEG are quite far from the original WTEG, for instance, a WTEG which is blocking may have a fluid approximation which is alive. In [START_REF] Hamaci | On modeling and control of discrete timed event graphs with multipliers using (min,+) algebra[END_REF][START_REF] Hamaci | Modeling and control of hybrid timed event graphs with multipliers using (min, +) algebra[END_REF], "just-in-time" control for WTEGs are studied in a similar dioid of operators, called pD min vδw , ', bq. In [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF][START_REF] Cottenceau | Modeling of time-varying (max,+) systems by means of weighted timed event graphs[END_REF], a slightly different dioid is introduced to describe the dynamic behavior of WTEGs. This dioid is denoted pErrδss, ', bq and based on the operators tγ ν , δ τ , µ m , β b u. In these works, an important subclass of WTEGs -the class of WTEGs where parallel paths have balanced weights -are studied. This class is therefore called Weight-Balanced Timed Event Graphs (WBTEGs). It is shown that the input-output behavior of WBTEGs can be described by ultimately cyclic series in this dioid. Subsequently, based on these series an interpretation of the impulse response for WBTEGs is given [START_REF] Cottenceau | Modeling of time-varying (max,+) systems by means of weighted timed event graphs[END_REF] and some model matching control problems for WBTEGs are addressed [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF][START_REF] Trunk | Model decomposition of weight-balanced timed event graphs in dioids: Application to control synthesis[END_REF].

Synchronous Data-Flow (SDF) Graphs

In the field of computer science, an equivalent graphical representation for WTEGs is known as SDF Graphs [START_REF] Sriram | Embedded Multiprocessors: Scheduling and Synchronization[END_REF]. In this model, edges are associated with places, actors are associated with transitions and data exchange between actors are associated with tokens. These graphs were introduced in [START_REF] Lee | Synchronous data flow[END_REF][START_REF] Lee | Static scheduling of synchronous data flow programs for digital signal processing[END_REF] to model data flow in digital signal processing applications. They are useful tools to obtain, optimize and verify scheduling algorithms for parallel processing [START_REF] Geilen | Synchronous dataflow scenarios[END_REF]. Moreover, SDF Graphs are suitable to obtain performance bounds for the underlying systems. Clearly, an important performance indicator is the throughput of a sys-tem, i.e., the maximal rate at which a system produces an output. Unsurprisingly, lots of research focuses on throughput analysis of SDF Graphs. In [START_REF] Ghamarian | Throughput analysis of synchronous data flow graphs[END_REF][START_REF] Stuijk | Exploring trade-offs in buffer requirements and throughput constraints for synchronous dataflow graphs[END_REF], an algorithm is introduced to explore the state space of an SDF Graph. The basic idea is to obtain the throughput based on the simulation of the SDF Graph. In [START_REF] De Groote | Max-plus algebraic throughput analysis of synchronous dataflow graphs[END_REF], an approach is presented based on the (max,+)-algebra. Buffer size minimization, with respect to throughput constraints, for SDF Graphs have been studied in [START_REF] Geilen | Minimising buffer requirements of synchronous dataflow graphs with model checking[END_REF]. Clearly, minimizing buffer size is important for embedded systems due to the high costs for memory.

Time-variant Timed Event Graphs

Time-varying DESs have been studied in [START_REF] Brat | Analyzing non-deterministic real-time systems with (max,+) algebra[END_REF][START_REF] Brat | A (max,+) Algebra for Non-Stationary Periodic Timed Discrete Event Systems[END_REF][START_REF] Cofer | A generalized max-algebra model for performance analysis of timed and untimed discrete event systems[END_REF][START_REF] Komenda | Max-plus algebra and discrete event systems[END_REF]. The models considered in these works are TEGs in which holding times of places change periodically based on event sequences. Therefore, these systems can describe event-variant time behaviors. For these TEGs, places must respect a first-in-first-out (FIFO) behavior, in other words, tokens must not overtake each other. In [START_REF] Komenda | Max-plus algebra and discrete event systems[END_REF], optimal feedforward control problems for these systems are studied. In [START_REF] Cottenceau | Modeling of time-varying (max,+) systems by means of weighted timed event graphs[END_REF], it is shown that the input-output behavior of these systems can be represented by WTEGs. Another class of time-variant DESs has been discussed in [START_REF] David-Henriet | Modeling and control for max-plus systems with partial synchronization[END_REF][START_REF] David-Henriet | Discrete event systems with standard and partial synchronizations[END_REF]. There, TEGs are extended by allowing a weaker form of synchronization, called partial synchronization (PS). PS of a transition means that the transition can only fire when it is enabled by an external signal S : Z Ñ t0, 1u. S enables the firing of the transition at times t P Z where Sptq " 1. Such time-variant behaviors occurring in TEG under PS can be modeled as a (max,+)-linear systems under additional constraints [START_REF] David-Henriet | Modeling and control for (max, +)-linear systems with set-based constraints[END_REF]. In the case where such signals are predefined and ultimately periodic, it is possible to obtain transfer functions for TEGs under PS [START_REF] David-Henriet | Modeling and control for (max, +)-linear systems with set-based constraints[END_REF][START_REF] David-Henriet | Discrete event systems with standard and partial synchronizations[END_REF]. Moreover, some control problems for TEGs under PS have been tackled in [START_REF] David-Henriet | Modeling and control for (max, +)-linear systems with set-based constraints[END_REF][START_REF] David-Henriet | Model predictive control for discrete event systems with partial synchronization[END_REF]. A similar extension was introduced in [START_REF] Schutter | Mpc for discrete-event systems with soft and hard synchronization constraints[END_REF], where TEGs with hard and soft synchronization are studied.

Contribution

The main contribution of this work relates to modeling and control of extended TEGs, namely Weighted Timed Event Graphs (WTEGs) and Periodic Time-variant Event Graphs (PTEGs), in dioids. First based on dioid theory, a decomposition model for consistent WTEGs is introduced, in which the event-variant and the event-invariant parts are separated. It is shown that the event-variant part is "invertible", thus many tools developed for analysis and control of ordinary TEGs can be directly applied to the more general class of consistent WTEGs. In particular, based on this model decomposition, optimal feedforward control and model matching control for TEGs are generalized to WTEGs. Second, to describe the time-variant behavior of some DESs, Periodic Time-variant Event Graphs (PTEGs) are introduced. PTEGs are an alternative model to TEGs under PS to describe periodic time-variant behaviors. In PTEGs, holding times of places depending on the firing times of their upstream transitions. More precisely, the holding time Hptq is time-variant and immediately periodic, 1. Introduction i.e. Hpt ̀ωq " Hptq. The current delay is then determined by the firing time t of the corresponding upstream transition. In contrast to FIFO TEGs considered in [START_REF] Komenda | Max-plus algebra and discrete event systems[END_REF], which are event-variant, PTEGs have a time-variant behavior. However, in PTEGs places must respect a FIFO behavior as well which implies a constraint on holding time values. A comparison between TEGs under PS and PTEGs is provided. The input-output behavior of PTEGs can be described by ultimately cyclic series in a new dioid denoted pT rrγss, ', bq. Similarly, it is shown how TEGs under periodic PS can be modeled in this dioid pT rrγss, ', bq.

As for consistent WTEGs with a dioid model in pErrδss, ', bq, a decomposition for series in T rrγss is introduced, where the time-invariant part can be separated from the time-variant part. The time-variant part is "invertible", therefore many problems concerning performance analysis and control synthesis for PTEGs (resp. TEGs under periodic PS) can be reduced to the case of an ordinary TEG, and solved efficiently by applying the already established tools for TEGs. Especially, optimal feedforward control and model reference control for PTEGs (resp. TEGs under periodic PS) are studied. Based on the dioids pErrδss, ', bq and pT rrγss, ', bq similarities between WTEGs and PTEGs (resp. TEGs under periodic PS) are investigated. Finally, the results for WTEGs and PTEGs (resp. TEGs under periodic PS) can be combined, so that a class of periodic time-variant and event-variant TEGs can be handled in a new dioid structure. These TEGs can model synchronization, time delay, batch/split and also some periodic time-variant behavior which, for instance, arises in traffic light control.

Outline

This thesis is structured in two parts, Chapter 2, Chapter 3, Chapter 4 and Chapter 5, introducing the dioids pM ax in vγ, δw , ', bq, pErrδss, ', bq, pT rrγss, ', bq and pET , ', bq, respectively. In Chapter 6 and Chapter 7, these dioids are then applied to the modeling and the control of WTEGs, TEGs under PS and PTEGs.

Part 1 Algebraic Tools

Chapter 2 summarizes fundamentals of dioids and residuation theory. The chapter begins with explaining the general properties of dioids and recalls the (max,+)-and (min,+)-algebra. Then more sophisticated dioid structures such as dioids of formal power series are given. Moreover, residuation theory is introduced to give an approximate inverse of some mappings defined over complete dioids. Finally, the particular dioid pM ax in vγ, δw , ', bq is recalled, which is useful to analyze TEGs and plays a key role in this thesis.

Chapter 3 introduces the dioid pErrδss, ', bq. This dioid is based on the operators tγ ν , δ τ , µ m , β b u. Moreover, in Section 3.3, it is shown that under some conditions all relevant operations p', b, z, {q on elements in Errδss can be reduced to operations on matrices with entries in M ax in vγ, δw.

Chapter 4 introduces the dioid pT rrγss, ', bq. This dioid is comprised of the basic operators tγ ν , δ τ , ∆ ω|ω u. This dioid is used to model the time-variant behavior of PTEGs, and TEGs under PS. As for the dioid pErrδss, ', bq, it is shown that under some conditions all relevant operations p', b, z, {q on elements in T rrγss can be reduced to operations on matrices with entries in M ax in vγ, δw.

Chapter 5 combines the results obtained in Chapter 3 and Chapter 4. The dioid pET , ', bq is introduced, which can be seen as the combination of the dioids pErrδss, ', bq and pT rrγss, ', bq. This permits the description of event-variant and time-variant behaviors in the same dioid structure. Therefore, it is applicable for the modeling and the control of WTEG under PS.

Part 2 Modeling and Control

Chapter 6 shows how the earliest behavior of TEGs, WTEGs, PTEGs, and TEGs under PS can be modeled in a dioid structure. In particular, the input-output behavior of a WTEG can be modeled by a matrix where the entries are ultimately cyclic series in Errδss. These transfer function matrices are used to compute the output for a given input of a system. Subsequently, the relation between the transfer function and the impulse response of a system is elaborated. Similar to WTEGs, the input-output behavior of PTEGs and TEGs under PS are modeled by ultimately cyclic series in T rrγss. Moreover, an interpretation of the impulse response is given for these systems. In the last part of this chapter, the modeling of WTEGs under PS in the dioid pET , ', bq is addressed.

Chapter 7 generalizes some control approaches already introduced for ordinary TEGs to the more general classes of WTEGs, PTEGs, and TEGs under PS. The control problems are stated in a dioid framework and are efficiently solved by applying residuation theory. In particular, optimal control and model reference control are investigated.

2

Mathematical Preliminaries

This chapter introduces the basic mathematical concepts needed to understand this thesis. In particular, dioid and residuation theory are recalled. Dioids are suitable to obtain linear models for particular DESs where dynamic behaviors are only governed by synchronization and saturation phenomena. Furthermore, residuation theory has an application in the controller design process and the performance evaluation of DESs modeled in a dioid setting. Most of the following results are taken from the literature, especially from [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF]. For a broader overview on dioids and residuation theory, see [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Blyth | Residuation theory. International series of monographs in pure and applied mathematics[END_REF][START_REF] Cohen | Algèbres Max-Plus et applications en informatique et automatique[END_REF][START_REF] Cohen | Algebraic tools for the performance evaluation of discrete event systems[END_REF][START_REF] Heidergott | Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications[END_REF].

Dioid Theory

Definition 1 (Monoid). A monoid is a set M endowed with a binary associative operation ànd an identity element 0 such that @a P M, a ̀0 " 0 ̀a " a. A monoid is denoted by pM, ̀, 0q.

A monoid pM, ̀, 0q is said to be commutative if the binary operation ̀is commutative. And a commutative monoid is said to be idempotent if ̀is idempotent, i.e., @a P M, àa " a.

Definition 2 (Dioid). A dioid is a set D endowed with two binary operations, denoted ' (called addition) and b (called multiplication), such that -' is associative, commutative and idempotent, i.e. @a P D, a ' a " a, moreover ' admits a neutral element denoted ε. b is associative, distributive over ' and b admits a neutral element denoted e.

ε is absorbing for b, i.e., @a P D, a b ε " ε b a " ε. Moreover, ε is called the zero element and e is called the unit element of D. A dioid is denoted by pD, ', bq.

Clearly, let pD, ', bq be a dioid, then pD, ', εq is a commutative idempotent monoid and pD, b, eq is a monoid. If multiplication b is commutative, then dioid pD, ', bq is said to be commutative. Note that, as in conventional algebra, the multiplication symbol b is often omitted.

Example 1 ((max,+)-algebra pZ max , ', bq). The (max,+)-algebra is the set Z max :" Z Y t́8u endowed with max as addition ' and ̀as multiplication b, e.g., 5 b 4 ' 7 " maxp5 4, 7q " 9. Moreover, the zero element is ε " ́8 and the unit element is e " 0, respectively.

Example 2 ((min,+)-algebra pZ min , ', bq). Conversely, the (min,+)-algebra is the set Z min :" Z Y t8u endowed with min as addition ' and ̀as multiplication b, e.g., 5 b 4 ' 7 " minp5 ̀4, 7q " 7. The zero element is ε " 8 and the unit element is e " 0, respectively.

Example 3 (Boolean Dioid pB, ', bq). The set B " tε, eu, consisting of the zero and the unit element, with the two binary operations addition ' and multiplication b constitute the Boolean dioid. Since the zero element ε is absorbing for b and neutral for ', the operations ' and b are defined by ε b e " e b ε " ε and ε ' e " e ' ε " e.

Definition 3 (D-Semimodule [START_REF] Nola | Semiring and semimodule issues in mv-algebras[END_REF]). Let pD, ', bq be a dioid with unit element e and zero element ε. A D-semimodule is a commutative monoid pM, ̀, 0q with an external operation ̈: D̂M Ñ M, pa, xq Þ Ñ äx, called scalar-multiplication, such that the following conditions hold @a, b P D and @x, y P M pa b bq ̈x " a ̈pb ̈xq, a ̈px ̀yq " pa ̈xq ̀pa ̈yq, pa ' bq ̈x " pa ̈xq ̀pb ̈yq, ε ̈x " a ̈0 " 0, e ̈x " x.

Subdioids

Definition 4 (Subdioid). Let pD, ', bq be a dioid with unit element e and zero element ε, then a subset S of D is a subdioid of pD, ', bq if e, ε P S and S is closed for b and ', that is @a, b P S, a ' b P S and a b b P S.

Example 4. Consider the dioid pZ max , ', bq, the dioid pN max , ', bq with N max " N 0 Ý8, is a subdioid of pZ max , ', bq.

Order Relation in Dioids

An order relation ĺ on a set S is a binary relation which is reflexive, i.e., @a P S, a ĺ a, transitive, i.e., @a, b, c P S, a ĺ b and b ĺ c ñ a ĺ c and anti-symmetric, i.e., @a, b P S, a ĺ b and b ĺ a ñ a " b. A set S is called totally ordered if for every pair of elements a, b P S we can either write a ľ b or a ĺ b. Moreover, if a pair of elements a, b P S exists, for which a ń b, a ł b, the set S is called partially ordered.

The idempotent characteristic of the addition induces a canonical order relation on dioids. Let pD, ', bq be a dioid, then @a, b P D, the relation ĺ defined by

@a, b P D, a ' b " b ô a ĺ b, (2.1)
is an order relation. In general in a dioid pD, ', bq with a, b P D, the sum a ' b is not equal to either a or b. Thus, general dioids are only partially ordered, i.e., a ń b, a ł b.

However, the sum a ' b P D gives a natural upper bound for the set ta, bu. Therefore, with ε as bottom element, i.e. @a P D, a ľ ε a dioid is an ordered set.

Complete Dioids

Definition 5 (Complete Dioid). A dioid pD, ', bq is said to be complete if it is closed for infinite sums and if b distributes over infinite sums, i.e., for all subsets S of D and for all a P D,

a b ̃à bPS b ̧" à bPS pa b bq, ̃à bPS b ̧b a " à bPS pb b aq. Remark 1.
Similarly, an idempotent commutative monoid pM, ', εq is said to be complete if it is closed for infinite sums.

A complete dioid pD, ', bq admits a top element J " À aPD a P D which is given by the sum over all elements in the dioid. Furthermore, in a complete dioid the infimum operator is defined as, a, b P D,

a ^b " à tx P D|x ' a ĺ a, x ' b ĺ bu.
The ^operator is associative, commutative, idempotent and admits J as neutral element, i.e., @a P D, a ^J " a. Then, for complete dioids the ^operation defines a lower bound for the set ta, bu. Thus for a complete dioid pD, ', bq with a, b P D,

a ľ b ô a " a ' b ô b " a ^b.
One can show that a complete dioid equipped with ^and J is a complete lattice, for a more exhaustive description see [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Blyth | Lattices and ordered algebraic structures[END_REF]. Note that in general for a partially ordered dioid pD, ', bq multiplication is not distributive over ^, but one can show that for a, b, c P D, cpa ^bq ĺ ca ^cb and pa ^bqc ĺ ac ^bc.

(2.2) Furthermore, distributivity of ^with respect to ' and conversely ' with respect to ^is not given either. However, for a, b, c P D, the following inequalities are satisfied, pa ^bq ' c ĺ pa ' cq ^pb ' cq, pa ' bq ^c ľ pa ^cq ' pb ^cq.

Example 5. The (max,+)-algebra extended with the top element J " 8 is a complete dioid. Since the zero element ε is absorbing for multiplication one has, J b ε " ε or differently ́8 b 8 " ́8. This dioid is denoted by (Z max , ', b), with Z max " Z Y t́8, ̀8u.

Conversely, the (min,+)-algebra with J " ́8 is a complete dioid, denoted by (Z min , ', b), with Z min " Z Y t́8, ̀8u.

Example 6. The Boolean dioid pB, ', bq is a complete dioid where the top element is equal to the unit element, i.e., J " e.

Mathematical Preliminaries

Kleene Star Definition 6. Let pD, ', bq be a complete dioid, the Kleene star of an element a P D is defined as, a ̊" 8 à i"0 a i , where a 0 " e and a ì1 " a b a i .

Theorem 2.1 ([1]

). In a complete dioid pD, ', bq with a, b P D, x " a ̊b is the least solution of the implicit equation x " ax ' b.

The Kleene Star satisfies the following relations, for a complete dioid pD, ', bq with a, b P D

pa ̊q̊" a ̊, (2.3)
a ̊å" a ̊, (2.4
)

apbaq ̊" pabq ̊a, (2.5)
pa ' bq ̊" pa ̊bq ̊å" b ̊pab ̊q̊, (2.6)

pab ̊q̊" e ' apa ' bq ̊. (2.7)
Furthermore, for a commutative complete dioid pD, ', bq, with a, b P D, ab " ba, pa ' bq ̊" a ̊b̊.

(2.8)

For the proofs of these relations see [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF].

Rational Closure

Definition 7 (Rational closure). Let S be a subset of a complete dioid pD, ', bq, such that S contains the zero and unit elements ε and e. The rational closure of S, denoted by S ̊, is the least subdioid of pD, ', bq containing all finite combinations of sums, products, and Kleene stars over S. The subset S is rationally closed if S " S ̊.

Matrix Dioids

Addition ' and multiplication b can be extended to matrices with entries in a dioid pD, ', bq. For matrices A, B P D m̂n , C P D n̂q and a scalar λ P D, matrix addition and multiplication are defined by pA ' Bq i,j :" pAq i,j ' pBq i,j , (2.9)

pA b Cq i,j :" n à k"1 ̀pAq i,k b pCq k,j ̆, (2.10
)

pλ b Aq i,j :" λ b pAq i,j .
The order relation in the matrix case coincides with the element-wise order, i.e., for A, B P D m̂n , A ľ B iff @i, j pAq i,j ľ pBq i,j . The identity matrix, denoted by I, is a square matrix with e on the diagonal and ε elsewhere. The zero matrix, denoted by ε, has only ε entries.

Proposition 1 ([1]

). The set of square matrices, denoted D n̂n , with entries in a dioid pD, ', bq, endowed with (2.9) as addition and (2.10) as multiplication is a dioid denoted by pD n̂n , ', bq.

The unit and zero element is I and ε, respectively. Moreover, if pD, ', bq is complete then pD n̂n , ', bq is complete.

Remark 2. Note that non-square matrices can be included by adding additional rows (resp. columns) with ε.

Furthermore, if we assume that pD, ', bq is a complete dioid the Kleene star can be extended to square matrices A P D n̂n . For this, A P D n̂n is partitioned into sub-matrices as follows,

A " « B C D E ff ,
where B P D n 1 ̂n1 , C P D n 1 ̂n2 , D P D n 2 ̂n1 and E P D n 2 ̂n2 and n " n 1 ̀n2 . Then A can be written as

A ̊" « B ̊' B ̊CpDB ̊C ' Eq ̊DB ̊B̊C pDB ̊C ' Eq pDB ̊C ' Eq ̊DB ̊pDB ̊C ' Eq ̊ff . (2.11)
Clearly, if we assume A P D 2̂2 , then B, C, D, and E are scalars in D and the Kleene star of the matrix A is obtained by sum, product, and Kleene star operations between scalars. Thus for a square matrix A P D n̂n with arbitrary dimension, the star A ̊can be obtained in a recursive way. Additionally, for pD, ', bq a complete dioid the infimum operation is extended to matrices as follows, for A, B P D m̂n , pA ^Bq i,j " pAq i,j ^pBq i,j .

(2.12)

Quotient Dioids

Definition 8 (Congruence [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF]). A congruence relation in a dioid pD, ', bq is an equivalence relation R which satisfies @a, b, c P D,

aRb ñ $ ' ' ' & ' ' ' % pa ' cqRpb ' cq, pa b cqRpb b cq, pc b aqRpc b bq.
For a dioid pD, ', bq with an equivalence relation R the equivalence class of a P D is defined by ras R :" tb P D|aRbu.

Proposition 2 ([1]

). The quotient of a dioid pD, ', bq by a congruence relation R is again a dioid, denoted by pD R , ', bq, with addition and multiplication given by, ras R ' rbs R " ra ' bs R and ras R b rbs R " ra b bs R .

The zero element ε and unit element e in D R correspond to the equivalence classes rεs R and res R of D.

Remark 3. Let pD, ', bq be a complete (resp. commutative) dioid, then pD R , ', bq is a complete (resp. commutative) dioid.

Dioid of Formal Power Series

Definition 9 (Formal Power Series [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF](Chap. 4.7)). A formal power series in p commutative variables with coefficients in a dioid pD, ', bq is a mapping from Z p into D, i.e., s : Z p Ñ D. The variables are denoted by z 1 , ̈̈̈, z p and @k " pk 1 , . . . , k p q P Z p , spkq represents the coefficient of z k

1 1 . . . z kp p . An equivalent compact representation of s is s " à kPZ p spkqz k 1 1 . . . z kp p .
Definition 10 (Support, Degree, and Valuation). Support (supp), degree (deg) and valuation (val) of a formal power series s are defined as -supppsq " tk P Z p |spkq ‰ εu, -degpsq is the least upper bound of supppsq, -valpsq is the greatest lower bound of supppsq. A polynomial (resp. monomial) is a formal power series with finite support (resp. the support is reduced to only one element).

The set of formal power series with coefficients in a dioid pD, ', bq and variables z 1 , ̈̈̈, z p is denoted by D vz 1 , ̈̈̈, z p w. On this set addition ' is defined as, for s 1 , s 2 P D vz 1 , ̈̈̈, z p w, @k P Z p , ps 1 ' s 2 qpkq " s 1 pkq ' s 2 pkq.

(2.13)

Additionally, multiplication b is defined by the Cauchy product, thus

@k P Z p , ps 1 b s 2 qpkq " à ìj"k s 1 piq b s 2 pjq. (2.14) Proposition 3 ([1]
). Let pD, ', bq be a complete dioid, then the set D vz 1 , ̈̈̈, z p w, endowed with addition and multiplication defined by (2.13) and (2.14) is a complete dioid.

In [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF] it is shown that in general Prop. 3 holds only for complete dioids since the definition of the product (2.14) includes infinite sums. In the dioid pD vz 1 , ̈̈̈, z p w , ', bq the zero element εpkq is defined by, @k P Z p , εpkq " ε. Likewise, the unit element epkq in D vz 1 , ̈̈̈, z p w is defined as epkq "

$ & % e for k " 0 (the zero vector), ε otherwise.

The top element Jpkq in D vz 1 , ̈̈̈, z p w is defined by Jpkq " J, @k P Z p .

Since pD vz 1 , ̈̈̈, z p w , ', bq is a complete dioid the greatest lower bound of two series s 1 , s 2 P D vz 1 , ̈̈̈, z p w is given by @k P Z p , ps 1 ^s2 qpkq " s 1 pkq ^s2 pkq.

Moreover, if the dioid pD, ', bq is commutative and the variables z 1 , ̈̈̈, z p also commute, then the dioid pD vz 1 , ̈̈̈, z p w , ', bq is commutative as well.

Proposition 4 ([19]

). Let pS, ', bq be a complete subdioid of a complete dioid pD, ', bq, then pS vz 1 , ̈̈̈, z p w , ', bq is a complete subdioid of pD vz 1 , ̈̈̈, z p w , ', bq.

Mappings over Dioids

Definition 11. On a dioid pD, ', bq the identity mapping, denoted by Id D , is a mapping from D into itself defined as, @a P D, Id D paq " a.

Definition 12. Let f : D Ñ C be a mapping from a dioid pD, ', bq into a dioid pC, ', bq, then f is a '-morphism if @a, b P D, fpa ' bq " fpaq ' fpbq and fpεq " ε.

Definition 13. Let f : D Ñ C be a mapping from a dioid pD, ', bq into a dioid pC, ', bq, then f is a b-morphism if @a, b P D, fpa b bq " fpaq b fpbq and fpeq " e.

A mapping f is said to be a homomorphism if it is both a '-morphism and a b-morphism. A homomorphism f : D Ñ D is called an endomorphism. Furthermore, if f is a homomorphism and the inverse of f is defined and itself a homomorphism then f is called an isomorphism.

Definition 14 (Isotony).

A mapping f from a complete dioid pD, ', bq into a complete dioid pC, ', bq is called isotone (or order preserving) if @a, b P D, a ľ b ñ fpaq ľ fpbq.

Definition 15 (Antitony).

A mapping f from a complete dioid pD, ', bq into a complete dioid pC, ', bq is called antitone (or order reversing) if @a, b P D, a ľ b ñ fpaq ĺ fpbq.

Definition 16 (Lower semi-continuity). A mapping f from a complete dioid pD, ', bq into a complete dioid pC, ', bq is called lower semi-continuous if

@S Ď D, f ̃à aPS a ̧" à aPS fpaq.
Definition 17 (Upper semi-continuity). A mapping f from a complete dioid pD, ', bq into a complete dioid pC, ', bq is called upper semi-continuous if

@S Ď D, f ̃ľ aPS a ̧" ľ aPS fpaq.
A mapping f which is both, upper semi-continuous and lower semi-continuous is called continuous. A lower semi-continuous mapping f such that fpεq " ε is a '-morphism. Moreover, f is a '-morphism implies that f is an isotone mapping. Note that in general the opposite is not true, however, an isotone mapping f : D Ñ C satisfies @a, b P D, fpa ' bq ľ fpaq ' fpbq. In the particular case where f : D Ñ C is an isotone mapping and the dioid pD, ', bq is a totally ordered set, i.e., for a, b P D the sum a ' b is either equal to a or b, f is a '-morphism.

In analogy with the definition of endomorphism for dioids one can define endomorphism for a monoid pM, ', εq and lower semi-continuity for complete monoids. Definition 18. A mapping f : M Ñ M, from a monoid pM, ', εq into itself, is called an endomorphism if, @a, b P M, fpa ' bq " fpaq ' fpbq and fpεq " ε.

Definition 19. A mapping f : M Ñ M, from a complete monoid pM, ', εq into itself, is called lower semi-continuous if, @S Ď M, f ̃à aPS a ̧" à aPS fpaq.
Proposition 5 ([52]). Let pM, ', εq be a commutative monoid and S be the set of its endomorphisms. The set S endowed with addition and multiplication defined by

f 1 , f 2 P S, @x P M : pf 1 ' f 2 qpxq " f 1 pxq ' f 2 pxq, f 1 , f 2 P S, @x P M : pf 1 b f 2 qpxq " f 1 ̀f2 pxq ̆,
is a dioid. The zero and unit element are given by the mappings @x P M, εpxq " ε and @x P M, epxq " x, respectively.

Residuation Theory

In general, the product b in a dioid is not invertible. However, since a compete dioid is a complete lattice, then residuation theory, see e.g. [START_REF] Blyth | Residuation theory. International series of monographs in pure and applied mathematics[END_REF][START_REF] Cohen | Algèbres Max-Plus et applications en informatique et automatique[END_REF], is applicable to define an approximate mapping inverse for particular mappings defined between compete dioids. More precisely this theory yields the greatest solution of the inequality fpaq ĺ b, with a, b are elements in a complete dioid. By defining the product b in a complete dioid as a mapping, i.e., R a : x Þ Ñ a b x, residuation theory is in particular useful to obtain an approximate inverse of the product. In other words, we can determine the greatest solution for x of the inequality a b x ĺ b (note that a solution always exists, ε at least). In this section, we give the conditions under which mappings between complete dioids are residuated and recall some useful properties of residuation theory.

Definition 20 (Residuated Mapping).

A mapping f : D Ñ C, with pD, ', bq and pC, ', bq complete dioids, is said to be residuated if 1. f is isotone and, 2. for all y P C, the inequality fpxq ĺ y has a greatest solution in D.

Theorem 2.2 ( [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Cohen | Algèbres Max-Plus et applications en informatique et automatique[END_REF]). Let f : D Ñ C be a residuated mapping from a complete dioid pD, ', bq into a complete dioid pC, ', bq then, there exists a unique mapping f 7 from C into D which satisfies,

f ̋f7 ĺ Id C (Id C identity mapping in pC, ', bq), (2.15)
f 7 ̋f ľ Id D (Id D identity mapping in pD, ', bq).

(2.16)

The mapping f 7 : C Ñ D is called the residual of f. Remark 4. From (2.15) and (2.16) it follows that @x P D and @y P C,

x ĺ f 7 ̀fpxq ̆, y ľ f ̀f7 pyq ̆, (2.17)
fpxq " f ́f7 ̀fpxq ̆̄, f 7 pyq " f 7 ́f̀f 7 pyq ̆̄.

(2.18)

Conversely, one can define dual residuation which yields the least solution of the inequality fpaq ľ b, where a, b are elements in a complete dioid.

Definition 21 (Dually Residuated Mapping).

A mapping f : D Ñ C, with pD, ', bq and pC, ', bq complete dioids, is said to be dually residuated if 1. f is isotone and, 2. for all y P C, the inequality fpxq ľ y has a least solution in D.

Theorem 2.3 ([1]

). Let f : D Ñ C be a dually residuated mapping from a complete dioid pD, ', bq into a complete dioid pC, ', bq then, there exists a unique mapping f 5 from C into D which satisfies,

f ̋f5 ľ Id C (Id C identity mapping in pC, ', bq), (2.19)
f 5 ̋f ĺ Id D (Id D identity mapping in pD, ', bq).

(2.20)

The mapping f 5 : C Ñ D is called the dual residual of f.

Remark 5. From (2. [START_REF] David-Henriet | Discrete event systems with standard and partial synchronizations[END_REF]) and (2.20) it follows that @x P D and @y P C,

x ľ f 5 ̀fpxq ̆, y ĺ f ̀f5 pyq ̆, (2.21)
fpxq " f ́f5 ̀fpxq ̆̄, f 5 pyq " f 5 ́f̀f 5 pyq ̆̄.

(2.22)

The following theorems give a link between the lower (rep. upper) semi-continuous property and the residuated (rep. dually residuated) property of a mapping.

Theorem 2.4 ([1]

). A mapping f : D Ñ C, with pD, ', bq and pC, ', bq complete dioids, is residuated, iff fpεq " ε and f is lower semi-continuous. Furthermore, the corresponding residual f 7 is upper semi-continuous.

Theorem 2.5 ([1]

). A mapping f : D Ñ C, with pD, ', bq and pC, ', bq complete dioids, is dually residuated iff fpJq " J and f is upper semi-continuous. Furthermore, the corresponding dual residual f 5 is lower semi-continuous.

Clearly, Theorem 2.4 and Theorem 2.5 implies that the residual f 7 of a mapping f is dually residuated and thus pf 7 q 5 " f. Conversely, the dual residual g 5 of a mapping g is residuated and thus pg 5 q 7 " g.

Residuation of Multiplication

On a complete dioid the mappings R a : x Þ Ñ xa, (right multiplication by a) and L a :

x Þ Ñ ax (left multiplication by a) are lower semi-continuous and therefore residuated. The residual mappings are denoted R 7 a pbq " b{a " À tx|xa ĺ bu (right division by a) and L 7 a pbq " a zb " À tx|ax ĺ bu (left division by a). An alternative notation for the left and right division by a are b a and b a , respectively. The following two relations give some useful properties of left and right division in combination with the Kleene star.

a " a ̊ô a " a za " pa zaq ̊a " a ̊ô a " a{a " pa{aq ̊(2.23)

Additionally, for pD, ', bq a complete dioid left-division and right-division are extended to matrices as follows, for A P D m̂n , B P D m̂q , C P D n̂q ,

pA zBq i,j " m ľ k"1 pAq k,i zpBq k,j , pB{Cq i,j " q ľ k"1 pBq i,k {pCq j,k . (2.24)
In Appendix A we provide a list with some basic relations of left and right division in complete dioids. A more detailed presentation can be found in [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF].

In general, in a complete dioid pD, ', bq, left and right division do not distribute over ', however for a, b, x P D

x zpa ' bq ľ x za ' x zb, pa ' bq{x ľ a{x ' b{x,
see [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF]. Moreover, when we deal with dioids of power series the following proposition provides a useful result for division between power series.

À i hpiqz i fpmqz m " à i hpiq fpmq z ím , À i hpiqz i fpmqz m " à i hpiq fpmq z ím .

Residuation of the Canonical Injection

Definition 22. Let pS, ', bq be a complete subdioid of a complete dioid pD, ', bq. The canonical injection, from pS, ', bq into pD, ', bq is a mapping defined by, Inj : S Ñ D, @x P S, Injpxq " x.

Clearly, the canonical injection is lower-semi continuous and therefore it is residuated. Conversely, if pS, ', bq and pD, ', bq have the same top element J the canonical injection Inj : S Ñ D is dually residuated. Moreover, for the dual residual Inj 5 the following conditions hold 1. Inj 5 ̋Inj 5 " Inj 5 , 2. Inj 5 ľ Id D , 3. x P S ô Inj 5 pxq " x.

Dioid of two Dimensional Power Series M ax in rrγ, δss

The dioid pM ax in vγ, δw , ', bq is useful for modeling and control of some DESs, e.g. [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF], and plays a major role in this thesis. Here we briefly introduce the dioid pM ax in vγ, δw , ', bq and we give some basic results. These results are mainly based on [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF]. For a more comprehensive representation, the reader is invited to consult [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Cohen | Algebraic tools for the performance evaluation of discrete event systems[END_REF].

pM ax in vγ, δw , ', bq is a quotient dioid of formal power series in two variables γ and δ and Boolean coefficients. We first introduce the dioid pB vγ, δw , ', bq and then develop pM ax in vγ, δw , ', bq by introducing a congruence relation on pB vγ, δw , ', bq.

Definition 23 (Dioid pB vγ, δw , ', bq). We denote by pB vγ, δw , ', bq the dioid of formal power series in the two commutative variables γ and δ with Boolean coefficients, i.e., B " te, εu and exponents in Z. An element s P B vγ, δw is represented as s " À ν,τPZ spν, τqγ ν δ τ , with spν, τq P te, εu. The zero element is ε " À ν,τPZ εγ ν δ τ and the unit element e " eγ 0 δ 0 .

Moreover, we write only the elements of a series s " À ν,τPZ spν, τqγ ν δ τ , for which spν, τq " e, therefore a monomial m P B vγ, δw is represented as γ ν 1 δ τ 1 . Since, pB, ', bq is a complete dioid and due to Prop. 3 the dioid pB vγ, δw , ', bq is complete as well. Moreover, since the variable γ and δ commute and pB, ', bq is a commutative dioid, the dioid pB vγ, δw , ', bq is a commutative dioid.

Example 7. A series s P B vγ, δw has a natural graphical representation in the Z 2 -plane. For instance, the series s " γ 1 δ 1 ' γ 2 δ 3 ' γ 3 δ 4 is shown in Figure 2.1. The zero and unit element in M ax in vγ, δw are equal to the zero and unit element in B vγ, δw, and thus ε " À ν,τPZ εγ ν δ τ and e " eγ 0 δ 0 , respectively. Due to Remark 3 the dioid pM ax in vγ, δw , ', bq inherits the commutative and completeness properties from the dioid pB vγ, δw , ', bq.

Two series s 1 , s 2 P M ax in vγ, δw belong to the same equivalence class if γ ̊̀δ ́1̆̊s

1 " γ ̊̀δ ́1̆̊s 2 . A canonical representative of an equivalence class is defined to the series of the class with minimal support. Differently speaking the series in the equivalence class with the minimal number of elements is the canonical representative of the equivalence class. For instance consider the following two series s 1 , s 2 P M ax in vγ, δw

s 1 " γ 1 δ 1 ' γ 2 δ 3 , s 2 " γ 1 δ 1 ' γ 2 δ 3 ' γ 3 δ 1 ,
both series belong to the same equivalence class but s 1 is the canonical representative of the class since s 1 has minimal support. This equivalence relation has a graphical interpretation in the Z 2 -plane, unlike to B vγ, δw where a monomial represents a point in the Z 2 -plane, a monomial in M ax in vγ, δw represents the south-est cone of a point in the Z 2 -plane. Respectively, a series in M ax in vγ, δw represents the union of the south-est cones of its elements. If two series cover the same area in the Z 2 -plane, then they belong to the same equivalence class. For instance, the series s 1 and s 2 , shown in Figure 2.2, cover the same area. Note that The series s 2 " γ 1 δ 1 ' γ 2 δ 3 ' γ 3 δ 1 belongs to the same equivalence class, since both series s 1 , s 2 cover the same area in the Z 2 -plane. γ 2 δ 3 dominates γ 3 δ 1 , since

γ 2 δ 3 pγ 1 q ̊pδ ́1q ̊"γ 2 δ 3 ' γ 3 δ 3 ' γ 4 δ 3 ' ̈̈' γ 2 δ 2 ' γ 3 δ 2 ' γ 4 δ 2 ' ̈̈' γ 2 δ 1 ' γ 3 δ 1 ' γ 4 δ 1 ' ̈̈T
herefore, this equivalence relation leads to the following simplification rules for monomials in M ax in vγ, δw,

δ τ 1 ' δ τ 2 " δ maxpτ 1 ,τ 2 q , (2.25) γ ν 1 ' γ ν 2 " γ minpν 1 ,ν 2 q . (2.26)
The order relation on the dioid pM ax in vγ, δw , ', bq, induced by the ' operation, is partial. This can be illustrated on monomial. Let γ ν 1 δ τ 1 , γ ν 2 δ τ 2 P M ax in vγ, δw then γ ν 1 δ τ 1 ľ γ ν 2 δ τ 2 if and only if τ 1 ě τ 2 and ν 1 ď ν 2 . For instance, consider the monomials γ 1 δ 1 , γ 2 δ 3 , γ 3 δ 1 P M ax in vγ, δw, γ 1 δ 1 ľ γ 3 δ 1 , and γ 2 δ 3 ľ γ 3 δ 1 but γ 1 δ 1 ń γ 2 δ 3 and γ 1 δ 1 ł γ 2 δ 3 . Moreover, multiplication b, addition ', and the infimum operation ^between monomial in M ax in vγ, δw satisfy the following relations

γ ν 1 δ τ 1 b γ ν 2 δ τ 2 " γ ν 1 ̀ν2 δ τ 1 ̀τ2 , (2.27
)

γ ν δ τ 1 ' γ ν δ τ 2 " γ ν δ maxpτ 1 ,τ 2 q ,
(2.28)

γ ν 1 δ τ ' γ ν 2 δ τ " γ minpν 1 ,ν 2 q δ τ , (2.29)
γ ν 1 δ τ 1 ^γν 2 δ τ 2 " γ maxpν 1 ,ν 2 q δ minpτ 1 ,τ 2 q . (2.30)
Recall that a polynomial is a series with finite support, i.e., a polynomial in M ax in vγ, δw can be written as a finite sum À I i"0 γ ν i δ τ i , with I P N. Definition 25 (Ultimately Cyclic Series). A series s " À i γ ν i δ τ i P M ax in vγ, δw is called ultimately cyclic if s can be written as s " p ' qpγ ν δ τ q ̊, where p and q are polynomials in M ax in vγ, δw and ν, τ P N. The asymptotic slope of s is defined by σpsq " τ{ν. The polynomial p (resp. q) is called transient (resp. cyclic-pattern) and the monomial pγ ν δ τ q is called growingterm.

Example 8. Consider the following ultimately cyclic series s " pe ' γ 1 δ 1 ' γ 2 δ 3 q ' pγ 4 δ 4 ' γ 5 δ 6 qpγ 2 δ 3 q ̊in M ax in vγ, δw. The asymptotic slope σpsq " 3{2, the transient part is given by pe ' γ 1 δ 1 ' γ 2 δ 3 q and the cyclic-pattern is pγ 4 δ 4 ' γ 5 δ 6 q, which is repeated by a shift of 2 units in the γ-domain and 3 units in the δ-domain.

" pe ' γ 1 δ 1 ' γ 2 δ 3 q ' pγ 4 δ 4 ' γ 5 δ 6 qpγ 2 δ 3 q ̊in M ax in vγ, δw.
In the following theorem, we give the basic results for calculations with ultimately cyclic series in M ax in vγ, δw.

Theorem 2.6 ([1]). Let s 1 " p 1 ' q 1 pγ ν 1 δ τ 1 q ̊and s 2 " p 2 ' q 2 pγ ν 2 δ τ 2 q ̊be two ultimately cyclic series in M ax in vγ, δw, where p 1 , q 1 , p 2 , q 2 are polynomials in M ax in vγ, δw and ν 1 , ν 2 , τ 1 , τ 2 P N. Furthermore, s 1 ‰ ε, s 2 ‰ ε and the asymptotic slope of s 1 is defined by σps 1 q " τ 1 {ν 1 (resp. σps 2 q " τ 2 {ν 2 ), then s 1 ' s 2 is an ultimately cyclic series such that σps 1 ' s 2 q " maxpσps 1 q, σps 2 qq.

s 1 b s 2 is an ultimately cyclic series such that σps 1 b s 2 q " maxpσps 1 q, σps 2 qq.

ps 1 q ̊is an ultimately cyclic series.

s 1 ^s2 is an ultimately cyclic series such that σps 1 ^s2 q " minpσps 1 q, σps 2 qq.

s 2 zs 1 (resp. s 1 {s 2 ) is an ultimately cyclic series such that s 2 zs 1 " s 1 {s 2 " ε if σps 1 q ă σps 2 q and σps 2 zs 1 q " σps 1 {s 2 q " σps 1 q otherwise.

Definition 26 (Causal Series in M ax in vγ, δw [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF], [START_REF] Brunsch | Modeling and control of complex systems in a dioid framework[END_REF]). A series s P M ax in vγ, δw is said to be causal if s P ε or both val γ psq ě 0 and s ľ γ valγpsq δ 0 , where val γ psq refers to the valuation in γ of series s. The set of causal series, denoted by M axì n vγ, δw, is a complete subdioid of pM ax in vγ, δw , ', bq denoted by pM axì n vγ, δw , ', bq.

Remark 6 ([7]

). The canonical injection Inj : M axì n vγ, δw Ñ M ax in vγ, δw is residuated and its residual is called causal projection, which is denoted by Pr ̀: M ax in vγ, δw Ñ M axì n vγ, δw. Therefore, Pr ̀psq is the greatest causal series less than or equal to s P M ax in vγ, δw.

Example 9. Consider the series s " γ ́3δ ́4 ' γ ́2δ 1 ' γ 3 δ 4 P M ax in vγ, δw, then the causal projection Pr ̀psq " γ 0 δ 1 ' γ 3 δ 4 P M axì n vγ, δw. In Figure 2.4a and Figure 2.4b the causal projection of this series s is illustrated. Remark 7. In [START_REF] Brunsch | Modeling and control of complex systems in a dioid framework[END_REF]

3

Dioids pE, ', bq and pErrδss, ', bq

In the first part of this chapter, Section 3.1, the dioid pErrδss, ', bq is recalled. It was introduced in [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF] and is useful to model Weighted Timed Event Graphs (WTEGs). In particular, the transfer function of a single-input and single-output (SISO) WTEG corresponds to an ultimately cyclic series s P Errδss. In Section 3.2 it is shown that the dioid pM ax in vγ, δw , ', bq introduced in Section 2.3 is a subdioid of pErrδss, ', bq. Moreover, particular mappings between Errδss and M ax in vγ, δw are studied -which have an application in optimal control of WTEGs. Some first results of this section have been published in [START_REF] Trunk | Output reference control for weightbalanced timed event graphs[END_REF]. In the third part of this chapter, Section 3.3, it is shown that under some conditions all relevant operations p', b, z, {q on Errδss can be reduced to operations between matrices with entries in M ax in vγ, δw, some results of this section have previously appeared in [START_REF] Trunk | Model decomposition of weight-balanced timed event graphs in dioids: Application to control synthesis[END_REF].

Dioid pErrδss, ', bq

The firings of a transition in a WTEG can be naturally described by a counter function x : Z Ñ Z min , with xptq is the accumulated number of firings up to a time t. Let us recall that the order in Z min is reverse to the natural order, i.e., let x 1 , x 2 P Z min , then x 1 ľ x 2 ô x 1 ď x 2 . Subsequently, counter functions are antitone mappings. In the following the dioid pErrδss, ', bq is defined as a set of operators on counter functions.

The set of antitone mappings from Z into Z min is denoted by Σ. On this set addition is defined to be the pointwise addition in the dioid (Z min , ', b), thus for x 1 , x 2 P Σ, @t P Z, ̀x1 ' x 2 ̆ptq :" x 1 ptq ' x 2 ptq " minpx 1 ptq, x 2 ptqq.

(3.1)

Moreover, scalar multiplication is defined as, for λ P Z min , @t P Z, ̀λ b x 1 ̆ptq :" λ ̀x1 ptq.

The zero and top mappings on Σ, denoted by ε resp. J, are defined by @t, εptq :" ε (recall that in Z min , ε " 8 ), @t, Jptq :" J (recall that in Z min , J " ́8 ).

Note that equipped with the operation ' and the scalar multiplication b the set Σ is a Z minsemimodule (see Definition 3), where pΣ, ', εq is an idempotent commutative monoid. Moreover, by including the top mapping J, pΣ, ', εq is a complete monoid.

Dioids pE, ', bq and pErrδss, ', bq

The order relation on Σ, naturally induced by ', is the order in the dioid pZ min , ', bq, i.e., @x 1 , x 2 P Σ,

x 1 ĺ x 2 ô x 1 ' x 2 " x 2 , (3.3)
ô x 1 ptq ' x 2 ptq " x 2 ptq, @t P Z, ô min ̀x1 ptq, x 2 ptq ̆" x 2 ptq, @t P Z, ô x 1 ptq ě x 2 ptq, @t P Z.

The infimum (^operator) on the set Σ is defined by @t P Z, px 1 ^x2 qptq :" x 1 ptq ^x2 ptq " maxpx 1 ptq, x 2 ptqq.

Definition 27 (Operator). An operator is a lower semi-continuous mapping f : Σ Ñ Σ from the set Σ into itself, such that fpεq " ε. Including the property fpεq " ε implies that f is an endomorphism. The set of these operators is denoted by O.

Proposition 8 ([16]

). The set of operators O, equipped with multiplication and addition as follows,

f 1 , f 2 P O, @x P Σ ̀f1 ' f 2 ̆pxq :" f 1 pxq ' f 2 pxq, (3.4)
f 1 , f 2 P O, @x P Σ ̀f1 b f 2 ̆pxq :" f 1 ̀f2 pxq ̆, (3.5)
is a complete dioid.

Proof. This proof is based on a slightly different version given in [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][Chap. 4, Lemma 4.46] and [START_REF] David-Henriet | Discrete event systems with standard and partial synchronizations[END_REF][Chap. 2, Proposition 5]. There, the set of lower semi-continuous mappings from a complete dioid into itself is studied. First, due to Prop. 5 the set of endomorphisms S over the monoid pΣ, ', εq is a dioid with the zero mapping and unit mapping given by @x P Σ, εpxq :" ε, êpxq :" x.

Furthermore, the set of operators O (lower semi-continuous mapping over Σ), such that @f P O, fpεq " ε, is a subset of S which contains the zero and unit mapping. We have to show that pO, ', bq is a complete subdioid of pS, ', bq. O is closed for addition and multiplication, since the lower semi-continuous property is preserved for both operations, i.e., for f 1 , f 2 P O and X Ď Σ, for addition: To simplify notation we sometimes omit the multiplication symbol b, e.g., for f 1 , f 2 P O, x P Σ, f 1 pf 2 pxqq " pf 1 b f 2 qpxq we write f 1 f 2 pxq. Moreover, for f P O, x P Σ we sometimes write fx instead of fpxq. Due to (2.1) the ' operation induces a partial order relation on O, defined by

pf 1 ' f 2 q ̀à xPX x ̆" f 1 ̀à xPX x ̆' f 2 ̀à xPX x ̆due to (3.4) " à xPX f 1 pxq ' à xPX f 2 pxq f 1 ,
f 1 ľ f 2 ô f 1 ' f 2 " f 1 ,
ô ̀f1 x ̆ptq ' ̀f2 x ̆ptq " ̀f1 x ̆ptq, @x P Σ, @t P Z, ô min ́̀f 1 x ̆ptq, ̀f2 x ̆ptq ̄" ̀f1 x ̆ptq @x P Σ, @t P Z.

(3.7)

Subsequently, two operators f 1 , f 2 P O are equal iff @x P Σ, @t P Z: pf 1 xqptq " pf 2 xqptq . Since pO, ', bq is a complete dioid the top mapping is given by, @x P Σ,

Ĵpxq " $ & % ε for x " ε, J otherwise, (3.8)
and the infimum is defined as, for f 1 , f 2 P O,

f 1 ^f2 " à tf 3 P O|f 3 ' f 1 ĺ f 1 , f 3 ' f 2 ĺ f 2 u.
Proposition 9. The following operators are both endomorphisms and lower semi-continuous mappings, and thus operators in O.

m P N µ m : @x P Σ, t P Z ̀µm pxq ̆ptq " m ̂xptq, (3.9)

b P N β b : @x P Σ, t P Z ̀βb pxq ̆ptq " Y xptq b ] , (3.10)
ν P Z γ ν : @x P Σ, t P Z ̀γν pxq ̆ptq " ν ̀xptq.

(3.11)

Note that tau denotes the greatest integer smaller than or equal to a.

Proof. The mapping µ m is an endomorphism, first, recall that @t P Z, εptq " 8 and m P N is a finite positive integer, therefore, @t P Z, pµ m pεqqptq " m ̂εptq " m ̂8 " 8, and thus pµ m pεqqptq " εptq. Second @t P Z: ́µm ̀x1 ' x 2 ̆̄ptq " m ̂̀x 1 ' x 2 ̆ptq, due to (3.9)

" m ̂min ̀x1 ptq, x 2 ptq ̆, due to (

" min ̀m ̂x1 ptq, m ̂x2 ptq ̆,

" min ́̀µ m px 1 q ̆ptq, ̀µm px 2 q ̆ptq ̄, due to (3.9)

" ́µm ̀x1 ̆̄ptq ' ́µm ̀x2 ̆̄ptq, again due to (3.1).

Of course, this extends to all finite and infinite subsets X Ď Σ, i.e., ). The operators γ ν , µ m and β b introduced in Prop. 9 satisfy the following relations,

́µm
γ ν γ ν 1 " γ ν̀ν 1 , γ ν ' γ ν 1 " γ minpν,ν 1 q , ( 3.12
)

µ m γ n " γ n̂m µ m , γ n β b " β b γ n̂b . (3.13)
Proof. For the proof of (3.12), recall (3.5) and (3.11), then @x P Σ, @t P Z,

pγ ν γ ν 1 xqptq " ̀γν pγ ν 1 xq ̆ptq " ν ̀pγ ν 1 xqptq " ν ̀ν1 ̀xptq " ̀γν̀ν 1 x ̆ptq,
and since (3.4), (3.1) and (3.11), then @x P Σ, @t P Z,

̀pγ ν ' γ ν 1 qx ̆ptq " ̀γν x ' γ ν 1 x ̆ptq " min ̀pγ ν xqptq, pγ ν 1 xqptq ̆,
" min ̀ν ̀xptq, ν 1 ̀xptq ̆" minpν, ν 1 q ̀xptq, " ̀γminpν,ν 1 q x ̆ptq.

For the proof of (3.13), since (3.9) and (3.11), then @x P Σ, @t P Z, ̀pµ m γ n qx ̆ptq " m ̂̀n ̀xptq ̆" mn ̀m ̂xptq " ̀pγ n̂m µ m qx ̆ptq, and, since (3.10) and (3.11), then @x P Σ, @t P Z,

̀pγ n β b qx ̆ptq " n ̀Y xptq b ] " Y xptq ̀nb b ]
" ̀pβ b γ n̂b qx ̆ptq.

Dioid of Event Operators

Definition 28 (Dioid of Event Operators, [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]). The dioid of event operators, denoted by pE, ', bq, is defined by sums and compositions over the set tê, ε, µ m , β b , γ ν , Ĵu with m, b P N, ν P Z, equipped with addition and multiplication defined in (3.4) and (3.5), respectively.

An element w P E is called E-operator (E for event) in the sequel. The dioid pE, ', bq is a complete subdioid of pO, ', bq [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]. Note that the dioid pE, ', bq is not commutative, i.e., in general for w 1 , w 2 P E, w 1 w 2 ‰ w 2 w 1 . For instance, consider the operators µ 2 and γ 1 , according to (3.9) and (3.11), pµ 2 γ 1 xqptq " 2 ̂p1 ̀xptqq and pγ 1 µ 2 xqptq " 1 ̀2 ̂xptq, these two expressions are clearly not equal for arbitrary x P Σ.

Again, the ' operation induces a partial order relation on E, defined by

w 1 ľ w 2 ô w 1 ' w 2 " w 1 ,
ô ̀w1 x ̆ptq ' ̀w2 x ̆ptq " ̀w1 x ̆ptq, @x P Σ, @t P Z, ô min ́̀w 1 x ̆ptq, ̀w2 x ̆ptq ̄" ̀w1 x ̆ptq @x P Σ, @t P Z.

( Definition 29 ((C/C)-Function [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]). The function F w : Z min Ñ Z min , k i Þ Ñ k o maps counter-value to counter-value and is defined by an E-operator w P E such that @k i P Z min , F w pk i q :" ̀wpxq ̆ptq, for xptq " k i and x P Σ.

In other words xptq is replaced by k i in the expression pwpxqqptq.

There is an isomorphism between the set of E-operators and the set of (C/C)-functions. Thus, the order relation over the dioid pE, ', bq, see (3.14), corresponds to the order induced by the min operation on (C/C)-functions, @w 1 , w 2 P E, w 1 ľ w 2 ô w 1 ' w 2 " w 1 , ô F w 1 pkq ľ F w 2 pkq, @k P Z min , ô min ́Fw 1 pkq, F w 2 pkq ̄" F w 1 pkq, @k P Z min , ô F w 1 pkq ď F w 2 pkq, @k P Z min .

(3.15)

Note that the order in Z min is the reverse of the natural order. The (C/C)-functions provide a graphical representation of E-operators in Z 2 min , which is useful to compare E-operators. In this graphical representation the horizontal axis is labeled by I-count and the vertical axis is labeled by O-count, which stand for input counter-value and output counter-value, respectively.

Example 10. Let us consider the following operator γ 3 µ 2 β 3 γ 1 'γ 2 µ 2 β 2 γ 1 with a corresponding (C/C)-function

F γ 3 µ 2 β 3 γ 1 'γ 2 µ 2 β 2 γ 1 pkq " min ́3 ̀2Y k ̀1 3 ] , 2 ̀2Y k ̀1 2 ]̄.
This function is shown in Figure 3.1b and is the minimum of the functions F γ 3 µ 2 β 3 γ 1 and F γ 2 µ 2 β 2 γ 1 , see Figure 3.1a. In Figure 3.1b the operators γ 7 µ 2 β 2 and γ 3 µ 2 β 3 γ 1 ' γ 2 µ 2 β 2 γ 1 are compared. The gray area in Figure 3.1b corresponds to the domain of (C/C)-functions less than or equal to F γ 3 µ 2 β 3 γ 1 'γ 2 µ 2 β 2 γ 1 or equivalently to operators w P E less than or equal to γ 3 µ 2 β 3 γ 1 ' γ 2 µ 2 β 2 γ 1 .

F γ 3 µ 2 β 3 γ 1 F γ 2 µ 2 β 2 γ 1 I-count O-count -4 -2 2
F γ 3 µ 2 β 3 γ 1 and F γ 2 µ 2 β 2 γ 1 . F γ 2 µ 2 β 2 γ 1 'γ 3 µ 2 β 3 γ 1 F γ 7 µ 2 β 2 I-count O-count -4 -2 2
F γ 7 µ 2 β 2 and F γ 3 µ 2 β 3 γ 1 'γ 2 µ 2 β 2 γ 1 . Figure 3.1. -In (a) minpF γ 3 µ 2 β 3 γ 1 , F γ 2 µ 2 β 2 γ 1 q is equal to the function F γ 3 µ 2 β 3 γ 1 'γ 2 µ 2 β 2 γ 1
given in (b). In (b) the (C/C)-function F γ 7 µ 2 β 2 lies in the gray area shaped by the

F γ 3 µ 2 β 3 γ 1 'γ 2 µ 2 β 2 γ 1 function, thus F γ 7 µ 2 β 2 ą F γ 3 µ 2 β 3 γ 1 'γ 2 µ 2 β 2 γ 1 , in (min,+) F γ 7 µ 2 β 2 ă F γ 3 µ 2 β 3 γ 1 'γ 2 µ 2 β 2 γ 1 and thus γ 7 µ 2 β 2 ă γ 3 µ 2 β 3 γ 1 ' γ 2 µ 2 β 2 γ 1 .
Periodic E-operators Definition 30. An E-operator w P E is said to be pm, bq-periodic if Dm, b P N such that, @x P Σ, @t P Z, pwpb b xqqptq " m b pwpxqqptq. The set of pm, bq-periodic E-operators is denoted by E m|b .

Definition 31. A (C/C)-function F is said to be quasi pm, bq-periodic if Dm, b P N such that Fpk b bq " m b Fpkq, @k P Z min , (Fpk ̀bq " m ̀Fpkq, @k P Z min ).

Recall that the b operation in the dioid pZ min , ', bq corresponds to the standard ̀operation. In the sequel, both representations are used. Remark 8. Since the periodic property does only depend on the value xptq we can neglect the time t for examining the periodic property of an E-operator. Therefore, an E-operator w P E is pm, bq-periodic if and only if the corresponding (C/C)-function F w is quasi pm, bq-periodic.

Definition 32. The gain of an pm, bq-periodic operator w P E m|b , denoted by Γ pwq, is defined by the ratio Γ pwq " m{b.

Example 11. The γ ν operator, with ν P Z is p1, 1q-periodic, since @k P Z min , F γ ν pkq " k̀ν and therefore, F γ ν pk ̀1q " pk ̀1q ̀ν " 1 ̀Fγ ν pkq. The γ 2 β 3 γ 1 µ 2 operator is p2, 3qperiodic, for which the corresponding (C/C)-function is illustrated in Figure 3.2. In contrast, the γ 3 µ 2 β 3 γ 1 ' γ 2 µ 2 β 2 γ 1 operator, shown in Figure 3.1b, is not periodic. In the following E-operators of the form [START_REF] Cohen | Second order theory of minlinear systems and its application to discrete event systems[END_REF], thus γ ν µ m β b γ ν 1 can be written such that 0 ď ν 1 ă b. This form is particularly useful to check the ordering of E-operators. Given two E-operators

À I i"1 γ ν i µ m β b γ ν 1 i are studied. Recall that γ m µ m β b " µ m β b γ b (3.
γ ν 1 µ m β b γ ν 1 1 , γ ν 2 µ m β b γ ν 1 2 P E m|b , with 0 ď ν 1 1 , ν 1 2 ă b, then γ ν 1 µ m β b γ ν 1 1 ľ γ ν 2 µ m β b γ ν 1 2 ô $ & % ν 1 ď ν 2 and ν 1 1 ď ν 1 2 , or ν 1 ́m ď ν 2 .
(3.16) Proposition 11 ([16]). A periodic E-operator w P E m|b has a canonical form, which is a finite sum w " À I i"1 γ ν i µ m β b γ ν 1 i such that 0 ď ν 1 i ă b, ν i P Z and I ď minpm, bq.

Proof. Let us define an operator w " À b́1 i"0 wi , with wi " γ Fwpiq µ m β b γ b́1́i . Then, first we show that any pm, bq-periodic operator w P E m|b can be expressed by w P E m|b , i.e., w " w. Recall the isomorphism between an E-operator and the (C/C)-function thus it is equivalent to show that F w " F w. The (C/C)-function to wi is

F wi " Y k̀pb́1q́i b
] m Fw piq and therefore the (C/C)-function F wpkq can be written as

F w " min ́Y k ̀pb ́1q b ] m ̀Fw p0q, Y k ̀pb ́2q b ] m̀F w p1q, ̈̈, Y k b ] m ̀Fw pb ́1q ̄.
Let us recall that F w is an isotone function and satisfies F w p0q ď F w p1q ď ̈̈̈ď F w pb ́1q ď m ̀Fw p0q ď ̈̈̈.

(3.17)

Since F w and F w are quasi pm, bq-periodic functions it is sufficient to show that F w pkq " F wpkq for all k P t0, ̈̈̈, b ́1u. We now evaluate F wpkq for k " 0,

F wp0q " min ́Y pb ́1q b ] m ̀Fw p0q, Y pb ́2q b ] m ̀Fw p1q, ̈̈, Y 0 b ] m ̀Fw pb ́1q ̄,
" min ́Fw p0q, F w p1q, ̈̈̈, F w pb ́1q " F w p0q, since F w is isotone, see (3.17).

Similarly we can show that for k P t1, ̈̈̈, b ́1u,

F wp1q " min ́Y b b ] m ̀Fw p0q, Y b ́1 b ] m ̀Fw p1q, ̈̈̈, Y 1 b ] m ̀Fw pb ́1q ̄,
" min ́m ̀Fw p0q, F w p1q, ̈̈̈, F w pb ́1q ̄" F w p1q, see (3.17),

̈̈F

wpb ́1q " min ́m ̀Fw p0q, ̈̈̈, m ̀Fw pb ́2q, F w pb ́1q ̄" F w pb ́1q.

The canonical form can then be obtained by removing redundant terms according to (3.16).

Example 12. Consider the γ 2 β 3 γ 1 µ 2 operator with a (C/C)-function shown in Figure 3.2. This operator is p2, 3q-periodic. Moreover, the (C/C)-function F γ 2 β 3 γ 1 µ 2 evaluated on t leads to,

F γ 2 β 3 γ 1 µ 2 p0q " 2, F γ 2 β 3 γ 1 µ 2 p1q " 3, F γ 2 β 3 γ 1 µ 2 p2q " 3, F γ 2 β 3 γ 1 µ 2 p3q " F γ 2 β 3 γ 1 µ 2 p0q ̀2 " 4, F γ 2 β 3 γ 1 µ 2 p4q " F γ 2 β 3 γ 1 µ 2 p1q ̀2 " 5,
̈̈T herefore, the operator γ 2 β 3 γ 1 µ 2 can be written as,

γ 2 µ 2 β 3 γ 2 ' γ 3 µ 2 β 3 γ 1 ' γ 3 µ 2 β 3 γ 0 .
Since, γ 3 µ 2 β 3 γ 1 ' γ 3 µ 2 β 3 γ 0 " γ 3 µ 2 β 3 pγ 1 ' γ 0 q " γ 3 µ 2 β 3 γ 0 , this expression is simplified to

γ 2 µ 2 β 3 γ 2 ' γ 3 µ 2 β 3 γ 0 ,
which is the canonical representation of γ 2 β 3 γ 1 µ 2 . Figure 3.3 shows the (C/C)-functions F γ 3 µ 2 β 3 and F γ 2 µ 2 β 3 γ 2 of the operators γ 3 µ 2 β 3 and γ 2 µ 2 β 3 γ 2 , respectively. The intersection of the area beneath F γ 3 µ 2 β 3 and F γ 2 µ 2 β 3 γ 2 is equal to the area beneath the (C/C)-function

F γ 2 β 3 γ 1 µ 2 shown in Figure 3.2. Thus, minpF γ 3 µ 2 β 3 , F γ 2 µ 2 β 3 γ 2 q " F γ 3 µ 2 β 3 'γ 2 µ 2 β 3 γ 2 " F γ 2 β 3 γ 1 µ 2 . F γ 2 µ 2 β 3 γ 2 F γ 3 µ 2 β 3 I-count O-count -1 1
F γ 2 µ 2 β 3 γ 2 . One has minpF γ 3 µ 2 β 3 , F γ 2 µ 2 β 3 γ 2 q " F γ 3 µ 2 β 3 'γ 2 µ 2 β 3 γ 2 .
Or in other words, the intersection of the area beneath F γ 3 µ 2 β 3 and F γ 2 µ 2 β 3 γ 2 is equal to the area beneath

F γ 2 β 3 γ 1 µ 2 " F γ 3 µ 2 β 3 'γ 2 µ 2 β 3 γ 2 .
Remark 9. Clearly an pm, bq-periodic operator is also pnm, nbq-periodic. Thus, an pm, bqperiodic operator w P E m|b can be represented in a pnm, nbq-periodic form given by w " nb́1 à i"0 γ Fwpiq µ nm β nb γ nb́1́i . Proposition 12 ([16]). The pm, bq-periodic µ m β b operator can be expressed in the following pnm, nbq-periodic form

µ m β b " ń1 à i"0 γ im µ nm β nb γ pń1́iqb . (3.18)
Proof. Recall that the (C/C)-function of the µ m β b operator is given by F µmβ b " tk{bum.

F µ 1 β 2 is equal to minpF µ 3 β 6 γ 4 , F γ 1 µ 3 β 6 γ 2 , F γ 2 µ 3 β 6 q.
Due to Remark 9 the pnm, nbq-periodic representation of this operator is given by

µ m β b " nb́1 à k"0 γ tk{bum µ nm β nb γ nb́1́k , " ń1 à i"0 b́1 à j"0 γ tpib̀jq{bum µ nm β nb γ nb́1́pib̀jq , with k " ib ̀j, " ń1 à i"0 b́1 à j"0
γ im µ nm β nb γ nb́1́pib̀jq , since for j P t0, ̈̈̈, b ́1u, tpib ̀jq{bu " i.

Due to the order relation for monomials in E, see (3.16), we have b́1 à j"0 γ im µ nm β nb γ nb́1́pib̀jq " γ im µ nm β nb γ nb́1́pib̀b́1q " γ im µ nm β nb γ pń1́iqb and thus

µ m β b " ń1 à i"0 γ im µ nm β nb γ pń1́iqb .
Example 13. For instance, with n " 3, the operator µ 1 β 2 can be written as µ 3 β 6 γ 4 ' γ 1 µ 3 β 6 γ 2 ' γ 2 µ 3 β 6 . Clearly µ 1 β 2 P E 1|2 and µ 1 β 2 P E 3|6 as well. Figure 3.4 illustrates this extension of the µ 1 β 2 operator.

Definition 33. The minimal representative of a periodic operator w P E m|b rrδss is defined such that w is expressed in a canonical form and the period pm, bq is minimal.

In the algorithm 1 we show how to obtain this form. In this algorithm, we check for all common divisors n of m and b if an pm, bq-periodic operator w P E m|b rrδss can be represented in an pm{n, b{nq-periodic form.

Input: Operator w P E m|b rrδss Output: Minimal form of w P E m|b rrδss Calculate the set S :" tn P N|m{n P N and b{n P Nu of all common divisors of pm, bq. Store the set S in a vector k in descending order. j " 0; do m t " m{krjs ; b t " b{krjs ; a " À b t

́1

i"0 γ Fwpiq µ m t β b t γ b t ́1́i ; j " j ̀1; while w ‰ a; return a; Algorithm 1: Minimal representative of a periodic operator w P E m|b rrδss.

Proposition 13. Given two periodic operators

w 1 P E m 1 |b 1 , w 2 P E m 2 |b 2 such that w 1 ‰ ε, w 2 ‰ ε and m 1 b 1 ą m 2 b 2 .
Then, w 1 and w 2 are not ordered, i.e., w 1 ń w 2 and w 1 ł w 2 . Proof. Due to Remark 9 and by choosing b " lcmpb 1 , b 2 q we can represent w 1 P E m1 | b as an p m1 , bq-periodic operator and w 2 P E m2 | b as an p m2 , bq-periodic operator with corresponding quasi periodic (C/C)-functions

F w 1 pk ̀bq " F w 1 pkq ̀m 1 , F w 2 pk ̀bq " F w 2 pkq ̀m 2 .
Then by evaluating the functions for k " j b, j P Z we obtain F w 1 pj bq " F w 1 p0q ̀j m1 , F w 2 pj bq " F w 2 p0q ̀j m2 .

Since F w 1 p0q and F w 2 p0q are finite and m1 ą m2 there exists a positive integer j such that F w 1 pj bq ą F w 2 pj bq and a negative integer j such that F w 1 pj bq ă F w 2 pj bq. Thus, the operators w 1 and w 2 are not ordered.

Example 14. Consider the p2, 3q-periodic operator γ 3 µ 2 β 3 γ 1 and the p2, 2q-periodic operator γ 2 µ 2 β 2 γ 1 . In the graphical representation of the corresponding (C/C)-function, Figure 3.5, one can see that these two operators are not ordered, for instance, for all k ă 0 one has

F γ 2 µ 2 β 2 γ 1 pkq ă F γ 3 µ 2 β 3 γ 1 pkq and for all k ą 3 one has F γ 2 µ 2 β 2 γ 1 pkq ą F γ 3 µ 2 β 3 γ 1 pkq.

Dioid of Formal Power Series pErrδss, ', bq

Besides E-operators introduced in the last section, we now define the time-shift operator δ τ as a mapping over Σ as follows τ P Z δ τ : @x P Σ, t P Z ̀δτ x ̆ptq " xpt ́τq.

(3.19)

F γ 3 µ 2 β 3 γ 1 F γ 2 µ 2 β 2 γ 1 I-count O-count -4 -2 2
F γ 2 µ 2 β 3 γ 1 . For k ă 0: F γ 2 µ 2 β 2 γ 1 pkq ă F γ 3 µ 2 β 3 γ 1 pkq and for k ą 3: F γ 2 µ 2 β 2 γ 1 pkq ą F γ 3 µ 2 β 3 γ 1 pkq.
Clearly, the δ τ mapping is lower-semi continuous, since for all finite and infinite subsets Furthermore, pδ τ pεqqptq " εpt́τq and since εptq " 8, @t P Z and τ P Z then pδ τ pεqqptq " εptq, thus δ τ is an endomorphism. Consequently, the time-shift operator δ τ P O. Moreover, the time-shift operator commutes with all E-operators [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF], i.e., @w P E, wδ τ " δ τ w.

̀pδ τ wqx ̆ptq " ̀δτ pwxq ̆ptq, due to (3.5), w P E m|b . A polynomial in E m|b rrδss is a finite sum of monomials p " À I i"1 w i δ τ i such that @i P t1, ̈̈̈, Iu, w i P E m|b . For instance, µ 2 β 3 γ 1 δ 2 ' µ 2 β 3 γ 2 δ 3 P E 2|3 rrδss, but the polynomial µ 2 β 3 γ 1 δ 2 'µ 3 β 4 γ 2 δ 3 R E m|b rrδss. Moreover, the gain of an element s P E m|b rrδss is defined to the gain of its coefficients spτq, i.e., Γ psq " Γ pspτqq, for instance, Γ pµ 2 β 3 γ 1 δ 2 q " Γ pµ 2 β 3 γ 1 q " 2{3.

"

Graphical Representation

An element s P Errδss can be graphically represented in Z min ̂Zmin ̂Z. For a series s " À iPZ w i δ i P Errδss this graphical representation is constructed by depicting for every i the corresponding (C/C)-function F w i of the coefficient w i in the (I-count/O-count)-plane of i.

Example 15. For the graphical representation of p " pµ 3 β 3 γ 2 ' γ 1 µ 3 β 3 γ 1 qδ 2 ' µ 3 β 3 γ 2 δ 3 P E 3|3 rrδss, respectively its representative ppδ ́1q ̊see in Figure 3.6, with the (I-count/O-count)plane for t ď 2 (resp. t " 3) shown in Figure 3.7a (resp. Figure 3.7b). To improve readability, the graphical representation for elements s P Errδss has been truncated to non-negative values in Figure 3.6.

The ordering of two monomials w 1 δ τ 1 , w 2 δ τ 2 P E m|b rrδss can be checked by w 1 δ τ 1 ľ w 2 δ τ 2 ô τ 1 ě τ 2 and w 1 ľ w 2 .

(3.21) Proposition 14 ([16]). Let p P E m|b rrδss, then p has a canonical form p " À J j"1 w 1 j δ t 1 j such that the pm, bq-periodic E-operator w 1 j is in canonical form of Prop. 11, and coefficients and exponents are strictly ordered, for j P t1, ̈̈̈, J ́1u, t 1 j ă t 1 j̀1 and w 1 j ą w 1 j̀1 . t-shift Proof. Without loss of generality, we can assume that p " À I i"1 w i δ t i , with t i ă t ì1 for i " 1, ̈̈̈I ́1. As p P Errδss, we can identify s with their maximal representative spδ ́1q ̊, we can also identify p and

I -c o u n t O -c o u n t Figure 3.6. -3D representation of polynomial pµ 3 β 3 γ 2 ' γ 1 µ 3 β 3 γ 1 qδ 2 ' µ 3 β 3 γ 2 δ 3 . F µ 3 β 3 γ 2 'γ 1 µ 3 β 3 γ 1 t " 2 I-count O-count 1
p 1 " I à i"1 ́I à j"i w j lo omo on w 1 i ̄δt i as ppδ ́1q ̊" p 1 pδ ́1q ̊. Therefore, w 1 i ľ w 1 ì1 . If w 1 i " w 1 ì1 we can write w 1 i δ n i ' w 1 ì1 δ n ì1 " w 1 i pδ n i ' δ n ì1 q " w 1 i δ n ì1 .
For this reason, we can write p 1 as

À J j"1 w 1 j δ t 1 j
with w 1 j ą w 1 j̀1 and J ď I.

Definition 35 (Ultimately Cyclic Series).

A series s P E m|b rrδss is said to be ultimately cyclic if it can be written as s " p ' qpγ ν δ τ q ̊, where ν, τ P N 0 and p, q are polynomials in E m|b rrδss, i.e., p and q have the same period. The expression pγ ν δ τ q ̊is called growing term.

Proposition 15 ([16]). An ultimately cyclic series s P E m|b rrδss has a left-and right-cyclic form given by: s " p ' pγ ν l δ τ l q ̊ql , (left-cyclic form)

s " p ' q r pγ νr δ τr q ̊, (right-cyclic form)

where p, q l , q r P E m|b rrδss are polynomials and τ l , ν l , τ r , ν r P N 0 . The left-and right-asymptotic slopes are respectively defined by σ l psq " τ l {ν l and σ r psq " τ r {ν r . The asymptotic slopes of an ultimately cyclic series s P E m|b rrδss satisfy the following property m{b " σ r psq{σ l psq.

Proof. Consider an ultimately cyclic series s " p ' q r pγ νr δ τr q ̊P E m|b rrδss in a rightcyclic form. Since, the dioid pErrδss, ', bq is not commutative in general the growing term pγ νr δ τr q ̊does not commute with the q r polynomial, i.e., q r pγ νr δ τr q ̊‰ pγ νr δ τr q ̊qr . However, due to (3.13), for specific growing terms given by pγ nb δ τ q ̊with n P N 0 we have q r pγ nb δ τ q ̊" pγ nm δ τ q ̊qr . For an arbitrary series s " p ' q r pγ νr δ τr q ̊P E m|b rrδss in a right-cyclic form we can rewrite q r and pγ νr δ τr q ̊such that the conversion is possible. With nν r " lcmpb, ν r q the growing term can be expressed as pγ νr δ τr q ̊" pe ' γ νr δ τr ' γ 2νr δ 2τr ' ̈̈̈' γ pń1qνr δ pń1qτr qpγ nνr δ nτr q " qpγ nνr δ nτr q ̊.

Since nν r is a multiple of b, we have q r qpγ nνr δ nτr q ̊" pγ pnνr{bqm δ nτr q ̊qr q.

Then by choosing q l " q r q, ν l " pnν r {bqm and nτ r " τ l the series s can be represented in a left-cyclic form s " p ' pγ ν l δ τ l q ̊ql " p ' pγ pnνr{bqm δ τ l q ̊qr q.

Furthermore, σ r psq " τ r {ν r and σ l psq " pnτ r q{ppnν r {bqmq and thus σ r psq σ l psq "

τr νr nτr pnνr{bqm " m b .
The conversion of an ultimately cyclic series from a left-cyclic form into a right-cyclic one can be shown analogously.

Example 16. Consider the following series s " γ 1 µ 3 β 2 γ 1 δ 2 'pγ 3 µ 3 β 2 γ 1 'γ 5 µ 3 β 2 qδ 3 pγ 1 δ 1 q in a right-cyclic form. By extending the "growing-term" pγ 1 δ 1 q ̊" pe'γ 1 δ 1 qpγ 2 δ 2 q ̊the series can be expressed in a left-cyclic form as follows

s " γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 δ 2 q ̊̀pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 ' pγ 6 µ 3 β 2 ' γ 5 µ 3 β 2 γ 1 qδ 4 ̆.
The left-and right asymptotic slopes are σ l psq " 2{3 and σ r psq " 1{1, respectively. This series has a graphical representation given in Figure 3.8, with the left-asymptotic slope indicated by the red stairs in the (O-count/t-shift)-plane (I-count value ́1) and the right asymptotic slope indicated by the blue stairs in the (I-count/t-shift)-plane (O-count value 15).

" γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 pγ 1 δ 1 q ̊.
Clearly, a polynomial p " À I i"1 w i δ τ i can be considered as a specific ultimately cyclic series such that s " p À I i"1 w i δ τ i qpγ 0 δ 0 q ̊. Let us note that the set of pb, bq-periodic operators, i.e. the set E b|b rrδss, endowed with ' and b is a complete subdioid of pErrδss, ', bq, but in general the set E m|b rrδss endowed with the ' and b is not a dioid since it is not closed for the b-operation. For instance, consider the operator µ 1 β 2 γ 1 δ 2 P E 1|2 rrδss the product

µ 1 β 2 γ 1 δ 2 b µ 1 β 2 γ 1 δ 2 " µ 1 β 2 γ 1 δ 2 b pµ 2 β 4 γ 3 ' γ 1 µ 2 β 4 γ 1 qδ 2 since µ 1 β 2 " µ 2 β 4 γ 2 ' γ 1 µ 2 β 4 see Prop. 12 " µ 1 β 2 γ 1 µ 2 β 4 γ 3 δ 4 ' µ 1 β 2 γ 2 µ 2 β 4 γ 1 δ 4 " µ 1 β 4 γ 3 δ 4 ' µ 1 β 4 γ 5 δ 4
" µ 1 β 4 γ 3 δ 4 due to (3.21) this operator is (1,4)-periodic and therefore in E 1|4 rrδss and not in E 1|2 rrδss. Clearly since an element s P E m|b rrδss is also an element in Errδss, addition, and multiplication between elements in E m|b rrδss are defined, however, the result is not necessarily in E m|b rrδss. In the following proposition, we summarize the conditions under which sum, product, and infimum of ultimately cyclic series in E m|b rrδss are again ultimately cyclic series in E m|b rrδss. The proofs for these propositions are given later in Section 3.3.

Proposition 16 ([16])

. Let s 1 P E m 1 |b 1 rrδss, s 2 P E m 2 |b 2 rrδss be two ultimately cyclic series with equal gain, i.e. Γ ps 1 q " Γ ps 2 q " m 1 {b 1 " m 2 {b 2 , then ps 1 ' s 2 q P E m|b rrδss is an ultimately cyclic series with gain Γ ps 1 ' s 2 q " Γ ps 1 q " Γ ps 2 q.

Proposition 17 ([16]). Let s 1 P E m 1 |b 1 rrδss, s 2 P E m 2 |b 2 rrδss be two ultimately cyclic series with equal gain, i.e. Γ ps 1 q " Γ ps 2 q " m 1 {b 1 " m 2 {b 2 , then ps 1 ^s2 q P E m|b rrδss is an ultimately cyclic series with gain Γ ps 1 ^s2 q " Γ ps 1 q " Γ ps 2 q.

Proposition 18 ([16]). Let s 1 P E m 1 |b 1 rrδss and s 2 P E m 2 |b 2 rrδss be two ultimately cyclic series then ps 1 b s 2 q P E m 1 m 2 |b 1 b 2 rrδss is an ultimately cyclic series. Moreover, since Γ ps 1 q " m 1 {b 1 and Γ ps 2 q " m 2 {b 2 the gain Γ ps 1 b s 2 q " pm 1 m 2 q{pb 1 b 2 q " Γ ps 1 q ̂Γ ps 2 q. Proposition 19 ([16]). Let s P E b|b rrδss be an ultimately cyclic series then s ̊P E b|b rrδss is an ultimately cyclic series.

Division in pErrδss, ', bq

Recall Section 2.2, since pE, ', bq (resp. pErrδss, ', bq) is a complete dioid right and left multiplication are residuated. We obtain the following results for the left (resp. right) division of periodic elements. Again the proofs of the following propositions are provided in Section 3.3.

Proposition 20 ([16])

. Let s 1 P E m|b 1 rrδss and s 2 P E m|b 2 rrδss be two ultimately cyclic series then ps 2 zs 1 q P E b 2 |b 1 rrδss is an ultimately cyclic series. Moreover, since Γ ps 1 q " m{b 1 and Γ ps 2 q " m{b 2 the gain Γ ps 2 zs 1 q " b 2 {b 1 " Γ ps 1 q{Γ ps 2 q. Proposition 21 ([16]). Let s 1 P E m 1 |b rrδss and s 2 P E m 2 |b rrδss be two ultimately cyclic series then ps 1 {s 2 q P E m 1 |m 2 rrδss is an ultimately cyclic series. Moreover, since Γ ps 1 q " m 1 {b and Γ ps 2 q " m 2 {b the gain Γ ps 2 {s 1 q " m 1 {m 2 " Γ ps 1 q{Γ ps 2 q.

pM ax

in vγ, δw , ', bq as a Subdioid of pErrδss, ', bq

Let us recall the dioid pM ax in vγ, δw , ', bq introduced in Section 2.3. The dioid pM ax in vγ, δw , ', bq is a subdioid of pErrδss, ', bq. More precisely M ax in vγ, δw is the set E 1|1 rrδss, i.e., the set of p1, 1q-periodic series. Then according to Definition 22 the canonical injection from M ax in vγ, δw into Errδss is defined by Inj : M ax in vγ, δw Ñ Errδss, x Þ Ñ Injpxq " x. For instance, Injpγ 1 δ 2 q " γ 1 µ 1 β 1 δ 2 " γ 1 δ 2 . In the following example, we give a graphical interpretation of this injection.

Example 17. Consider the series s " γ 1 δ 2 ' ̀γ3 δ 3 ' γ 5 δ 4 ̆pγ 3 δ 2 q ̊P M ax in vγ, δw, the graphical representation of s is shown in Figure 3.9a. Moreover, the graphical representation of the canonical injection Injpsq P Errδss is shown in Figure 3.9b. The series s P M ax in vγ, δw (Figure 3.9a) corresponds to the (O-count/t-shift)-plane for the (I-count) value 0 of the 3D representation of the series Injpsq P Errδss (Figure 3.9b). Moreover, the canonical injection Injpsq P Errδss is (1,1)-periodic, therefore the (O-count/t-shift)-plane for the (I-count) value 1 corresponds to the series γ 1 s P M ax in vγ, δw and for the (I-count) value 2 to the series γ 2 s P M ax in vγ, δw, etc. Observe that the left-cyclic form and the right-cyclic form are the same since pM ax in vγ, δw , ', bq is commutative. The canonical injection Inj : M ax in vγ, δw Ñ Errδss is continuous and thus it is both residuated and dually residuated, see the following propositions.

γ (O-count) δ (t-shift)
Lemma 1. Let wδ τ P E b|b rrδss be a pb, bq-periodic monomial. Then residual Inj 7 pwδ τ q and dual residual Inj 5 pwδ τ q are given by

Inj 7 pwδ τ q " γ max b́1 k"0 pFwpkq́kq δ τ , (3.22)
Inj 5 pwδ τ q " γ min b́1 k"0 pFwpkq́kq δ τ . (3.23)
Proof. By definition, the residuated mapping Inj 7 pwδ τ q is the greatest solution x of the following inequality

wδ τ ľ Injpxq " Inj ́ài γ ν i δ ζ i ̄" à i γ ν i δ ζ i , (3.24)
where À i γ ν i δ ζ i P M ax in vγ, δw. Clearly, the greatest ζ i such that the inequality (3.24) holds is τ and thus,

wδ τ ľ à i pγ ν i δ τ q " γ ν δ τ , see, (2.29). (3.25)
Since wδ τ ľ γ ν δ τ ô w ľ γ ν , it remains to find the least ν such that (3.25) holds. By considering the isomorphism between E-operators and (C/C)-functions, see (3.14), this is equivalent to F w pkq ľ F γ ν pkq pF w pkq ď F γ ν pkqq, @k P Z min . Note that in Z min the order is reverse to the natural order. By using F γ ν pkq " ν ̀k, see (3.12), we obtain

F w pkq ď ν ̀k ô ν ě F w pkq ́k, @k P Z min . (3.26)
Since F w is a quasi pb, bq-periodic function it is sufficient to evaluate the function for @k P t0, ̈̈̈, b ́1u. Therefore, the least ν such that (3.26) (resp. (3.25)) holds is

ν " b́1 max k"0 ̀Fw pkq ́k̆.
Similarly, for (3.23), Inj 5 pwδ τ q is the least solution x of the inequality

wδ τ ĺ Injpxq " Inj ́ài γ ν i δ ζ i ̄" à i γ ν i δ ζ i . (3.27)
Then, the least ζ i such that the inequality (3.27) holds is τ and thus,

wδ τ ĺ à i pγ ν i δ τ q " γ ν δ τ , see, (2.29). (3.28)
Again since wδ τ ĺ γ ν δ τ ô w ĺ γ ν , it remains to find the greatest ν such that (3.28) holds. Therefore, @k P Z min Example 18. For the monomial γ 1 µ 3 β 3 γ 1 δ 2 P E 3|3 rrδss, see Figure 3.10b, the residual

Inj 7 pγ 1 µ 3 β 3 γ 1 δ 2 q " γ max 2 i"0 ̀Fγ 1 µ 3 β 3 γ 1 piq́i ̆δ2 " γ maxp1,0,2q δ 2 " γ 2 δ 2 .
We now compare γ 1 µ 3 β 3 γ 1 δ 2 to Inj ̀Inj 7 pγ 1 µ 3 β 3 γ 1 δ 2 q ̆and show that s ľ InjpInj 7 psqq is satisfied, see Remark 4. The canonical injection Inj ̀Inj 7 pγ 1 µ 3 β 3 γ 1 δ 2 q ̆" Injpγ 2 δ 2 q " γ 2 δ 2 is shown in Figure 3.10a. Clearly, Injpγ 2 δ 2 q " γ 2 δ 2 ĺ γ 1 µ 3 β 3 γ 1 δ 2 this is illustrated in Figure 3.11a where the (C/C)-functions

F γ 1 µ 3 β 3 γ 1 and F γ 2 , are shown. Obviously, F γ 2 ĺ F γ 1 µ 3 β 3 γ 1 pF γ 2 ě F γ 1 µ 3 β 3 γ 1 q, in particular, F γ 2 is the greatest quasi (1,1)-periodic (C/C)- function which is less than F γ 1 µ 3 β 3 γ 1 .
Therefore, γ 2 δ 2 is the greatest operator in E 1|1 rrδss which is less than γ 1 µ 3 β 3 γ 1 δ 2 . The dual residual of the monomial γ 1 µ 3 β 3 γ 1 δ 2 P E 3|3 rrδss is given by Inj 5 pγ 1 µ 3 β 3 γ 1 δ 2 q " γ minp1,0,2q δ 2 " γ 0 δ 2 " δ 2 .

Again we compare γ 1 µ 3 β 3 γ 1 δ 2 to Inj ̀Inj 5 pγ 1 µ 3 β 3 γ 1 δ 2 q ̆and show that s ĺ InjpInj 5 psqq is satisfied, see Remark 5. The canonical injection Inj ̀Inj 5 pγ 1 µ 3 β 3 γ 1 δ 2 q ̆" Injpδ 2 q " δ 2 is shown in Figure 3.10c. Clearly,

Injpδ 2 q " δ 2 ľ γ 1 µ 3 β 3 γ 1 δ 2 this is illustrated in Fig- ure 3.11b where the (C/C)-functions F γ 1 µ 3 β 3 γ 1 and F γ 0 , are shown. Obviously, F γ 1 µ 3 β 3 γ 1 ĺ F γ 0 pF γ 1 µ 3 β 3 γ 1 ě F γ 0 q, in particular, F γ 0 is the least quasi (1,1)-periodic (C/C)-function which is greater than F γ 1 µ 3 β 3 γ 1 and therefore γ 0 δ 2 is the least operator in E 1|1 rrδss which is greater than γ 1 µ 3 β 3 γ 1 δ 2 . 10 9 8 7 6 5 4 3 2 1 1 2 3 -1 0 1 2 3 4 5 6 7 8 t-shift I -c o u n t O -c o u n t (a) Inj ̀Inj 7 pγ 1 µ 3 β 3 γ 1 δ 2 q " γ 2 µ 1 β 1 δ 2 " γ 2 δ 2 10 9 8 7 6 5 4 3 2 1 1 2 3 -1 0 1 2 3 4 5 6 7 8 t-shift I -c o u n t O -c o u n t (b) γ 1 µ 3 β 3 γ 1 δ 2 10 9 8 7 6 5 4 3 2 1 1 2 3 -1 0 1 2 3 4 5 6 7 8 t-shift I -c o u n t O -c o u n t (c) Inj ̀Inj 5 pγ 1 µ 3 β 3 γ 1 δ 2 q " γ 0 µ 1 β 1 δ 2 " γ 0 δ 2 Figure 3.10. -Graphical comparison of γ 1 µ 3 β 3 γ 1 δ 2 , Inj ̀Inj 7 pγ 1 µ 3 β 3 γ 1 δ 2 q ̆and Inj ̀Inj 5 pγ 1 µ 3 β 3 γ 1 δ 2 q
̆. For all t P Z the slices in the (I/O-count)-planes of γ 1 µ 3 β 3 γ 1 δ 2 cover the slices of Inj ̀Inj 7 pγ 1 µ 3 β 3 γ 1 δ 2 q ̆, but are covered by the slices of Inj ̀Inj 5 pγ 1 µ 3 β 3 γ 1 δ 2 q ̆, see Figure 3.11.

Proposition 22. Let s " À i w i δ τ i P E b|b rrδss be a pb, bq-periodic series in the canonical representation, see Prop. 14, extended to infinite sums, then

Inj 7 psq " Inj 7 ́ài w i δ τ i ̄" à i γ max b́1 k"0 pFw i pkq́kq δ τ i , (3.30)
Second, for s P Errδss but s R E b|b rrδss,

Inj 7 psq " ε. (3.31)
Proof. For (3.30): Consider s " À i w i δ τ i in the canonical form, such that τ i ă τ ì1 and w i ą w ì1 and let F w i be the (C/C)-function associated with w i . Recall that Inj 7 psq is the greatest solution x in M ax in vγ, δw of inequality Injpxq ĺ s. This is given by (a) Graphical illustration of Inj 7 pγ 1 µ 3 β 3 γ 1 δ 2 q " γ 2 δ 2 in the (I/O-count)-planes for t ď 2

À i γ n i δ τ i F γ 2 F γ 1 µ 3 β 3 γ 1 I-count O-count 1
F γ 0 F γ 1 µ 3 β 3 γ 1 I-count O-count 1
µ 3 β 3 γ 1 δ 2 q " γ 0 δ 2 in the (I/O-count)-planes for t ď 2 Figure 3.11. -Comparison of Inj 7 pγ 1 µ 3 β 3 γ 1 δ 2 q and Inj 5 pγ 1 µ 3 β 3 γ 1 δ 2 q in the (I/O-count)-planes for t ď 2.
In (a) the (C/C)-function F γ 2 lies in the gray area shaped by the

F γ 1 µ 3 β 3 γ 1 function, thus F γ 2 ĺ F γ 1 µ 3 β 3 γ 1 and γ 2 ĺ γ 1 µ 3 β 3 γ 1 . In (b) the (C/C)-function F γ 1 µ 3 β 3 γ 1 lies in the gray area shaped by the F γ 0 function, thus F γ 1 µ 3 β 3 γ 1 ĺ F γ 0 and γ 1 µ 3 β 3 γ 1 ĺ γ 0 .
where n i is the greatest integer such that γ n i ĺ w i . Repeating the first step of the proof of Lemma 1, this is given by n i " max b́1 k"0 pF w i pkq ́kq. To prove (3.31), recall that @s P Errδss we must satisfy the following inequality, see (2.17) in Remark 4,

s ľ Inj ̀Inj 7 psq ̆. (3.32)
Now let us consider two series s 1 P E m 1 |b 1 rrδss and

s 2 P E m 2 |b 2 rrδss such that s 1 ‰ ε, s 2 ‰ ε and m 1 b 1 ‰ m 2 b 2 .
Then s 1 and s 2 are not ordered, i.e., s 1 ń s 2 and s 1 ł s 2 (see Prop. 13). The canonical injection Injp sq of an arbitrary series s P M ax in vγ, δw is p1, 1q-periodic, i.e., Injp sq P E 1|1 rrδss. Thus, for s R E b|b rrδss, s and Injp sq are not ordered and (3.32) holds if and only if Inj 7 psq " ε. Proposition 23. Let s " À i w i δ τ i P E b|b rrδss be a pb, bq-periodic series in the canonical representation, see Prop. 14, extended to infinite sums, then

Inj 5 psq " Inj 5 ́ài w i δ τ i ̄" à i γ min b́1 k"0 pFw i pkq́kq δ τ i , (3.33)
Second, for s P Errδss but s R E b|b rrδss,

Inj 5 psq " ε. (3.34)
Proof. The proof is similar to the proof of Prop. 22.

Example 19. Consider the polynomial p " γ 1 µ 3 β 3 γ 1 δ 2 'µ 3 β 3 γ 2 δ 3 P E 3|3 rrδss with a canonical form p " pµ 3 β 3 γ 2 ' γ 1 µ 3 β 3 γ 1 qδ 2 ' µ 3 β 3 γ 2 δ 3 and a graphical representation given in Figure 3.12a. Then, Inj 7 ppq " γ 1 δ 2 ' γ 2 δ 3 and Inj ̀Inj 7 ppq ̆" γ 1 δ 2 ' γ 2 δ 3 are shown in Figure 3.12b. Moreover, Figure 3.13a illustrates Inj 7 ppµ 3 β 3 γ 2 ' γ 1 µ 3 β 3 γ 1 qδ 2 q " γ 1 δ 2 for the (I/O-count)-plane at t " 2 and Figure 3.13b illustrates Inj 7 pµ 3 β 3 γ 2 δ 3 q " γ 2 δ 3 for the (I/Ocount)-plane at t " 3, respectively. t-shift

I -c o u n t O -c o u n t (a) p " pµ 3 β 3 γ 2 ' γ 1 µ 3 β 3 γ 1 qδ 2 ' µ 3 β 3 γ 2 δ 3 10 9 8 7 6 5 4 3 2 1 1 2 3 4 -1 0 1 2 3 4 5 6 7 8 t-shift I -c o u n t O -c o u n t (b) Inj ̀Inj 7 ppq ̆" γ 1 δ 2 ' γ 2 δ 3 Figure 3.12. -Graphical comparison of the polynomial p " pµ 3 β 3 γ 2 ' γ 1 µ 3 β 3 γ 1 qδ 2 ' µ 3 β 3 γ 2 δ 3 and
Inj ̀Inj 7 ppq ̆. For all t P Z the slices in the (I/O-count)-planes of p cover the slices of Inj ̀Inj 7 ppq ̆, see Figure 3.13. Zero slice Mapping Ψ m|b : E m|b rrδss Ñ M ax in vγ, δw In addition to the canonical injection Inj : M ax in vγ, δw Ñ Errδss, we define a mapping:

F γ 1 F µ 3 β 3 γ 2 'γ 1 µ 3 β 3 γ 1 t " 2 I-count O-count
F γ 2 F µ 3 β 3 γ 2 t " 3 I-count O-count
Ψ m|b : E m|b rrδss Ñ M ax in vγ, δw.
Definition 36. Let s " À i w i δ t i P E m|b rrδss be an pm, bq-periodic series, then

Ψ m|b psq " Ψ m|b ́ài w i δ t i ̄" à i γ Fw i p0q δ t i . (3.35)
This mapping Ψ m|b has a graphical interpretation. If we take the 3D representation of a series s P E m|b rrδss the series s " Ψ m|b psq P M ax in vγ, δw corresponds to the slice in the (Ocount/t-shift)-plane of the 3D representation at the I-count value 0, therefore this mapping is also called zero-slice mapping.

Example 20. Consider the following series s P E 3|2 rrδss,

s " γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 δ 2 q ̊̀pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 ' pγ 6 µ 3 β 2 ' γ 5 µ 3 β 2 γ 1 qδ 4 ̆.
with a graphical representation given in Example 16 in Figure 3.8. 4 The series Ψ 3|2 psq P M ax in vγ, δw corresponds to the slice ((O-count/t-shift)-plane) for the I-count value 0 of the 3D representation of s, see Figure 3.14a and Figure 3.14b. Moreover, the asymptotic slope of Ψ 3|2 psq P M ax in vγ, δw is the same as the left-asymptotic slope of s P E 3|2 rrδss, i.e., σpΨ 3|2 psqq " σ l psq " 2{3. The mapping Ψ m|b is by definition lower-semicontinuous, see Definition 36, therefore Ψ m|b is residuated.

Ψ 3|2 psq " γ 1 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 δ 3 ' γ 5 δ
Proposition 24. Let s " À i γ ν i δ τ i P M ax in vγ, δw.
The residual Ψ 7 m|b psq P E m|b rrδss of s is a series defined by

Ψ 7 m|b ́ài γ ν i δ τ i ̄" à i γ ν i δ τ i µ m β b " sµ m β b . (3.36) 50
3.2. pM ax in vγ, δw , ', bq as a Subdioid of pErrδss, ', bq

Proof. By definition of the residuated mapping, Ψ 7 m|b p À i γ ν i δ τ i q P E m|b rrδss is the greatest solution of the following inequality

s " à i γ ν i δ τ i ľ Ψ m|b pxq " Ψ m|b ́àj w j δ ζ j ̄, (3.37)
where x " À j w j δ ζ j P E m|b rrδss. First, we show that (3.36) satisfies (3.37) with equality.

Ψ m|b ́à i γ ν i δ τ i µ m β b ̄" à i γ F γ ν i µmβ b p0q δ τ i " à i γ ν i δ τ i , since F γ ν i µmβ b p0q " ν i ̀t0{bum " ν i , see (3.11
), (3.9) and (3.10). Taking into account that Ψ m|b is isotone, it remains to show that

À i γ ν i δ τ i µ m β b is the greatest solution of à i γ ν i δ τ i " Ψ m|b pxq " Ψ m|b ́à j w j δ ζ j ̄" à j γ Fw j p0q δ ζ j . (3.38)
Clearly, to achieve equality we need ζ j " τ i and F w j p0q " ν i . Furthermore, we are looking for the greatest w j P E m|b rrδss, such that ν i " F w j p0q. Due to the canonical form Prop. 11 we can write an pm, bq-periodic E-operator as

À b i"1 γ n i µ m β b γ n 1 i with 0 ď n 1 i ă b. This operator corresponds to the (C/C)-function Fpkq " b min i"1 ́ni ̀Z n 1 i ̀k b ^m̄.
Now we examine Fpkq for k " 0, thus

Fp0q " b min i"1 ́ni ̀Z n 1 i b ^m̄.
Recall that 0 ď n 1 i ă b, hence F w j pkq " ν i ̀tp0̀kq{bum is the least quasi pm, bq-periodic (C/C)-function such that (3.38) holds, i.e., F w j p0q " F γ ν i µmβ b p0q " ν i ̀t0{bum " ν i . This function corresponds to the operator γ ν i µ m β b .

Example 21. Recall Example 20 with,

s " γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 δ 2 q ̊̀pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 ' pγ 6 µ 3 β 2 ' γ 5 µ 3 β 2 γ 1 qδ 4 ̆, s " Ψ 3|2 psq " γ 1 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 δ 3 ' γ 5 δ 4 ̆.

The residual Ψ 7

3|2 p sq is given by

Ψ 7 3|2 p sq " ́γ1 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 δ 3 ' γ 5 δ 4 ̆̄µ 3 β 2 , " γ 1 µ 3 β 2 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 µ 3 β 2 δ 3 ' γ 5 µ 3 β 2 δ 4 ̆.
In Figure 3.15a and Figure 3.15b, s and Ψ 7 3|2 pΨ 3|2 psqq are compared, as required s ĺ Ψ 7 3|2 ̀Ψ3|2 psq ̆, see (2.17). t-shift t-shift Clearly, for all I-count values k P Z min the corresponding slice of Ψ 7 3|2 ̀Ψ3|2 psq ̆covers the corresponding slice of s, therefore as required s ĺ Ψ 7 3|2 ̀Ψ3|2 psq ̆.

I -c o u n t O -c o u n t (a) s " γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 δ 2 q ̊̀pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 ' pγ 6 µ 3 β 2 ' γ 5 µ 3 β 2 γ 1 qδ 4 ̆.
I -c o u n t O -c o u n t (b) Ψ 7 3|2 ̀Ψ3|2 psq ̆" γ 1 µ 3 β 2 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 µ 3 β 2 δ 3 ' γ 5 µ 3 β 2 δ 4
Proposition 25. Let s " À i γ ν i δ τ i P M ax in vγ, δw. The dual residual Ψ 5 m|b psq P E m|b rrδss of s is a series defined by

Ψ 5 m|b ́ài γ ν i δ τ i ̄" à i γ ν i δ τ i µ m β b γ b́1 " sµ m β b γ b́1 . (3.39)
Proof. The proof is similar to the proof of Prop. 24, with the difference that instead of finding the greatest solution we are now looking for the least solution, denoted by Ψ 5 m|b p À i γ ν i δ τ i q P E m|b rrδss, of the following inequality

s " à i γ ν i δ τ i ĺ Ψ m|b pxq " Ψ m|b ́àj w j δ ζ j ̄. (3.40)
Again we show that (3.39) satisfies (3.40) with equality.

Ψ m|b ́à i γ ν i δ τ i µ m β b γ b́1 ̄" à i γ F γ ν i µmβ b γ b́1 p0q δ τ i " à i γ ν i δ τ i , since F γ ν i µmβ b γ b́1 p0q " ν i ̀tpb ́1q{bum " ν i , see (3.11
), (3.9) and (3.10). Taking into account that Ψ m|b is isotone, it remains to show that

À i γ ν i δ τ i µ m β b γ b́1 is the least solution of à i γ ν i δ τ i " Ψ m|b pxq " Ψ m|b ́à j w j δ ζ j ̄" à j γ Fw j p0q δ ζ j . (3.41)
Clearly, to achieve equality we need ζ j " τ i and F w j p0q " ν i . Furthermore, we are looking for the smallest w j P E m|b rrδss, such that ν i " F w j p0q. Due to the canonical form Prop. 11

an pm, bq-periodic E-operator can be written as

À b i"1 γ n i µ m β b γ n 1 i with 0 ď n 1 i ă b. This operator corresponds to a (C/C)-function Fpkq " b min i"1 ́ni ̀Z n 1 i ̀k b ^m̄.
Now we examine Fpkq for k " 0, thus

Fp0q " b min i"1 ́ni ̀Z n 1 i b ^m̄.
Recall that 0 ď n 1 i ă b, hence F w j pkq " ν i ̀tppb ́1q ̀kq{bum is the smallest (i.e. smallest in the order in Z min , hence greatest in the natural order in Z ) quasi pm, bq-periodic (C/C)function such that (3.41) holds, i.e., F w j p0q " F γ ν i µmβ b γ b́1 p0q " ν i ̀tpb ́1q{bum " ν i . This function corresponds to the operator

γ ν i µ m β b γ b́1 . Example 22. Recall Example 20 with, s " γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 δ 2 q ̊̀pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 ' pγ 6 µ 3 β 2 ' γ 5 µ 3 β 2 γ 1 qδ 4 ̆, s " Ψ 3|2 psq " γ 1 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 δ 3 ' γ 5 δ 4 ̆.

The dual residual Ψ 5

3|2 p sq is given by

Ψ 5 3|2 p sq " ́γ1 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 δ 3 ' γ 5 δ 4 ̆̄µ 3 β 2 γ 1 ,
" γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 µ 3 β 2 γ 1 δ 3 ' γ 5 µ 3 β 2 γ 1 δ 4 ̆.

See Figure 3.16 for a graphical comparison of the series s and the series Ψ 5 3|2 ̀Ψ3|2 psq ̆.

Core Decomposition of Elements in E m|b rrδss

This section focuses on a specific decomposition of series in E m|b rrδss. This decomposition is a factorization of an element in E m|b rrδss, where the core part is a matrix in M ax in vγ, δw. Based on this decomposition it is shown that operations on ultimately cyclic series in E m|b rrδss can be reduced to operations on matrices with entries in M ax in vγ, δw. A series s P E m|b rrδss can always be represented as m m Qb b , where Q is a matrix with entries in M ax in vγ, δw, called core matrix, of size m ̂b. m m is a row vector defined by t-shift t-shift Clearly, for all I-count values k P Z min the corresponding slice of s covers the corresponding slice of

m m :" " µ m γ 1 µ m ̈̈̈γ ḿ1 µ m ı ,
I -c o u n t O -c o u n t (a) s " γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 δ 2 q ̊̀pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 ' pγ 6 µ 3 β 2 ' γ 5 µ 3 β 2 γ 1 qδ 4 ̆.
I -c o u n t O -c o u n t (b) Ψ 5 3|2 ̀Ψ3|2 psq ̆" γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 δ 2 q ̊̀γ 3 µ 3 β 2 γ 1 δ 3 ' γ 5 µ 3 β 2 γ 1 δ 4 ̆.
Ψ 5 3|2 ̀Ψ3|2 psq ̆, therefore as required s ľ Ψ 5 3|2 ̀Ψ3|2 psq ̆, see (2.21).
The index b (resp. m) determines the division (resp. multiplication) coefficient and gives the dimension of the vector. First, we illustrate how to obtain this representation on a small example and then provide a formal proof.

Example 23. Consider the following series s P E 2|2 rrδss,

s " γ 1 µ 2 β 2 ' pγ 2 δ 2 q ̊pµ 2 β 2 γ 1 ' γ 2 µ 2 β 2 δ 2 q.
Due to (3.13), γ m̂n µ m " µ m γ n , this series can be written as s " γ 1 µ 2 e lo omo on

M 1 β 2 ' µ 2 pγ 1 δ 2 q loomoon S 1 β 2 γ 1 ' µ 2 γ 1 δ 2 pγ 1 δ 2 q loooooomoooooon S 2 β 2 .
Clearly, M 1 , S 1 , S 2 P M ax in vγ, δw. Furthermore, in this form the entries of the m 2 -vector and b 2 -vector appear on the left and on the right of M 1 , S 1 , S 2 . We now can write s in the core-form m 2 Qb 2 as follows,

s " " µ 2 γ 1 µ 2 ı looooomooooon m 2 « pγ 1 δ 2 q ̊γ1 δ 2 pγ 1 δ 2 q ε e ff loooooooooooooomoooooooooooooon Q « β 2 γ 1 β 2 ff looomooon b 2 .
It is easy to check that this expression m 2 Qb 2 , indeed represents the series s, since

m 2 Qb 2 " " µ 2 pγ 1 δ 2 q ̊' γ 1 µ 2 ε µ 2 γ 1 δ 2 pγ 1 δ 2 q ̊' γ 1 µ 2 e ı « β 2 γ 1 β 2 ff , " " pγ 2 δ 2 q ̊µ2 γ 2 δ 2 pγ 2 δ 2 q ̊µ2 ' γ 1 µ 2 ı « β 2 γ 1 β 2 ff , " pγ 2 δ 2 q ̊µ2 β 2 γ 1 ' γ 2 δ 2 pγ 2 δ 2 q ̊µ2 β 2 ' γ 1 µ 2 β 2 , " γ 1 µ 2 β 2 ' pγ 2 δ 2 q ̊pµ 2 β 2 γ 1 ' γ 2 µ 2 β 2 δ 2 q " s.
Proposition 26. Let s " À i w i δ i P E m|b rrδss be an pm, bq-periodic series, then s can be written as s " m m Qb b , where Q P M ax in vγ, δw m̂b .

Proof. s being an pm, bq-periodic series implies that all coefficients w i of s are pm, bqperiodic E-operators. Then due to Prop. 11 all coefficients can be expressed in canonical form w i "

À J i j"1 γ ν i j µ m β b γ ν 1
i j with J i ď minpm, bq and 0 ď ν 1 i j ă b. Therefore, s can be rewritten as

s " à i ̀Ji à j"1 γ ν i j µ m β b γ ν 1 i j ̆δi .
Due to (3.13) and the fact that @w P E, wδ " δw, the series s can be written as

s " à i ̀Ji à j"1 γ νi j µ m γ νi j δ i β b γ ν 1 i j ̆,
where 0 ď νi j " ν i j ́tν i j {mum ă m and νi j " tν i j {mu. Observe that 0 ď νi j ă m an 0 ď ν 1 i j ă b, hence s is expressed by

s " " µ m γ 1 µ m ̈̈̈γ ḿ1 µ m ı à i ̀Ji à j"1 Q i j ̆» - - - - - β b γ b́1 ̈̈β b γ 1 β b fi ffi ffi ffi ffi fl ,
where the entry pQ i j q 1̀ν i j ,b́ν 1 i j " γ νi j δ i and all other entries of Q i j are equals ε. Finally s

is in the required form s " m m Qb b , where Q " À i ̀ÀJ i j"1 Q i j ̆.
For the particular case, where s P E m|b rrδss is a periodic ultimately cyclic series the coreform can be obtained as follows. Given an ultimately cyclic series s "

À K k"1 w k δ t k ' À L l"1 w 1 l δ t 1 l pγ ν δ τ q ̊P E
m|b rrδss, we can always write s such that all coefficients w k , w 1 l are in the pm, bq-periodic canonical form of Prop. 11, i.e., s "

I à i"1 γ n i µ m β b γ n 1 i δ t i ' J à j"1 γ N j µ m β b γ N 1 j δ T j ̀γν δ τ ̆̊.
Recall that γ m µ m " µ m γ 1 and β b γ b " γ 1 β b , see (3.13). Moreover, we can always represent an ultimately cyclic series s P E m|b rrδss such that ν is a multiple of b, i.e., we can extend pγ νδ τq ̊such that, ν " νl " lcmp ν, bq

pγ νδ τq ̊" pe ' γ νδ τ ' ̈̈̈' γ pĺ1q νδ pĺ1q τqpγ l νδ l τq ̊, " pe ' γ νδ τ ' ̈̈̈' γ pĺ1q νδ pĺ1q τqpγ ν δ τ q ̊.
Therefore, in the following we assume ν{b P N and thus β b pγ ν δ τ q ̊" pγ ν{b δ τ q ̊βb . It follows that s can be written as, s "

I à i"1 γ ñi µ m γ n i δ t i lo omo on M i β b γ ñ 1 i ' J à j"1 γ Ñj µ m γ N j δ T j pγ ν{b δ τ q loooooooomoooooooon S j β b γ Ñ 1 j , (3.42)
where 0 ď ñ1 i , Ñ1 j ă b and 0 ď ñi , Ñj ă m. Clearly, in this representation, M i are monomials and S j are series in the dioid pM ax in vγ, δw , ', bq. Moreover, the entries of the b bvector appear on the right and the entries of the m m -vector appear on the left of monomial M i (resp. series S j ). For a given s we denote the set of monomials by M " tM i , ̈̈̈, M I u and the set of series by S " tS j , ̈̈̈, S J u. Furthermore, the subsets M l,k (resp. S l,k ) are defined as @l P t0, ̈̈̈, m ́1u, @g P t0, ̈̈̈, b ́1u, M l,g :"

tM i P M| γ l µ m M i β b γ g P I à i"1 γ ñi µ m M i β b γ ñ 1 i u, S l,g :" tS j P S| γ l µ m S j β b γ g P J à j"1 γ Ñj µ m S j β b γ Ñ 1 j u.
The element pQq l̀1,b́g of the core matrix is then obtained by

pQq l̀1,b́g " à MPM l,g M ' à SPS l,g S.
In other words, monomial M i and series S j are "dispatched" in Q depending on the left factor γ i µ m and the right factor β b γ j of each term of s in (3.42).

Remark 10. Note that, for s " m m Qb b be an ultimately cyclic series in E m|b rrδss, the entries of Q are ultimately cyclic series in M ax in vγ, δw. Example 24. Consider the following series

s 1 " γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 pγ 1 δ 1 q ̊P E 3|2 rrδss.
We first extend pγ 1 δ 1 q ̊" pe ' γ 1 δ 1 qpγ 2 δ 2 q ̊, because in this example b " 2. This leads to

s 1 "γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 pe ' γ 1 δ 1 qpγ 2 δ 2 q "γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 µ 3 β 2 γ 1 δ 3 ' γ 5 µ 3 β 2 δ 3 ' γ 3 µ 3 β 2 γ 2 δ 4 ' γ 5 µ 3 β 2 γ 1 δ 4 qpγ 2 δ 2 q "γ 1 µ 3 β 2 γ 1 δ 2 ' ppγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 ' pγ 6 µ 3 β 2 ' γ 5 µ 3 β 2 γ 1 qδ 4 qpγ 2 δ 2 q ̊.
Now every term in the sum is rewritten as follows

γ 1 µ 3 β 2 γ 1 δ 2 " γ 1 µ 3 δ 2 β 2 γ 1 , γ 3 µ 3 β 2 γ 1 δ 3 pγ 2 δ 2 q ̊" µ 3 γ 1 δ 3 pγ 1 δ 2 q ̊β2 γ 1 , γ 5 µ 3 β 2 δ 3 pγ 2 δ 2 q ̊" γ 2 µ 3 γ 1 δ 3 pγ 1 δ 2 q ̊β2 , γ 6 µ 3 β 2 δ 4 pγ 2 δ 2 q ̊" µ 3 γ 2 δ 4 pγ 1 δ 2 q ̊β2 , γ 5 µ 3 β 2 γ 1 δ 4 pγ 2 δ 2 q ̊" γ 2 µ 3 γ 1 δ 4 pγ 1 δ 2 q ̊β2 γ 1 .
Therefore, s 1 can be rephrased as,

s 1 "γ 1 µ 3 δ 2 lo omo on M 1 β 2 γ 1 ' µ 3 ́γ2 δ 4 pγ 1 δ 2 q ̊loooooooomoooooooon S 1 β 2 ' µ 3 ́γ1 δ 3 pγ 1 δ 2 q ̊loooooooomoooooooon S 2 β 2 γ 1 ' γ 2 µ 3 ́γ1 δ 3 pγ 1 δ 2 q ̊loooooooomoooooooon S 3 β 2 ' γ 2 µ 3 ́γ1 δ 4 pγ 1 δ 2 q ̊loooooooomoooooooon S 4 β 2 γ 1 .
For this series we obtain the following subsets

M 1,1 " tδ 2 u, M 0,0 " M 0,1 " M 1,0 " M 2,0 " M 2,1 " tεu, S 0,0 " tγ 2 δ 4 pγ 1 δ 2 q ̊u, S 0,1 " tγ 1 δ 3 pγ 1 δ 2 q ̊u, S 2,0 " tγ 1 δ 3 pγ 1 δ 2 q ̊u, S 2,1 " tγ 1 δ 4 pγ 1 δ 2 q ̊u, S 1,0 " S 1,1 " tεu.
The core-form of the series s 1 is given by m 3 Qb 2 where

Q " » - - γ 1 δ 3 pγ 1 δ 2 q ̊γ2 δ 4 pγ 1 δ 2 q δ2 ε γ 1 δ 4 pγ 1 δ 2 q ̊γ1 δ 3 pγ 1 δ 2 q ̊fi ffi fl .
Properties of the m m -vector and the b b -vector

In the following, we elaborate some properties of the m m -vector and the b b -vector, which are useful for computations for series s P E m|b rrδss. Consider the m i -vector and the b i -vector with same index i, i.e., m i -vector and b i -vector have the same length. The scalar product m i b b i is the identity e, since Prop. 12,

m i b b i " µ i β i γ í1 ' γ 1 µ i β i γ í2 ' ̈̈̈' γ í1 µ i β i " e.
(3.43)

The dyadic product b i b m i is a square matrix in M ax in vγ, δw, denoted by E.

E " b i b m i " » - - - - - - - - β i γ í1 µ i γ 1 β i µ i γ 1 β i γ 1 µ i ̈̈̈γ 1 β i γ í2 µ i β i γ í2 µ i β i γ í1 µ i γ 1 β i µ i ̈̈̈γ 1 β i γ í3 µ i . . . . . . . . . . . . β i γ 1 µ i β i γ 2 µ i β i γ 3 µ i ̈̈̈γ 1 β i µ i β i µ i β i γ 1 µ i β i γ 2 µ i ̈̈̈β i γ í1 µ i fi ffi ffi ffi ffi ffi ffi ffi fl , " » - - - - - - e γ 1 ̈̈̈γ 1 . . . . . . . . . . . . . . . . . . γ 1 e ̈̈̈̈̈̈e fi ffi ffi ffi ffi ffi fl , (3.44)
since β i γ n µ i " e for 0 ď n ă i. If necessary, the dimension of E is stated as an index, e.g., E i " b i m i P te, γ 1 u îi .

Proposition 27. For the E matrix, the following relations hold

E i b E i " E i , (3.45)
E i b b i " b i , (3.46)
m i b E i " m i . (3.47)
Proof. Because of m i b i " e, see (3.43), we have

E i b E i " b i b m i b b i b m i " b i b e b m i " E i , E i b b i " b i b m i b b i " b i b e " b i , m i b E i " m i b b i b m i " e b m i " m i .
Corollary 1. Observe that I ' E " E and E " EE, as a consequence, Proof.

E " I ' E ' EE ' EEE ' ̈̈" E ̊. ( 3
E zpEDq " pb m m m q zpEDq, " m m zpb m zpEDqq, since pabq zx " b zpa zxq, see (A.5) in Appendix A " m m zpm m EDq, since (3.50) " b m pm m EDq, since (3.49)
" EED " ED.

The proof of the right division pDEq{E " DE is analogous.

Greatest Core-Form

Given a series s " m m Qb b P E m|b rrδss, in general, the core-matrix Q is not unique, i.e., s " m m Qb b " m m Q 1 b b , where Q ‰ Q 1 . In the following, we prove that s admits a unique greatest core, denoted Q P M ax in vγ, δw m̂b (greatest with respect to the order relation in the dioid M ax in vγ, δw, i.e., Q ľ Q and Q ľ Q 1 ). Proposition 29. Let s " m m Qb b P E m|b rrδss be a decomposition of s P E m|b rrδss. The greatest core matrix is given by,

Q " E m QE b .
(3.51)

Proof. Consider the inequality m m Xb b ĺ m m Qb b " s. Then because of Prop. 28, the greatest solution for X is

m m zm m Qb b {b b " b m m m Qb b m b " E m QE b " Q. Furthermore, because of m m " m m E m (3.47) and b b " E b b b (3.46), m m Qb b " m m E m QE b b b " m m Qb b " s.
Remark 11. The greatest core matrix Q has the following properties. Since E b E " E, then E Q " EEQE " Q and QE " EQEE " Q. As a consequence, E QE " Q.

Remark 12. Due to the order of the entries in the E matrix, the left and right multiplications of the core matrix with the E matrix induce ordering of the entries in the greatest core Q. More precisely, in every row the entries are in descending order, i.e. @i P t1, ̈̈̈, mu, @j P t1, ̈̈̈, b ́1u p Qq i,j ľ p Qq i,j̀1 and in every column the entries are in an ascending order, i.e. @i P t1, ̈̈̈, m ́1u, @j P t1, ̈̈̈, bu p Qq i,j ĺ p Qq ì1,j . Furthermore, all entries of the greatest core have the same asymptotic slope. Thus, the greatest core is highly redundant. When we think about software tools it is desirable to reduce memory usage. Therefore, for implementation the order in Q can be used to define a lean representation of Q.

Example 25. The greatest core of the series considered in Example 24 is given by

Q " E 3 QE 2 , " » - - e γ 1 γ 1 e e γ 1
e e e fi ffi fl » --

γ 1 δ 3 pγ 1 δ 2 q ̊γ2 δ 4 pγ 1 δ 2 q δ2 ε γ 1 δ 4 pγ 1 δ 2 q ̊γ1 δ 3 pγ 1 δ 2 q ̊fi ffi fl « e γ 1 e e ff , " » - - γ 1 δ 3 pγ 1 δ 2 q ̊γ2 δ 4 pγ 1 δ 2 q δ2 ' γ 1 δ 3 pγ 1 δ 2 q ̊γ1 δ 2 pγ 1 δ 2 q δ2 pγ 1 δ 2 q ̊γ1 δ 3 pγ 1 δ 2 q ̊fi ffi fl .
Note that all entries have the same asymptotic slope pγ 1 δ 2 q ̊. Moreover, observe that the entries in Q are ordered, e.g., p Qq 1,1 ľ p Qq 1,2 , as γ 1 δ 3 pγ 1 δ 2 q ̊ľ γ 2 δ 4 pγ 1 δ 2 q ̊, respectively p Qq 1,2 ĺ p Qq 2,2 , as γ 2 δ 4 pγ 1 δ 2 q ̊ĺ γ 1 δ 2 pγ 1 δ 2 q ̊, etc.

Calculation with the Core Decomposition

To perform addition between two series

s 1 " m m 1 Q1 b b 1 P E m 1 |b 1 rrδss, s 2 " m m 2 Q2 b b 2 P E m 2 |b 2
rrδss with equal gain, i.e. m 1 {b 1 " m 2 {b 2 , in the core-form it is necessary to express the core matrices Q1 P M ax in vγ, δw m 1 ̂b1 and Q2 P M ax in vγ, δw m 2 ̂b2 with equal dimensions. This is possible by expressing both series with their least common period m " lcmpm 1 , m 2 q, see the following proposition.

Q1 " » - - - - - β n γ ń1 Qµ n β n γ ń1 Qγ 1 µ n ̈̈̈β n γ ń1 Qγ ń1 µ n β n γ ń2 Qµ n β n γ ń2
Proposition 31. Let s " m m Qb b , s 1 " m m Q 1 b b P E m|b rrδss be two ultimately cyclic series, the sum s ' s 1 " m m Q 2 b b P E m|b rrδss is an ultimately cyclic series, where Q 2 " pQ ' Q 1 q.
Proof.

s ' s 1 " m m Qb b ' m m Q 1 b b " m m pQ ' Q 1 qb b " m m Q 2 b b
Clearly, the entries of the core matrices Q and Q 1 are ultimately cyclic series in M ax in vγ, δw. Because of Theorem 2.6 the sum of two ultimately cyclic series in M ax in vγ, δw is again an ultimately cyclic series. Therefore, Q 2 is composed of ultimately cyclic series in M ax in vγ, δw and thus s '

s 1 " m m Q 2 b b is an ultimately cyclic series in E m|b rrδss. Corollary 3. Let s " m m Qb b , s 1 " m m Q1 b b P E m|b rrδss be two ultimately cyclic series,
with Q, Q1 are greatest cores, the sum s ' s 1 " m m Q2 b b P E m|b rrδss is an ultimately cyclic series, where Q2 " p Q ' Q1 q is again a greatest core.

Proof.

s ' s 1 " m m Qb b ' m m Q1 b b " m m pE QE ' E Q1 Eqb b " m m Ep Q ' Q1 qE loooooomoooooon Q 2 b b To perform multiplication between two series s 1 " m m 1 Q1 b b 1 P E m 1 |b 1 rrδss, s 2 " m m 2 Q2 b b 2 P E m 2 |b 2
rrδss in the core-form it is necessary to express the core matrices with appropriate dimensions. Due to Prop. 30 and by choosing b " lcmpb 1 , m 2 q we can express

s 1 , s 2 such that s 1 " m m 1 1 Q1 1 b b and s 2 " m b Q1 2 b b 1 2 , with m 1 1 " m 1 ̂lcmpb 1 , m 2 q{b 1 and b 1 2 " b 2 ̂lcmpb 1 , m 2 q{m 2 . Proposition 32. Let s " m m Qb b P E m|b rrδss and s 1 " m b Q 1 b b 1 P E b|b 1 rrδss be two ulti- mately cyclic series, the product s b s 1 " m m Q 2 b b 1 P E m|b 1 rrδss is an ultimately cyclic series, where Q 2 " QEQ 1 .
Proof.

s b s 1 " m m Qb b m b Q 1 b b 1 " m m QEQ 1 b b 1 " m m Q 2 b b 1
Moreover, the entries of the core matrices Q and Q 1 are ultimately cyclic series in M ax in vγ, δw. Because of Theorem 2.6 the sum and product of ultimately cyclic series in M ax in vγ, δw are again ultimately cyclic series in M ax in vγ, δw. Therefore, entries of Q 2 are ultimately cyclic series in M ax in vγ, δw and the product Proof. Again due to Theorem 2.6 the Kleene star, sum, and product of ultimately cyclic series in M ax in vγ, δw are ultimately cyclic series in M ax in vγ, δw and therefore, s ̊" m b pQEq ̊bb is an ultimately cyclic series in E b|b rrδss. Remark 13. Let s " m b Qb b P E b|b rrδss be an ultimately cyclic series expressed with a greatest core. Then, s ̊" m b Q̊b b P E b|b rrδss is an ultimately cyclic series. However, in general, Q̊ĺ E Q̊E as:

s b s 1 " m m Q 2 b b 1 is an ultimately cyclic series in E m|b 1 rrδss.
s b s 1 " m m Qb b m b Q1 b b 1 " m m QE Q1 b b 1 " m m Q Q1 b b 1 , Furthermore: Q Q1 " E QEE Q1 E " Q2 .
Q̊" I ' Q ' Q2 ̈̈" I ' E QE ' E Q2 E ̈̈̈.
Whereas,

E Q̊E " EIE ' E QE ' E Q" E ̈̈" E ' Q ' Q2 ̈̈̈.
However, E Q̊E " pE Q̊E q ̊as E " E ' I and E Q̊E E Q̊E " E Q̊E . For an illustration, consider the star of the zero element ε, clearly pεq ̊" e. In the core-from, with m " b " 2, this can be written as

pεq ̊" m 2 « ε ε ε ε ff ̊b2 " m 2 « e ε ε e ff b 2 .
Note that in this case, I is not the greatest core, i.e. I ă EIE " E.

In general, for complete partially ordered dioids, such as pErrδss, ', bq, multiplication is not distributive over ^, see (2.2). However, in the following lemmas, we show that distributivity holds for left multiplication by the m m -vector and right multiplication by the b m -vector for specific matrices with entries in Errδss.

Lemma 2. Let Q 1 , Q 2 P Errδss m̂b , then m m pEQ 1 ^EQ 2 q " m m EQ 1 ^mm EQ 2 .
Proof. The proof is similar to the proof of Lemma 4.36 in [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][Chap 4.3.]. There distributivity is proven for cpa ^bq " ca ^cb for the case that c admits a left and right inverse. For this proof, recall that e " m m b m (3.43), E " b m m m (3.44) and E " EE (3.45). Moreover, recall that inequality cpa ^bq ĺ ca ^cb holds for a, b, c elements in a partially ordered dioid, see (2.2). Now let us define q 1 " m m EQ 1 and q 2 " m m EQ 2 , then q 1 ^q2 " epq 1 ^q2 q " m m b m pq 1 ^q2 q ĺ m m pb m q 1 ^bm q 2 q.

Inserting q 1 " m m EQ 1 and q 2 " m m EQ 2 lead to,

m m pb m q 1 ^bm q 2 q " m m pb m m m EQ 1 ^bm m m EQ 2 q, " m m pEEQ 1 ^EEQ 2 q, " m m pEQ 1 ^EQ 2 q.
Finally, m m pEQ 1 ^EQ 2 q ĺ m m EQ 1 ^mm EQ 2 " q 1 ^q2 .

Hence, equality holds throughout.

Lemma 3. Let Q 1 , Q 2 P Errδss m̂b , then pQ 1 E ^Q2 Eqb b " Q 1 Eb b ^Q2 Eb b .
Proof. The proof is similar to the proof of Lemma 2. q is again a greatest core.

Proof. Let us recall that Q " E QE, then according to Lemma 2 and Lemma 3 we can write

s ^s1 " m m Qb b ^mm Q1 b b " m m E QEb b ^mm E Q1 Eb b " m m pE QE ^E Q1 Eqb b " m m p Q ^Q 1 qb b " m m p Q2 qb b .
It remains to be shown that

Q2 " p Q ^Q 1 q is a greatest core. Clearly, E " E ̊" I ' E implies that Q2 ĺ E Q2 E.
Then, according to Lemma 2 and Lemma 3,

E Q2 E " Ep Q ^Q 1 qE " b m m m p Q ^Q 1 qb b m b " b m pm m Qb b ^mm Q1 b b qm b .
Recall, cpa ^bq ĺ ca ^cb and pa ^bqc ĺ ac ^bc (2.2), therefore

b m pm m Qb b ^mm Q1 b b qm b ĺ b m m m Qb b m b ^bm m m Q1 b b m b " Q ^Q 1 " Q2 .
Hence, equality holds throughout. Moreover, note that due to Theorem 2.6 Q2 is a matrix where entries are ultimately cyclic series in M ax in vγ, δw, hence s ^s1 " d ω Q2 p ω is an ultimately cyclic series in E m|b rrδss.

Division in the Core Form

Proposition 35. Let s " m m Qb b P E m|b rrδss, s 1 " m m
Q1 b b 1 P E m|b 1 rrδss be two ultimately cyclic series. Then,

s 1 zs " m b 1 p Q1 z Qqb b " m b 1 Q2 b b , is an ultimately cyclic series in E b 1 |b rrδss, where Q2 " Q1 z Q is again a greatest core.
Proof. First, it is shown that

Q1 z Q " E b 1 p Q1 z QqE b . (3.52)
For this, Due to Theorem 2.6, the quotient Q z Q1 is a matrix composed of ultimately cyclic series in M ax in vγ, δw and therefore the quotient

́Eb 1 ́Q 1 z Q̄̄E b " ́Eb 1 z ́Eb 1 ́Q 1 z Q̄̄̄E b , since: Corollary 2 " ́Eb 1 z ́Eb 1 ́́Q 1 E b 1 ̄z Q̄̄̄E b , since: Q " QE " ́Eb 1 z ́Eb 1 ́Eb 1 z ́Q 1 z Q̄̄̄̄E b , since: pabq zx " b z pa zxq (A.5) " ́Eb 1 z ́Q 1 z Q̄̄E b , since: a z pa pa zxqq " a zx (A.4) " ́́Q 1 E b 1 ̄z Q̄E b " ́Q 1 z Q̄E b , since: pabq zx " b z pa zxq (A.5) and Q " QE " ́́Q 1 z ̀Q{E b ̆̄E b ̄{E b , since: Corollary 2 twice " ́́́Q 1 z Q̄{ E b ̄Eb ̄{E b , since: pa zxq{b " a zpx{bq (A.6) " ́Q 1 z Q̄{ E b , since: ppx{aqaq{a " x{a (A.4) " Q1 z ̀Q{E b ̆" Q1 z Q, since: pa zxq{b " a zpx{bq (A.
s 1 zs " m b 1 p Q1 z Qqb b is an ultimately cyclic series in E b 1 |b rrδss. Proposition 36. Let s " m m Qb b P E m|b rrδss, s 1 " m m 1 Q1 b b P E m 1 |b
rrδss be two ultimately cyclic series. Then,

s{s 1 " m m p Q{ Q1 qb m 1 " m m Q2 b m 1 ,
is an ultimately cyclic series in E m|m 1 rrδss, where Q2 " Q{ Q1 is again a greatest core.

Proof. The proof is analogous to the proof of Prop. 35.

Remark 14. Let us note that to compute the infimum and the quotient of two series in the core-form both series must be represented with their greatest cores.

Minimal Core-Form

In contrast, to extending a core, see Prop. 30, we can check if a series s P E m|b rrδss can be represented by a core-matrix with smaller dimensions. In the following, we prove that a series s P E m|b rrδss can be uniquely represented by a greatest core with minimal dimension. Proposition 37. An ultimately cyclic series s P E m|b rrδss has the minimal core-form s " m m Qb b , where Q P M ax in vγ, δw m̂b is a canonical matrix of minimal dimensions m ̂b.

Proof. In the following, we give an algorithm to obtain the minimal core-form. Given a series s " m m Qb b P E m|b rrδss, with K " tn P N|m{n P N and b{n P Nu is the set of all common divisors of m and b. where for @i, j P t1, ̈̈̈nu,

Q " » - - - - - Q 11 Q 12 ̈̈̈Q 1n Q 21 Q 22 ̈̈̈Q 2n . . . . . . . . . Q n1 Q 2n ̈̈̈Q nn fi ffi ffi ffi ffi fl ) m{n ) m{n ) m{n , / / / /
Q ij P M ax in vγ, δw m{n̂b{n . Then a core candidate Q is computed based on the matrices Q 11 , Q 12 , ̈̈̈, Q 1n as follows, s " " µ ln ̈̈̈γ ĺ1 µ ln ı Q 11 » - - - β ng γ nǵ1
. . .

β ng γ nǵn fi ffi ffi fl ' ̈̈' " µ ln ̈̈̈γ ĺ1 µ ln ı Q 1n » - - - β ng γ ǵ1 . . . β ng fi ffi ffi fl , " m l µ n Q 11 β n γ ń1 b g ' ̈̈̈' m l µ n Q 1n β n b g , " m l ́µn Q 11 β n γ ń1 ' ̈̈̈' µ n Q 1n β n ̄bg , " m l Qb g .
Definition 37 (Causal Series in E m|b rrδss). A series s " À iPZ w i δ i P E m|b rrδss, with w ì1 ĺ w i , is said to be causal, if s " ε or for all i ă 0, w i " ε and for all i ě 0, w i ĺ µ m β b . The subset of causal pm, bq-periodic series of E m|b rrδss is denoted by E m|b rrδss. where Pr ̀p Qq P M axì n vγ, δw m̂b is the causal projection of the greatest core Q in the dioid M ax in vγ, δw, see Remark 6.

Example 26. Consider the operator γ ́1δ 0 P E 1|1 rrδss, clearly, this operator is not causal since the exponent of γ is ́1, i.e., µ 1 β 1 " e " γ 0 ĺ γ ́1. The causal projection Pr 1|1 pγ ́1δ 0 q " γ 0 δ 0 " e. Therefore, e is the greatest p1, 1q-periodic causal operator smaller than γ ́1δ 0 . This coincides with the causal projection of the operator γ ́1δ 0 P M ax in vγ, δw, i.e., Pr ̀pγ ́1δ 0 q " γ 0 δ 0 " e, see Remark 6. However, if we express γ ́1δ 0 in its p2, 2q-periodic form, i.e., pγ ́1µ 2 β 2 γ 1 'µ 2 β 2 qδ 0 , and then perform the causal projection, i.e.,

Pr 2|2 ́pγ ́1µ 2 β 2 γ 1 ' µ 2 β 2 qδ 0 ̄" µ 2 β 2 δ 0
we obtain µ 2 β 2 δ 0 . Observe that µ 2 β 2 δ 0 ą e " pµ 2 β 2 γ 1 ' γ 1 µ 2 β 2 qδ 0 and hence Pr 2|2 pγ ́1δ 0 q ą Pr 1|1 pγ ́1δ 0 q. µ 2 β 2 δ 0 is the greatest p2, 2q-periodic causal operator smaller than γ ́1δ 0 .

As shown in Example 26, for s P E m|b rrδss, the causal projection Pr m|b psq only provides the greatest causal pm, bq-periodic series such that Pr m|b psq ĺ s, in general, there might be a causal pnm, nbq-periodic series s 1 such that Pr m|b psq ă s 1 ĺ s. Remark 16. Let s " À iPZ w i δ i P E m|b rrδss, with w ì1 ĺ w i , and for all i ě 0, w i ĺ µ m β b , then the causal pm, bq-periodic series Pr m|b psq is the greatest causal series such that Pr m|b psq ĺ s.

Matrices with entries in E m|b rrδss

In the last section the core decomposition for series in E m|b rrδss was introduced. This section extends the core representation to matrices with entries in E m|b rrδss. We first give the decomposition for a trivial example and then show how arbitrary matrices A P E m|b rrδss ĝp can be handled. However, the focus of this section lies on a particular subclass of matrices with entries in E m|b rrδss, called consistent matrices. The study of this subclass is motivated by the modeling of consistent WTEGs in the dioid pErrδss, ', bq, see Section 6.2. Similarly to Section 3.3 it is shown that all relevant operations on consistent matrices with entries in E m|b rrδss can be reduced to operations on matrices with entries in M ax in vγ, δw. ff "

« m m Q 11 b b m m Q 12 b b m m Q 21 b b m m Q 22 b b ff .
This matrix can be represented in the following decomposed form

A " « m m ε ε m m ff « Q 11 Q 12 Q 21 Q 22 ff loooooomoooooon Q « b b ε ε b b ff .
Clearly, Q is a matrix with entries in M ax in vγ, δw of size 2m ̂2b.

In general, for matrices with entries in E m|b rrδss, the entries may have different periods. For instance, consider the matrix

A " « a 11 a 12 a 21 a 22 ff " « m 3 Q 11 b 2 m 2 Q 12 b 3 m 4 Q 21 b 5 m 3 Q 22 b 3 ff .
For this matrix, the decomposition as shown in Example 27 is not possible. However, we can decompose an arbitrary matrix A P E m|b rrδss M̂N as follows,

A " » - - - a 11 ̈̈̈a 1N . . . . . . a M1 ̈̈̈a MN fi ffi ffi fl " » - - - m m 11 Q 11 b b 11 ̈̈̈m m 1N Q 1N b b 1N . . . . . . m m M1 Q M1 b b M1 ̈̈̈m m MN Q MN b b MN fi ffi ffi fl " M L QB R
where, In this form, Q is a block diagonal matrix again with entries in M ax in vγ, δw.

M L " » - - - - - » - - - - - m m 11 ̈̈̈m m 1N ε ̈̈̈ε . . . . . . . . . ε ̈̈̈ε fi ffi ffi ffi ffi fl ̈̈̈» - - - - - ε ̈̈̈ε . . . . . . . . . ε ̈̈̈ε m m M1 ̈̈̈m m MN fi ffi ffi ffi ffi fl fi ffi ffi ffi ffi fl B R " » - - - - - - - - - - - - - - - - - - - - » - - - - - - b b 11 ε ̈̈̈ε ε . . . . . . . . . . . . . . . . . . ε ε ̈̈̈ε b b 1N fi ffi ffi ffi ffi ffi fl . . . » - - - - - - b b M1 ε ̈̈̈ε ε . . . . . . . . . . . . . . . . . . ε ε ̈̈̈ε b b MN fi ffi ffi ffi ffi ffi fl fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl Q " » - - - - - - - - - - - - - - Q 11 ε ̈̈̈̈̈̈̈̈̈̈̈̈ε ε . . . . .

Decomposition of Consistent Matrices

Definition 38. The gain of a matrix A P E m|b rrδss p̂g , denoted by Γ pAq P Q p̂g , is defined by pΓ pAqq i,j :" Γ ppAq i,j q .

Remark 17. Note that, because for an pm, bq-periodic element a P E m|b rrδss, m and b are strictly positive integers, therefore the entries of Γ pAq P Q p̂g are again strictly positive.

Definition 39. A matrix A P E m|b rrδss p̂g is called consistent, if rankpΓ pAqq " 1, i.e., the rank of its corresponding gain matrix is 1.

Remark 18. When we consider matrices with entries in E m|b rrδss and some entries equal to the zero element ε, the gain to these elements can be freely chosen to any positive value in Q.

Recall that @k P Z min , F ε pkq " 8 and therefore @k P Z min , @m, b P N, F ε pk ̀bq " m ̀Fε pkq " 8. Hence, we can choose any period for the ε operator (Remark 8). Now recall that for s P E m|b rrδss, Γ psq " m{b (Definition 32), therefore the gain Γ pεq can be chosen to any value in Q. Thus, if we check (minimize) the rank of the matrix Γ pAq the entries pAq i,j equal to ε are variables.

Example 28. Consider the following matrix

A P E m|b rrδss 2̂2 , A " « e µ 2 β 3 δ 2 ε µ 4 β 1 δ 3
ff .

The corresponding gain matrix Γ pAq is

Γ pAq " « 1 2 3 a 4 ff ,
where a P Q, a ą 0 is variable. Clearly, for a " 6, the matrix Γ pAq has rank 1 and thus the matrix A is consistent.

We use the adjective consistent for matrices with entries in E m|b rrδss since a consistent WTEG admits a consistent transfer function matrix H P E m|b rrδss, this is shown in Section 6, Prop. 95. In the sequel, we elaborate the core decomposition for consistent matrices with entries in E m|b rrδss. Furthermore, we give the conditions under which the sum, product, and quotient of consistent matrices are again consistent matrices. Theorem 3.1 ([41](0.4.6(e))). Let N P Q p̂g be a matrix of rank 1, then N can be written as a product: N " LR where L P Q p̂1 and R P Q 1̂g .

Remark 19. The full-rank factorization of N is not unique. Therefore, given a matrix N P Q p̂g be of rank 1, then N can be written as N " LR, where L P Z p̂1 and R " r1{r 1 ̈̈̈1{r g s P Q 1̂g , where @i P t1, ̈̈̈, gu, r i P Z.

Remark 20. Recall the fact that the gain of an element a P E m|b rrδss is a strictly positive value. Therefore, given a consistent matrix A P E m|b rrδss p̂g we can express the gain Γ pAq P Q p̂g by the product a c a r where a c P Q p̂1 is a column vector with strictly positive entries and a r P Q 1̂g is a row vector with strictly positive entries.

In general, a consistent matrix A P E m|b rrδss ĝp can be decomposed into a M ax in vγ, δw matrix (core), a matrix M w and a matrix B w 1 , which are block diagonal matrix where the entries are m m -vectors and b b -vectors, i.e.,

M w " » - - - - - - m m 1 ε ̈̈̈ε ε . . . . . . . . . . . . . . . . . . ε ε ̈̈̈ε m mp fi ffi ffi ffi ffi ffi fl , B w 1 " » - - - - - - b b 1 ε ̈̈̈ε ε . . . . . . . . . . . . . . . . . . ε ε ̈̈̈ε b bg fi ffi ffi ffi ffi ffi fl
.

The indices w " rm 1 ̈̈̈m p s and w 1 " rb 1 ̈̈̈b g s are vectors, the entries of which represent the multiplication and division coefficients.

Example 29.

M r3 2s " » - " µ 3 γ 1 µ 3 γ 2 µ 3 ı ε ε " µ 2 γ 1 µ 2 ı fi fl B r2 3s " » - - - - - - - « β 2 γ 1 β 2 ff ε ε » - - β 3 γ 2 β 3 γ 1 β 3 fi ffi fl fi ffi ffi ffi ffi ffi ffi fl
Similarly to the scalar case, where m i b i " e and b i m i " E, the product M w B w 1 and B w 1 M w are specific matrices. Let us consider a specific matrix M w and a specific matrix B w 1 such that w " w 1 , thus both matrices have the same weight vector w. Then, by recalling that m m i b m i " e (3.43), Proof. Due to Theorem 3.1 one can represent all entries of a row pAq i,: with the same m ivector. Similarly one can represent all entries of a column pAq :,i with the same b i -vector.

M w B w " » - - - - - - m m 1 b m 1 ε ̈̈̈ε ε . . . . . . . . . . . . . . . . . . ε ε ̈̈̈ε m mp b mp fi ffi ffi ffi ffi ffi fl " » - - - - - - e ε ̈̈̈ε ε . . . . . . . . . . . . . . . . . . ε ε ̈̈̈ε e fi ffi ffi ffi ffi ffi fl " I. Moreover, because of b m i m m i " E m i (3.44), B w M w " » - - - - - - b m 1 m m 1 ε ̈̈̈ε ε . . . .
Then the m i -vector are factored out on the left in the M w -matrix and the b i -vector are factored out on the right in the B w 1 -matrix.

Example 30. Consider the following matrix

A P E m|b rrδss 2̂2 , A " « pµ 3 β 2 γ 1 ' γ 2 µ 2 β 3 qδ 1 pγ 1 δ 1 q ̊µ3 β 2 δ 2 µ 4 β 1 µ 4 β 1 δ 3 ff .
Expressing all elements of the matrix in the core form leads to,

A " » - - - - - - - - - - - - m 3 » - - δ 1 pγ 1 δ 2 q ̊γ1 δ 2 pγ 1 δ 2 q ε ε δ 2 pγ 1 δ 2 q ̊δ1 pγ 1 δ 2 q ̊fi ffi fl b 2 m 3 » - - ε δ 2 ε ε ε ε fi ffi fl b 2 m 4 » - - - - - e ε ε ε fi ffi ffi ffi ffi fl b 1 m 4 » - - - - - δ 3 ε ε ε fi ffi ffi ffi ffi fl b 1 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl
.

The gain matrix Γ pAq of matrix A is,

Γ pAq " « 3 2 3 2 4 4 ff " « 3 8 ff " 1 2 1 2 ı .
Clearly, Γ pAq has rank 1, thus A is consistent. Moreover, Γ pAq has a rank 1 factorization given by the vectors

" 3 8 ı T and " 1{2 1{2
ı . According to the entries of

" 3 8
ı T all entries of the first row of matrix A are expressed with the m 3 -vector and all entries of the second row with the m 8 -vector. Respectively, according to the entries of

" 1{2 1{2
ı all entries of the first column of matrix A are expressed with the b 2 -vector and all entries of the second column with the b 2vector. This is achieved by extending the core-matrices of the entries pAq 1,2 and pAq 2,2 , Prop. 30 and leads to, Note that in this form the entries are expressed with their greatest core-matrices. This matrix can now be written as a product in the following form,

A " » - - - - - - - - - - - - - - - - - - - - - - - m 3 » - - δ 1 pγ 1 δ 2 q ̊γ1 δ 2 pγ 1 δ 2 q δ1 pγ 1 δ 2 q ̊γ1 δ 2 pγ 1 δ 2 q δ2 pγ 1 δ 2 q ̊δ1 pγ 1 δ 2 q ̊fi ffi fl b 2 m 3 » - - δ 2 δ 2 δ 2 δ 2 δ 2 δ 2 fi ffi fl b 2 m 8 » - - - - - - - - - - - - - - - e γ 1
A " « m 3 ε ε m 8 ff looooomooooon M r3 8s » - - - - - - - - - - - - - - - - - - - - - - - δ 1 pγ 1 δ 2 q ̊γ1 δ 2 pγ 1 δ 2 q ̊δ2 δ 2 δ 1 pγ 1 δ 2 q ̊γ1 δ 2 pγ 1 δ 2 q ̊δ2 δ 2 δ 2 pγ 1 δ 2 q ̊δ1 pγ 1 δ 2 q ̊δ2 δ 2 e γ 1 δ 3 γ 1 δ 3 e γ 1 δ 3 γ 1 δ 3 e γ 1 δ 3 γ 1 δ 3 e γ 1 δ
« b 2 ε ε b 2 ff loooomoooon B r2 2s
.

Clearly in this form Q is a matrix with entries in M ax in vγ, δw.

Greatest Core-Matrix in the Matrix Case

As in the scalar case, where Q " EQE is the greatest core of the series s " m m Qb b P E m|b rrδss, it can be shown that E w QE w 1 is the greatest core of the consistent matrix A " M w QB w 1 P E m|b rrδss p̂g . Proposition 40. Let A " M w QB w 1 P E m|b rrδss p̂g be the decomposition of A P E m|b rrδss p̂g . Then the greatest core matrix is given by

Q :" E w QE w 1 .
(3.56)

Proof. Consider the inequality M w XB w 1 ĺ M w QB w 1 " A. The greatest solution for X is

M w zpM w QB w 1 q{B w 1 " B w M w QB w 1 M w 1 " E w QE w 1 " Q. Furthermore, Q is indeed a core of A P E m|b rrδss p̂g , as M w " M w E w and B w 1 " E w 1 B w 1 , therefore M w QB w 1 " M w E w QE w 1 B w 1 " M w QB w 1 " A.
Prop. 40 implies that E w Q " QB w 1 " Q. Similarly to the core extension in Prop. 30 the core Q can be extended as follows. .

Proof. @i P t1, ̈̈̈, pu, @j P t1, ̈̈̈, gu, Q1 ij can be obtained by extending Qij , see Prop. 30.

Calculation with Matrices

Sum and Product in the Matrix Case Proposition 42. Let A, P P E m|b rrδss m̂p be two consistent matrices, then the sum A ' P is a consistent matrix if and only if Γ pAq " Γ pPq.

Proof. This follows from Prop. 16, all entries pA'Pq i,j must satisfy Γ ppA'Pq i,j q " Γ ppAq i,j q " Γ ppPq i,j q.

Recall that due to Prop. 41, by extending the core-form if necessary, two consistent matrices with equal gain can be expressed with their least common period.

Proposition 43 (Sum of Matrices). Let A " M w QB w 1 , P " M w Q1 B w 1 P E m|b rrδss m̂p be two consistent matrices satisfying Prop. 42, then the sum

A ' P " M w Q2 B w 1 , where Q2 " Q '
Q1 is again a greatest core.

Proof.

M w QB w 1 ' M w Q1 B w 1 " M w pE w QE w 1 ' E w Q 1 E w 1 qB w 1 " M w E w pQ ' Q 1 qE w 1 loooooooomoooooooon Q 2 B w 1
Clearly, the product of two consistent matrices is not necessarily consistent itself. In the following proposition, the conditions are given under which the product of two consistent matrices is again consistent. Proposition 44. Let A P E m|b rrδss p̂g and P P E m|b rrδss ĝl be two consistent matrices, then the product A b P is consistent if and only if @k P t2, ̈̈̈, gu, ̀Γ pAq ̆1,k ̀Γ pPq ̆k,1 " ̀Γ pAq ̆1,1 ̀Γ pPq ̆1,1 .

(3.57)

Proof. Recall ̀A b P ̆i,j " À g k"1 ̀pAq i,k b pPq k,j ̆(2.10), this sum is in E m|b rrδss iff @k P t1, ̈̈̈, gu, Γ ̀pAq i,k ̆Γ ̀pPq k,j ̆" Γ ̀pAq i,1 ̆Γ ̀pPq 1,j ̆,
see Prop. 18. It is sufficient to check this property for i " j " 1, i.e., for the first row of matrix Γ pAq and first column of matrix Γ pPq, since both matrices have rank 1 and therefore all rows/columns are linearly dependent.

Corollary 5. Let A P E m|b rrδss p̂g and P P E m|b rrδss ĝl be consistent matrices satisfying Prop. 44. Then, Γ pAq " a c a r and Γ pPq " p c p r , where a c P Q p̂1 , a r P Q 1̂g , p c P Q ĝ1 and p r P Q 1̂l (Remark 20). Then, a r is linearly dependent to every row of Γ pAq and p c is linearly dependent to every column of Γ pPq. Therefore, (3.57) can be written as, pa r q 1 pp c q 1 " pa r q k pp c q k , @k P t1, ̈̈̈, gu.

Then gain matrix Γ pAPq is given by Γ pAPq " a c ̀pa r q 1 pp c q 1 ̆pr .

(3.59)

Proof. Form (3.57) follows that, pΓ pAPqq i,j " pΓ pAqq i,1 pΓ pPqq 1,j

" pa c q i pa r q 1 pp c q 1 pp r q j .

Hence Γ pAPq " a c ̀pa r q 1 pp c q 1 ̆pr .

Proposition 45 (Product of Matrices). Let A " M w QB w 1 , P " M w 1 Q1 B w 2 P E m|b rrδss be two consistent matrices satisfying Prop. 44, then the product AP " M w Q2 B w 2 , where Q2 " Q Q1 is again a greatest core.

Proof. Because of, B w 1 M w 1 " E w 1 and QE w 1 " Q,

M w QB w 1 M w 1 Q1 B w 2 " M w QE w 1 Q1 B w 2 " M w Q Q1 B w 2 , Furthermore: Q Q1 " E w QE w 1 E w 1 Q 1 E w 2 " Q2 .
Proposition 46. Let A P E m|b rrδss n̂n be a consistent matrix, then the Kleene star A ̊is a consistent matrix if and only if Γ pAq " a c a r , where a c P Q n̂1 and a r P Q 1̂n such that pa c q i " ppa r q i q ́1, @i P t1, ̈̈̈, nu.

Proof. The Kleene star of matrix A is computed by

A ̊" I ' A ' AA ' ̈̈Ä
ccording to Prop. 42 and Prop. 44 we need 1. Γ pIq " Γ pAq,

2. Γ pAq = Γ pAAq.
To satisfy (1) the diagonal entries of Γ pAq must be equal to 1, i.e., @i P t1, ̈̈̈nu, Γ pAq i,i " pa c q i ̂pa r q i " 1. Clearly, this condition is satisfied if pa c q i " ppa r q i q ́1. Moreover, (3.58) and Prop. 44 is satisfied as well. Then for (2) recall Corollary 5, thus Γ pAAq " a c ̀pa r q 1 pa c q 1 ̆ar " a c a r " Γ pAq, since pa r q 1 pa c q 1 " pa r q 1 ppa r q 1 q ́1 " 1.

Corollary 6. Let A P E m|b rrδss p̂g be a consistent matrix satisfying Prop. 46, then Γ pAq " Γ pA ̊q.

Proposition 47 (Kleene Star of a Matrix). Let A " M w QB w P E m|b rrδss n̂n be a consistent matrix satisfying Prop. 46, then A ̊" M w Q̊B w .

Proof. Note that, M w -matrix and the B w -matrix having the same weight vector w, implies that Q is a square matrix. Then since, M w B w " I, B w M w " E w and E w Q " Q,

A ̊" I ' M w QB w ' M w QB w M w QB w ' ̈̈̈. " M w B w ' M w QB w ' M w Q2 B w ' ̈̈̈, " M w pI ' Q ' Q2 ' ̈̈̈qB w , " M w Q̊B w .
Again not that Q̊ĺ E w Q̊E w , hence Q̊i s not the greatest core of A ̊.

Division in the Matrix Case

Proposition 48. Let A P E m|b rrδss p̂g and P P E m|b rrδss p̂l be consistent matrices, then the left division A zP is consistent iff Dc P Q, c ą 0 such that, c ̀Γ pAq ̆k,1 " ̀Γ pPq ̆k,1 , @k P t1, ̈̈̈, pu.

Proof. Let us recall that pA zPq i,j " Ź p k"1 ppAq k,i zpPq k,j q (2.24), this infimum is in E m|b rrδss iff @k P t1, ̈̈̈, pu, Γ ̀pAq k,i zpPq k,j ̆" Γ ̀pAq 1,i zpPq 1,j ̆.

Moreover, recall Γ ps 2 zs 1 q " Γ ps 1 q{Γ ps 2 q (Prop. 20), thus @k P t1, ̈̈̈, pu, Γ ̀pAq k,i zpPq k,j ̆" Γ ̀pAq 1,i zpPq 1,j ̆ô Γ ppPq k,j q Γ ppAq k,i q " Γ ppPq 1,j q Γ ppAq 1,i q ô ̀Γ pPq ̆k,j ̀Γ pAq ̆k,i "

̀Γ pPq ̆1,j ̀Γ pAq ̆1,i .

Finally, @k P t1, ̈̈̈, pu,

̀Γ pPq ̆k,j " ̀Γ pPq ̆1,j ̀Γ pAq ̆1,i ̀Γ pAq ̆k,i " c ij ̀Γ pAq ̆k,i ,
where c ij " pΓ pPqq 1,j {pΓ pAqq 1,i P Q. Since the equation above must hold @k P t1, ̈̈̈, pu this condition can be expressed by, ̀Γ pPq ̆:,j " c ij ̀Γ pAq ̆:,i .

Recall that Γ pAq and Γ pPq have rank 1, therefore all columns of Γ pAq (resp. Γ pPq) are linearly dependent. Hence, it is sufficient to consider ̀Γ pPq ̆:,1 " c 11 ̀Γ pAq ̆:,1 .

Or differently, ̀Γ pPq ̆k,1 " c 11 ̀Γ pAq ̆k,1 , @k P t1, ̈̈̈, pu.

Differently stated, the left division A zP is consistent if and only if every column of matrix Γ pAq is linearly dependent to every column of matrix Γ pPq. Note that since both matrices have rank 1 it is sufficient to check linear dependence for the first column of matrix Γ pAq and Γ pPq. Corollary 7. Let A P E m|b rrδss p̂g and P P E m|b rrδss p̂l be consistent matrices satisfying Prop. 48. Moreover, Γ pAq " a c a r and Γ pPq " p c p r , with a c , p c P Q p̂1 , a r P Q 1̂g and p r P Q 1̂l . The gain matrix Γ pA zPq is given by

Γ pA zPq " āc pa c q 1 pp c q 1 p r , (3.60)
where āc " rppa r q 1 q ́1 ppa r q 1 q ́1 ̈̈̈ppa r q g q ́1s T .

Proposition 49. Let A P E m|b rrδss p̂g and P P E m|b rrδss l̂g be consistent matrices, then the right division A{P is consistent iff, Dc P Q, c ą 0 such that c ̀Γ pAq ̆1,k " ̀Γ pPq ̆1,k , @k P t1, ̈̈̈, gu.

Proof. The proof is analogous to the proof of Prop. 48.

Differently stated, the right division A{P is consistent if and only if every row of matrix Γ pAq is linearly dependent to every row of matrix Γ pPq. Then again since Γ pAq and Γ pPq have rank 1 it is sufficient to check linear dependence for the first row of matrices Γ pAq and Γ pPq. Corollary 8. Let A P E m|b rrδss p̂g and P P E m|b rrδss l̂g be consistent matrices, satisfying Prop. 49. Moreover, Γ pAq " a c a r and Γ pPq " p c p r , with a r , p r P Q 1̂g , a c P Q p̂1 and p c P Q l̂1 . The gain matrix Γ pA{Pq is given by Γ pA{Pq " a c pa r q 1 pp r q 1 pr ,

where pr " rpp c q 1 q ́1 pp c q 2 q ́1 ̈̈̈pp c q l q ́1s.

Proposition 50 (Left Division of Matrices). Let A P E m|b rrδss p̂g and P P E m|b rrδss p̂l be consistent matrices satisfying Prop. 48. The quotient P zA is computed based on their core-forms, i.e. A " M w QB w 1 , P " M w Q1 B w 2 , in the following way

P zA " M w 2 p Q1 z QqB w 1 .
Proof. The proof is analogous to Prop. 35.

Proposition 51 (Right Division of Matrices). Let A P E m|b rrδss p̂g and P P E m|b rrδss l̂g be consistent matrices, satisfying Prop. 49. The quotient P{A is computed based on their core-forms, i.e. A " M w QB w 1 , P " M w 2 Q1 B w 1 in the following way

A{P " M w p Q{ Q1 qB w 2 .
Proof. The proof is analogous to Prop. 35.

4

Dioids pT , ', bq and pT rrγss, ', bq

In this chapter, the dioids pT , ', bq and pT rrγss, ', bq are introduced. These dioids have an application in the modeling and the control of Periodic Time-variant Event Graphs (PTEGs) (resp. Timed Event Graphs (TEGs) under partial synchronization (PS)). The dioids pT , ', bq and pT rrγss, ', bq are the counterpart to the dioids pE, ', bq and pErrδss, ', bq studied in Chapter 3. In contrast to pErrδss, ', bq, which consists of specific event-variant operators, the dioid pT rrγss, ', bq consists of specific time-variant operators. Therefore, many results are similar to the results obtained for the dioid pErrδss, ', bq in Chapter 3. Specifically, in Section 4.2 the core-form for periodic elements in T rrγss is similar to the core-form for periodic elements in E m|b rrδss, see Section 3.3. It is shown that for periodic elements in T rrγss all relevant operations p', b, z, {q in T rrγss can be reduced to operations between matrices with entries in M ax in vγ, δw. The presented results in this chapter have partially been published in [START_REF] Trunk | Model decomposition of timed event graphs under partial synchronization in dioids[END_REF][START_REF] Trunk | Modelling and control of periodic time-variant event graphs in dioids. Discrete Event Dynamic Systems[END_REF][START_REF] Trunk | Output reference control of timed event graphs under partial synchronization[END_REF].

Dioid pT rrγss, ', bq

The firing of a transition in a PTEG, respectively in a TEG under PS can be naturally described by a dater function x : Z Ñ Z max . For these functions, xpkq represents the time of the pk ̀1q st firing of the associated transition. Note that dater functions are isotone. In the following the dioid pT rrγss, ', bq is introduced as a set of operators on dater functions. We denote by Ξ the set of isotone mappings from Z into Z max . This set Ξ is a Z max -semimodule equipped with addition, defined to the pointwise addition in the dioid (Z max , ', b), thus for x 1 , x 2 P Ξ @k P Z, ̀x1 ' x 2 ̆pkq :" x 1 pkq ' x 2 pkq " maxpx 1 pkq, x 2 pkqq, (

and a scalar multiplication defined by, for λ P Z max and x 1 P Ξ,

@k P Z, ̀λ b x 1 ̆pkq :" λ ̀x1 pkq. (4.2)
The zero and top mapping on Ξ, denoted by ε resp. J, are defined by @k P Z, εpkq :" ε (Recall that in Z max , ε " ́8 ), @k P Z, Jpkq :" J (Recall that in Z max , J " 8 ).

Clearly, pΞ, ', εq is a complete idempotent commutative monoid, see Definition 3. The order relation on Ξ coincides with the order in the dioid pZ max , ', bq, i.e., the standard order on Z. Thus, for x 1 , x 2 P Ξ,

x 1 ĺ x 2 ô x 1 ' x 2 " x 2 , (4.3)
ô x 1 pkq ' x 2 pkq " x 2 pkq, @k P Z, ô max ̀x1 pkq, x 2 pkq ̆" x 2 pkq, @k P Z, ô x 1 pkq ď x 2 pkq, @k P Z.

The infimum (^operator) on the set Ξ is defined by @k P Z, px 1 ^x2 qpkq :" x 1 pkq ^x2 pkq " minpx 1 pkq, x 2 pkqq.

Definition 40 (Operator). An operator is a lower semi-continuous mapping f : Ξ Ñ Ξ from the set Ξ into itself, such that fpεq " ε. Including the property fpεq " ε implies that f is an endomorphism. The set of these operators is denoted by O.

Proposition 52 ([16]). The set of operators O, equipped with multiplication and addition as follows,

f 1 , f 2 P O, @x P Ξ ̀f1 ' f 2 ̆pxq :" f 1 pxq ' f 2 pxq, (4.4)
f 1 , f 2 P O, @x P Ξ ̀f1 b f 2 ̆pxq :" f 1 ̀f2 pxq ̆, (4.5)
is a complete dioid.

Proof. The proof is equivalent to the proof of Prop. 8 in Section 3.1.

Recall Prop. 5, therefore the zero and unit element of O are given by, @x P Ξ, εpxq :" ε and êpxq :" x. Again, to simplify notation the multiplication symbol b is often omitted and we write usually fx instead of fpxq. Due to (2.1) the ' operation induces a partial order relation on O, defined by

f 1 ľ f 2 ô f 1 ' f 2 " f 1 ,
ô ̀f1 x ̆pkq ' ̀f2 x ̆pkq " ̀f1 x ̆pkq, @x P Ξ, @k P Z, ô min ́̀f 1 x ̆pkq, ̀f2 x ̆pkq ̄" ̀f1 x ̆pkq @x P Ξ, @k P Z.

Subsequently, two operators f 1 , f 2 P O are equal iff @x P Ξ, @k P Z: pf 1 xqpkq " pf 2 xqpkq . Moreover, pO, ', bq is a complete dioid, thus the top mapping is given by, @x P Ξ,

Ĵpxq " $ & % ε for: x " ε, J otherwise, (4.7)
and the infimum is defined as, for f 1 , f 2 P O,

f 1 ^f2 " à tf 3 P O|f 3 ' f 1 ĺ f 1 , f 3 ' f 2 ĺ f 2 u.
Proposition 53. The following operators are both endomorphic and lower semi-continuous and thus in O.

τ P Z, δ τ : @x P Ξ, pδ τ xqpkq " xpkq ̀τ,

ω, ϖ P N, ∆ ω|ϖ : @x P Ξ, p∆ ω|ϖ xqpkq " rxpkq{ϖsω, (4.9

)
where ras is the least integer greater than or equal to a.

Proof. The mapping δ τ is an endomorphism and it is lower semi-continuous. First, since τ P Z is an integer @k P Z, ̀δτ pεq ̆pkq " τ ̀εpkq " τ ̀ṕ8q " ́8, thus ̀δτ pεq ̆pkq " εpkq.

Moreover, for all finite and infinite subsets X Ď Ξ,

́δτ ̀à xPX x ̆̄pkq " τ ̀́à xPX x ̄pkq " τ ̀max xPX ̀xpkq ̆" max xPX ̀τ ̀xpkq " ́à xPX δ τ x ̄pkq,
which proves the lower semi-continuity of δ τ . For the mapping ∆ ω|ϖ again ω, ϖ P N are finite positive integers, therefore @k P Z, ̀∆ω|ϖ pεq ̆pkq " rεpkq{ϖsω " rṕ8q{ϖsω " ́8 and ̀∆ω|ϖ pεq ̆pkq " εpkq. Moreover, for all finite and infinite subsets X Ď Ξ,

́∆ω|ϖ ̀à xPX x ̆̄pkq " S ̀ÀxPX x ̆pkq ϖ W ω " S max xPX ̀xpkq π W ω, " max xPX ̃S xpkq ϖ W ω ̧" max xPX ́̀∆ ω|ϖ x ̆pkq ̄, " ́à xPX ∆ ω|ϖ x ̄pkq.
Proposition 54. The operators δ τ and ∆ ω|ϖ introduced in Prop. 53 satisfy the following relations

δ τ δ τ 1 " δ τ̀τ 1 , δ τ ' δ τ 1 " δ maxpτ,τ 1 q , (4.10)
∆ ω|ϖ δ ϖ " δ ω ∆ ω|ϖ . (4.11)

Proof. For the proof of δ τ δ τ 1 " δ τ̀τ 1 , since (4.5) and (4.8), then @x P Ξ, @k P Z, ̀δτ δ τ 1 x ̆pkq " ̀δτ pδ τ 1 xq ̆pkq " τ ̀pδ τ 1 xqpkq " τ ̀τ1 ̀xpkq " ̀δτ̀τ 1 x ̆pkq.

For the proof of δ τ ' δ τ 1 " δ maxpτ,τ 1 q , since (4.4), (4.1) and (4.8), then @x P Ξ, @k P Z,

̀pδ τ ' δ τ 1 qx ̆pkq " ̀δτ x ' δ τ 1 x ̆pkq " max ̀pδ τ xqpkq, pδ τ 1 xqpkq "
max ̀τ ̀xpkq, τ 1 ̀xpkq ̆" maxpτ, τ 1 q ̀xpkq " ̀δmaxpτ,τ 1 q x ̆pkq.

For the proof of (4.11), recall (4.8) and (4.9), then @x P Ξ, @k P Z,

p∆ ω|ϖ δ ϖ xqpkq " Q xpkq ̀ϖ ϖ U ω " Q xpkq ϖ ̀1U ω " Q xpkq ϖ U ω ̀ω " pδ ω ∆ ω|ϖ xqpkq.
Remark 21. (4.11) implies that for ́b ă τ ď 0, ∆ ω|b δ τ ∆ b|ϖ " ∆ ω|ϖ , since,

p∆ ω|b δ τ ∆ b|ϖ xqpkq " S rxpkq{ϖsb ̀τ b W ω, " S Q xpkq ϖ U ̀τ b W ω, " Q xpkq ϖ U ω, since ́1 ă τ b ď 0,
" p∆ ω|ϖ xqpkq.

Dioid of Time Operators

Definition 41 (Dioid of Time Operators). The dioid of time operators, denoted by pT , ', bq, is defined by sums and compositions over the set tê, ε, Ĵ, δ τ , ∆ ω|ϖ u with ω, ϖ P N, τ P Z, equipped with addition and multiplication defined in (4.4) and (4.5), respectively.

Clearly pT , ', bq is a complete subdioid of pO, ', bq. Similarly to the dioid pE, ', bq, introduced in Section 3.1.1, the dioid pT , ', bq is not commutative, i.e. let v 1 , v 2 P T , then in general v 1 v 2 ‰ v 2 v 1 . The order on T , naturally induced by ' is as follows. Let v 1 , v 2 P T then @x P Ξ, @k P Z,

v 1 ľ v 2 ô v 1 ' v 2 " v 1 , ô v 1 x ' v 2 x " v 1 x, ô ̀v1 x ̆pkq ' ̀v2 x ̆pkq " ̀v1 x ̆pkq, ô max ́̀v 1 x ̆pkq, ̀v2 x ̆pkq ̄" ̀v1 x ̆pkq.
Recall that x : Z Ñ Z max is an isotone mapping, an operator v P T only manipulates the value of the mapping x. Therefore, we can associate a function R v : Z max Ñ Z max to a T-operator v P T . This function R v is obtained by replacing xpkq by t in the expression vpxqpkq. For example pp∆ 3|4 δ 1 ' δ 2 ∆ 3|3 qxqpkq " maxprpxpkq ̀1q{4s3, 2 ̀rxpkq{3s3q and therefore R ∆ 3|4 δ 1 'δ 2 ∆ 3|3 ptq " maxprpt ̀1q{4s3, 2 ̀rt{3s3q. We denote by R the set of functions generated by all operators in T . Since T-operators are lower-semi continuous, then functions in R are lower-semi continuous and isotone. For a reason explained later on in Section 6.1.3, we call functions in R release-time function. Clearly, the set R and the set of T-operators T are isomorphic, therefore the order relation over the dioid pT , ', bq corresponds to the order induced by the max operation on R. For v 1 , v 2 P T ,

v 1 ľ v 2 ô v 1 ' v 2 " v 1 ô pv 1 xqpkq ' pv 2 xqpkq " pv 1 xqpkq @x P Ξ, @k P Z, ô R v 1 ptq ľ R v 2 ptq @t P Z max , ô R v 1 ptq ě R v 2 ptq, @t P Z max . (4.12)
The release-time function R v provides a graphical representation of a T-operator v P T . Moreover, the order relation on T has a graphical interpretation which is shown in the following example.

Example 31.

R δ 2 ∆ 4|4 δ -1 (resp. δ 2 ∆ 4|4 δ ́1). Con- sider now the release-time function R δ 1 ∆ 4|4 δ -2 associated to the operator δ 1 ∆ 4|4 δ ́2. R δ 1 ∆ 4|4 δ -2 lies in the area shaped by R δ 2 ∆ 4|4 δ -1 (R δ 1 ∆ 4 δ -2 is beneath R δ 2 ∆ 4 δ -1 ) and therefore δ 1 ∆ 4|4 δ ́2 ĺ δ 2 ∆ 4|4 δ ́1.
In contrast, consider the operators δ ́3∆ 4|4 and ∆ 4|4 δ ́1 with corresponding releasetime functions shown in Figure 4.1b. R δ ́3∆ 4|4 does not cover and is not covered by

R ∆ 4|4 δ ́1 , therefore δ ́3∆ 4|4 ł ∆ 4|4 δ ́1 and δ ́3∆ 4|4 ń ∆ 4|4 δ ́1. R δ 2 ∆ 4|4 δ -1 R δ 1 ∆ 4|4 δ -2 t Rptq -4 -2 2 4 6 8 -2 2 4 6 8 10 12 (a) R δ 2 ∆ 4|4 δ -1 ą R δ 1 ∆ 4|4 δ -2 R ∆ 4|4 δ -1 R δ -3 ∆ 4|4 t Rptq -2 2
(b) R δ -3 ∆ 4|4 '∆ 4|4 δ -1 . Figure 4.1. -(a) R δ 2 ∆ 4|4 δ -1 covers R δ 1 ∆ 4|4 δ -2 . (b) R δ -3 ∆ 4|4 does not cover and is not covered by R ∆ 4|4 δ -1 .
Periodic T-operators Definition 42. A T-operator v P T is said to be ω-periodic if Dω P N such that, @x P Ξ, @k P Z, pvpω b xqqpkq " ω b pvpxqqpkq. The set of ω-periodic T-operators is denoted by T ω . Moreover, the set of periodic operators is defined by T per " Ť ωPN T ω .

Definition 43. A release-time function R : Z max Ñ Z max is called quasi ω-periodic if Dω P N such that @t P Z max , R v pt ̀ωq " ω ̀Rv ptq.

Remark 22. Since the periodic property does only depend on the value xpkq (the time) we can neglect the argument k for examining the periodic property of a T-operator. Therefore, a T-operator v P T is ω-periodic if its corresponding release-time function R v is quasi ω-periodic.

Example 32. The δ τ operator, with τ P Z is p1q-periodic since R δ τ ptq " t ̀τ one has R δ τ pt ̀1q " pt ̀1q ̀τ " 1 ̀Rδ τ ptq. For example, see Figure 4.2a for the graphical representation of the δ 3 operator. The δ 2 ∆ 2|2 δ ́1 operator is p2q-periodic, with a graphical illustration given in Figure 4.2b. In contrast, the ∆ 2|1 operator, shown in Figure 4.2c, is according to Definition 42 not periodic since R ∆ 2|1 ptq " rt{1s2 and therefore @t P Z max , R ∆ 2|1 pt ̀1q " 2 ̀R∆ 2|1 ptq.

(b) R δ 2 ∆ 2|2 δ -1 t Rptq -2 2 4 6 -2 2 4 6 (c) R ∆ 2|1 Figure 4.2. -In (a) the function R δ 3 is quasi p1q-periodic. In (b) the function R δ 2 ∆ 2|2 δ -1 is quasi p2q-periodic. (c) the function R ∆ 2|1 is not quasi ω-periodic.
Proposition 55 (Canonical form of an ω-periodic T-operator). An ω-periodic T-operator v P T per has a canonical form given by a finite sum

À I i"1 δ τ i ∆ ω|ω δ τ 1 i .
Moreover, the sum is strictly ordered such that @i P t1, ̈̈̈, I ́1u, τ i ă τ ì1 and 1 ́ω ă τ 1 ď 0.

Proof. We first show that an ω-periodic T-operator v P T per can be represented as

v " ώ1 à i"0 δ Rvṕiq ∆ ω|ω δ íὼ1 . (4.13)
For this, we consider the operator w " À ώ1 i"0 w i with w i " δ Rvṕiq ∆ ω|ω δ íὼ1 . The release-time function associated to w i is

R w i ptq " R v ṕiq ̀Q t ̀i ́ω ̀1 ω U ω.
Hence, R w is given by

R w ptq " max ́Rv p0q ̀Q t ́ω ̀1 ω U ω, R v ṕ1q ̀Q t ́ω ̀2 ω U ω, ̈̈R v p1 ́ωq ̀Q t ω U ω ̄. (4.14)
Clearly, R w is a quasi ω-periodic function. To prove that v can be represented as (4.13) we have to show that R w ptq " R v ptq. Because R w and R v are both quasi ω-periodic functions it is sufficient to check R w ptq " R v ptq for t " t1 ́ω, ̈̈̈, 0u. Let us remark that R v is isotone and thus,

̈̈̈ď R v p0q ́ω ď R v p1 ́ωq ď ̈̈̈ď R v p0q ď R v p1 ́ωq ̀ω ď ̈̈Ẅ
e evaluate (4.14) for t " 0, this leads to

R w p0q " max ́Rv p0q ̀Q 1 ́ω ω U ω, R v ṕ1q ̀Q 2 ́ω ω U ω, ̈̈R v p1 ́ωq ̀Q 0 ω U ω " max ̀Rv p0q, R v ṕ1q, ̈̈̈, R v p1 ́ωq "R v p0q.
Similarly, one can show that for t P t1 ́ω, ̈̈̈, ́1u, R w ptq " R v ptq. For this, recall (4.14)

R w ptq " max ́Rv p0q ̀Q t ̀1 ́ω ω U ω, R v ṕ1q ̀Q t ̀2 ́ω ω U ω, ̈̈R v p1 ́ωq ̀Q t ω U ω ̄.
For 1 ď j ď ω and 1 ́ω ď t ď ́1 observe that,

Q t ̀j ́ω ω U ω " $ & % ́ω, for t ̀j ă 0 0, for t ̀j ě 0, therefore, R w ptq " max ̀Rv p0q ́ω, ̈̈̈, R v pt ̀1q ́ω, R v ptq, ̈̈, R v p1 ́ωq ̆, " R v ptq,
and v " w " À ώ1 i"0 w i " À ώ1 i"0 δ Rvṕiq ∆ ω|ω δ íὼ1 . The canonical representation is the one obtained by removing redundant w i according to the order relation given in (4.12). Remark 23. Clearly, an ω-periodic operator is also nω-periodic. Therefore, an ω-periodic operator v is represented in a nω-periodic form given by v "

nώ1 à i"0 δ Rvṕiq ∆ nω|nω δ ínὼ1 .
Proposition 56. The ω-periodic ∆ ω|ω operator can be represented in an expended nω-periodic form by the sum

∆ ω|ω " ń1 à i"0 δ ́iω ∆ nω|nω δ ́pń1́iqω .
Proof. See Section C.2.1 in the appendix. Example 33. The 1-periodic identity operator e " ∆ 1|1 is represented in a 3-periodic form given by e "

∆ 3|3 δ ́2 ' δ ́1∆ 3|3 δ ́1 ' δ ́2∆ 3|3 , see Figure 4.3. R ∆ 3|3 δ ́2 R δ ́1 ∆ 3|3 δ ́1 R δ ́2 ∆ 3|3
∆ 3|3 δ ́2 ptq, R δ ́1 ∆ 3|3 δ ́1 ptq, R δ ́2 ∆ 3|3 ptqq.
Proposition 57. The set of periodic operators T per equipped with addition and multiplication defined in (4.4) and (4.5) is a complete subdioid of pT , ', bq.

Proof. Clearly, the unit, zero and top element e, ε and J belong to T per . Moreover, due to Definition 4 one has to show that the set of periodic operators T per are closed for addition and multiplication. Given two periodic operators v 1 , v 2 P T per , due to Remark 23, v 1 and v 2 are expressed with their least common period ω in the following form

v 1 " I à i"1 δ τ i ∆ ω|ω δ τ 1 i , v 2 " J à j"1 δ t j ∆ ω|ω δ t 1 j .
Then the sum, v 1 ' v 2 " À I i"1 δ τ i ∆ ω|ω δ τ 1 i ' À J j"1 δ t j ∆ ω|ω δ t 1 j is clearly an ω-periodic operator. This also holds for infinite sums. The product v 1 b v 2 is as well ω-periodic, recall that ∆ ω|ω δ τ ∆ ω|ω " ∆ ω|ω for ́ω ă τ ď 0 (Remark 21), hence,

v 1 b v 2 " ́I à i"1 δ τ i ∆ ω|ω δ τ 1 i ̄b ́J à j"1 δ t j ∆ ω|ω δ t 1 j ̄, " I à i"1 J à j"1 δ τ i ∆ ω|ω δ τ 1 i δ t j ∆ ω|ω δ t 1 j , " I à i"1 J à j"1 δ τ i ̀rpτ i ̀tj q{ωsω ∆ ω|ω δ t 1 j .
The distributivity of left and right multiplication over infinite sums are carried over from the dioid pT , ', bq.

Corollary 10. The set of ω-periodic operators T ω equipped with addition and multiplication defined in (4.4) and (4.5) is a complete subdioid of pT , ', bq and pT per , ', bq.

Causal T-Operators

Definition 44. A T-operator v P T is said to be causal if v " ε or if its corresponding releasetime function satisfies, @t P Z max ,

R v ptq ě t. (4.15)
Clearly, the least causal operator in T (except ε) is the unit operator e with the release-time function, R e ptq " t.

Dioid of Formal Power Series pT rrγss, ', bq

The event-shift operator γ η is defined as a mapping over Ξ as follows, η P Z γ η : @x P Ξ, k P Z ̀γη x ̆pkq " xpk ́ηq.

(4.16)

Clearly, the γ η mapping is lower-semi continuous, since for all finite and infinite subsets X Ď Ξ Furthermore, pγ η εqpkq " εpk ́ηq and since @k P Z, εpkq " ́8, then η P Z, @k P Z, pγ η εqpkq " εpḱηq " εpkq " ́8. Therefore, the event-shift operator is an endomorphism, i.e., γ η P O. Moreover, the event-shift operator commutes with all T-operators, i.e., @v P T , vγ η " γ η v, since, ̀pγ η vqx ̆pkq " ̀γη pvxq ̆pkq, since (4.5),

" ̀vx ̆pk ́ηq, since (4.16),

" ̀vpγ η xq ̆pkq, again (4.16),

" ̀pvγ η qx ̆pkq, again (4.5).

Definition 45. (Dioid pT rrγss, ', bq) We denote by pT rrγss, ', bq the quotient dioid in the set of formal power series in one variable γ with exponents in Z and coefficients in the noncommutative complete dioid pT , ', bq induced by the equivalence relation @s P T rrγss, s " pγ 1 q ̊s " spγ 1 q ̊. (4.17)

Hence we identify two series s 1 , s 2 with the same equivalence class if s 1 γ ̊" s 2 γ ̊. It is helpful to think of sγ ̊as the representative of the equivalence class of s. Note that we can interpret elements in T rrγss as isotone functions s : Z Ñ T , where spηq refers to the coefficient of γ η . Hence, @η P Z, spηq ĺ spη ̀1q. The quotient structure (4.17) allows assimilating the variable γ to the event-shift operator γ P O, defined in (4.16). Recall the definition for addition and multiplication on formal power series (2.13) and (2.14), respectively. Therefore we obtain the following definition for addition and multiplication in the dioid pT rrγss, ', bq. Definition 46. Let s 1 , s 2 P T rrγss, then addition and multiplication are defined by

s 1 ' s 2 " à ηPZ ̀s1 pηq ' s 2 pηq ̆γη , s 1 b s 2 " à ηPZ ̃à ǹn 1 "η ̀s1 pnq b s 2 pn 1 q ̧̆γ η .
As before, ' defines an order on T rrγss, i.e., a, b P T rrγss : a ' b " b ô a ĺ b.

Remark 24. Recall that pT per , ', bq and pT ω , ', bq are complete subdioids of pT , ', bq, then from Prop. 4 it follows that pT per rrγss, ', bq and pT ω rrγss, ', bq are complete subdioids of pT rrγss, ', bq. Moreover, pT ω rrγss, ', bq is a complete subdioid of pT per rrγss, ', bq.

Monomial, Polynomial and ultimately cyclic Series in T rrγss

A monomial in T rrγss is defined by vγ η , where v P T . A polynomial is a finite sum of monomials, i.e., À I i"1 v i γ η i . The ordering of two periodic monomials v 1 γ η 1 , v 2 γ η 2 P T rrγss can be checked as follows,

v 1 γ η 1 ĺ v 2 γ η 2 ô $ & % v 1 ĺ v 2 , η 1 ě η 2 .
(4.18) Proposition 58. Let p P T per rrγss, then p has a canonical form p " À J j"1 v 1 j γ η 1 j such that @j P t1, ̈̈̈, Ju, the ω-periodic T-operator v 1 j is in the canonical form of Prop. 55, and coefficients and exponents are strictly ordered, i.e., for j P t1, ̈̈̈, J ́1u, η 1 j ă η 1 j̀1 and v 1 j ă v 1 j̀1 .

Proof. Without loss of generality, we can assume that p " À I i"1 v i γ η i , with η i ă η ì1 , i " 1, ̈̈̈I ́1. As p P T per rrγss, we identify all elements s with their maximal representation sγ ̊, we can also identify p and

p 1 " I à i"1 ́i à j"1 v j lo omo on v 1 i ̄γη i as pγ ̊" p 1 γ ̊. Hence, v 1 i ĺ v 1 ì1 . If v 1 i " v 1 ì1 we can write v 1 i γ η i ' v 1 ì1 γ η ì1 " v 1 i pγ η i ' γ η ì1 q " v 1 i γ η i .
For that we can write p 1 as

À J j"1 v 1 j γ η 1 j with v 1 j ă v 1 j̀1 and J ď I.
Definition 47. (Ultimately Cyclic Series in T rrγss ): A series s P T rrγss is said to be ultimately cyclic if it can be written as s " p ' qpγ η δ τ q ̊, where η, τ P N 0 and p, q are polynomials in T rrγss.

Note that a polynomial p " À I i"0 v i γ n i can be considered as a specific ultimately cyclic series s " ε ' ppγ 0 δ 0 q ̊where η " 0 and τ " 0.

Similarly to Errδss, an element s P T rrγss has a graphical representation in the Z max Ẑmax ̂Z. Given a series s " À i v i γ i P T rrγss, this graphical representation is obtained by depicting for every i the release-time function R v i of the coefficient v i in the (input-time / output-time)-plane of i.

Example 34. For the graphical representation of the polynomial p " pδ 1 ∆ 4|4 δ ́1'δ ́2∆ 4|4 qγ 0 ' pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 2 ' pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ 4 P T per rrγss, respectively its representative pγ ̊see e v e n ts h i f t k i n p u t -t i m e t

∆ 4|4 δ ́1 ' δ ́2∆ 4|4 qγ 0 ' pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 2 ' pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ 4 .
(a) R δ 1 ∆ 4|4 δ ́1 'δ ́2 ∆ 4|4 , (b) R δ 5 ∆ 4|4 δ ́1 'δ 2 ∆ 4|4 and (c) R δ 5 ∆ 4|4 'δ 6 ∆ 4|4 δ ́1
Similarly to Section 3.3, in this section, a specific decomposition of series in T per rrγss is proposed. It is shown that a periodic series s P T per rrγss can always be represented as s " d ω Qp ω where Q is a square matrix in M ax in vγ, δw of size ω ̂ω, d ω is a row vector defined as

d ω :" " ∆ ω|1 δ ́1∆ ω|1 ̈̈̈δ 1́ω ∆ ω|1 ı ,
and p ω is a column vector defined as

p ω :" " ∆ 1|ω δ 1́ω ̈̈̈∆ 1|ω δ ́1 ∆ 1|ω ı T .
The index ω determines the dimension of the vectors. It is important to note that in this form the core matrix Q is a matrix with entries in M ax in vγ, δw. An advantage of this representation is that all relevant operations on periodic series s P T per rrγss can be reduced to operations on square matrices with entries in M ax in vγ, δw. In the following, this decomposition is first demonstrated on a small example.

Example 35. Consider the following series in T per rrγss,

s " ∆ 2|2 ' δ 1 ∆ 2|2 δ ́1 ' δ 2 ∆ 2|2 γ 2 pδ 2 γ 2 q
Because of ∆ ω|ϖ " ∆ ω|b ∆ b|ϖ (Remark 21), δ ω ∆ ω|ϖ " ∆ ω|ϖ δ ϖ (4.11) and @v P T , vγ " γv, this series can be rewritten as s " ∆ 2|1 e lo omo on

M 1 ∆ 1|2 ' δ ́1∆ 2|1 δ 1 lo omo on M 2 ∆ 1|2 δ ́1 ' ∆ 2|1 δ 1 γ 2 pδ 1 γ 2 q loooooomoooooon S 1 ∆ 1|2 .
Clearly M 1 , M 2 and S 1 are elements in M ax in vγ, δw. We now can rewrite s in the core representation, s "

" ∆ 2|1 δ ́1∆ 2|1 ı « ε e ' δ 1 γ 2 pδ 1 γ 2 q δ1 ε ff « ∆ 1|2 δ ́1 ∆ 1|2 ff , due to e ' δ 1 γ 2 pδ 1 γ 2 q ̊" pδ 1 γ 2 q ̊, s " " ∆ 2|1 δ ́1∆ 2|1 ı loooooooomoooooooon d 2 « ε pδ 1 γ 2 q δ1 ε ff looooooomooooooon Q « ∆ 1|2 δ ́1 ∆ 1|2 ff loooomoooon p 2 ,
which is in the required form.

Proposition 59. Let s " À i v i γ i P T per rrγss be an ω-periodic series, then s can be written as s " d ω Qp ω , where Q P M ax in vγ, δw ω̂ω .

Proof. s being an ω-periodic series implies that all coefficients v i of s are ω-periodic Toperators. Then due to Prop. 55 all coefficients can be expressed in canonical form v i "

À J i j"1 δ τ i j ∆ ω|ω δ τ 1
i j with J i ď ω and ́ω ă τ 1 i j ď 0. Then s can be rewritten as

s " à i ̀Ji à j"1 δ τ i j ∆ ω|ω δ τ 1 i j ̆γi .
By using ∆ ω|ω " ∆ ω|1 ∆ 1|ω (Remark 21), δ ω ∆ ω|1 " ∆ ω|1 δ 1 (4.11) and vγ " γv, @v P T , the series s is written as

s " à i ̀Ji à j"1 δ τi j ∆ ω|1 δ τi j γ i ∆ 1|ω δ τ 1 i j ̆,
where ́ω ă τi j " τ i j ́rτ i j {ωsω ď 0 and τi j " rτ i j {ωs. Observe that ́ω ă τi j , τ 1 i j ď 0 hence we can express s by s "

" ∆ ω|1 δ ́1∆ ω|1 ̈̈̈δ 1́ω ∆ ω|1 ı à i ̀Ji à j"1 Q i j ̆» - - - - - ∆ 1|ω δ 1́ω ̈̈∆ 1|ω δ ́1 ∆ 1|ω fi ffi ffi ffi ffi fl
, where the entry pQ i j q 1́τ i j ,ὼτ 1 i j " δ τi j γ i and all other entries of Q i j are equals ε. Finally s is in the required form s " d ω Qp ω , where Q "

À i ̀ÀJ i j"1 Q i j ̆.
For the particular case of an ultimately cyclic series s P T per rrγss, the core-matrix Q is obtained as follows. The ultimately cyclic series s " À I i v i γ n i ' À J j v 1 j γ n 1 j pδ τ γ ν q ̊P T per rrγss is written, such that all coefficients v i and v 1 j are represented with their least common period (Remark 23), i.e., s "

L à l"1 δ t l ∆ ω|ω δ t 1 l γ n l ' K à k"1 δ ξ k ∆ ω|ω δ ξ 1 k γ n k pδ τ γ ν q ̊.
Recall that ∆ ω|ϖ " ∆ ω|b ∆ b|ϖ (Remark 21) therefore,

s " L à l"1 δ t l ∆ ω|1 ∆ 1|ω δ t 1 l γ n l ' K à k"1 δ ξ k ∆ ω|1 ∆ 1|ω δ ξ 1 k γ n k pδ τ γ ν q ̊.
Note that the δ ω operator commutes with ∆ ω|ω , i.e., δ ω ∆ ω|ω " ∆ ω|ω δ ω (4.11). Moreover, we can always represent an ultimately cyclic series s P T per rrγss such that τ is a multiple of ω, i.e., we can extend pγ ν δ τq ̊such that, τ " l τ " lcmp τ, ωq pγ ν δ τq ̊" pe ' γ ν δ τ ' ̈̈̈' γ pĺ1qν δ pĺ1q τqpγ lν δ l τq " pe ' γ ν δ τ ' ̈̈̈' γ pĺ1qν δ pĺ1q τqpγ lν δ τ q ̊.

Therefore, in the following we assume τ{ω P N, thus ∆ 1|ω pδ τ γ ν q ̊" pδ τ{ω γ ν q ̊∆1|ω . This leads to

s " L à l"1 δ tl ∆ ω|1 δ tl γ n l lo omo on M l ∆ 1|ω δ t 1 l ' K à k"1 δ ξk ∆ ω|1 δ ξk γ n k pδ τ{ω γ ν q looooooooomooooooooon S k ∆ 1|ω δ ξ 1 k ,
with ́ω ă tl , t1 l , ξk , ξ1 k ď 0. In this representation M l are monomials and S k are series in M ax in vγ, δw. Moreover, the entries of the p ω -vector appear on the right and the entries of the d ω -vector appear on the left of monomial M l (resp. series S k ). For a given s we denote the set of monomials by M " tM 1 , ̈̈̈, M L u and the set of series by S " tS 1 , ̈̈̈, S K u. Furthermore, the subsets M i,j (resp. S i,j ) are defined as, @i, j P t0, ̈̈̈, ω ́1u

M i,j :" tM l P M| δ ́i∆ ω|1 M l ∆ 1|ω δ ́j P L à l"1 δ tl ∆ ω|1 M l ∆ 1|ω δ t 1 l u, S i,j :" tS k P S| δ ́i∆ ω|1 S k ∆ 1|ω δ ́j P K à k"1 δ ξk ∆ ω|1 S k ∆ 1|ω δ ξ 1 k u.
The entry pQq ì1,ώj of the core matrix is then given by pQq ì1,ώj "

à MPM i,j M ' à SPS i,j S.
Remark 25. Note that, for series s " d ω Qp ω P T per rrγss be an ultimately cyclic series, the entries of Q are ultimately cyclic series in M ax in vγ, δw.

Properties of d ω and p ω

In the following, we elaborate some properties of the d ω -vector and p ω -vector, which are necessary for computations in the core-from. The scalar product d ω b p ω of these two vectors is the identity e:

d ω b p ω "δ 0 ∆ ω|1 ∆ 1|ω δ 1́ω ' ̈̈̈' δ 1́ω ∆ ω|1 ∆ 1|ω δ 0 "δ 0 ∆ ω|ω δ 1́ω ' ̈̈̈' δ 1́ω ∆ ω|ω δ 0 " e, ( 4.19)
where the latter inequalities hold because of ∆ ω|1 ∆ 1|ω " ∆ ω|ω (Remark 21) and Corollary 9.

For an illustration see Example 33. The dyadic product p ω b d ω is a square matrix with entries in M ax in vγ, δw denoted by N. For i, j P t1, ̈̈̈, ωu, the entry pp ω b d ω q i,j is given by, pNq i,j " pp ω b d ω q i,j " ∆ 1|ω δ píjq̀p1́ωq ∆ ω|1 .

Then, because of ∆ 1|ω δ ́ω " δ ́1∆ 1|ω and ∆ 1|ω δ ́n∆ ω|1 " ∆ 1|1 " e for ́ω ă ́n ď 0, see Remark 21, Proof. Recall, that N " N ̊(4.24) and that a ̊zpa ̊xq " a ̊x (resp. pa ̊xq{a ̊" xa ̊), see (A.9), which completes the proof.

N " p ω b d ω " » - - - - - - e δ ́1 ̈̈̈δ
Greatest Core of a Series s P T per rrγss

In general a series s P T per rrγss may have several core-representations. In the following it is shown that a series s P T per rrγss admits a unique greatest core, denoted Q, i.e., s " d ω Qp ω and Q ľ Q for all core matrices Q such that s " d ω Qp ω . Note that the greatest core is referred to the order relation in the dioid pM ax in vγ, δw , ', bq.

Proposition 63. Let s " d ω Qp ω P T per rrγss be a decomposition of s P T per rrγss. The greatest core matrix is given by

Q " N ω QN ω . (4.27)
Proof. Consider the inequality d ω Xp ω ĺ d ω Qp ω " s. Then, because of Prop. 61 the greatest solution for X is

d ω zd ω Qp ω {p ω " p ω d ω Qp ω d ω " N ω QN ω " Q.
Furthermore, because of d ω " d ω N ω and p ω " N ω p ω (Prop. 60),

d ω Qp ω " d ω N ω QN ω p ω " d ω Qp ω " s.
Remark 26. The greatest core matrix Q has the following properties. Since:

N b N " N, N Q " NNQN " Q; QN " NQNN " Q.
Example 36. The greatest core of the series considered in Example 35 is given by

Q " NQN " « e δ ́1 e e ff « ε pδ 1 γ 2 q δ1 ε ff « e δ ́1 e e ff " « pδ 1 γ 2 q ̊pδ 1 γ 2 q δ1 ' δ 1 γ 2 pδ 1 γ 2 q ̊pδ 1 γ 2 q ̊ff .

Calculation with the Core Decomposition

Sum and Product of Periodic Series in T per rrγss

In this section, it is shown that operations on ultimately cyclic series in T per rrγss can be reduced to operations on matrices with entries in M ax in vγ, δw. To perform addition and multiplication of two series s 1 " m ω 1 Q1 b ω 1 , s 2 " m ω 2 Q2 b ω 2 P T per rrγss in the core-form it is necessary to express the core matrices Q1 P M ax in vγ, δw ω 1 ̂ω1 and Q2 P M ax in vγ, δw ω 2 ̂ω2 with equal dimensions. This is possible by expressing both series with their least common period ω " lcmpω 1 , ω 2 q, see the following proposition.

" d ω Q 2 p ω , where Q 2 " Q ' Q 1 ,
is again an ultimately cyclic series in T per rrγss.

Proof.

s ' s 1 " d ω Qp ω ' d ω Q 1 p ω " d ω pQ ' Q 1 qp ω " d ω Q 2 p ω .
Clearly, the entries of the core matrices Q and Q 1 are ultimately cyclic series in M ax in vγ, δw. Because of Theorem 2.6, the sum of two ultimately cyclic series in M ax in vγ, δw is again an ultimately cyclic series. Therefore, Q 2 is composed of ultimately cyclic series in M ax in vγ, δw and thus s ' s 1 " d ω Q 2 p ω is an ultimately cyclic series in T per rrγss. Corollary 12. Let s " d ω Qp ω , s 1 " d ω Q1 p ω P T per rrγss be two ultimately cyclic series, with Q, Q1 are greatest cores, the sum s ' s 1 " d ω Q2 p ω P T per rrγss is an ultimately cyclic series, where Q2 " p Q ' Q1 q is again a greatest core.

Proof.

s ' s 1 " d ω Qp ω ' d ω Q1 p ω " d ω pN QN ' N Q1 Nqp ω " d ω Np Q ' Q1 qN loooooomoooooon Q 2 p ω
Proposition 66. Let s " d ω Qp ω , s 1 " d ω Q 1 p ω be two ultimately cyclic series in T per rrγss, the product s b s 1 " d ω Q 2 p ω , where Q 2 " QNQ 1 , is again an ultimately cyclic series in T per rrγss.

Proof. Recall that p ω d ω " N (4.20), then

s b s 1 " d ω Qp ω d ω Q 1 p ω " d ω QNQ 1 p ω " d ω Q 2 p ω .
Moreover, the entries of the core matrices Q and Q

s ̊" d ω p ω ' d ω QNp ω ' d ω QNNQNp ω ' ̈̈" d ω pI ' QN ' pQNq 2 ' ̈̈̈qp ω " d ω pQNq ̊pω .
Again, due to Theorem 2.6 the Kleene star, sum, and product of ultimately cyclic series in M ax in vγ, δw are ultimately cyclic series in M ax in vγ, δw and therefore, s ̊" d ω pQNq ̊pω is an ultimately cyclic series in T per rrγss. Remark 27. Let s " d ω Qp ω P T per rrγss be an ultimately cyclic series, where Q is a greatest core, i.e., Q " N QN. Then, s ̊" d ω Q̊p ω P T per rrγss is an ultimately cyclic series. However, in general, Q̊i s not the greatest core of the series s ̊.

Q̊" I ' Q ' Q2 ̈̈"

I ' N QN ' N Q2 N ̈̈̈.
Whereas,

N Q̊N " NIN ' N QN ' N Q2 N ̈̈" N ' Q ' Q2 ̈̈̈. Moreover, N Q̊N " pN Q̊N q ̊, since N Q̊N " I ' N Q̊N and N Q̊N N Q̊N " N Q̊N .
In general, multiplication does not distribute with respect to ^in the dioid pT rrγss, ', bq. However, as shown for the dioid pErrδss, ', bq in Lemma 2 and Lemma 3, distributivity holds for left multiplication by the d ω -vector and right multiplication by the b ω -vector for specific matrices with entries in T rrγss.

Lemma 4. Let Q 1 , Q 2 P T rrγss ω̂ω , then d ω pNQ 1 ^NQ 2 q " d ω NQ 1 ^dω NQ 2 , pQ 1 N ^Q2 Nqp ω " Q 1 Np ω ^Q2 Np ω .
Proof. The proof is similar to the proof of Lemma 2. Recall that e " d ω p ω (4.19), N " p ω d ω (4.20) and N " NN Prop. 60. Moreover, recall that inequality cpa ^bq ĺ ca ^cb holds for a, b, c elements in a complete dioid, see (2.2). Now let us define q 1 " d ω NQ 1 and q 2 " d ω NQ 2 , then q 1 ^q2 " epq 1 ^q2 q " d ω p ω pq 1 ^q2 q ĺ d ω pp ω q 1 ^dω q 2 q.

Inserting q 1 " d ω NQ 1 and q 2 " d ω NQ 2 leads to,

d ω pp ω q 1 ^dω q 2 q " d ω pp ω d ω NQ 1 ^pω d ω NQ 2 q, " d ω pNNQ 1 ^NNQ 2 q, " d ω pNQ 1 ^NQ 2 q.
Finally,

d ω pNQ 1 ^NQ 2 q ĺ d ω NQ 1 ^dω NQ 2 " q 1 ^q2 .
Hence, equality holds throughout. The proof for pQ 1 N ^Q2 Nqp ω " Q 1 Np ω ^Q2 Np ω is similar.

Proposition 68. Let s " d ω Qp ω , s 1 " d ω Q1 p ω P T per rrγss be two ultimately cyclic series, then s ^s1 " d ω Q2 p ω P T per rrγss is an ultimately cyclic series, where Q2 " p Q ^Q 1 q is again a greatest core.

Proof. Again, this proof is similar to the proof of Prop. 34. Let us recall that Q " N QN, then according to Lemma 4 we can write

s ^s1 " d ω Qp ω ^dω Q1 p ω " d ω N QNp ω ^dω N Q1 Np ω " d ω pN QN ^N Q1 Nqp ω " d ω p Q ^Q 1 qp ω .
It remains to be shown that

Q2 " p Q ^Q 1
q is a greatest core. First, N " N ̊, therefore, I ' N " N, and Q2 ĺ N Q2 N. Then, according to Lemma 4,

N Q2 N " Np Q ^Q 1 qN " p ω d ω p Q ^Q 1 qp ω d ω " p ω pd ω Qp ω ^dω Q1 p ω qd ω .
Recall, cpa ^bq ĺ ca ^cb and pa ^bqc ĺ ac ^bc (2.2), therefore

p ω pd ω Qp ω ^dω Q1 p ω qd ω ĺ p ω d ω Qp ω d ω ^pω d ω Q1 p ω d ω " Q ^Q 1 " Q2 .
Hence, equality holds throughout. Moreover, note that due to Theorem 2.6 Q2 is a matrix where entries are ultimately cyclic series in M ax in vγ, δw, hence s ^s1 " d ω Q2 p ω is an ultimately cyclic series in T per rrγss.

Division of Series in T per rrγss

Proposition 69. Let s " d ω Qp ω , s 1 " d ω Q1 p ω be two ultimately cyclic series in T per rrγss. Then,

s 1 zs " d ω p Q1 z Qqp ω , s{s 1 " d ω p Q{ Q1 qp ω ,
are ultimately cyclic series in T per rrγss.

Proof. First, we show that

Q1 z Q " Np Q1 z QqN. (4.29)
For this, ́N ́Q

1 z Q̄̄N " ́N z ́N ́Q 1 z

Q̄̄̄N

, because of Prop. 62

" ́N z ́N ́́Q 1 N ̄z Q̄̄̄N , because of Q " QN " ́N z ́N ́N z ́Q 1 z
Q̄̄̄̄N , since: pabq zx " b z pa zxq (A.5)

" ́N z ́Q 1 z

Q̄̄N

, because of a z pa pa zxqq " a zx (A.4) Definition 48 (Causal Series in T per rrγss). A series s " À iPZ v i γ i P T per rrγss, with v i ĺ v ì1 , is said to be causal, if s " ε or for all i ă 0, v i " ε and for all i ě 0, v i ĺ e. The subset of causal periodic series of T per rrγss is denoted by T per rrγss.

" ́́Q 1 N ̄z Q̄N " ́Q 1

Matrices with entries in pT per rrγss, ', bq

Remark 28. The causal projection Pr ̀: T per rrγss Ñ T per rrγss, is given by, for s " À iPZ v i γ i P T per rrγss, with v i ĺ v ì1 ,

Pr ̀psq " Pr ̀́à iPZ v i γ i ̄" à iPZ s ̀piqγ i
where,

s ̀piq " $ & % v i , if i ě 0 and v i ľ e, i.e., v i is a causal T-operator, ε, otherwise.

Matrices with entries in pT per rrγss, ', bq

Recall that the sum, product, Kleene star as well as left and right division of ultimately cyclic series in T per rrγss are again ultimately cyclic series in T per rrγss. Therefore, the extension of the basic operations p', b, z, {q to matrices with entries in T per rrγss is straightforward. Additionally, the core representation of series in T per rrγss is extended to the matrix case. Therefore, consider a matrix A P T per rrγss n̂m where the entries are in the core-form, i.e., Due to Prop. 64 all entries of A can be represented with a common d ω -vector and a common p ω -vector, where ω " lcmpω 1,1 , ̈̈̈, ω n,m q. This leads to,

A " » - - - d ω 1,1 Q1,1 p ω 1,1 ̈̈̈d ω 1,m Q1,m p ω 1,m . . . . . .
A " » - - - d ω Q1 1,1 p ω ̈̈̈d ω Q1 1,m p ω . . . . . . d ω Q1 n,1 p ω ̈̈̈d ω Q1 n,m p ω fi ffi ffi fl , " » - - - - - - d ω ε ̈̈̈ε ε . . . . . . . . . . . . . . . . . . ε ε ̈̈̈ε d ω fi ffi ffi ffi ffi ffi fl loooooooooooomoooooooooooon Dw » - - - Q1 1,1 ̈̈̈Q 1 1,m . . . . . . Q1 n,1 ̈̈̈Q 1 n,m fi ffi ffi fl looooooooooomooooooooooon Q » - - - - - - p ω ε ̈̈̈ε ε . . . . .

Subdioids of pT per rrγss, ', bq

Recall that pT ω rrγss, ', bq is a complete subdioid of pT per rrγss, ', bq (Remark 24). The subdioid pT 1 rrγss, ', bq of pT per rrγss, ', bq, i.e. the set of 1-periodic series endowed with addition and multiplication, is the dioid pM ax in vγ, δw , ', bq. Moreover, pM ax in vγ, δw , ', bq is a subdioid of pT ω rrγss, ', bq, e.g., a subdioid of pT 3 rrγss, ', bq, pT 4 rrγss, ', bq etc. Due to the subdioid structure of pT per rrγss, ', bq, one can define the canonical injection Inj : M ax in vγ, δw Ñ T per rrγss, x Þ Ñ Injpxq " x. For a graphical illustration of this canonical injection see the following example.

Example 38. Recall the series s " γ 1 δ 2 ' ̀γ3 δ 3 ' γ 5 δ 4 ̆pγ 3 δ 2 q ̊P M ax in vγ, δw (Example 17) with a graphical representation of s given in Figure 4.7a. Then, the graphical representation of the canonical injection Injpsq P T per rrγss is shown in Figure 4.7b. The series s P M ax in vγ, δw (Figure 4.7a) corresponds to the (event-shift/output-time)-plane for the (input-time) value 0 of the 3D representation of the series Injpsq P T per rrγss (Figure 4.7b). Moreover, the canonical injection Injpsq P T per rrγss is (1)-periodic, therefore the (event-shift/output-time)-plane for the (input-time) value 1 corresponds to the series δ 1 s P M ax in vγ, δw and for the (input-time) value 2 to the series δ 2 s P M ax in vγ, δw, etc. in vγ, δw Ñ T per rrγss of the series s " γ 1 δ 2 ' ̀γ3 δ 3 ' γ 5 δ 4 ̆pγ 3 δ 2 q ̊P M ax in vγ, δw.

Lemma 5. Let vγ n P T ω rrγss be an ω-periodic monomial. Then residual Inj 7 pvγ n q and the dual residual Inj 5 pvγ n q are given by Inj 7 pvγ n q " δ min ώ1 t"0 pRvptq́tq γ n , (4.31) Inj 5 pvγ n q " δ max ώ1 t"0 pRvptq́tq γ n . (4.32)

Proof. By definition, the residuated mapping Inj 7 pvγ n q is the greatest solution x of the following inequality

vγ n ľ Injpxq " Inj ́ài γ η i δ ζ i ̄" à i γ η i δ ζ i , (4.33)
where À i γ η i δ ζ i P M ax in vγ, δw. Clearly, the least η i such that the inequality (4.33) holds is n and thus, Similarly, for (4.32), Inj 5 pvγ n q is the least solution x of the inequality

vγ n ľ à i pγ n δ ζ i q " γ n δ τ ,
vγ n ĺ Injpxq " Inj ́ài δ ζ i γ η i ̄" à i γ η i δ ζ i . (4.36)
Then, the greatest η i such that the inequality (4.36) holds is n and thus, Proof. For (4.39): Consider s " À i v i γ n i in the canonical form, such that n i ă n ì1 and v i ă v ì1 and let R v i be the release-time function associated to the operator v i . Recall that Inj 7 psq is the greatest solution x P M ax in vγ, δw of inequality Inj 7 pxq ĺ s. This is given by À i δ τ i γ n i where τ i is the greatest integer such that δ τ i ĺ v i . Repeating the first step of Lemma 5, this is given by τ i " min ώ1 t"0 pR v i ptq ́tq. The proof of (4.40) is analogous.

vγ n ĺ à i pγ n δ ζ i q " γ n δ τ ,
Example 39. Recall the polynomial p " pδ 1 ∆ 4|4 δ ́1'δ ́2∆ 4|4 qγ 0 'pδ 5 ∆ 4|4 δ ́1'δ 2 ∆ 4|4 qγ 2 ' pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ 4 P T per rrγss with a graphical representation given in Figure 4.8a. Moreover, recall the function

R δ 1 ∆ 4|4 δ ́1'δ ́2∆ 4|4 (resp. R δ 5 ∆ 4|4 δ ́1'δ 2 ∆ 4|4 and R δ 5 ∆ 4|4 'δ 6 ∆ 4|4 δ ́1 )
shown in Figure 4.5a (resp. Figure 4.5b and Figure 4.5c). The residual of the canonical injection is Inj 7 ppq " δ 1 γ 0 ' δ 2 γ 2 ' δ 5 γ 4 , which is shown in Figure 4.8b. In Figure 4.8 and Figure 4.9 the polynomial p is compared to InjpInj 7 ppqq, as required p ľ InjpInj 7 ppqq (2.17).

R(t) e v e n t -s h i f t k i n p u t -t i m e t (a) 3D representation of polynomial p " pδ 1 ∆ 4|4 δ ́1 'δ ́2∆ 4|4 qγ 0 'pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 2 ' pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ 4 .
R(t) e v e n t -s h i f t k i n p u t -t i m e t (b) 3D representation of InjpInj 7 ppqq " δ 1 γ 0 ' δ 2 γ 2 ' δ 5 γ 4 . Figure 4.8. -Comparison of the polynomial p " pδ 1 ∆ 4|4 δ ́1 ' δ ́2∆ 4|4 qγ 0 ' pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 4 '
pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ 6 and InjpInj 7 ppqq. For all k P Z the slices in the (input-time/outputtime)-plane of p cover the slices of InjpInj 7 ppqq, see Figure 4.9. (a) k " t0, 1u Zero slice Mapping Ψ ω : T ω rrγss Ñ M ax in vγ, δw Recall that pM ax in vγ, δw , ', bq is a subdioid of pT ω rrγss, ', bq, hence we define a specific projection from T ω rrγss into M ax in vγ, δw as follows.

R δ 1 R δ 1 ∆ 4|4 δ ́1 'δ ́2 ∆ 4|4 t Rptq 0 2 4
R δ 5 R δ 5 ∆ 4|4 δ ́1 'δ 2 ∆ 4|4 t Rptq 0 2 4
Definition 49. Let s " À i v i γ n i P T ω rrγss be an ω-periodic series, then

Ψ ω psq " Ψ ω ́ài v i γ n i ̄" à i γ n i δ Rv i p0q . (4.41)
This projection Ψ ω has a graphical interpretation, for a given s P T ω rrγss the series s " Ψ ω psq P M ax in vγ, δw corresponds to the slice in the (event/output-time)-plane of the 3D representation of s P T ω rrγss at the input-time value 0, thus this projection is also called zeroslice mapping. Note that in contrast to the zero-slice mapping Ψ m|b : E m|b rrδss Ñ M ax in vγ, δw defined in Section 3.2, the mapping Ψ ω is a projection because M ax in vγ, δw is a subset of T ω rrγss and Ψ ω satisfies Ψ ω " Ψ ω ̋Ψω . However, this is not the case for the set E m|b rrδss, for instance, M ax in vγ, δw is not a subset of E 3|2 rrδss and therefore the concatenation of the mappings Ψ 3|2 ̋Ψ3|2 is not defined (possible).

Example 40. Recall the polynomial p " pδ 1 ∆ 4|4 δ ́1'δ ́2∆ 4|4 qγ 0 'pδ 5 ∆ 4|4 δ ́1'δ 2 ∆ 4|4 qγ 2 ' pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ 4 P T per rrγss with a graphical representation given in Figure 4.4. Then,

Ψ 4 ppq " δ 1 γ 0 ' δ 5 γ 2 ' δ 6 γ 4 .
The series Ψ 4 ppq corresponds to the slice in the (event-shift/output-time)-plane for the input-time value t " 0 in the 3D representation of p, see The projection Ψ ω is by definition lower-semicontinuous, see Definition 49, therefore Ψ ω is residuated. Proposition 71. Let s " À i γ n i δ τ i P M ax in vγ, δw. The residual Ψ 7 ω psq P T ω rrγss of s is a series defined by

Ψ 7 ω ́ài γ n i δ τ i ̄" à i γ n i δ τ i ∆ ω|ω " s∆ ω|ω . (4.42)
Proof. By definition of the residuated mapping, Ψ 7 ω p À i γ n i δ τ i q P T ω rrγss is the greatest solution of the following inequality

à i γ n i δ τ i ľ Ψ ω pxq " Ψ ω ́àj v j γ η j ̄, (4.43)
where x " À j v j γ η j P T ω rrγss. First we show that (4.42) satisfies (4.43) with equality.

Ψ ω ́à i γ n i δ τ i ∆ ω|ω ̄" à i γ n i δ R δ τ i ∆ ω|ω p0q " à i γ n i δ τ i ,
since R δ τ i ∆ ω|ω p0q " τ i ̀r0{ωsω " τ i , see (4.8) and (4.9). Taking into account that Ψ ω is isotone, it remains to show that

À i γ n i δ τ i ∆ ω|ω is the greatest solution of à i γ n i δ τ i " Ψ ω pxq " Ψ ω ́à j v j γ η j ̄" à j γ η j δ Rv j p0q . (4.44)
Clearly, to achieve equality we need η j " n i and R v j p0q " τ i . Furthermore, we are looking for the greatest v j P T ω , such that τ i " R v j p0q. Due to the canonical form Prop. 55 we can write an ω-periodic T-operator as

À ω i"1 δ ζ i ∆ ω|ω γ ζ 1 i with ́ω ă ζ 1 i ď 0.
This operator corresponds to the release-time function

Rptq " ω max i"1 ́ζi ̀R ζ 1 i ̀t ω V ω ̄.
Now we examine Rptq for t " 0, thus Rp0q "

ω max i"1 ́ζi ̀R ζ 1 i ω V ω ̄.
Recall that ́ω ă ζ 1 i ď 0, hence R v j ptq " τ i ̀rp0̀tq{ωsω is the greatest quasi ω-periodic release-time function such that (4.44) holds, i.e., R v j p0q " R δ τ i ∆ ω|ω p0q " τ i ̀r0{ωsω " τ i . This function corresponds to the operator δ τ i ∆ ω|ω . Proposition 72. Let s " À i γ n i δ τ i P M ax in vγ, δw. The dual residual Ψ 5 ω psq P T ω rrγss of s is a series defined by

Ψ 5 ω ́ài γ n i δ τ i ̄" à i γ n i δ τ i ∆ ω|ω δ 1́ω " s∆ ω|ω δ 1́ω . (4.45)
Proof. The proof is similar to the proof of Prop. 71, with the difference that instead of finding the greatest solution we are now looking for the least solution, denoted by Ψ 5 ω p À i γ n i δ τ i q P T ω rrγss, of the following inequality

à i γ n i δ τ i ĺ Ψ ω pxq " Ψ ω ́àj v j γ η j ̄. (4.46)
Again, we show that (4.45) satisfies (4.46) with equality.

Ψ ω ́à i γ ν i δ τ i ∆ ω|ω δ 1́ω ̄" à i γ R δ τ i ∆ ω|ω δ 1́ω p0q δ τ i " à i γ ν i δ τ i ,
since R δ τ i ∆ ω|ω δ 1́ω p0q " τ i ̀rp1 ́ωq{ωsω " τ i , see (4.8) and (4.9). Taking into account that Ψ ω is isotone, it remains to show that À i γ n i δ τ i ∆ ω|ω δ 1́ω is the least solution of

à i γ n i δ τ i " Ψ ω pxq " Ψ ω ́à j v j γ η j ̄" à j γ η j δ Rv j p0q . (4.47)
Clearly, to achieve equality we need η j " n i and R v j p0q " τ i . Furthermore, we are looking for the least v j P T ω rrγss, such that τ i " R v j p0q. Due to the canonical form Prop. 55 we can write an ω-periodic T-operator as

À ω i"1 δ ζ i ∆ ω|ω δ ζ 1 i with ́ω ă ζ 1 i ď 0.
This operator corresponds to the release-time function

Rptq " ω max i"1 ́ζi ̀R ζ 1 i ̀t ω V ω ̄.
Now we examine Rptq for t " 0, thus Rp0q "

ω max i"1 ́ζi ̀R ζ 1 i ω V ω ̄.
Let us recall that ́ω ă ζ 1 i ď 0, hence R v j ptq " τ i ̀rpp1 ́ωq ̀tq{ωsω is the least ω-periodic release-time function such that (3.41) holds, i.e., R v j p0q " R δ τ i ∆ ω|ω δ 1́ω p0q " τ i ̀rp1 ́ωq{ωsω " τ i . This function corresponds to the operator δ τ i ∆ ω|ω δ 1́ω .

Example 41. Let us consider the polynomial p " pδ 1 ∆ 4|4 δ ́1 ' δ ́2∆ 4|4 qγ 0 ' pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 2 'pδ 5 ∆ 4|4 'δ 6 ∆ 4|4 δ ́1qγ 4 P T per rrγss with a projection Ψ 4 ppq " δ 1 γ 0 'δ 5 γ 2 'δ 6 γ 4 . The residual of the projection Ψ 4 ppq, is given by

Ψ 7
4 pΨ 4 ppqq " pδ 1 γ 0 ' δ 5 γ 2 ' δ 6 γ 4 q∆ 4|4 .

See Figure 4.11 and Figure 4.12 for a comparison of p and Ψ 7 4 pΨ 4 ppqq, as required p ĺ Ψ 7 4 pΨ 4 ppqq (2.17). 6 and InjpInj 7 ppqq. For all k P Z the slices in the (input-time/outputtime)-plane of Ψ 7 4 pΨ 4 ppqq cover the slices of p, see Figure 4.12. (a) k " t0, 1u (b) k " t2, 3u 111

R(t) e v e n t -s h i f t k i n p u t -t i m e t (a) 3D representation of polynomial p " pδ 1 ∆ 4|4 δ ́1 'δ ́2∆ 4|4 qγ 0 'pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 2 ' pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ 4 .
∆ 4|4 δ ́1 ' δ ́2∆ 4|4 qγ 0 ' pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 4 ' pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ
R δ 1 ∆ 4|4 R δ 1 ∆ 4|4 δ ́1 'δ ́2 ∆ 4|4 t Rptq 0 2 4
R δ 5 ∆ 4|4 R δ 5 ∆ 4|4 δ ́1 'δ 2 ∆ 4|4 t Rptq 0 2 4
R δ 6 ∆ 4|4 R δ 5 ∆ 4|4 'δ 6 ∆ 4|4 δ ́1 t Rptq 0 2 4
In this chapter, the dioid pET , ', bq is introduced. This dioid is used for the modeling and the control of Weighted Timed Event Graphs under partial synchronization. The dioid pET , ', bq consists of specific event-variant and time-variant operators, in other words, it is a composition of the dioids pErrδss, ', bq and pT rrγss, ', bq introduced in Chapter 3 and Chapter 4. Note that many results are similar to the results obtained for the dioid pErrδss, ', bq and pT rrγss, ', bq. In particular, just as for periodic elements in Errδss and T rrγss, a core decomposition is introduced for periodic elements in ET . Again, it is shown that all relevant operations p', b, z, {q on periodic elements in ET can be reduced to operations on matrices with entries in M ax in vγ, δw.

Dioid ET

Let us first recall some results from Section 3.1. The set of antitone mappings Σ : Z Ñ Z min is a idempotent commutative monoid, denoted pΣ, ', εq. An operator is defined as a lower semi-continuous mapping from the set Σ into itself, see Definition 27. The set of operators O is a complete dioid denoted pO, ', bq, see Prop. 8. On this dioid the order introduced by ' is partial and given by, for f 1 , f 2 P O

f 1 ľ f 2 ô f 1 ' f 2 " f 1 ,
ô ̀f1 x ̆ptq ' ̀f2 x ̆ptq " ̀f1 x ̆ptq, @x P Σ, @t P Z, ô min ́̀f 1 x ̆ptq, ̀f2 x ̆ptq ̄" ̀f1 x ̆ptq @x P Σ, @t P Z.

Then, two operators f 1 , f 2 P O are equal iff @x P Σ, @t P Z: pf 1 xqptq " pf 2 xqptq. In the following proposition, some specific operators in O are recalled and the ∆ ω|ϖ operator is redefined.

Proposition 73. The following elementary operators are endomorphisms and lower semi-con-tinuous mappings and therefore operators in O.

m, b P N ∇ m|b : @x P Σ, t P Z ̀∇m|b pxq ̆ptq " m

̂Y xptq b ] , (5.1)
ω, ϖ P N ∆ ω|ϖ : @x P Σ, t P Z ̀∆ω|ϖ pxq ̆ptq " x ́ϖ

̂Y t ́1 ω ] ̀1̄, (5.2)
ν P Z γ ν : @x P Σ, t P Z ̀γν pxq ̆ptq " ν ̀xptq, (5.3) τ P Z δ τ : @x P Σ, t P Z ̀δτ pxq ̆ptq " xpt ́τq.

(5.4)

Proof. For the proof of (5.3) see the proof of Prop. 9. For the proof of (5.4) see (3.19) and the following paragraph in Section 3.1.2. Note that the ∇ m|b operator is nothing but the composition µ m β b , with µ m and β b defined in Prop. 9. The mapping ∇ m|b is a '-morphism, since first @t P Z, εptq " 8 and m, b P N are finite positive integers, thus p∇ m|b pεqqptq " m ̂tεptq{bu " εptq. Moreover, for all finite and infinite subsets X Ď Σ,

́∇m|b ̀à xPX x ̆̄ptq " m ̂Y ̀ÀxPX x ̆ptq b ] " m ̂Y min xPX ̀xptq b ] , " min xPX ̀m ̂Y xptq b ] ̆" min xPX ́̀∇ m|b pxq ̆ptq ̄, " ́à xPX ∇ m|b pxq ̄ptq,
which proves the lower semi-continuity property. Note that in contrast to Prop. 53 in Section 4.1 here the ∆ ω|ϖ operator is defined on the set Σ instead of Ξ, i.e., the set of isotone mappings from Z into Z max . In the current form, it manipulates the domain Z of a mapping x : Z Ñ Z min whereas for mappings x P Ξ, x : Z Ñ Z max the ∆ ω|ϖ operator manipulates the codomain Z max of x, see Prop. 53. The ∆ ω|ϖ operator defined in (5.2) is lower semi-continuous and endomorphic. First, we have to prove that, ∆ ω|ϖ pεq " ε. Clearly, since ω, ϖ P N are finite positive integers then @t P Z, ωtpt ́1q{ϖu ̀1 P Z. Then @t P Z, εptq " 8 and therefore @t P Z, p∆ ω|ϖ pεqqptq " ὲω tpt ́1q{ϖu ̀1̆" 8. Second, for all finite and infinite subsets X Ď Σ and @t P Z, Note that the identity operator e : pexqptq " xptq can be written as ∆ 1|1 and ∇ 1|1 , i.e., ̀∆1|1 x ̆ptq " xp1 ̂tpt ́1q{1u ̀1q " xptq and ̀∇1|1 x ̆ptq " 1 ̂txptq{1u " xptq.

́∆ω|ϖ ̀à xPX x ̆̄ptq " ́à xPX x ̄́ϖ ̂Y t ́1 ω ] ̀1̄d ue to (5.2), " à xPX x ́ϖ ̂Y t ́1 ω ] ̀1̄d ue

Dioid ET

Remark 29. Note that in analogy with Section 4.1, these operators can be defined on the set Ξ in the following form, m, b P N ∇ m|b : @x P Ξ, k P Z ̀∇m|b pxq ̆pkq "

x ́b ̂Q k ̀1 m U ́1̄, (5.5
)

ω, ϖ P N ∆ ω|ϖ : @x P Ξ, k P Z p∆ ω|ϖ xqpkq " Q xpkq ϖ U ω, (5.6)
ν P Z γ ν : @x P Ξ, k P Z ̀γν pxq ̆pkq " xpk ́νq, (5.7)

τ P Z δ τ : @x P Ξ, k P Z ̀δτ pxq ̆pkq " xpkq ̀τ.

(5.8)

Proposition 74. The elementary operators satisfy the following relations

γ ν ' γ ν 1 " γ minpν,ν 1 q , γ ν γ ν 1 " γ ν̀ν 1 , (5.9)
δ τ ' δ τ 1 " δ maxpτ,τ 1 q , δ τ δ τ 1 " δ τ̀τ 1 , (5.10)

∆ ω|ϖ δ ϖ " δ ω ∆ ω|ϖ ∇ m|b γ b " γ m ∇ m|b .
(5.11)

Proof. For the proof of (5.9) see Prop. 10. For the proof of δ τ ' δ τ 1 " δ maxpτ,τ 1 q , recall (3.4), (3.1) and (5.4), then @x P Σ, @t P Z,

̀pδ τ ' δ τ 1 q ̆ptq " ̀pδ τ xq ' pδ τ 1 xq ̆ptq " pδ τ xqptq ' pδ τ 1 xqptq
" min ̀xpt ́τq, xpt ́τ1 q ̆" xpt ́maxpτ, τ 1 qq " ̀δmaxpτ,τ 1 q x ̆ptq.

For the proof of δ τ δ τ 1 " δ τ̀τ 1 , recall (3.5) and (5.4), then @x P Σ, @t P Z, ̀pδ τ δ τ 1 qx ̆ptq " ̀pδ τ pδ τ 1 xq ̆ptq " ̀δτ 1 x ̆pt ́τq " xpt ́pτ ̀τ1 qq " ̀δτ̀τ 1 x ̆ptq.

For the proof of ∆ ω|ϖ δ ϖ " δ ω ∆ ω|ϖ , recall (3.5), (5.2) and (5.4), then first @x P Σ, @t P Z,

̀∆ω|ϖ δ ϖ x ̆ptq " ̀∆ω|ϖ pδ ϖ xq ̆ptq " pδ ϖ xq ́ϖY t ́1 ω ] ̀1" x ́ϖY t ́1 ω ] ́ϖ ̀1̄.
Second,

x ́ϖY t ́1 ω ] ́ϖ ̀1̄" x ́ϖ́Y t ́1 ω ] ́1̄̀1̄" x ́ϖY t ́ω ́1 ω ] ̀1" ̀δω ∆ ω|ϖ x ̆ptq. For the proof of ∇ m|b γ b " γ m ∇ m|b , recall that ∇ m|b " µ m β b , γ m µ m " µ m γ 1 and γ 1 β b " β b γ b (3.13), therefore µ m β b γ b " µ m γ 1 β b " γ m µ m β b and ∇ m|b γ b " γ m ∇ m|b .
Remark 30. (5.11) implies that for 0 ď n ă i, ∇ m|i γ n ∇ i|b " ∇ m|b , since,

p∇ m|i γ n ∇ i|b xqptq " [ txptq{bui ̀n i _ m, " [ Y xptq b ] ̀n i _ m, " Y xptq b ] m, since 0 ď n i ă 1.
Moreover, for ́i ă τ ď 0, ∆ ω|i δ τ ∆ i|ϖ " ∆ ω|ϖ , since

p∆ ω|i δ τ ∆ i|ϖ xqptq " pδ τ ∆ i|ϖ xq ́iY t ́1 ω ] ̀1̄, " p∆ i|ϖ xq ́iY t ́1 ω ] ́τ ̀1̄, " x ̃ϖ[ itpt ́1q{ωu ́τ ̀1 ́1 i _ ̀1̧, " x ̃ϖ[ Y t ́1 ω ] ́τ i _ ̀1̧, since 0 ď ́τ i ă 1, " x ́ϖY t ́1 ω ] ̀1̄,
" p∆ ω|ϖ xqptq.

In general mappings (operators) in O do not commute, i.e., f 1 , f 2 P O and x P Σ in general f 1 pf 2 pxqq ‰ f 2 pf 1 pxqq, however, the following proposition gives some properties regarding the commutation of the elementary operators.

Proposition 75. The operators introduced in Prop. 73 commute according to the following rules,

δ 1 γ 1 " γ 1 δ 1 , ∆ ω|ϖ ∇ m|b " ∇ m|b ∆ ω|ϖ , (5.12
)

∇ m|b δ 1 " δ 1 ∇ m|b , ∆ ω|ϖ γ 1 " γ 1 ∆ ω|ϖ .
(5.13)

Proof. For the proof of δ 1 γ 1 " γ 1 δ 1 , recall (3.5), (5.3) and (5.4), then @x P Σ, @t P Z, ̀pδ 1 γ 1 qx ̆ptq " ̀δ1 pγ 1 xq ̆ptq " ̀γ1 x ̆pt ́1q " 1 ̀xpt ́1q " 1 ̀̀δ 1 x ̆ptq, " ̀pγ 1 δ 1 qx ̆ptq.

The proofs for the right equation of (5.12) and the equations of (5.13) are similar.

Dioid ET

Proposition 76. The ∇ m|b and the ∆ ω|ϖ operator are expressed in the following forms

∇ m|b " ń1 à i"0 γ im ∇ nm|nb γ pń1́iqb , (5.14) ∆ ω|ϖ " ń1 à i"0 δ ́iω ∆ nω|nϖ δ ́pń1́iqϖ .
(5.15)

Proof. For the proof of (5.14) see Prop. 12 and the proof of (5.15) is similar to the proof of Prop. 56 in Appendix Section C.2.1.

Example 42. The identity operator e " ∇ 1|1 ∆ 1|1 is represented with n " 2 in an extended form

∇ 1|1 ∆ 1|1 " p∇ 2|2 γ 1 ' γ 1 ∇ 2|2 qp∆ 2|2 δ ́1 ' δ ́1∆ 2|2 q, " ∇ 2|2 ∆ 2|2 γ 1 δ ́1 ' δ ́1∇ 2|2 ∆ 2|2 γ 1 ' γ 1 ∇ 2|2 ∆ 2|2 δ ́1 ' γ 1 δ ́1∇ 2|2 ∆ 2|2 .
Definition 50 (Dioid ET). The dioid pET, ', bq is defined by sums and compositions over the set tê, ε, Ĵ, ∇ m|b , γ ν , ∆ ω|ϖ , δ τ u with m, b, ω, ϖ P N, ν, τ P Z and addition and multiplication defined in (3.4) and (3.5).

The dioid pET, ', bq is a complete subdioid of pO, ', bq. Again the ' operation defines a natural order on ET, therefore for a, b P ET, a ' b " a ô a ľ b. Note that, in contrast to Errδss and T rrγss, an element s P ET does not have the structure of a formal power series, see Definition 9. However, a basic element in pET, ', bq is defined as γ n δ τ ∇ m|b ∆ ω|ϖ γ n 1 δ τ 1 . A basic sum is defined as a finite sum of basic elements in ET, i.e.,

À I i"0 γ ν i δ τ i ∇ m i |b i ∆ ω i |ϖ i γ n 1 i δ τ 1
i and an infinite sum

À i γ ν i δ τ i ∇ m i |b i ∆ ω i |ϖ i γ n 1 i δ τ 1 i is called a series.
Proposition 77. A basic element γ n δ τ ∇ m|b ∆ ω|ϖ γ n 1 δ τ 1 P ET has a canonical form such that 0 ď n 1 ă b and ́ϖ ă τ 1 ď 0.

Proposition 78. [Standard Form] All elements s P ET can be expressed by a finite or infinite sum of basic elements, i.e., s "

À i γ ν i δ τ i ∇ m|b i ∆ ω|ϖ i γ n 1 i δ τ 1 i
, such that all basic element have the same m and ω indices, are in the canonical form of (Prop. 77) and are not ordered.

Proof. See Section C.3.1.

The standard form is used to check the ordering of two basic sums. Consider two sums

s 1 " À i γ ν 1 i δ τ 1 i ∇ m 1 |b 1 i ∆ ω 1 |ϖ 1 i γ n 1 1 i δ τ 1 1 i and s 2 " À j γ ν 2 j δ τ 2 j ∇ m 2 |b 2 j ∆ ω 2 |ϖ 2 j γ n 1 2 j δ τ 1
2 j in the standard form (Prop. 78). Due to (5.14), (5.15) and by choosing ω " lcmpω 1 , ω 2 q and m " lcmpm 1 , m 2 q, s 1 and s 2 can be rewritten as

s 1 " à k γ ν 1 k δ τ 1 k ∇ m|b 1 k ∆ ω|ϖ 1 k γ n 1 1 k δ τ 1 1 k , (5.16
)

s 2 " à l γ ν 2 l δ τ 2 l ∇ m|b 2 l ∆ ω|ϖ 2 l γ n 1 2 l δ τ 1 2 l .
(5.17)

Then the sum s 1 is greater than or equal to the sum s 2 if and only if, every basic element in (5.17) is smaller than or equal to at least one basic element in (5.16). Clearly, two sums s 1 , s 2 P ET are equal if s 1 ĺ s 2 and s 2 ĺ s 1 .

Definition 51. An element s P ET is called pm, b, ωq-periodic if its standard form is written as

À i γ ν i δ τ i ∇ m|b ∆ ω|ω γ ν 1 i δ τ 1 i ,i.
e., all basic elements in the sum have the same m, b, ω indices. Furthermore, the gain of s is then defined by Γ psq " m{b.

The set of periodic operators, denoted by ET per , is a subset of ET.

Definition 52 (Ultimately cyclic series in ET per ). A series s P ET per is said to be ultimately cyclic if it can be written as p ' qpγ ν δ τ q ̊where ν, τ P N 0 and p, q are pm, b, ωq-periodic finite basic sums in ET per (p and q must have the same period).

Core Decomposition of Series in ET per

This section introduces the core-form of series in ET per . This core-form is orthogonal to the core-forms of series s P E m|b rrδss and series s 1 P T per rrγss introduced in Section 3.3 and Section 4.2. Hence, the following results are orthogonal to the results obtained in Section 3.3 and Section 4.2. However, to improve the readability of this section again all propositions with proofs in the introduced notation are provided. Note that most of the presented propositions and proof are similar to those given in Section 3.3 and Section 4.2. Recall that an ultimately cyclic series s P E m|b rrδss can always be expressed as m m Qb b with Q a matrix in M ax in vγ, δw and m m :"

" ∇ m|1 γ 1 ∇ m|1 ̈̈̈γ ḿ1 ∇ m|1 ı , (5.18) b b :" " ∇ 1|b γ b́1 ̈̈̈∇ 1|b γ 1 ∇ 1|b ı T . (5.19)
Respectively, an ultimately cyclic series s P T per rrγss can always be expressed as d ω Qp ω again with Q a matrix in M ax in vγ, δw and

d ω :" " ∆ ω|1 δ ́1∆ ω|1 ̈̈̈δ 1́ω ∆ ω|1 ı , p ω :" " ∆ 1|ω δ 1́ω ̈̈̈∆ 1|ω δ ́1 ∆ 1|ω ı T .
Similarly to the core representation of s P E m|b rrδss and s 1 P T per rrγss, in this section, a core representation for series s P ET per is introduced. It is shown that an ultimately cyclic series s P ET per can always be written as a product m m,ω Qb b,ω where Q is a matrix in M ax in vγ, δw and b b,ω :"

" ∆ 1|ω δ 1́ω b T b ̈̈̈∆ 1|ω b T b ı T , (5.20)
m m,ω :"

" ∆ ω|1 m m ̈̈̈δ 1́ω ∆ ω|1 m m ı . (5.21)
Based on this representation all operations on series s P ET per can be reduced to operations on matrices in M ax in vγ, δw. For an illustration of this core-form, see the following example.

Example 43. Consider a series s " δ 2 ∇ 3|2 ∆ 2|2 γ 1 δ ́1 ' pγ 3 δ 3 ∇ 3|2 ∆ 2|2 δ ́1qpγ 1 δ 2 q ̊. By using pγ 1 δ 2 q ̊" pe ' γ 1 δ 2 qpγ 2 δ 4 q ̊, this series is rephrased as,

s "δ 2 ∇ 3|2 ∆ 2|2 γ 1 δ ́1 ' ̀γ3 δ 3 ∇ 3|2 ∆ 2|2 δ ́1 ' γ 3 δ 5 ∇ 3|2 ∆ 2|2 γ 1 δ ́1̆p γ 2 δ 4 q ̊.
Because of ∆ 2|2 " ∆ 2|1 ∆ 1|2 and ∇ 3|2 " ∇ 3|1 ∇ 1|2 (Remark 30) one has,

s " δ 2 ∇ 3|1 ∇ 1|2 ∆ 2|1 ∆ 1|2 γ 1 δ ́1 ' ̀γ3 δ 3 ∇ 3|1 ∇ 1|2 ∆ 2|1 ∆ 1|2 δ ́1' γ 3 δ 5 ∇ 3|1 ∇ 1|2 ∆ 2|1 ∆ 1|2 γ 1 δ ́1pγ 2 δ 4 q ̊.
Recall, ∇ 2|1 γ 1 " γ 2 ∇ 2|1 , ∆ 2|1 δ 1 " δ 2 ∆ 2|1 (5.11) and the commutation laws for elementary operators (Prop. 75), therefore s can be rephrased as

s " ∇ 3|1 ∆ 2|1 δ 1 lo omo on M 1 ∇ 1|2 ∆ 1|2 γ 1 δ ́1'δ ́1∇ 3|1 ∆ 2|1 γ 1 δ 2 pγ 1 δ 2 q loooooomoooooon S 1 ∇ 1|2 ∆ 1|2 δ ́1' δ ́1∇ 3|1 ∆ 2|1 γ 1 δ 3 pγ 1 δ 2 q loooooomoooooon S 2 ∇ 1|2 ∆ 1|2 γ 1 δ ́1.
Observe that M 1 , S 1 and S 2 are elements in M ax in vγ, δw, and that the entries of the m 3,2 -vector (resp. b 2,2 -vector) appear on the left (resp. right) of M 1 , S 1 , S 2 . The series s is now expressed in the core representation m 3,2 Qb 2,2 where,

Q " » - - - - - - - - - - δ 1 ε ε ε ε ε ε ε ε ε ε ε γ 1 δ 3 pγ 1 δ 2 q ̊γ1 δ 2 pγ 1 δ 2 q ̊ε ε ε ε ε ε ε ε ε ε fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl
.

A formal method to obtain the core representation for an arbitrary ultimately cyclic series s P ET per is given in the following.

Core-Equation for a Series in ET per

The decomposition of an ultimately cyclic series s P ET per is carried out according to the following equation s " m m,ω Xb b,ω .

( 5.22) This equation is called core-equation. Then, Q P M ax in vγ, δw mω̂bω is called a core of s P ET per , if Q is a solution of (5.22), i.e., s " m m,ω Qb b,ω . In general, there exists several cores Q which solve (5.22). A solution Q for an arbitrary ultimately cyclic s P ET per can be obtained as follows. A series s " p ' qpγ ν δ τ q ̊P ET per can be expressed as s "

I à i"1 γ n i δ σ i ∇ m|1 ∆ ω|1 γ ni δ σi loomoon M i ∇ 1|b ∆ 1|ω γ n 1 i δ σ 1 i ' J à j"1 γ N j δ t j ∇ m|1 ∆ ω|1 γ Nj δ tj pγ ν δ τ q looooooomooooooon S j ∇ 1|b ∆ 1|ω γ N 1 j δ t 1 j .
where M i is a monomial and S j is a series in M ax in vγ, δw. Furthermore 0 ď n i , N j ă m, 0 ď n 1 i , N 1 j ă b and ́ω ă σ i , σ 1 i , t j , t 1 j ď 0. In this form, the entries of the m m,ω -vector appear on the left of monomials M i and series S j . Respectively, the entries of the b b,ωvector appear on the right of monomials M i and series S j . Note that in general, the growingterm pγ ν δ τ q ̊of a series s P ET per does not commute with the ∇ 1|b ∆ 1|ω (resp. ∇ m|1 ∆ ω|1 ) operator. To bring the growing-term pγ ν δ τ q ̊of a series to the left-hand side of the ∇ 1|b ∆ 1|ω operator ν must be a multiple of b and τ must be a multiple of ω, see (5.11). However, any series s P ET per can be rewritten such that the growing-term commutes with ∇ 1|b ∆ 1|ω by extending pγ ν δ τ q ̊such that, l " lcmpl 1 , l 2 q with l 1 " lcmpν, b, q{ν, l 2 " lcmpτ, ωq{τ pγ ν δ τ q ̊" pe ' γ ν δ τ ' ̈̈̈' γ pĺ1qν δ pĺ1qτ qpγ lν δ lτ q ̊.

For an illustration see Example 43. We denote the set of monomials by M " tM 1 , ̈̈̈, M I u and the set of series by S " tS 1 , ̈̈̈, S J u. Furthermore, the subsets M l,k,g,p (resp. S l,k,g,p ) are defined as @l P t0, ̈̈̈, m ́1u, @g P t0, ̈̈̈, b ́1u, @k, p P t0, ̈̈̈, ω ́1u, M l,k,g,p " tM i P M| γ l δ ́k∇ m|1 ∆ ω|1 M i ∇ 1|b ∆ 1|ω γ g δ ́p P

I à i"1 γ n i δ σ i ∇ m|1 ∆ ω|1 M i ∇ 1|b ∆ 1|ω γ n 1 i δ σ 1 i u, S l,k,g,p " tS j P S| γ l δ ́k∇ m|1 ∆ ω|1 S j ∇ 1|b ∆ 1|ω γ g δ ́p P J à j"1 γ N j δ t j ∇ m|1 ∆ ω|1 S j ∇ 1|b ∆ 1|ω γ N 1 j δ t 1 j u.
(5.23)

The entry pQq mk̀l̀1,bpώpq́g of the core matrix is then obtained by

pQq mk̀l̀1,bpώpq́g " à MPM l,k,g,p M ' à SPS l,k,g,p S. ( 5
m i,ω b i,ω " m i b i p∆ ω|1 ∆ 1|ω δ 1́ω ' δ ́1∆ ω|1 ∆ 1|ω δ 2́ω ' ̈̈' δ 1́ω ∆ ω|1 ∆ 1|ω q " m i b i p∆ ω|ω δ 1́ω ' ̈̈̈' δ 1́ω ∆ ω|ω q " e, ( 5.25)
The dyadic product b i,ω m i,ω is a particular matrix in M ax in vγ, δw of size iω ̂iω denoted by E. Recall that,

E " b m m m " » - - - - - - e γ
E " b i,ω b m i,ω " » - - ∆ 1|ω δ 1́ω b i ̈̈∆ 1|ω b i fi ffi fl " ∆ ω|1 m i ̈̈̈δ 1́ω ∆ ω|1 m i ı , " » - - - ∆ 1|ω δ 1́ω ∆ ω|1 b i m i ̈̈̈∆ 1|ω δ 1́ω δ 1́ω ∆ ω|1 b i m i . . . . . . ∆ 1|ω ∆ ω|1 b i m i ̈̈̈∆ 1|ω δ 1́ω ∆ ω|1 b i m i fi ffi ffi fl , " » - - - - - - E δ ́1E ̈̈̈δ
E i,ω b b i,ω " b i,ω , (5.28)
m i,ω b E i,ω " m i,ω .
(5.29)

Proof.

E i,ω b E i,ω " b i,ω m i,ω b i,ω m i,ω " b i,ω b e b m i,ω " E i,ω , E i,ω b b i,ω " b i,ω m i,ω b i,ω " b i,ω b e " b i,ω , m i,ω b E i,ω " m i,ω b iω m i,ω " e b m i,ω " m i,ω .
Corollary 14. Note that E " E ̊, because of EE " E and E " I ' E.

Due to m i,ω b iω " e (5.25) and EE " E (Prop. 79), under some conditions the left and right product of matrices with entries in ET by the m m,ω -vector and the b b,ω -vector are invertible, see the following proposition.

m,ω m m,ω b X 1 " E b X 1 ĺ b m,ω D Furthermore, X 1 ĺ EX 1 as E " I ' E. Hence, X 1 ĺ b m,ω D and therefore, X 1 " b m,ω D.
This proofs that b m,ω D is indeed the greatest solution of (5.32). Similarly, X " P b m b,ω solves Xb b,ω ĺ P with equality. Assume that X 1 " Pbm b,ω is a solution, i.e., X 1 bb b,ω ĺ P.

Multiplication by m b,ω gives

X 1 ĺ X 1 E ĺ P b m b,ω .
Therefore

m m,ω Qb b,ω " m m,ω E m,ω QE b,ω b b,ω " m m,ω Qb b,ω " s.
Remark 31. Since, E b E " E (Prop. 79) the greatest core Q satisfies the following relations,

E Q " EEQE " Q, QE " EQEE " Q. Alternative Core-Form
An alternative core form is defined by replacing the m m,ω -vector and b b,ω -vector by

d ω,m :" " ∇ m|1 d ω ̈̈̈γ ḿ1 ∇ m|1 d ω ı , (5.34)
p ω,b :"

" ∇ 1|b γ b́1 p T ω ̈̈̈∇ 1|b p T ω ı T . (5.35)
Observes that, the difference between the vectors m m,ω and d ω,m (resp. b b,ω and p ω,b ) is just the ordering of its entries. Thus, the alternative core equation for an ultimately cyclic series s P ET per is

s " d ω,m Xp ω,b . (5.36)
A solution of (5.36) is denoted by U. Note again that U P M ax in vγ, δw mω̂bω . Recall the sets M l,k,g,p and S l,k,g,p for an ultimately cyclic series s P ET per (5.23). A solution of (5.36) for s is then obtained by @l P t0, ̈̈̈, m ́1u, @g P t0, ̈̈̈, b ́1u, @k, p P t0, ̈̈̈, ω ́1u,

pUq ωl̀k̀1,ωpb́gq́p " à MPM l,k,g,p M ' à SPS l,k,g,p S. (5.37)
This alternative core form is sometimes preferable over the core form m m,ω Qb b,ω for calculations with s P ET per . Consider a d ω,i -vector and the p ω,i -vector with same indices, i.e., the d ω,i -vector and the p ω,i -vector the have same size. The scalar product,

d ω,i p ω,i " d ω p ω pµ i β i γ í1 ' γ 1 µ i β i γ í2 ' ̈̈' γ í1 µ i β i q " e. ( 5.38)
Recall that, Proof. The proof is analogous to the proof of Prop. 80.

N " p ω d ω " » - - - - - - e δ ́1 ̈̈̈δ

Core Transformation

The transformation between the two core representations is achieved by reordering the entries in the core matrix Q (respectively U). The relation between the two cores Q and U is given by (5.24) and (5.37). Hence, let s " m m,ω Qb b,ω P ET per , then s is written as d ω,m Up ω,b , where @l P t0, ̈̈̈, m ́1u, @g P t0, ̈̈̈, b ́1u, @k, p P t0, ̈̈̈, ω ́1u, pUq ωl̀k̀1,ωpb́gq́p " pQq mk̀l̀1,bpώpq́g .

By choosing

k " i ́1 ́Z i ́1 ω ^ω, l " Z i ́1 ω ^, p " ωb ́j ́Z ωb ́j ω ^ω, g " Z ωb ́j ω ^,
one can establish for i P t1, ̈̈̈, mωu, j P t1, ̈̈̈, bωu,

pUq i,j " pQq mpí1́t í1 ω uωq̀t í1 ω ù1,bpώωb̀j̀t ωb́j ω uωq́t ωb́j ω u . (5.40)
Conversely, let s " d ω,m Up ω,b P ET per , then s is written as m m,ω Qb b,ω where for i P t1, ̈̈̈, mωu, j P t1, ̈̈̈, bωu

pQq i,j " pUq ωpí1́t í1 m umq̀t í1 m ù1,ωpb́ωb̀j̀t ωb́j b ubq́t ωb́j b u . (5.41)
Example 44. Recall the series s "

δ 2 ∇ 3|2 ∆ 2|2 γ 1 δ ́1 ' pγ 3 δ 3 ∇ 3|2 ∆ 2|2 δ ́1qpγ 1 δ 2 q ̊of Exam- ple 43. The alternative core-form of s is d 2,3 Up 2,2
, where

U " » - - - - - - - - - - - - δ 1 ε ε ε γ 1 δ 3 pγ 1 δ 2 q ̊ε γ 1 δ 2 pγ 1 δ 2 q ̊ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl .
Let s " m m,ω Qb b,ω P ET per be an ultimately cyclic series. Clearly, since d ω,m p ω,b " e, we can express s as

s " d ω,m p ω,m loooomoooon e m m,ω Qb b,ω d ω,b p ω,b looomooon e .
Clearly, p ω,m m m,ω Qb b,ω d ω,b is a solution of the alternative core equation (5.36). Moreover, it can be shown that

Û " p ω,m m m,ω looooomooooon T QU 1 Q b b,ω d ω,b looomooon T QU 2 ,
is the greatest solution of (5.36), for details see Section C.3.2. T QU 1 " p ω,m m m,ω and T QU 2 " b b,ω d ω,b are matrices with entries in M ax in vγ, δw, see Section C.3.2. Respectively,

Q " b b,ω p ω,b looomooon T UQ 1 U d ω,m m m,ω looooomooooon T UQ 2
.

Again, T UQ 1 and T UQ 2 are matrices with entries in M ax in vγ, δw, see Section C.3.2. The transformation between the core-matrices Q and Û is necessary to express an ultimately cyclic series s P ET per with a multiple period in the core form.

Proposition 83. A series s " m m,ω Qb b,ω P ET per can be expressed with a multiple period pm, b, nωq by extending the core matrix Q, i.e., s " m m,ω Qb b,ω " m mn,ω Q1 b bn,ω , where Q1 P M ax in vγ, δw mnω̂bnω and is given by

Q1 " » - - - ∆ 1|n δ 1́ω Q∆ n|1 ̈̈̈∆ 1|n δ 1́ω
Û1 " » - - - ∇ 1|n γ ń1 Û∇ n|1 ̈̈̈∇ 1|n γ ń1 Ûγ ń1 ∇ n|1 . . . . . . ∇ 1|n Û∇ n|1 ̈̈̈∇ 1|n Ûγ ń1 ∇ n|1 fi ffi ffi fl .
Proof. The proof is analogous to the proof of Prop. 83, given in Section C.3.2.

Therefore, a series s " m m,ω Qb b,ω can be expressed as m n 1 m,n 2 ω Qb n 1 b,n 2 ω with a multiple period pn 1 m, n 1 b, n 2 ωq.

Calculation with the Core Decomposition

This section illustrates how to perform the basic operations p', b, z, {q on series in ET per , based on the core decomposition.

Sum and Product of Series in ET per

Due to Prop. 83 and Prop. 84, by extending the core-form if necessary, two ultimately cyclic series s, s 1 P ET per with equal gain can be expressed with their least common period, i.e., s " m m,ω Qb b,ω , s 1 " m m,ω Q1 b b,ω . Then observe that matrices Q and Q1 have equal dimensions.

Proposition 85 (Sum of Series). Let s " m m,ω Qb b,ω , s 1 " m m,ω Q1 b b,ω P ET per , be two ultimately cyclic series, then the sum s ' s 1 " m m,ω Q2 b b,ω , where Q2 " Q ' Q1 , is again an ultimately cyclic series in ET per .

Proof.

s ' s 1 " m m,ω Qb b,ω ' m m,ω Q1 b b,ω " m m,ω pE QE ' E Q1 Eqb b,ω " m m,ω Ep Q ' Q1 qE loooooomoooooon Q 2 b b,ω
Clearly, the entries of the core matrices Q, Q1 are ultimately cyclic series in M ax in vγ, δw and because of Theorem 2.6 the sum of two ultimately cyclic series in M ax in vγ, δw is again an ultimately cyclic series. Therefore, Q2 is composed of ultimately cyclic series in M ax in vγ, δw and thus s ' s 1 " d ω Q2 p ω is an ultimately cyclic series in ET per .

Again, because of Prop. 83 and Prop. 84, two ultimately cyclic series s, s 1 P ET per can be written such that s is pm, b, ωq-periodic and s 1 is pb, b 1 , ωq-periodic, i.e., s " m m,ω Qb b,ω and s 1 " m b,ω Qb b 1 ω , where Q P M ax in vγ, δw mω̂bω and Q P M ax in vγ, δw bω̂b 1 ω . Proposition 86 (Product of Series). Let s " m m,ω Qb b,ω P ET per and s 1 " m b,ω Qb b 1 ω P ET per , be two ultimately cyclic series, then the product s b s 1 " m m,ω Q2 b b 1 ,ω , where Q2 " Q Q1 , is again an ultimately cyclic series in ET per .

Proof. Recall that b b,ω m b,ω " E (5.26) and QE " Q (Remark 31), then

s b s 1 " m m,ω Qb b,ω m b,ω Q1 b b 1 ,ω " m m,ω QE Q1 b b 1 ,ω " m m,ω Q Q1 b b 1 ,ω .
Furthermore, because of EE " E (Prop. 79),

Q Q1 " E QEE Q1 E " Q2 .
Recall that, the entries of the core matrices Q, Q1 are ultimately cyclic series in M ax in vγ, δw and because of Theorem 2.6 the sum and product of ultimately cyclic series in M ax in vγ, δw are again ultimately cyclic series in M ax in vγ, δw. Therefore, entries of the matrix Q2 are ultimately cyclic series in M ax in vγ, δw and the product s b s 1 is an ultimately cyclic series in ET per .

Proposition 87. Let s " m m,ω Qb m,ω P ET per be an ultimately cyclic series, then

s ̊" m m,ω Q̊b m,ω . (5.42)
is again an ultimately cyclic series in ET per .

Proof. In this case, Γ psq " 1 means that Q is a square matrix. The Kleene star of a series in the core representation can be written as,

s ̊" e ' m m,ω Qb m,ω ' m m,ω Qb m,ω m m,ω Qb m,ω ' ̈̈R ecall that, e " m m,ω b m,ω (5.25), E " b m,ω m m,ω (5.26) and E Q " Q Remark 31, s ̊" m m,ω b m,ω ' m m,ω Qb m,ω ' m m,ω Q2 b m,ω ' ̈̈" m m,ω pI ' Q ' Q2 ' ̈̈̈qb m,ω " m m,ω Q̊b m,ω
Again, due to Theorem 2.6 the Kleene star, sum, and product of ultimately cyclic series in M ax in vγ, δw are ultimately cyclic series in M ax in vγ, δw and therefore, s ̊" m m,ω Q̊b m,ω is an ultimately cyclic series in ET per .

Note that

Q̊i s not the greatest core of s ̊as Q̊ĺ E Q̊E . In general, multiplication does not distribute with respect to ^in the dioid pET , ', bq. However, as shown for the dioid pErrδss, ', bq in Lemma 2 and Lemma 3, distributivity holds for left multiplication by the m m,ω -vector and right multiplication by the b b,ω -vector for specific matrices with entries in ET . Lemma 6. Let Q 1 , Q 2 be two matrices of appropriate dimension, then m m,ω pEQ 1 ^EQ 2 q " m m,ω EQ 1 ^mm,ω EQ 2 ,

pQ 1 E ^Q2 Eqb b,ω " Q 1 Eb b,ω ^Q2 Eb b,ω .
Proof. The proof is similar to the proof of Lemma 2. Recall that e " m m,ω b m,ω (4. [START_REF] David-Henriet | Discrete event systems with standard and partial synchronizations[END_REF]), E " b m,ω m m,ω (4.20) and E " EE Prop. 60. Moreover, recall that inequality cpa^bq ĺ ca^cb holds for a, b, c elements in a complete dioid, see (2.2). Now let us define q 1 " m m,ω EQ 1 and q 2 " m m,ω EQ 2 , then q 1 ^q2 " epq 1 ^q2 q " m m,ω b m,ω pq 1 ^q2 q ĺ m m,ω pb m,ω q 1 ^mm,ω q 2 q.

Inserting q 1 " m m,ω EQ 1 and q 2 " m m,ω EQ 2 lead to,

m m,ω pb m,ω q 1 ^mm,ω q 2 q " m m,ω pb m,ω m m,ω EQ 1 ^bm,ω m m,ω EQ 2 q, " m m,ω pEEQ 1 ^EEQ 2 q, " m m,ω pEQ 1 ^EQ 2 q.
Finally, m m,ω pEQ 1 ^EQ 2 q ĺ m m,ω EQ 1 ^mm,ω EQ 2 " q 1 ^q2 .

Hence, equality holds throughout. The proof for pQ 1 E^Q 2 Eqb b,ω " Q 1 Eb b,ω ^Q2 Eb b,ω is similar.

Proposition 88. Let s " m m,ω Qb b,ω , s 1 " m m,ω
Q1 b b,ω P ET per be two ultimately cyclic series, then s^s 1 " m m,ω Q2 b b,ω P ET per is an ultimately cyclic series, where Q2 " p Q^Q 1 q is again a greatest core.

Proof. Again, this proof is similar to the proof of Prop. 34. Let us recall that Q " E QE, then according to Lemma 4 we can write

s ^s1 " m m,ω Qb b,ω ^mm,ω Q1 b b,ω " m m,ω E QEb b,ω ^mm,ω E Q1 Eb b,ω " m m,ω pE QE ^E Q1 Eqb b,ω " m m,ω p Q ^Q 1 qb b,ω .
It remains to be shown that

Q2 " p Q ^Q 1
q is a greatest core. First, E " E ̊, therefore, I ' E " E, and Q2 ĺ E Q2 E. Then, according to Lemma 4,

E Q2 E " Ep Q ^Q 1 qE " b m,ω m m,ω p Q ^Q 1 qb b,ω m b,ω " b m,ω pm m,ω Qb b,ω ^mm,ω Q1 b b,ω qm b,ω .
Recall, cpa ^bq ĺ ca ^cb and pa ^bqc ĺ ac ^bc (2.2), therefore

b m,ω pm m,ω Qb b,ω ^mm,ω Q1 b b,ω qm b,ω ĺ b m,ω m m,ω Qb b,ω m b,ω ^bm,ω m m,ω Q1 b b,ω m b,ω " Q ^Q 1 " Q2 .
Hence, equality holds throughout.

Division in ET per

Proposition 89. Let s " m m,ω Qb b,ω , s 1 " m m,ω Q1 b b 1 ,ω be ultimately periodic series in ET per where s is pm, b, ωq-periodic and s 1 is pm, b 1 , ωq-periodic then

s 1 zs " m b 1 ,ω p Q1 z Qqb b,ω ,
is an ultimately cyclic series in ET per .

Proof. First, it is shown that

Q1 z Q " E b 1 ,ω p Q1 z QqE b,ω . (5.43)
For this,

́Eb 1 ,ω ́Q 1 z Q̄̄E b,ω " ́Eb 1 ,ω z ́Eb 1 ,ω ́Q 1 z Q̄̄̄E b,ω , since: E zpE Qq " E Q " ́Eb 1 ,ω z ́Eb 1 ,ω ́́Q 1 E b 1 ,ω ̄z Q̄̄̄E b,ω , since: Q " QE " ́Eb 1 ,ω z ́Eb 1 ,ω ́Eb 1 ,ω z ́Q 1 z Q̄̄̄̄E b,ω , since: pabq zx " b z pa zxq (A.5) " ́Eb 1 ,ω z ́Q 1 z Q̄̄E b,ω , since: a z pa pa zxqq " a zx (A.4) " ́́Q 1 E b 1 ,ω ̄z Q̄E b,ω " ́Q 1 z Q̄E b,ω , since: pabq zx " b z pa zxq (A.5) and Q " QE " ́́Q 1 z ̀Q{E b,ω ̆̄E b,ω ̄{E b,ω , since: p QEq{E " QE " ́́́Q 1 z Q̄{ E b,ω ̄Eb,ω ̄{E b,ω ,
since: pa zxq{b " a zpx{bq (A.6)

" ́Q 1 z Q̄{ E b,ω , since: ppx{aqaq{a " x{a (A.4) " Q1 z ̀Q{E b,ω ̆" Q1 z Q, since: pa zxq{b " a zpx{bq (A.6) and Q{E " QE " Q . Second, ́mm,ω Q1 b b 1 ,ω ̄z ̀mm,ω Qb b,ω ̆" ́Q 1 b b 1 ,ω ̄z ̀mm,ω zpm m,ω Qb b,ω q ̆,
because of (A.5),

" ́Q 1 b b 1 ,ω ̄z ̀bm,ω m m,ω Qb b,ω ̆,
because of (5.30),

" ́Q 1 b b 1 ,ω ̄z ̀Qb b,ω ̆, as b m,ω m m,ω Q " Q Remark 31, " ́Q 1 b b 1 ,ω ̄z ̀Q{m b ̆,
from (5.31) and Remark 31,

" b b 1 ,ω z ́Q 1 zp Q{m b q ̄, because of (A.5), " b b 1 ,ω z ́p Q1 z Qq{m b ̄, because of (A.6), " m b 1 ,ω p Q1
z Qqb b,ω , because of (5.31) and (5.43).

Due to Theorem 2.6, the quotient Q z Q1 is a matrix composed of ultimately cyclic series in M ax in vγ, δw and therefore the s 1 zs " m m,ω p Q1 z Qqb b,ω is an ultimately cyclic series in ET per .

Proposition 90. Let s " m m,ω Qb b,ω , s 1 " m m 1 ,ω
Q1 b b,ω be ultimately cyclic series in ET per where s is pm, b, ωq-periodic and s 1 is pm 1 , b, ωq-periodic then

s{s 1 " m m,ω p Q1 { Qqb m 1 ,ω ,
is an ultimately cyclic series in ET per .

Proof. The proof is analogous to the proof of Prop. 89.

Matrices with entries in ET per

In analogy with Section 3.4 the operations p', b, ̊, z, {q can be generalized to matrices with entries in ET per .

6

Model of Discrete Event Systems

In this chapter, Timed Event Graphs (TEGs) and their weighted extension, Weighted Timed Event Graphs (WTEGs) are studied. TEGs and WTEGs are a subclass of timed Petri nets, which are commonly used to model timed Discrete Event Systems (DESs), where the dynamic behavior is only governed by synchronization and saturation effects. Whereas the behavior of TEGs is event-invariant, due to the weight on the arcs in WTEGs, WTEGs exhibit eventvariant behavior. In the first part of this chapter Petri nets, TEGs and WTEGs are recalled. Next, time-variant TEGs are studied. Two time-variant extensions of TEGs are considered in this chapter. First, TEGs are expanded with a specific form of synchronization, which is referred to as partial synchronization (PS) [START_REF] David-Henriet | Modeling and control for max-plus systems with partial synchronization[END_REF] and is associated with transitions in TEGs. Second, the time-variant behavior is modeled with a time-variant holding time of places in TEGs. This leads to the introduction of Periodic Time-variant Event Graphs (PTEGs). The second part of this chapter focuses on dioid models for TEGs, WTEGs, TEGs under PS, and PTEGs. Clearly, the earliest functioning of TEGs can be described by linear equations over some dioids, e.g., the (max,+)-algebra. Due to the event-variant (resp. time-variant) behavior, this is not the case for WTEGs (resp. PTEGs). However, the input-output behavior of WTEGs can be described by ultimately cyclic series in the dioid pErrδss, ', bq, respectively for PTEGs and TEGs under periodic PS in the dioid pT per rrγss, ', bq). In Section 6.2, the modeling process of TEGs and WTEGs in the dioids pM ax in vγ, δw , ', bq and pErrδss, ', bq is presented. This section is mainly based on [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF][START_REF] Cottenceau | Modeling of time-varying (max,+) systems by means of weighted timed event graphs[END_REF][START_REF] Trunk | Model decomposition of weight-balanced timed event graphs in dioids: Application to control synthesis[END_REF][START_REF] Trunk | Output reference control for weightbalanced timed event graphs[END_REF]. Section 6.2.4 studies Timed Event Graphs under Partial Synchronization, which were first introduced in [START_REF] David-Henriet | Modeling and control for max-plus systems with partial synchronization[END_REF]. In this section it is shown how the earliest functioning of a TEG under periodic PS can be modeled in the dioid pT per rrγss, ', bq. In Section 6.2.6, partial synchronization is introduced for WTEGs. Again, it is shown that under some constraints the earliest functioning of WTEGs under periodic PS can be modeled in the dioid pET , ', bq. Some ideas, results, and figures presented in this chapter have appeared previously in [START_REF] Trunk | Output reference control for weightbalanced timed event graphs[END_REF][START_REF] Trunk | Model decomposition of weight-balanced timed event graphs in dioids: Application to control synthesis[END_REF][START_REF] Trunk | Model decomposition of timed event graphs under partial synchronization in dioids[END_REF][START_REF] Trunk | Modelling and control of periodic time-variant event graphs in dioids. Discrete Event Dynamic Systems[END_REF][START_REF] Trunk | Output reference control of timed event graphs under partial synchronization[END_REF].

Petri Nets and Timed Event Graphs

In the following, necessary facts on Petri nets TEGs and WTEGs are restated, for a comprehensive overview for Petri nets in general see, e.g., [START_REF] Murata | Petri nets: Properties, analysis and applications[END_REF][START_REF] Cassandras | Introduction to Discrete Event Systems[END_REF] and in particular for TEGs [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Heidergott | Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications[END_REF], for WTEGs respectively [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF][START_REF] Marchetti | Complexity results for weighted timed event graphs[END_REF][START_REF] Teruel | On weighted t-systems[END_REF]. Note also that, equivalent graphical models for WTEGs are known as SDF graphs. SDF graphs are used in the field of computer science, for instance to model data flow applications in embedded systems. For a detailed description on SDF graphs see, e.g., [START_REF] Geilen | Synchronous dataflow scenarios[END_REF][START_REF] Lee | Synchronous data flow[END_REF][START_REF] Sriram | Embedded Multiprocessors: Scheduling and Synchronization[END_REF]. Definition 53. A Petri net graph is a directed bipartite graph N " pP, T, wq, where:

-P " tp 1 , p 2 , . . . , p n u is the finite set of places.

-T " tt 1 , t 2 , . . . , t m u is the finite set of transitions.

w : pP ̂T q Y pT ̂Pq Ñ N 0 is the weight function.

A :" tpp i , t j q|wpp i , t j q ą 0u Y tpt j , p i q|wpt j , p i q ą 0u is the arc set, and W P Z n̂m , where pWq i,j " wpt j , p i q ́wpp i , t j q, is the incidence matrix of the Petri net graph N . Furthermore, -' p i :" tt j P T |pt j , p i q P Au is the set of upstream transitions of p i , p ' i :" tt j P T |pp i , t j q P Au is the set of downstream transitions of place p i . Conversely, -' t j :" tp i P P|pp i , t j q P Au is the set of upstream places of transition t j , t ' j :" tp i P P|pt j , p i q P Au is the set of downstream places of transition t j . A Petri net consists of a Petri net graph N and a vector of initial markings M 0 P N n 0 , i.e. an initial distribution of tokens over places in N . A transition t j can fire, iff @p i P ' t j , pMq i ě wpp i , t j q. If a transition t j fires, the marking is changing according to M 1 " M ̀pWq :,j , where M and M 1 are the markings before and after the firing of t j . A potential firing sequence can be encoded by a vector t P N m 0 (called Parikh vector), where ptq i gives the number of firings of t i in the sequence. E.g., for the Petri net shown in Figure 6.1, a firing sequence t 1 t 2 t 2 t 3 is described by t " r1 2 1s T . If the encoded firing sequence can actually occur when marking is M, the new marking is obtained as M 1 " M ̀Wt. A Petri net is said to be bounded if the marking in all places is bounded. Moreover, a Petri net is said to be live if any transition can ultimately fire from any reachable marking [START_REF] Teruel | On weighted t-systems[END_REF]. The structural properties of a Petri net can be analyzed by linear algebraic techniques. In particular, the right and left null spaces of the incidence matrix W reveal invariants of the net structure. Definition 54. A vector ξ is called T(ransition)-semiflow if ξ P N m̂1 and Wξ " 0, where 0 denotes the zero vector.

Note that a T-semiflows is a strictly positive integer vector. A T-semiflow, therefore, describes a firing sequence which involves all transitions in the Petri net and, if it can occur at marking M, leaves the latter invariant, i.e., M " M ̀Wξ. It can then of cause be repeated indefinitely and is therefore also called repetitive vector.

Example 45. Consider the Petri net shown in Figure 6.1. The incidence matrix of the corresponding Petri net graph is

W " » - - 2 ́1 0 0 1 ́2 ́1 0 1 fi ffi fl .
Then the vector ξ " r1 2 1s T is a T-semiflow for the Petri net shown in Figure 6.1, since Wξ " 0 and all entries of ξ are strictly positive integers. The initial marking of the Petri net is M 0 " r0 0 1s. Clearly, the firing of ξ, i.e. the firing of the sequence t 1 t 2 t 2 t 3 , results again in the marking M 0 " r0 0 1s. A timed Petri net with holding times is a triple pN , M 0 , ϕq, where pN , M 0 q is a Petri net and ϕ P N n 0 represents the holding times of the places, i.e., pϕq i is the time a token has to remain in place p i before it contributes to the firing of a transition in p ' i . We can divide the set of transitions of a Petri net into input, output and internal transitions. Input transitions are transitions without upstream places. Output transitions are transitions without downstream places and internal transitions are transitions with both upstream and downstream places. A single-input and single-output (SISO) Petri net has exactly one input and one output transition. If a Petri net has several input or output transitions it is referred to as multiple-input and multiple-output (MIMO) Petri net.

p 1 p 2 p 3 2 2 t 1 t 3 t 2

Weighted Timed Event Graphs

Definition 55. A timed Petri net (N , M 0 , ϕ) is called Weighted Timed Event Graph, if every place has exactly one upstream and one downstream transition i.e., @p i P P :

|p ' i | " | ' p i | " 1.
Definition 56. An (ordinary) Timed Event Graph is a WTEG, where all arcs have weight 1, i.e., @pp i , t j q, pt j , p i q P A, wpp i , t j q " wpt j , p i q " 1.

Definition 57 (Earliest Functioning Rule). A WTEG is operating under the earliest functioning rule if all internal and output transitions fire as soon as they are enabled.

Let t i and t i be the unique upstream respectively downstream transition of place p i , i.e., tt i u " ' p i and tt i u " p ' i .

Definition 58. t i Ñ p i Ñ t i is said to be a basic directed path, denoted by π i . The gain of π i is Γ pπ i q " wpt i , p i q wpp i , t i q .

Hence, the gain of a basic directed path is a positive rational number. It is interpreted as follows: if the upstream transition t i fires wpp i , t i q times, this deposits wpt i , p i q ̂wpp i , t i q tokens in place p i , and this, in turn, leads to wpt i , p i q firings of the downstream transition t i . Definition 59. A directed path is a sequence π " π i 1 ̈̈̈π iq with i j ‰ i k , j, k P t1, ̈̈̈, qu, such that i j " i j̀1 , @j P t1, ̈̈̈, q ́1u. Its gain is the product of the gains of its constituent basic directed paths, i.e., Γ pπq "

q ź j"1 Γ pπ i j q.
It should be clear that every path in an ordinary TEG has gain 1. Definition 60. A WTEG is called -strongly connected, if @t j , t l P T there exists a directed path from t j to t l .

-consistent if there exists a T-semiflow.

In this thesis, only consistent WTEGs are considered since a non-consistent WTEG is either not live or not bounded [START_REF] Teruel | On weighted t-systems[END_REF].

Proposition 91. Let pN , M 0 , ϕq be a consistent WTEG with T-semiflow ξ. Then the diverted directed path π " π i 1 ̈̈̈π iq has gain Γ pπq " pξq iq pξq i 1 .

Proof. According to the definition of T-semiflows, ξ is a positive integer vector such that Wξ " 0, (

where W P Z n̂m is the incidence matrix of the WTEG. Lines i j , j P t1, ̈̈̈, qu of (6.1) read as follows: wpt i j , p i j qpξq i j ́wpp i j , t i j̀1 qpξq i j̀1 " 0, for j " 1, ̈̈̈, q ́1 wpt i j , p i j qpξq i j ́wpp i j , t i j qpξq i j " 0, for j " q.

Equivalently, pξq i j̀1 pξq i j " wpt i j , p i j q wpp i j , t i j̀1 q " Γ pπ i j q, for j " 1, ̈̈̈, q ́1 pξq iq pξq i q " wpt i q , p iq q wpp iq , t iq q " Γ pπ iq q.

Therefore:

Γ pπq " q ź j"1 Γ pπ i j q " pξq iq pξq i 1 .
A WBTEG, introduced in [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF], is defined as follows.

Definition 61 (Weight-Balanced Timed Event Graph). Two paths π " π i 1 ̈̈̈π iq and π 1 " π i 1 1 ̈̈̈π i 1 q are called parallel if they start and end in the same transition, i.e., if i 1 " i 1 1 and i q " i 1 q . A WTEG is called Weight-Balanced Timed Event Graph (WBTEG), if all parallel paths have identical gain.

Proposition 92. A consistent WTEG is a WBTEG.

Proof. Let π " π i 1 ̈̈̈π iq and π 1 " π i 1 1 ̈̈̈π i 1 q be parallel paths. Then according to Prop. 91 and Definition 61,

Γ pπq " pξq iq pξq i 1 " pξq i 1 q pξq i 1 1 " Γ pπ 1 q.
Remark 32. Note that in general, the opposite is not true, i.e. consistent WTEGs are a strict subclass of WBTEGs.

Example 46. Figure 6.2a shows a consistent WBTEG, where Figure 6.2b depicts a non-consistent one. Note that the only difference is the weight of arc pt 1 , p 2 q. In Figure 6.2a, wpt 1 , p 2 q " 4, while in Figure 6.2b wpt 1 , p 2 q " 1. In case (a) this leads to an incidence matrix

W " » - - - - - - - - - - 3 0 0 0 ́1 4 0 0 ́1 0 0 ́2 3 0 0 0 0 4 ́1 0 0 0 0 0 0 0 ́2 0 0 1 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl
W has rank 4 and the vector ξ " r2 3 2 8 6s T satisfies Wξ " 0 and is therefore a T-semiflow. It can be easily checked that the firing of ξ " r2 3 2 8 6s T results again in the initial marking M 0 " r0 0 0 0 3 1s T . In case (b) the incidence matrix is

W " » - - - - - - - - - - 3 0 0 0 ́1 1 0 0 ́1 0 0 ́2 3 0 0 0 0 4 ́1 0 0 0 0 0 0 0 ́2 0 0 1 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl
and rank W " 5. Therefore no T-semiflow exists and the WBTEG is not consistent. The operation of this WBTEG leads to an irreversible accumulation of tokens in the system, i.e. after the firing of any transition, the initial marking M 0 cannot be reached anymore.

t 1 t 5 t 3 t 2 t 4 p 5 p 3 p 2 p 6 p 1 p 4 3 2 1 1 4 3 3 2 2 4
(a) A consistent WBTEG.

Transformation of consistent WTEGs to TEGs

Particular properties of consistent WTEGs can be analyzed on the basis of standard TEGs. For this a transformation from a consistent WTEG into a TEG was introduced in [START_REF] Munier | Régime asymptotique optimal d'un graphe d'évènement temporisé généralisé: application à un problème d'assemblage[END_REF][START_REF] Nakamura | Cycle time computation in deterministically timed weighted marked graphs[END_REF]. Then specific performance indicators of the WTEG can be computed by using this transformation. For instance, in [START_REF] Nakamura | Cycle time computation in deterministically timed weighted marked graphs[END_REF] it was shown that the throughput, i.e., the maximal firing rate of transitions, of a consistent WTEG can be computed on the basis of the transformed TEG. The transformation is based on the T-semiflow of a consistent WTEG. Each transition in the WTEG is duplicated by its corresponding entry in the T-semiflow. A drawback of the transformation is that the number of transitions and places can significantly increase for the transformed TEG. More precisely, the number of transitions in the transformed TEG is |ξ| and the number of places is at most 2|ξ|, where |ξ| is the 1-norm of the T-semiflow of the original consistent WTEG. Moreover, note that the |ξ| can grow exponentially independent of the net size of the WTEG, for more details see [START_REF] Nakamura | Cycle time computation in deterministically timed weighted marked graphs[END_REF]. Also note that, as input and output transitions are duplicated, in general the corresponding TEG has a different input-output behavior as the original WTEG. For SDF graphs a similar transformation is given in [START_REF] Sriram | Embedded Multiprocessors: Scheduling and Synchronization[END_REF]. For an illustration of this transformation see the following example. The TEG was obtained based on the algorithm published in [START_REF] Sriram | Embedded Multiprocessors: Scheduling and Synchronization[END_REF].

Example 47. Consider the consistent WTEG shown in Figure 6.3. Its incidence matrix is

W " » - - - - - - - - - - 1 ́2 0 0 3 0 ́1 0 0 0 0 0 0 0 0 0 0 3 0 ́1 0 0 1 ́2fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl
.

The vector ξ " r2 1 6 3s T is a T-semiflow for the WTEG, since This WTEG can be transformed into the TEG shown in Figure 6.4. The transition t 1 in Fig-

Wξ " » - - - - - - - - - - 1 ́2 0 0 3 0 ́1 0 0 0 0 0 0 0 0 0 0 3 0 ́1 0 0 1 ́2fi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl » - - - - - 2
t 1 t 4 t 2 t 3 2 2 1 1 2 2 3 3 Figure 6.3.
-A simple consistent WTEG, example is taken from [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]. ure 6.3 is duplicated twice, since the first entry of ξ being 2. The transition t 1 corresponds to the transitions t 1 1 and t 1 2 in the corresponding TEG (Figure 6.4). Respectively, transition t 2 corresponds to transition t 2 1 , transition t 3 is duplicated 6 times and corresponds to the transitions t 3 1 , t 3 2 , t 3 3 , t 3 4 , t 3 5 , t 3 6 and transition t 4 is duplicated 3 times and corresponds to transitions t 4 1 , t 4 2 , t 4 3 . Clearly, this transformation significantly increases the number of transitions in the corresponding TEG. Observe that the WTEG is a SISO system, whereas the corresponding TEG is a MIMO system with 2 input and 3 output transitions. As a consequence, the input-output behavior of the WTEG is different from the input-output behavior of the corresponding TEG. However, the maximal throughput of the WTEG can be studied based on its corresponding TEG. For instance in this example, the maximal throughput of transition t 2 in the WTEG is equal to the maximal throughput of transition t 2 1 in the TEG. Similarly, the maximal throughput of transition t 3 in the WTEG is six times the maximal throughput of transition t 3 1 , respectively transitions t 3 2 , t 3 3 , t 3 4 , t 3 5 , t 3 6 , in the TEG.

Timed Event Graphs under Partial Synchronization

To express systems with time-variant behaviors, a new form of synchronization, called PS, has been introduced for TEGs [START_REF] David-Henriet | Modeling and control for max-plus systems with partial synchronization[END_REF][START_REF] David-Henriet | Modeling and control for (max, +)-linear systems with set-based constraints[END_REF][START_REF] David-Henriet | Model predictive control for discrete event systems with partial synchronization[END_REF]. Unlike exact synchronization, where two transitions t 1 , t 2 can only fire if both transitions are simultaneously enabled, PS of transition t 1 by transition t 2 means that t 1 can fire only when transition t 2 fires, but t 1 does not influence the firing of t 2 . TEGs under PS provide a suitable model for some time-variant discrete event systems. In the following, a brief introduction is given.

Considering the TEG in Figure 6.5a, assuming the earliest functioning rule, incoming tokens in place p 1 are immediately transferred to place p 2 by the firing of transition t 2 , as the holding time of place p 1 is zero. Note that zero holding times are, by convention, not indicated in visual illustrations of TEGs. In contrast, Figure 6.5b illustrates a TEG with PS of transition t 2 by transition t a . This means that t 2 can only fire if t a fires, but the firing of t a does not depend on t 2 . In this example, place p 3 (equipped with a holding time of ω) and transition t a , together with the corresponding arcs, constitute an autonomous TEG. Under the earliest functioning rule, the firing of transition t a generates a periodic signal S ω with a period ω P N. Therefore, the PS of t 2 by t a can also be described by a predefined signal S ω : Z Þ Ñ t0, 1u, enabling the firing of t 2 at times t where S ω ptq " 1. The signal S ω ptq " 1 if t P tjω, with j P Zu and 0 otherwise. Definition 62. A Timed Event Graph under Partial Synchronization is a TEG where some internal and output transitions are subject to partial synchronization.

Note that the assumption that only internal and output transitions are subject to PS is not restrictive since it is always possible to add new input transitions and extend the set of internal transitions by the former input transitions. In [START_REF] David-Henriet | Modeling and control for (max, +)-linear systems with set-based constraints[END_REF], ultimately periodic signals are considered for PS of transitions. It was shown that the behavior of a TEG under PS can be described by recursive equations in a state space form. This thesis focuses on (immediately) periodic signals for PS of transitions. In the following, only PS of transitions by periodic signals as given in Definition 63 are considered. Such a PS is called periodic PS. Considering only periodic PS allows us to model the earliest functioning of a Timed Event Graph under Partial Synchronization (TEGPS) in the dioid pT rrγss, ', bq, see Chapter 4. In particular, we can show that the transfer behavior of a TEG under periodic PS is described by a rational power series of an ultimately cyclic form in this dioid. Note that focusing on periodic signals for a PS of a transition is not overly restrictive as periodic schedules are common in many applications.

Example 49. Such periodic timing phenomena occur for instance in traffic networks. As an example, let us consider a crossroad which is controlled by a traffic light. A vehicle can only cross during the green phase. If it reaches the crossing during this phase, it can immediately proceed. But if it reaches the cross in the red phase, it has to wait for the next green phase. The vehicle is delayed by a time that depends on its time of arrival. Under the assumption that the behavior of the traffic light is periodic, the crossroad can be modeled as a TEGPS where the timing behavior of the traffic light is described by a periodic PS. For instance, the TEGPS given in Figure 6.6 with the signal, S 2 ptq " $ & % 1 if t P t0 ̀4j, 1 ̀4j |j P Zu, 0 otherwise, models such a time-variant behavior of a crossroad. According to the signal S 2 , at time instances t0, 1, 4, 5, ̈̈̈u the traffic light is green and the vehicle can proceed without being delayed. In contrast at time instances t2, 3, 6, 7, ̈̈̈u the traffic light is red and the vehicle is delayed by one or two time units.

Periodic Time-variant Event Graphs

An alternative way to model periodic time-variant behavior with TEGs is to consider timevariant holding times in places. Then holding times of places depend on the firing times of their upstream transitions. More precisely, the holding time Hptq is time-variant and immediately periodic, i.e. Hpt ̀ωq " Hptq. The current delay is then determined by the firing time t of the corresponding upstream transition. Such a time-variant holding time is described by a periodic function H : Z Ñ Z, called holding-time function, which is defined as follows.

Definition 64 (Holding-time function H). A holding-time function H : Z Ñ Z is an ωperiodic function, i.e. Dω P N, @t P Z : Hptq " Hpt ̀ωq.

4 ̀4j if t " 3 ̀4j.
In this example, place p 2 has a constant holding time, whereas the holding times of places p 1 and p 3 are changing periodically with period 4. R 1 , R 3 , respectively H 1 , H 3 , are illustrated in Figure 6.9a, respectively, Figure 6.9b. The place p 1 can be interpreted as the model of a traffic light which is green for time instants t0, 1, 4, 5, ̈̈̈u and red for time instants t2, 3, 6, 7, ̈̈̈u. Note that the transition subjected to PS by the signals S 0 , S 1 , S 2 , S 3 are placed in parallel paths, see Figure 6.13. Therefore, the path with the signal S 1 is redundant and can be removed. Then, the earliest functioning of the PTEG shown in Figure 6.14a is modeled by the TEG under periodic PS shown in Figure 6.14b. Indeed Remark 34 shows that the earliest functioning of a PTEG can be modeled by a TEG under periodic PS. However as indicated in Example 53, PTEGs can model certain timevariant behavior in a more compact form. Finally, let us note that PTEGs can be seen as the counterpart to Cyco-Weighted Timed Event Graphs (CWTEGs) [START_REF] Cottenceau | Weight-balanced timed event graphs to model periodic phenomena in manufacturing systems[END_REF]. CWTEGs are an extension of WTEGs, where weights on the arcs are changing periodically depending on firing sequences of transitions attached to these arcs [START_REF] Cottenceau | Weight-balanced timed event graphs to model periodic phenomena in manufacturing systems[END_REF]. A similar extension is known for additional firing. In contrast to dater function, counter functions are antitone rather than isotone. An impulse is represented as a specific counter function Iptq, defined as

Iptq " $ & % 0 (resp. e)
for t ď 0, ̀8 (resp. ε) for t ą 0.

(6.9)

Counter and Series in M ax in vγ, δw As for dater functions, counter functions can be represented as series in M ax in vγ, δw, see [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Cohen | Second order theory of minlinear systems and its application to discrete event systems[END_REF]. The counter functions c canonically associated with a series s c P M ax in vγ, δw is such that s c " ́à ttPZ|́8ăcptqằ8u γ cptq δ t ̄' ́à tkPZ|cptq"́8u pγ ́1q ̊δt ̄. (6.10)

Then the series in M ax in vγ, δw associated with the impulse Iptq, see (6.9), is again the unit element e " γ 0 δ 0 in the dioid pM ax in vγ, δw , ', bq.

Notation

Expressing counter and dater functions as series in M ax in vγ, δw is convenient for calculations with transfer function models of TEGs in M ax in vγ, δw. From now on a counter function is denoted by a small letter with a tilde and the associated series in M ax in vγ, δw by a small letter, e.g., x denotes the counter function canonically associated with the series x P M ax in vγ, δw. Respectively, a dater function is denoted by a small letter with a bar and the associated series in M ax in vγ, δw by a small letter, e.g., x denotes the dater function canonically associated with the series x P M ax in vγ, δw.

Dioid Model of ordinary Timed Event Graphs

In this section dioid models for TEGs are recalled. For a more detailed representation, see e.g., [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Heidergott | Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications[END_REF][START_REF] Hardouin | Control and state estimation for max-plus linear systems[END_REF]. For the purpose of modeling a TEG, a dater function x : Z Ñ Z max is associated with each transition. xpkq gives the time (or date) when the transition fires the pk ̀1q st time, recall that the first firing is numbered by 0, see Section 6.2.1.

Example 54. Consider the TEG of Figure 6.15. By assigning ū1 pkq (resp. ū2 pkq) to the input transition t 1 (resp. t 2 ), x1 pkq (resp. x2 pkq) to internal transition t 3 (resp. t 4 ) and ȳpkq to the output transition t 5 , the behavior of the TEG can be described by the following inequalities x1 pkq ě maxpx 2 pk ́2q, ū1 pkq ̀1, ū2 pk ́1q ̀3q, ȳpkq ě x2 pkq ě x1 pkq ̀2.

If the TEG operates under the earliest functioning rule, its behavior is described by equations Obviously, due to the max operation, these equations are nonlinear in conventional algebra. In the (max,+)-algebra, the system (6.11) is expressed as x1 pkq " x2 pk ́2q ' 1 ū1 pkq ' 3 ū2 pk ́1q, ȳpkq " x2 pkq " 2x 1 pkq. (6.12)

It is easy to see that the equations in (6.12) are linear. Therefore, the system in (6.12) is also called "max-plus linear system". With the event-shift operator γ and time shift operator δ, system (6.12) can be expressed by x1 " γ 2 x2 ' δ 1 ū1 ' γ 1 δ 3 ū2 , ȳ " x2 " δ 2 x1 . Or, equivalently, with

x " rx 1 x2 s T and ū " r ū1 ū2 s T , in matrix form x " Ax ' B ū; ȳ " Cx, where

A " « ε γ 2 δ 2 ε ff , B " « δ 1 γ 1 δ 3 ε ε ff , C " " ε e ı .
Due to Theorem 2.1, the least solution for the output ȳ is given by, ȳ " H ū, with transfer function matrix H " CA ̊B " " δ 3 pγ 2 δ 2 q ̊γ1 δ 5 pγ 2 δ 2 q ̊ı .

For some applications, it is more convenient to model the evolution of a TEG in the "timedomain" rather than in the "event-domain". Then a counter function x : Z Ñ Z min is associated with each transition of the TEG. Recall that the counter value xptq describes the accumulated number of firings strictly before time t. The earliest functioning of a TEG is then described by a linear model in the (min,+)-algebra instead of the (max,+)-algebra, see the following example. Again by considering the time-and event-shift operators, the system can be rephrased in the dioid pM ax in vγ, δw , ', bq. Let x " r x1 x2 s T and ũ " r ũ1 ũ2 s T , then the system (6.13) is represented in matrix form x " A x ' B ũ; ỹ " C x, where

A " « ε γ 2 δ 2 ε ff , B " « δ 1 γ 1 δ 3 ε ε ff , C " " ε e ı .
Note that the M ax in vγ, δw model of a TEG is the same in the counter and dater representation. Therefore, the transfer function matrix for the counter representation is again, H " CA ̊B " " δ 3 pγ 2 δ 2 q ̊γ1 δ 5 pγ 2 δ 2 q ̊ı .

Output Computation and Impulse Response of Timed Event Graphs

In the following, it is shown how to compute the output of a SISO TEG based on its transfer function h P M ax in vγ, δw. Note that the following results can be easily extended to MIMO TEGs. As a SISO TEG is a time-invariant and event-invariant system, its transfer function h satisfies γ 1 h " hγ 1 and δ 1 h " hδ 1 . Moreover, similarly to conventional systems theory, the system response hI to an impulse I describes the complete transfer behavior of the corresponding SISO TEG [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Cohen | Second order theory of minlinear systems and its application to discrete event systems[END_REF]. Therefore, the transfer function h P M ax in vγ, δw of the system is the series in M ax in vγ, δw corresponding to the impulse response hI, see Prop. 93. Then the output dater function ȳ induced by an input dater function ū is nothing but the (max,+)-convolution of the impulse response and the input, i.e., ȳpkq " à nPZ phIqpk ́nq ūpnq.

By expressing this input and output dater functions as series y, u P M ax in vγ, δw the output y induced by the input u is obtained by y " h b u, or equivalent, the output dater function ȳ is obtained by ȳpkq " ̀ph b uqI ̆pkq.

Dioid Model of Weighted Timed Event Graphs

In the last section, it was shown how the earliest functioning of an ordinary TEG can be modeled linearly in the (max,+)-algebra as well as in the (min,+)-algebra. Moreover, the transfer behavior of an ordinary TEG refers to an ultimately cyclic series in M ax in vγ, δw. Unfortunately, the weights on the arcs of WTEG lead to event-variant behavior. Therefore, the earliest functioning of a WTEG can in general not be modeled by linear equations in the (max,+)-algebra nor in the (min,+)-algebra. However, in [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF] it was shown that the inputoutput behavior of WTEGs is described by series in Errδss. See Chapter 3 for the definition of the dioid pErrδss, ', bq. In the following the modeling process of consistent WTEGs based on operators in Errδss is recalled. Moreover, let us recall the core decomposition of periodic elements in Errδss, Section 3.3. Based on this decomposition the dynamic behavior of a consistent WTEG can be decomposed into an event-variant and an event-invariant part. This event-invariant part is described by a matrix with entries in M ax in vγ, δw. As the event-variant part is invertible, see Prop. 28, the tools for performance evaluation introduced for ordinary TEGs in the dioid pM ax in vγ, δw , ', bq can be applied to the more general class of consistent WTEGs.

For the purpose of modeling a consistent WTEG in the dioid pErrδss, ', bq a counter function x : Z Ñ Z min is associated with each transition. Recall that an operator in Errδss is defined as a mapping from the set of counter functions into itself, Section 3.1. Then for a consistent WTEG operating under the earliest functioning rule, the firing relation between transitions can be described by operators in E m|b rrδss (the subset of periodic operators in Errδss). Consider a basic path π i : t i Ñ p i Ñ t i . The influence of transition t i on transition t i is described by the following operator, xi " β wpp i ,t i q δ pϕq i γ pM 0 q i µ wpt i ,p i q xi , (

where xi and xi refer to the counter functions of transition t i and t i , wpt i , p i q and wpp i , t i q are weights of the arcs pt i , p i q and pp i , t i q, pϕq i is the holding time of place p i and pM 0 q i is the initial marking of p i . As E-operators and the time-shift operator commute, β wpp i ,t i q δ pϕq i γ pM 0 q i µ wpt i ,p i q " β wpp i ,t i q γ pM 0 q i µ wpt i ,p i q δ pϕq i .

For instance, consider the following basic path, the firing relation between t 1 and t 2 corre-

p 1 3 t 1 t 2 2 1 Figure 6.16. -A basic path π 1 : t 1 Ñ p 1 Ñ t 2 .
sponds to an operator representation x2 " β 2 γ 1 µ 3 δ 1 x1 .

Remark 35. Observe that the gain of the path π 1 : t 1 Ñ p 1 Ñ t 2 coincides with the gain of the operator β 2 γ 1 µ 3 δ 1 , i.e., Γ pπ 1 q " Γ pβ 2 γ 1 µ 3 δ 1 q " 3{2. This holds for any path in a consistent WTEG [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF].

Based on the operator representation of a basic path (6.15), the firing relation between internal, input and output transitions in a consistent WTEG can be described by: [This example is taken from [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]] Consider the consistent WTEG in Figure 6.3. By assigning the counter function ũ1 to the input transition t 1 , the counter function x " r x1 x2 s T to the internal transition t 2 and t 3 and the counter function ỹ to the output transition t 4 , the firing relations are written down as,

x " A x ' B ũ, ỹ " C x, (6.16
x "

« γ 1 δ 2 ε ε γ 2 δ 1 ff x ' « β 2 δ 2 µ 3 ff ũ ỹ " " µ 3 β 2 γ 1 δ 1 ı x.
Solving the implicit equation leads to the following transfer function of the system.

h "µ 3 β 2 δ 2 ' pγ 2 µ 3 β 2 γ 1 ' γ 3 µ 3 β 2 qδ 3 ' γ 3 µ 3 β 2 δ 4 ' pγ 4 µ 3 β 2 γ 1 ' γ 6 µ 3 β 2 qδ 5

' pγ 5 µ 3 β 2 γ 1 ' γ 6 µ 3 β 2 qδ 6 ' pγ 1 δ 1 q ̊̀pγ 6 µ 3 β 2 γ 1 ' γ 8 µ 3 β 2 qδ 7 ̆(6.17)

This transfer function h describes the firing relation between input transition t 1 and output transition t 4 and has a graphical representation given in Figure 6.18a. For example, in the case where the consistent WTEG is describing a production line, this transfer function describes the relation between incoming raw materials and finished parts. The left asymptotic growth rate of this transfer series is pγ 1 δ 1 q ̊therefore the maximal throughput of the system is 1 piece per time unit. The gain of the transfer series is Γ phq " 3 2 and therefore in average 2 input pieces generate 3 output pieces.

Example 57. The core representation of the transfer function (6.17) obtained in Example 56 is given by h " m 3 Qb 2 , " m 3 » --γ 2 δ 7 pγ 1 δ 3 q ̊δ2 ' γ 1 δ 4 ' γ 2 δ 6 ' γ 3 δ 8 pγ 1 δ 3 q γ1 δ 5 pγ 1 δ 3 q ̊ϵ δ 3 γ 2 δ 7 pγ 1 δ 3 q ̊fi ffi fl b 2 .

This core representation is realized in the consistent WTEG shown in Figure 6.17. Note that the realization has two basic paths from the input transition t 1 to the first layer of internal transitions pt 2 , t 3 q. These two paths represent the b 2 -vector and both paths have gain 1{2. Furthermore, the realized WTEG has three basic paths between the last layer of internal transitions pt 8 , t 9 , t 10 q and the output transition t 11 . These three paths represent the m 3 -vector and all three paths have gain 3. Moreover, the core matrix Q is realized by the internal transitions and all paths between them. Clearly, the entries of the core matrix are elements in M ax in vγ, δw, therefore the internal transitions pt 2 , ̈̈̈, t 10 q together with the paths between them constitute an ordinary TEG. Moreover, observe that the event variant behavior of this WTEG is only modeled by the realization of the b 2 -and m 3 -vector and that holding times are only attached to places between internal transitions. Subsequently, the internal dynamics are modeled by an ordinary TEG.

Output Computation and Impulse Response of Consistent Weighted Timed Event Graphs

As shown in Section 6.2.2 the impulse response of an ordinary TEG describes its complete transfer behavior. However, this is not the case for a consistent WTEG, because they are event-variant systems. In general for a consistent WTEG with transfer function h P E m|b rrδss, γ 1 h ‰ hγ 1 . In [START_REF] Cottenceau | Modeling of time-varying (max,+) systems by means of weighted timed event graphs[END_REF] it is shown that the impulse response of a consistent WTEG with a transfer function h " À i w i δ t i P E m|b rrδss can be obtained by

à i w i δ t i Iptq " à i γ Fw i p0q δ t i Iptq " à i Ipt ́ti q b F w i p0q.
This impulse response is a sum of time-and event-shifted impulses and gives us partial information about the transfer behavior of the consistent WTEG. Indeed, it can be shown that the complete transfer behavior can be constructed from a finite set of event-shifted impulse responses, for a more exhaustive presentation see [START_REF] Cottenceau | Modeling of time-varying (max,+) systems by means of weighted timed event graphs[END_REF]. The following remark gives a link between the impulse response of a consistent SISO WTEG and the zero slice mapping, Ψ m|b : E m|b rrδss Ñ M ax in vγ, δw, introduced in Section 3.2. Remark 36. Given a transfer function h P E m|b rrδss, then Ψ m|b phq is the series in M ax in vγ, δw associated with the impulse response phIq, i.e., ̀hI ̆ptq " ̀Ψm|b phqI ̆ptq.

As consistent WTEGs are event-variant, the output ỹ induced by an arbitrary input ũ is not simply the (min,+)-convolution of the impulse response hI with the input ũ. To compute the response to an arbitrary input counter function ũ, this counter function is represented as a sum of time-and event-shifted impulses. The output of the system is then obtained by the sum of these time-and event-shifted impulses responses. Differently stated, let ũ be a counter function with a corresponding series u " À i γ ν i δ t Example 58. The series y I P M ax in vγ, δw corresponding to the impulse response of the consistent WTEG shown in Figure 6.2b with a transfer function (6.17) is given by y I " Ψ 3|2 ́µ3 β 2 δ 2 ' pγ 2 µ 3 β 2 γ 1 ' γ 3 µ 3 β 2 qδ 3 ' γ 3 µ 3 β 2 δ 4 ' pγ 4 µ 3 β 2 γ 1 ' γ 6 µ 3 β 2 qδ 5 ' pγ 5 µ 3 β 2 γ 1 ' γ 6 µ 3 β 2 qδ 6 ' pγ 1 δ 1 q ̊̀pγ 6 µ 3 β 2 γ 1 ' γ 8 µ 3 β 2 qδ 7 ̆"

δ 2 ' γ 2 δ 3 ' γ 3 δ 4 ' γ 4 δ 5 ' γ 5 δ 6 ' γ 6 δ 7 pγ 1 δ 1 q " δ 2 ' γ 2 δ 3 pγ 1 δ 1 q ̊.
This series y I corresponds to the slice at the (I-count) value 0 in the graphical representation of the transfer function h, see Figure 6.18b.

Example 59. Consider the input u " δ 1 ' γ 1 δ 4 pγ 2 δ 2 q ̊P M ax in vγ, δw for a consistent WTEG with a transfer series h " pµ 3 β 2 γ 1 ' γ 2 µ 3 β 2 qδ 1 pγ 1 δ 1 q ̊. For this input, the response y P M ax in vγ, δw of the WTEG is given by

y " Ψ 3|2 ph b Injpuqq " Ψ 3|2 ́pµ 3 β 2 γ 1 ' γ 2 µ 3 β 2 qδ 1 pγ 1 δ 1 q ̊b́δ 1 'γ 1 δ 4 pγ 2 δ 2 q ̊̄" Ψ 3|2 ̀̀pµ 3 β 2 γ 1 ' γ 2 µ 3 β 2 qδ 2 ' pγ 2 µ 3 β 2 γ 1 ' γ 3 µ 3 β 2 qδ 5 ̆pγ 1 δ 1 q ̊" pδ 2 ' γ 2 δ 5 qpγ 1 δ 1 q " δ 2 ' γ 1 δ 3 ' γ 2 δ 5 pγ 1 δ 1 q ̊.

Dioid Model of Timed Event Graphs under Partial Synchronization

Unlike ordinary TEGs, TEGs under PS are time-variant systems. Therefore, their earliest functioning cannot be modeled as a (max,+)-linear nor a (min,+)-linear system. However, the operators introduced in Chapter 4 are suitable to model the input-output behavior of TEGs under periodic PS. More precisely, the time-variant behavior caused by a periodic PS of a transition can be modeled in the dioid pT , ', bq, see Chapter 4. To show this, recall that a periodic signal S can be associated with a release-time function R S : Z max Ñ Z max , see (6.6). To prove that a periodic PS of a transition (i.e. the PS is specified by a periodic signal S) admits an operator representation in T , it has to be shown that an operator v P T exists, such that R v " R S .

Proposition 97. A periodic partial synchronization of a transition by signal S in Definition 63 has an operator representation in T , given by v "δ n 0 ∆ ω|ω δ ́nI ' δ n 1 ́ω∆ ω|ω δ ́n0 ' ̈̈̈' δ n I ́ω∆ ω|ω δ ́npÍ1q . (6.18)

Proof. Let us recall that a periodic signal S corresponds to a quasi-periodic function R S , see (6.6). Moreover, there is an isomorphism between the function R v and the T-operator v. It remains to show that R v " R S . The function R v is given by A TEG under periodic PS operating under the earliest functioning rule admits a representation in T per rrγss, given by,

R v ptq " max ̀n0 ̀Q t ́nI ω U ω, n 1 ́ω ̀Q t ́n0 ω U ω, ̈̈, n I ́ω ̀Q t ́npÍ1q ω U ω ̆. ( 6
x " Ax ' B ū, ȳ " Cx, (6.20)
where x (resp. ū, ȳ) refers to the vector of dater functions of internal (resp. input, output) transitions. The matrices A P T per rrγss n̂n , B P T per rrγss n̂g and C P T per rrγss p̂n describe the influence of transitions on each other, encoded by operators in T per rrγss. Let t i Ñ p i Ñ t i constitute a basic path. The influence of transition t i on transition t i is coded as an operator v t i δ pϕq i γ pM 0 q i where v t i is the operator representation of the signal S i corresponding to the PS of t i (see Example 60), pϕq i is the holding time of place p i and pM 0 q i is the initial marking of p i .

Example 61. Consider the TEGPS in Figure 6.20 with PS of transition t 2 by the signal

S 2 ptq " $ & % 1 if t P t1 ̀2j |j P Zu, 0 otherwise.
As ω " 2, I " 0, n 0 " 1 according to Prop. 98 v S 2 " v t 2 " δ 1 ∆ 2|2 δ ́1. For the path t 3 Ñ p 2 Ñ t 2 , the influence of t 3 on transition t 2 corresponds to an operator representation v t 2 δ 0 γ 2 " v t 2 γ 2 " δ 1 ∆ 2|2 δ ́1γ 2 . Moreover, by assigning a dater function ūpkq (resp. x1 pkq, x2 pkq, ȳpkq) to transition t 1 (resp. t 2 , t 3 , t 4 ), the earliest functioning of the TEGPS is described by x " Ax ' B ū; ȳ " Cx, where

A " « ε δ 1 ∆ 2|2 δ ́1γ 2 δ 1 ε ff , B " « δ 1 ∆ 2|2 δ ́1 ε ff , C " " ε δ 1 ı .

Dioid Model of Periodic Time-variant Event Graphs

As for TEGs under periodic PS the earliest functioning of PTEGs can be modeled in the dioid pT rrγss, ', bq.

Proposition 98. A release-time function Rptq, as given in (6.5), can be expressed by a Toperator v P T in the following form:

v "δ n 0 ∆ ω|ω δ 1́ω ' δ n 1 ́ω∆ ω|ω ' δ n 2 ́ω∆ ω|ω δ ́1' . . . ' δ n ώ1 ́ω∆ ω|ω δ 2́ω .

(6.21)

Proof. First recall that release-time functions are isotone, therefore in (6.5), n ώ1 ́ω ď n 0 ď n 1 ď ̈̈̈ď n ώ1 ď n 0 ̀ω. Moreover, recall that the release-time function R δ σ ∆ ω|ω δ σ 1 ptq of an operator δ σ ∆ ω|ω δ σ 1 is defined by

R δ σ ∆ ω|ω δ σ 1 ptq " σ ̀Q t ̀σ1 ω U ω,
where t " xpkq is a date. Thus, R v associated with (6.21) is

R v ptq " maxpn 0 ̀Q t ́pω ́1q ω U ω, n 1 ́ω ̀Q t ω U ω, ̈̈̈, n ω-1 ́ω ̀Q t ́pω ́2q ω U ωq. (6.22)
We can evaluate the expression (6.22) for all dates t. If we choose t " jω, @j P Z max , we can show that:

R v pjωq " maxpn 0 ̀Q jω ́pω ́1q ω U ω, n 1 ́ω ̀Q jω ω U ω,
̈̈̈, n ω-1 ́ω ̀Q jω ́pω ́2q ω U ωq " maxpn 0 ̀jω, n 1 ́ω ̀jω, ̈̈̈, n ω-1 ́ω ̀jωq " n 0 ̀jω.

Similarly, we can show, that @i " t1, ̈̈̈, pω ́1qu,

R v pi ̀jωq " max ́n0 ̀Q i ̀jω ́pω ́1q ω U ω, n 1 ́ω ̀Q i ̀jω ω U ω, ̈̈̈, n ω-1 ́ω ̀Q i ̀jω ́pω ́2q ω U ω " n i ̀jω.
associated with t i , t i . Thus, the relation between input, output and internal transitions of a general PTEG can be modeled by

x " Ax ' B ū, ȳ " Cx, where x (resp. ū, ȳ) refers to a vector of dater functions of the n internal (resp. m input, p output) transitions of the PTEG. The relations between internal transitions can be modeled by a system matrix A P T per rrγss n̂n , the relation between input and internal transitions by an input matrix B P T per rrγss n̂m , and the relation between internal and output transitions by an output matrix C P T per rrγss p̂n .

Example 63. Consider the PTEG in Figure 6.8 of Example 51. The firing relation between its transitions can be modeled by

x "

" p∆ 4|4 ' δ 1 ∆ 4|4 δ ́3qγ 2 ı x ' " δ ́3∆ 4|4 ' ∆ 4|4 δ ́1ı u, y " " δ 1 ı x,
where ∆ 4|4 ' δ

" CA ̊B " " ε δ 1 ı « ε δ 1 ∆ 2|2 δ ́1γ 2 δ 1 ε ff ̊«δ 1 ∆ 2|2 δ ́1 ε ff " δ 1 pA ̊q2,1 δ 1 ∆ 2|2 δ ́1,
where pA ̊q2,1 " pδ 2 ∆ 2|2 δ ́1γ 2 q ̊δ1 , see (2.11). To express h as an ultimately cyclic series we rewrite pA ̊q2,1 in the core-form and compute the Kleene star based on the core matrix Q P M ax in vγ, δw with the toolbox MinmaxGD [START_REF] Hardouin | Minmaxgd a Software Tools to Handle Series in (max,+) Algebra[END_REF]. Recall Prop. 33, therefore

pA ̊q2,1 " ̀d2 « γ 2 δ 1 ε ε ε ff p 2 ̆̊δ 1 " d 2 ̀N « γ 2 δ 1 ε ε ε ff N ̆̊p 2 δ 1 " d 2 « pγ 2 δ 1 q ̊γ2 pγ 2 δ 1 q γ2 δ 1 pγ 2 δ 1 q ̊e ' γ 4 δ 1 pγ 2 δ 1 q ̊ff p 2 δ 1 .
Then, after multiplication, h " δ 3 pγ 2 δ 2 q ̊∆2|2 δ ́1.

Example 65. Consider the PTEG in Figure 6.8 of Example 51. We can describe the firing relation between input transition t 1 and output transition t 3 by a transfer function in T per rrγss, i.e. ȳ " h ū, where

h " δ 1 rpδ 1 ∆ 4|4 δ ́3 ' ∆ 4|4 qγ 2 s ̊pδ ́3∆ 4|4 ' ∆ 4|4 δ ́1q " pγ 4 δ 4 q ̊́pδ 1 ∆ 4|4 δ ́1 ' δ ́2∆ 4|4 q ' pδ 1 ∆ 4|4 ' δ 2 ∆ 4|4 δ ́1qγ 2 ̄.

Impulse Responses of TEGs under periodic PS and PTEGs

As shown in Section 6.2.2, the impulse response of a TEG system provides complete knowledge of the input-output behavior [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF]. In contrast, the impulse response of a PTEG (resp. TEGPS) is not sufficient to describe its complete behavior, because it is a time-variant system. The moment when the impulse is applied matters. One single impulse gives only partial information. In order to obtain the complete knowledge, we need the system responses of ω consecutive time-shifted impulses, i.e. δ τ I, τ P t0, ̈̈̈, ω ́1u. Each single response corresponds then to one slice in the 3D representation of the transfer function. The impulse response for a SISO PTEG (resp. TEGPS) with a transfer function h " À i v i γ n i P T per rrγss is obtained by

̀hI ̆pkq " ̀à i v i γ n i I ̆pkq " ̀à i δ Rv i p0q γ n i I ̆pkq " à i ̀Ipk ́ni q b R v i p0q ̆.
Note that the impulse response is a sum of time-and event-shifted impulses. Moreover, recall the zero slice mapping Ψ ω : T per rrγss Ñ M ax in vγ, δw, Section 4.4, therefore the series Ψ ω phq P M ax in vγ, δw corresponds to the impulse response ̀hI ̆pkq of the system.

Example 66 (Transfer function and impulse response). Consider the PTEG in Figure 6.8 of Example 51 with a transfer function obtained in Example 65.

h "pγ 4 δ 4 q ̊́δ 1 ∆ 4|4 δ ́1 ' δ ́2∆ 4|4 ' pδ 1 ∆ 4|4 ' δ 2 ∆ 4|4 δ ́1qγ 2 " ́δ1 ∆ 4|4 δ ́1 ' δ ́2∆ 4|4 ̄γ0 ' ́δ1 ∆ 4|4 ' δ 2 ∆ 4|4 δ ́1̄γ 2 ' ́δ5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 ̄γ4 ' ́δ5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1̄γ 6 ' ̈̈T
his transfer function has a graphical representation, see Figure 6.22a. The response of an im- pulse at time 1, i.e. hδ 1 I, is pδ 2 ' δ 5 γ 2 qpγ 4 δ 4 q ̊I . This response corresponds to the slice at input-time 1 (event-shift/output-time)-plane in Figure 6.22b. Furthermore, the system response to an impulse at time 5 is pδ 5 ' δ 6 γ 2 qpγ 4 δ 4 q ̊I . Therefore, the 3D representation of a transfer function in h P T per rrγss is interpreted as the juxtaposition of its time-shifted impulse responses.

Output computation

Again, as PTEGs (resp. TEGs under PS) are time-variant systems, the output to an arbitrary input dater function cannot simply be obtained by the (max,+)-convolution of the impulse response and the input. To compute the output of a PTEG (resp. TEG under periodic PS) caused by input dater function ū, this input dater function ū is expressed as a series u P from Z into Z min . This redefinition of the ∆ ω|ϖ operator allows to model the event-and time-variant behavior of consistent WTEGs under periodic PS in the dioid pET , ', bq.

Example 68. Consider the simple TEGPS, shown in Figure 6. [START_REF] De Groote | Multi-rate equivalents of cyclo-static synchronous dataflow graphs[END_REF] As ω " 3, I " 1, n 0 " 1 and n 1 " 2 according to Prop. 99,

v S 2 " v t 2 " δ 1 ∆ 3|3 δ ́2 ' δ ́1∆ 3|3 δ ́1. Therefore, x2 " v t 2 ∇ 1|2 δ 5 γ 3 ∇ 3|1 x1 " pδ 1 ∆ 3|3 δ ́2 ' δ ́1∆ 3|3 δ ́1q∇ 1|2 δ 5 γ 3 ∇ 3|1 x1 " pδ 4 ∆ 3|3 ' δ 5 ∆ 3|3 δ ́2q∇ 1|2 ∇ 3|1 γ 1 x1 since, γ 3 ∇ 3|1 " ∇ 3|1 γ 1 , δ 1 ∆ 3|3 δ 3 " δ 4 ∆ 3|3 and δ ́1∆ 3|3 δ 4 " δ 5 ∆ 3|3 δ ́2 " pδ 4 ∆ 3|3 ' δ 5 ∆ 3|3 δ ́2qpγ 3 ∇ 3|2 ' γ 1 ∇ 3|2 γ 1 q x1 since, ∇ 1|2 ∇ 3|1 " p∇ 3|6 γ 4 ' γ 1 ∇ 3|6 γ 2 ' γ 2 ∇ 3|6 qp∇ 6|2 γ 1 ' γ 3 ∇ 6|2 q " ∇ 3|2 γ 1 ' γ 1 ∇ 3|2 " pδ 4 γ 3 ∆ 3|3 ∇ 3|2 ' δ 5 γ 3 ∆ 3|3 ∇ 3|2 δ ́2 ' δ 4 γ 1 ∆ 3|3 ∇ 3|2 γ 1 ' δ 5 γ 1 ∆ 3|3 ∇ 3|2 δ ́2γ 1 q x1 .
Observe that δ 4 γ 3 ∆ 3|3 ∇ 3|2 ' δ 5 γ 3 ∆ 3|3 ∇ 3|2 δ ́2 ' δ 4 γ 1 ∆ 3|3 ∇ 3|2 γ 1 ' δ 5 γ 1 ∆ 3|3 ∇ 3|2 δ ́2γ 1 is the standard form, which was introduced in Prop. 78. Clearly based on this operator representation for a basic path, the earliest functioning of a consistent WTEG under periodic PS can be described by

x " A x ' B ũ, ỹ " C x,
where x (resp. ũ, ỹ) refers to the vector of counter functions of internal (resp. input, output) transitions and A, B and C are matrices with entries in ET per of appropriate size. The earliest functioning of the system is modeled by

x " Ax ' Bu; y " Cx, (

where,

A " « ε δ 1 ∆ 2|2 δ ́1∇ 1|2 γ 3 ∇ 2|1 δ 1 ε ff , B " « δ 1 ∆ 2|2 δ ́1 ε ff , C " " ε δ 1 ı .
Solving the implicit equation (6.28) leads to the transfer function of the system,

h " CA ̊B " " ε δ 1 ı « ε δ 1 ∆ 2|2 δ ́1∇ 1|2 γ 3 ∇ 2|1 δ 1 ε ff ̊«δ 1 ∆ 2|2 δ ́1 ε ff " δ 1 pA ̊q2,1 δ 1 ∆ 2|2 δ ́1.
Let us recall (2.11), hence pA ̊q2,1 " pγ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1q ̊∇2|1 δ 1 . Then

pγ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1q ̊" e ' γ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1 ' γ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1γ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1 ' ̈̈R ecall, Remark 30 hence, pγ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1q ̊" e ' γ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1 ' γ 2 δ 2 γ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1 ' ̈̈1
and thus pγ 2 δ 2 ∇ 2|2 ∆ 2|2 γ 1 δ ́1q ̊" e ' γ 2 δ 2 pγ 2 δ 2 q ̊∇2|2 ∆ 2|2 γ 1 δ ́1. Finally,

h " δ 1 ̀e ' γ 2 δ 2 pγ 2 δ 2 q ̊∇2|2 ∆ 2|2 γ 1 δ ́1̆∇ 2|1 δ 1 δ 1 ∆ 2|2 δ ́1 " δ 3 ∇ 2|1 ∆ 2|2 δ ́1 ' γ 2 δ 5 pγ 2 δ 2 q ̊∇2|1 ∆ 2|2 δ ́1.
7

Control

In this chapter, some control problems for Weighted Timed Event Graphs (WTEGs), Periodic Time-variant Event Graphs (PTEGs) and Timed Event Graphs (TEGs) under periodic partial synchronization (PS) are addressed. Over the last three decades, several control strategies have been established for TEGs, among them are optimal feedforward control [START_REF] Cohen | Algebraic tools for the performance evaluation of discrete event systems[END_REF][START_REF] Menguy | Just-in-time control of timed event graphs: update of reference input, presence of uncontrollable input[END_REF], state feedback, output feedback control [START_REF] Cottenceau | Model reference control for timed event graphs in dioids[END_REF][START_REF] Gaubert | Resource optimization and (min,+) spectral theory[END_REF][START_REF] Maia | Optimal closed-loop control of timed event graphs in dioids[END_REF][START_REF] Hardouin | Discrete-Event Systems in a Dioid Framework: Control Theory[END_REF][START_REF] Maia | On the model reference control for max-plus linear systems[END_REF], and observer-based control [START_REF] Hardouin | Observer design for (max,+) linear systems[END_REF][START_REF] Hardouin | Observer-based controllers for max-plus linear systems[END_REF][START_REF] Hardouin | Control and state estimation for max-plus linear systems[END_REF]. In [START_REF] Menguy | Just-in-time control of timed event graphs: update of reference input, presence of uncontrollable input[END_REF], an optimal control strategy for TEGs has been studied. For this control strategy, an output reference signal for a system is assumed to be a priori known, and the controller aims to schedule the input events of the system as late as possible, but under the restriction that output events do not occur later than specified by the reference signal. In the context of manufacturing systems, this strategy is called "just-in-time" production. In [START_REF] Gaubert | Resource optimization and (min,+) spectral theory[END_REF], an output feedback strategy for TEGs is introduced which leads to a strongly connected closed-loop system. The controller inserts additional places to the system with a sufficient amount of initial tokens such that a given throughput of the closed-loop system can be guaranteed. In [START_REF] Cottenceau | Model reference control for timed event graphs in dioids[END_REF][START_REF] Libeaut | Model matching for timed event graphs[END_REF], model reference control was introduced for TEGs. The purpose of the controller is to modify the system dynamics such that the system matches as close as possible the behavior of the reference model. The key difference to optimal control, where an optimal input is computed and then is chosen directly as the control action, is that the (potentially unknown) input is first filtered and then applied to the system. In the following, optimal control, as well as model reference control, are generalized to the case of consistent WTEGs, PTEGs, and TEGs under periodic PS. Subsequently, it is shown that these control problems can be reduced to the case of ordinary TEGs. Therefore, the existing tools for control synthesis for ordinary TEGs can be directly applied to consistent WTEGs, PTEGs, and TEGs under periodic PS. Some ideas, results, and figures presented in this chapter have appeared previously in [START_REF] Trunk | Output reference control for weightbalanced timed event graphs[END_REF][START_REF] Trunk | Model decomposition of weight-balanced timed event graphs in dioids: Application to control synthesis[END_REF][START_REF] Trunk | Modelling and control of periodic time-variant event graphs in dioids. Discrete Event Dynamic Systems[END_REF][START_REF] Trunk | Output reference control of timed event graphs under partial synchronization[END_REF].

Optimal Control Optimal Control for WTEGs

For a consistent WTEG with a transfer function h P E m|b rrδss, the optimal control problem can be stated by the inequality zptq ľ ̀h ũ̆p tq, (7.1)

Figure 7.1 illustrates the reference output z and the system output ỹ resulting from the optimal input ũ. Note that in (min,+) the order is reversed, one can see that, in Figure 7.1 it is indeed true that z ľ ỹ. For all t, the number of outputs ỹptq is greater than the wanted outputs zptq.

In other words, if we number the events, then the pk ̀1q st output ȳ occurs before or at the time instant of the pk ̀1q st wanted output z. -Comparison between the reference output z and the system response ỹ to the optimal input ũ. As required, the condition z ľ ỹ is satisfied.

Optimal Control for TEGs under periodic PS

Similarly to optimal control of consistent WTEGs, for a TEG under periodic PS (resp. a PTEG) with a transfer function h P T per rrγss the optimal control problem can be stated by the inequality zpkq ľ ̀h ū̆p kq,

where z is a dater function describing the desired output schedule (a priori known signal) and ū is the unknown input schedule, which is supposed to be optimized under the "just-intime" criterion. Let us recall the calculation of a system output in (6.23) where the input and -Comparison between the reference output z and the system response ȳ to the optimal input ū. As required, the condition z ľ ȳ is satisfied.

Model Reference Control

In many applications, it is desirable to control the system such that a given reference model is matched. The control problem is then to modify the system dynamics such that for any input the output of the system matches as close as possible the output of the reference. In the following, a feedforward and an output feedback approach are presented to solve the problem of model reference control for consistent WTEGs (resp. PTEG, TEGsPS). For this, it is considered that the input/output behavior of the consistent WTEG (resp. PTEG, TEGPS) is described by a transfer function matrix H with entries in E m|b rrδss (resp. with entries in T per rrγss).

Feedforward

In Figure 7.3 an open-loop control structure is given. In this structure, a prefilter, described by a matrix P P E m|b rrδss ĝg , is placed at the input of the system H P E m|b rrδss p̂g . The control input is chosen to ũ " P ṽ, where ṽ denotes the external inputs. The transfer matrix of the overall system is then H b P and the output ỹ is, therefore The control problem is then to find a prefilter P for a plant model H such that the overall system HP satisfies,

HP ĺ G. (7.5)
Moreover, we are looking for the greatest possible prefilter P in order to guaranty the optimal behavior under the "just-in-time" criterion. As H and G are matrices with entries in E m|b rrδss and E m|b rrδss is a strict subset of the complete dioid pErrδss, ', bq, residuation theory is suitable to obtain the greatest solution for P in (7.5). Therefore, the optimal prefilter is

P opt " H zG. (7.6)
To realize the prefilter by a consistent WTEG and to guarantee that the overall system is again consistent, P must be designed such that P and HP are consistent matrices with entries in E m|b rrδss. Hence, the matrices H and P must satisfy Prop. 44. This leads to the following restrictions on the reference model G. In other words, all columns of Γ pGq must be linearly dependent to all columns of Γ pHq (recall that Γ pHq and Γ pGq have rank 1).

Proof. This follows immediately from Prop. 48.

Moreover, note that P opt may not be causal, i.e. the matrix is not realizable by a consistent WTEG. Hence the optimal causal pm, bq-periodic prefilter P òpt is obtained by

P òpt " Pr m|b ̀H zG ̆,
where Pr m|b : E m|b rrδss Ñ E m|b rrδss is the causal projection, see Remark 15. Note that as shown in Example 26 the obtained causal prefilter is in general only the greatest pm, bqperiodic causal prefilter. In the particular case, where the optimal non-causal prefilter satisfies Remark 16, the greatest pm, bq-periodic causal prefilter is the greatest causal prefilter which satisfies Pr m|b pP opt q ĺ P opt .

Example 73. Let us consider the consistent WTEG of Figure 6.2a with a transfer matrix H given by

H " « pµ 3 β 2 γ 1 ' γ 2 µ 3 β 2 qδ 1 pγ 1 δ 1 q ̊µ3 β 2 δ 2 µ 4 β 1 µ 4 β 1 δ 3
ff , with a gain matrix

ΓpHq "

« 3 2 3 2 4 4
ff .

The reference model is specified by the following matrix G " « δ 2 pγ 3 δ 2 q ̊µ3 β 4 δ 2 pγ 3 δ 2 q ̊µ3 β 2 δ 2 pγ 2 δ 2 q ̊µ2 β 1 δ 4 pγ 4 δ 2 q ̊µ4 β 1 ff with a gain matrix Γ pGq "

« 3 4 3 2 2 4
ff .

Clearly, Γ pGq has rank 1 and all columns of Γ pGq and Γ pHq are linearly dependent, since

2 ̂« 3 4 2 ff " « 3 2 4
ff .

Thus, the specification G satisfies the structural property, given by (7.7), and therefore it is an admissible reference model for the plant H. The optimal prefilter P opt is given by pP opt q 1,1 "β 2 γ 1 ' pγ 1 µ 2 β 4 γ 1 ' γ 2 µ 2 β 4 qδ 1 ' pγ 1 δ 1 q ̊pγ 2 µ 2 β 4 δ 2 q, pP opt q 1,2 "e ' pγ 1 δ 1 q ̊pγ 1 µ 2 β 2 δ 1 q, pP opt q 2,1 "β 2 γ 1 δ ́1'γ 1 β 2 'γ 2 µ 2 β 4 δ 1 'pγ 2 µ 2 β 4 γ 1 'γ 3 µ 2 β 4 qδ 2 'pγ 2 δ 2 q ̊pγ 4 µ 2 β 4 δ 4 q, pP opt q 2,2 "e ' pγ 2 δ 2 q ̊pγ 2 µ 2 β 2 δ 2 q.

The optimal causal pm, bq-periodic prefilter P òpt is given by

P òpt " Pr m|b ̀H zG ̆, with pP òpt q 1,1 "β 2 γ 1 ' pγ 1 µ 2 β 4 γ 1 ' γ 2 µ 2 β 4 qδ 1 ' pγ 1 δ 1 q ̊pγ 2 µ 2 β 4 δ 2 q,
pP òpt q 1,2 "e ' pγ 1 δ 1 q ̊pγ 1 µ 2 β 2 δ 1 q, pP òpt q 2,1 "γ 1 β 2 ' γ 2 µ 2 β 4 δ 1 ' pγ 2 µ 2 β 4 γ 1 ' γ 3 µ 2 β 4 qδ 2 ' pγ 2 δ 2 q ̊pγ 4 µ 2 β 4 δ 4 q, pP òpt q 2,2 "e ' pγ 2 δ 2 q ̊pγ 2 µ 2 β 2 δ 2 q.

Control

Note that in this case the greatest causal p2, 4q-periodic (resp. p2, 2q-periodic) prefilter, it is the greatest causal prefilter, since all coefficients of the optimal non-causal prefilter P opt are smaller than or equal to µ 2 β 4 , (resp. µ 2 β 2 ). A graphical representation of the overall system is given in Figure 7.4.

Remark 38. (Optimal prefilter for PTEGs and TEGs under periodic PS) Clearly, the design process for an optimal prefilter for a PTEG (resp. TEG under periodic PS) with a transfer function matrix H P T per rrγss p̂g is analogous. For these systems, the reference model is specified by a matrix G P T per rrγss p̂g . Note that in the case of PTEGs the reference model can be freely chosen to any matrix G P T per rrγss p̂g . There is no additional condition as in the case of consistent WTEG. Therefore, the optimal causal prefilter is obtained by, P òpt " Pr ̀̀H zG ̆.

Feedback

Feedback control allows the system to react on unforeseen disturbances during runtime. One approach is output feedback, which leads to the control structure shown in Figure 7.5. The closed-loop transfer function matrix to this control structure is given by H " HpFHq ̊P. (7.8) As in the feedforward case, the reference model is as well specified by a consistent transfer function matrix G P E m|b rrδss p̂g . The control problem is then to find an output feedback F and a prefilter P for a plant model H P E m|b rrδss p̂g such that the closed-loop system H satisfies H ĺ G. According to the definition of the Kleene star, the closed-loop system can be written as H " HpI ' FH ' pFHq 2 ' ̈̈̈qP this implies that the prefilter P must satisfy the following inequality HP ' HFHP ' HpFHq ̊P ̈̈̈ĺ G.

(7.9)

Clearly, P must satisfy the first element of the sum, i.e., HP ĺ G.

The greatest solution of (7.10) is given by P opt " H zG, see (7.6), furthermore in [START_REF] Hardouin | Discrete-Event Systems in a Dioid Framework: Control Theory[END_REF] it is shown that this P opt is also the greatest solution for P in (7.9). Therefore, the optimal prefilter is equivalent to the optimal prefilter in the feedforward case. Again, in order to guaranty that the overall system is consistent, the reference model G and the transfer function matrix H of the plant must satisfy (7.7). It remains to find the greatest feedback F such that HpFHq ̊Popt ĺ G. (7.11) Proposition 103 ( [START_REF] Hardouin | Discrete-Event Systems in a Dioid Framework: Control Theory[END_REF]). The greatest solution of the inequality HpFHq ̊Popt ĺ G is given by F opt " pP opt {P opt q{H.

v 1 v 2 u 1 u 2 y 1 y 2 Prefilter 1 1 2 2 2
Proof. By left division by H and right division by P opt the inequality HpFHq ̊Popt ĺ G can be written as pFHq ̊ĺ pH zGq{P opt " P opt {P opt .

Since P opt {P opt " pP opt {P opt q ̊we obtain FH ĺ P opt {P opt .

Therefore, the greatest solution F opt for the feedback F in (7.11) (resp. (7.9)) is

F opt " pP opt {P opt q{H.
Finally, we check whether F opt and the closed-loop transfer matrix HpF opt Hq ̊Popt are consistent matrices with entries in E m|b rrδss.

Proposition 104. The optimal feedback F opt " pP opt {P opt q{H, with P opt " H zG and the closed-loop system transfer matrix HpF opt Hq ̊Popt are consistent matrices with entries in E m|b rrδss, if and only if the transfer function matrix H and the reference model G satisfy (7.7).

Proof. Recall that Γ pGq " g c g r and Γ pHq " h c h r with g c , h c P Q p̂1 and g r , h r P Q 1̂m . Then because of (3.60), Γ pP opt q " Γ pH zGq " hc pg c q 1 ph c q 1 g r , where hc " rpph r q 1 q ́1 pph r q 2 q ́1 ̈̈̈pph r q m q ́1s T . Then Γ pP opt q " p c p r , where p c " hc and p r " pg c q 1 {ph c q 1 g r . Because of (3.61), Γ pP opt {P opt q " p c pp r q 1 pp r q 1 pr , where pr " rppp c q 1 q ́1 ppp c q 2 q ́1 ̈̈̈ppp c q m q ́1s. Clearly pp r q 1 {pp r q 1 " 1 and therefore Γ pP opt {P opt q " p c pr and as pr "

" ppp c q 1 q ́1 ppp c q 2 q ́1 ̈̈̈ppp c q m q ́1ı " " ppph r q 1 q ́1q ́1 ppph r q 2 q ́1q ́1 ̈̈̈ppph r q m q ́1q ́1ı " " ph r q 1 ph r q 2 ̈̈̈ph r q m ı " h r Then Γ pP opt {P opt q " hc h r . Therefore, the matrices P opt {P opt and H satisfy Prop. 49 and F opt is a consistent matrix with entries in E m|b rrδss. Furthermore, Γ pF opt q " Γ ppP opt {P opt q{Hq " hc hr and thus Γ pF opt q i,j " pΓ pHq j,i q ́1. Then recall (5), hence Γ pHF opt q " h c ph r q 1 p hc q 1 hr , " h c h c ph r q 1 pph r q 1 q ́1 hr " h c hr .

Second,

Γ pHF opt Hq " h c p hr q 1 ph c q 1 h r " h c pph c q 1 q ́1ph c q 1 h r " h c h r " Γ pHq.

This implies that the sum H ' HF opt H ' ̈̈̈is again a consistent matrix and therefore the closed-loop transfer matrix HpF opt Hq ̊Popt is consistent as well.

Again in order to guaranty that F opt is realizable by a consistent WTEG only the causal part is considered:

F òpt " Pr m|b ̀Fopt ̆" Pr m|b ̀pP opt {P opt q{H ̆.
Then again as indicated in Example 26 the obtained causal feedback is in general only the greatest pm, bq-periodic causal feedback. However, if the entries of F opt satisfy the condition laid out in Remark 16, then the greatest pm, bq-periodic causal feedback Pr m|b pF opt q is the greatest causal feedback which satisfies Pr m|b pF opt q ĺ F opt .

Control

Remark 39. (Neutral Feedback) A particular case of model reference control is to consider the transfer function matrix H as the reference model, i.e., G " H. The optimal pm, bq-periodic feedback F òpt " Pr m|b pH zH{Hq is the one which delays all firings of input transitions as much as possible while preserving the transfer behavior of the system. It is said neutral for this reason. This feedback minimizes internal stock without slowing down the system.

Example 74. Recall Example 73 with the reference model G " « δ 2 pγ 3 δ 2 q ̊µ3 β 4 δ 2 pγ 3 δ 2 q ̊µ3 β 2 δ 2 pγ 2 δ 2 q ̊µ2 β 1 δ 4 pγ 4 δ 2 q ̊µ4 β 1 ff , the transfer function matrix,

H " « pµ 3 β 2 γ 1 ' γ 2 µ 3 β 2 qδ 1 pγ 1 δ 1 q ̊µ3 β 2 δ 2 µ 4 β 1 µ 4 β 1 δ 3 ff .
and the optimal prefilter P opt with, pP opt q 1,1 "β 2 γ 1 ' pγ 1 µ 2 β 4 γ 1 ' γ 2 µ 2 β 4 qδ 1 ' pγ 1 δ 1 q ̊pγ 2 µ 2 β 4 δ 2 q, pP opt q 1,2 "e ' pγ 1 δ 1 q ̊pγ 1 µ 2 β 2 δ 1 q, pP opt q 2,1 "β 2 γ 1 δ ́1'γ 1 β 2 'γ 2 µ 2 β 4 δ 1 'pγ 2 µ 2 β 4 γ 1 'γ 3 µ 2 β 4 qδ 2 'pγ 2 δ 2 q ̊pγ 4 µ 2 β 4 δ 4 q, pP opt q 2,2 "e ' pγ 2 δ 2 q ̊pγ 2 µ 2 β 2 δ 2 q.

The optimal feedback F opt of the closed-loop system is computed by

F opt " pP opt {P opt q{H,
which results in pF opt q 1,1 " pγ 1 δ 1 q ̊pγ 1 µ 2 β 3 δ ́1q, pF opt q 1,2 " β 4 δ ́3 ' pγ 1 δ 1 q ̊pγ 1 µ 2 β 8 δ ́2q, pF opt q 2,1 " γ 1 µ 2 β 3 δ ́3 ' pγ 1 µ 2 β 3 γ 1 ' γ 2 µ 2 β 3 qδ ́2 ' pγ 2 δ 2 q ̊pγ 2 µ 2 β 3 q, pF opt q 2,2 " β 4 δ ́3 ' pγ 2 δ 2 q ̊pγ 2 µ 2 β 8 δ ́1q.

Then optimal causal feedback F òpt of the closed-loop system is

F òpt " Pr m|b ̀pP opt {P opt q{H " « γ 2 pγ 1 δ 1 q ̊µ2 β 3 γ 3 pγ 1 δ 1 q ̊µ2 β 8 γ 2 pγ 2 δ 2 q ̊µ2 β 3 γ 4 δ 1 pγ 2 δ 2 q ̊µ2 β 8 ff .
Again, note that for this example the greatest p2, 3q-periodic (resp. p2, 8q-periodic) causal feedback F òpt is the greatest causal feedback. The closed-loop system with the prefilter and feedback is shown in Figure 7.6.

v 1 v 2 u 1 u 2 y 1 y 2
Prefilter Feedback Example 75. Consider the PTEG given in Figure 6.8 with transfer function,

h " δ 1 rpδ 1 ∆ 4|4 δ ́3 ' ∆ 4|4 qγ 2 s ̊pδ ́3∆ 4|4 ' ∆ 4|4 δ ́1q.
For this system, the neutral "just-in-time" feedback is:

f opt " h zh{h " pγ 4 δ 4 q ̊ppδ ́3∆ 4|4 δ ́1 ' ∆ 4|4 δ ́2q ' p∆ 4|4 δ ́1 ' δ 1 ∆ 4|4 δ ́2qγ 2 q
After the causal projection,

f òpt " Pr ̀pf opt q " pγ 4 δ 4 q ̊pp∆ 4|4 δ ́1 ' δ 1 ∆ 4|4 δ ́2qγ 2 ' pδ 1 ∆ 4|4 δ ́1 ' δ 4 ∆ 4|4 δ ́2qγ 4 q.
Recall the control law u " f òpt y ' v. To realize the feedback f òpt , f òpt y is written as ρ " f òpt y " pγ 4 δ 4 q ̊"p∆ 4|4 δ ́1 ' δ 1 ∆ 4|4 δ ́2qγ 2 ' pδ 1 ∆ 4|4 δ ́1 ' δ 4 ∆ 4|4 δ ́2qγ 4 ı y.

The former expression is the solution of the following implicit equation ρ "

" γ 4 δ 4 ı ρ ' " p∆ 4|4 δ ́1 ' δ 1 ∆ 4|4 δ ́2qγ 2 ' pδ 1 ∆ 4|4 δ ́1 ' δ 4 ∆ 4|4 δ ́2qγ 4 ı y.
From this expression the feedback f òpt can be implemented by a PTEG as follows: The feedback has one transition, denoted by t c , associated with the dater-function ρ. Because of operator γ 4 δ 4 transition t c is attached with a self-loop, constituted by place p c1 with 4 initial tokens and a constant holding time of 4 time units. The polynomial p∆ 4|4 δ ́1 'δ 1 ∆ 4|4 δ ́2qγ 2 'pδ 1 ∆ 4|4 δ ́1 ' δ 4 ∆ 4|4 δ ́2qγ 4 describes the influence of the plant output transition t 3 onto the transition t c of the feedback. Observe that we have two monomials, therefore we obtain two parallel paths between t 3 and t c , each with one place. First, p∆ 4|4 δ ́1 ' δ 1 ∆ 4|4 δ ́2qγ 2 is described by the place p c2 and the arcs pt 3 , p c2 q and pp c2 , t c q. Because of the exponent of γ 2 the place p c2 contains 2 initial tokens. The holding-time function of p c2 is determined by the T-operator ∆ 4|4 δ ́1 ' δ 1 ∆ 4|4 δ ́2 as follows:

H pc 2 ptq " max ́R∆ 4|4 δ ́1 ptq, R δ 1 ∆ 4|4 δ ́2 ptq ̄́t, " max ̂R t ́1 4 V 4, 1 ̀R t ́2 4 V 4 ̇́t, " x1 0 2 2y
Respectively, pδ 1 ∆ 4|4 δ ́1 ' δ 4 ∆ 4|4 δ ́2qγ 4 is described by the place p c3 and the arcs pt 3 , p c3 q and pp c3 , t c q. Because of the exponent of γ The controller is connected to the plant input transition t 1 via the arcs pt c , p c4 q and pp c4 , t 1 q. Finally, transition t v is associated with the new input v and is connected to the plant input transition t 1 via the arcs pt v , p v q and pp v , t 1 q. Figure 7.7 illustrates the closed-loop system. The feedback keeps the number of tokens in places p 1 , p 2 as small as possible, while the throughput of the system is preserved. Clearly, model reference control can be generalized to consistent WTEGs under periodic PS. In this case the reference model is specified in the dioid pET , ', bq and must satisfy a similar condition as given in Prop. 102. Remark 40. Finally, note that an alternative interpretation for causality of transfer functions in M ax in vγ, δw was introduced in [START_REF] Brunsch | Modeling and control of complex systems in a dioid framework[END_REF]. In short, this causal transfer functions h P M ax in vγ, δw may contain monomials γ n δ τ , for which the exponents of γ are in Z, see Remark 7. Then to realize such a transfer function by a TEG, negative tokens must be introduced. A similar alternative interpretation can be given for transfer functions h P E m|b rrδss, then h " À iPZ w i δ i , with w i ľ w ì1 is a causal transfer function, if for all i ă 0, w i " ε. Hence, h may contain monomials, for which the coefficient w i ľ µ m β b , e.g., γ ́1µ m β b γ ́2. Again to give a realization of such a transfer function negative tokens must be considered.

Conclusion

Timed Event Graphs (TEGs) are a subclass of Discrete Event Systems (DESs) whose behaviors are solely described by synchronization phenomena. An advantage of TEGs is that they have linear expressions in some tropical algebras called dioids [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Heidergott | Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications[END_REF]. Therefore, TEGs are considered popular tools for analyzing systems governed by synchronization, such as complex manufacturing processes, transport networks, and computer systems. Over the last decades, a comprehensive linear system theory for TEGs has been developed where basic concepts of traditional system theory such as state space representation, spectral analysis and transfer functions have been adapted to TEGs [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Heidergott | Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications[END_REF]. However, many applications have event-variant and time-variant behavior, which cannot be described by an ordinary TEG. Therefore, TEGs have been extended by introducing integer weights on the arcs. This leads to Weighted Timed Event Graphs (WTEGs) which are suitable to model event-variant phenomena in DESs. Similarly, to express some time-variant behavior, TEGs were expanded by a weaker form of synchronization called partial synchronization (PS). Clearly, WTEGs and TEGs under PS can express a wider class of systems compared to ordinary TEGs, but cannot be described as a linear system in dioids anymore. Nevertheless, transfer functions were introduced for WTEGs and TEGs under PS in specific dioids. These dioids are based on a specific set of operators. In this thesis, WTEGs and TEGs under PS are studied in a dioid framework, in particular, the control of these systems in dioids is addressed.

The first contribution relates to the modeling of WTEGs in dioids. Based on the dioid pErrδss, ', bq a decomposition model is introduced for consistent WTEGs, in which the event-variant part and the event-invariant part are separated. The event-invariant part is modeled by a matrix with entries in M ax in vγ, δw. Moreover, it is shown that the event-variant part is "invertible", hence the problem of model reference control for consistent WTEGs can be reduced to the case of ordinary TEGs. Furthermore, it is shown that all relevant operations p', b, z, {q on periodic elements in the dioid pErrδss, ', bq can be reduced to operations on matrices with entries in M ax in vγ, δw. In analogy to consistent WTEGs, consistent matrices are defined in the dioid pErrδss, ', bq. A matrix with entries in Errδss is called consistent if its entries are periodic and its gain matrix has rank 1. It is shown that a consistent WTEG admits a consistent transfer function matrix with periodic entries in Errδss. Moreover, the conditions under which product, sum, and quotient of consistent matrices are again consistent matrices are elaborated. This is needed for control synthesis; e.g., when we compute a controller in the dioid pErrδss, ', bq, the computed matrix must be consistent in order to obtain a controller realizable by a consistent WTEG.

The second main contribution of this work is to show that the input/output behavior of Periodic Time-variant Event Graphs (PTEGs) (resp. TEGs under periodic PS) can be described by ultimately cyclic series in a new dioid denoted pT per rrγss, ', bq. Just like WTEGs, a decomposition model is introduced for PTEGs, where the transfer function is decomposed into a time-variant part and a time-invariant part. The time-variant part is "invertible" and therefore many tools for performance analysis and controller synthesis, developed for ordinary TEGs, can be directly applied to PTEGs. Moreover, in this work, the impulse response of a PTEG (resp. TEGs under periodic PS) and the relation to its transfer function is discussed. It is shown that the transfer function of the system can be interpreted as the juxtaposition of its time-shifted impulse responses. In general, for computations in the dioid pT per rrγss, ', bq, it is shown that all relevant operations p', b, z, {q on elements in T per rrγss can be reduced to operations on matrices with entries in M ax in vγ, δw. The third main contribution is motivated by modeling a class of event-variant and timevariant DESs in the same dioid setting. The dioid pET , ', bq was introduced which can be seen as the combination of the dioids pErrδss, ', bq and pT rrγss, ', bq. It was shown that the transfer behavior of WTEGs under periodic PS can be described by ultimately cyclic series in ET . Moreover, the decomposition model can be applied to consistent WTEG under periodic PS as well. Thus, many tools developed for TEGs can be applied to analyze and to control consistent WTEGs under periodic PS.

Finally, it is shown how this transfer function representation of WTEGs, PTEGs, and TEGs under periodic PS can be used to solve some control problems for these systems. Optimal control was studied in which a reference output is defined for the system and an optimal input is computed, which schedules all input events as late as possible under the constraint that the output events occur not later than defined by the reference. The second control approach which was extended to WTEGs, PTEGs, and TEGs under periodic PS is model reference control. Here the reference model is specified by a transfer function matrix in the dioid pErrδss, ', bq, respectively for PTEGs and TEGs under periodic PS in the dioid pT per rrγss, ', bq. The controller, based on this reference, modifies the system dynamics such that the system matches the behavior of the reference model as close as possible. To achieve this, an output feedback and a prefilter are computed and realized. For consistent WTEGs, the specified reference model must satisfy some additional conditions regarding its gain. This is needed to obtain an admissible prefilter and feedback which are realizable by consistent WTEGs. Note that this is not the case for ordinary TEGs.

In the following, some suggestions for further work are given. Second order theory for TEGs is useful to obtain tight bounds for the number of tokens in places and the sojourn times of tokens in places when TEGs are operating under the earliest functioning rule [START_REF] Cohen | Second order theory of minlinear systems and its application to discrete event systems[END_REF]. For this method the TEG is modeled in the dioid pM ax in vγ, δw , ', bq, then residuation theory is applied to obtain these bounds [START_REF] Cohen | Second order theory of minlinear systems and its application to discrete event systems[END_REF]. It is of interest to study second-order theory for WTEGs (resp. PTEGs) based on their dioid model in pErrδss, ', bq (resp. pT per rrγss, ', bq).

For consistent WTEGs the transfer function can be interpreted as a juxtaposition of its event-shifted impulse responses. Similarly, for TEGs under periodic PS, the transfer function can be interpreted as a juxtaposition of its time-shifted impulse responses. The relation of the impulse responses and the transfer function of a WTEG under periodic PS can be addressed in further works. For TEGs many control approaches beyond optimal control and model reference control, studied in this thesis, were investigated. Among them are robust control [START_REF] Lhommeau | Interval analysis and dioid: application to robust controller design for timed event graphs[END_REF], control of TEG under additional time constraints [START_REF] Maia | On the control of max-plus linear system subject to state restriction[END_REF][START_REF] Brunsch | Duality and interval analysis over idempotent semirings[END_REF][START_REF] Brunsch | Modeling and control of complex systems in a dioid framework[END_REF], and observer-based control [START_REF] Hardouin | Observer design for (max,+) linear systems[END_REF][START_REF] Hardouin | Observer-based controllers for max-plus linear systems[END_REF]. Based on the decomposition model, these control strategies can be generalized to consistent WTEGs, PTEGs, and TEGs under periodic PS in further works.

B

Formula for Floor and Ceil Operations

The following list provides some basic relations of floor and ceil operations for proofs and a more detailed list see [START_REF] Graham | Concrete Mathematics: A Foundation for Computer Science[END_REF]. For x P R, X txu \ " txu, rrxss " rxs.

For x P R, m P Z and n P N,

Z x ̀m n ^" Z txu ̀m n ^, R x ̀m n V " R rxs ̀m n V .
For m P Z and n P N, Recall that F w is isotone, therefore it is sufficient to consider k " pn ̀1qb ́1, i.e., the greatest argument in the interval. Then, F u pnq ě F w ̀pn ̀1qb ́1̆.

Y m n ] " R m ́n ̀1 n V , Q m n U " Z m ̀n ́1 n ^.
(C.4)

The smallest function such that (C.4) holds is therefore Proof. This proof is taken from [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]. For the proof of (C.5) and (C.6), the operators γ i and δ τ are invertible, since δ τ δ ́τ " γ i γ ́i " e. Moreover, to prove (C. Because of Prop. 105, for each inequality i P t1, ̈̈̈, nu we obtain

F
x 1,i ĺ µ m zd i " β m γ ḿ1 d i ,

x 2,i ĺ γ 1 µ m zd i " β m γ ḿ1 γ ́1d i " β m γ ḿ2 d i , . . .

x m,i ĺ pγ ḿ1 µ m q zd i " β m d i .

Rewriting the inequalities into matrix form leads to We obtain for each i P t1, ̈̈̈, nu the following inequalities

x i,1 ĺ p i {pβ b γ b́1 q " p i µ b ,

x i,2 ĺ p i {pβ b γ b́2 q " p i γ 1 µ b , . . . The extended core is a matrix with entries in M ax in vγ, δw, since β n γ ν µ n " γ tν{nun . Furthermore, the extended core Q 1 is a greatest core. For this, one has to show that Q2 " E Q1 E " Q1 .

Q2

" since @j P t0, ̈̈̈, ω ́1u, rṕiω ́jq{ωs " ́i.

Eb
Due to the order relation in T see (4.12) we have ώ1 à j"0 δ ́iω ∆ nω|nω δ iὼj́nὼ1 " δ ́iω ∆ nω|nω δ iὼώ1́nὼ1 " δ ́iω ∆ nω|nω δ ́pń1́iqω , and thus

∆ ω|ω " ń1 à i"0 δ ́iω ∆ nω|nω δ ́pń1́iqω .
Lemma 8. Let v P T , then:

∆ ω|ϖ zv " ∆ ϖ|ω δ 1́ω v, v{∆ ω|ϖ " vδ 1́ϖ ∆ ϖ|ω .

(C.12)

Proof. To prove (C.12), recall that by definition of the residuated mapping, ∆ ω|ϖ zv is the greatest solution of the inequality v ľ ∆ ω|ϖ x. This greatest solution is given by ∆ ω|ϖ zv " à tu P T |∆ ω|ϖ u ĺ vu " à tu P T |R ∆ ω|ϖ u ptq ď R v ptq, @t P Z max u.

Therefore, @t P Z max R ∆ ω|ϖ zv ptq " maxtR u ptq| rR u ptq{ϖsω ď R v ptqu

Observe that,

Q R u ptq ϖ U ω ď R v ptq ô Q R u ptq ϖ U ď R v ptq ω ô R u ptq ϖ ď Y R v ptq ω ] " Q R v ptq ́ω ̀1 ω U ô R u ptq ď Q R v ptq ́ω ̀1 ω U ϖ
where the equality above chain of equivalence follows from the basic properties of the "floor" and "ceil" operations listed in Appendix B. Consequently R ∆ ω|ϖ zv ptq ď Q R v ptq ́ω ̀1 ω U ϖ, @t P Z max ô ω zv " ∆ ϖ|ω δ 1́ω v.

The proof for v{∆ ω|ϖ " vδ 1́ϖ ∆ ϖ|ω is analogous.

Proposition 106. Let s be a series in T rrγss, then γ η zs " γ ́ηs, s{γ η " sγ ́η, (C.13) δ τ zs " δ ́τs, s{δ τ " sδ ́τ, (C.14)

∆ ω|ϖ zs " ∆ ϖ|ω δ 1́ω s, s{∆ ω|ϖ " sδ 1́ϖ ∆ ϖ|ω .

(C.15)

Proof. For the proof of (C.13) and (C.14), the operators δ τ and γ η are invertible, since δ τ δ ́τ " γ η γ ́η " e. Moreover, for the proof of (C.15), recall Lemma 8 ∆ ω|ϖ zv " ∆ ϖ|ω δ 1́ω v with v P T and Prop. 6. Therefore, for a series s " À i v i γ n i P T rrγss one has ∆ ω|ϖ zs " ∆ ω|ϖ γ 0 źà i v i γ n i ̄" à i ́∆ω|ϖ zv i ̄γn i ́0 " à i ∆ ϖ|ω δ 1́ω v i γ n i , " ∆ ϖ|ω δ 1́ω s.

The proof for s{∆ ω|ϖ " sδ 1́ϖ ∆ ϖ|ω is analogous.

C.2.2. Proof of Prop. 61

Proof. Note that this proof is similar to the proof of Prop. 28. For the proof of (4.25), by definition of the residual mapping d ω zA is the greatest solution of the following inequality This matrix inequality can be transformed into a set of n inequalities, ∆ ω|1 x 1,1 ' δ ́1∆ ω|1 x 2,1 ' ̈̈̈' δ 1́ω ∆ ω|1 x m,1 ĺ a 1 , ∆ ω|1 x 1,2 ' δ ́1∆ ω|1 x 2,2 ' ̈̈̈' δ 1́ω ∆ ω|1 x m,2 ĺ a 2 , . . . ∆ ω|1 x 1,n ' δ ́1∆ ω|1 x 2,n ' ̈̈̈' δ 1́ω ∆ ω|1 x mn ĺ a n .

Because of Prop. 106, for each inequality i P t1, ̈̈̈, nu we obtain

x 1,i ĺ ∆ ω|1 za i " ∆ 1|ω δ 1́ω a i ,

x 2,i ĺ δ ́1∆ ω|1 za i " ∆ 1|ω δ 1́ω δ 1 a i " ∆ 1|ω δ 2́ω a i , . . .

x m,i ĺ pδ 1́ω ∆ ω|1 q za i " ∆ 1|ω a i .

Since, ∆ 1|nω δ 1́nω " ∆ 1|n ∆ 1|ω δ ́ωpń1q δ 1́ω " ∆ 1|n δ 1́n ∆ 1|ω δ 1́ω then Proof. Recall that a basic element in ET is expressed as γ n δ τ ∇ m|b ∆ ω|ϖ γ n 1 δ τ 1 . Moreover, the unit operator can be written as e " γ 0 " δ 0 " ∇ 1|1 " ∆ 1|1 " γ 0 δ 0 ∇ 1|1 ∆ 1|1 γ 0 δ 0 . Then the elementary operators can be rephrased as follows, ̀∇m|b pxq ̆ptq " ̀γ0 δ 0 ∇ m|b ∆ 1|1 γ 0 δ 0 pxq ̆ptq, ̀∆ω|ϖ pxq ̆ptq " ̀γ0 δ 0 ∇ 1|1 ∆ ω|ϖ γ 0 δ 0 pxq ̆ptq, ̀γν pxq ̆ptq " ̀γν δ 0 ∇ 1|1 ∆ 1|1 γ 0 δ 0 pxq ̆ptq, ̀δτ pxq ̆ptq " ̀γ0 δ τ ∇ 1|1 ∆ 1|1 γ 0 δ 0 pxq ̆ptq.

p nω " » - - - - - - - - - - - - - - - - - - - » - - -
Lemma 10. The product of two basic elements in ET is a finite sum of basic elements in ET .

Proof. Consider the following product of two basic elements in ET .

γ ν 1 δ τ 1 ∇ m 1 |b 1 ∆ ω 1 |ϖ 1 γ ν 1 1 δ τ 1 1 b γ ν 2 δ τ 2 ∇ m 2 |b 2 ∆ ω 2 |ϖ 2 γ ν 1 2 δ τ 1 2 (C.18)
We chose ω " lcmpϖ 1 , ω 2 q, c 3 " ω{ϖ 1 , c 4 " ω{ω 2 and m " lcmpb 1 , m 2 q, c 1 " m{b 1 and c 2 " m{m 2 then due to (5.14) and (5.15) this product can be written as

γ ν 1 δ τ 1 ́c1 ́1 à i"0 γ im 1 ∇ c 1 m 1 |m γ pc 1 ́1́iqb 1 ̄́c 3 ́1 à l"0 δ ́lω 1 ∆ c 3 ω 1 |ω δ ́pc 3 ́1́lqϖ 1 ̄γν 1 1 δ τ 1 1 b γ ν 2 δ τ 2 ́c2 ́1 à j"0 γ jm 2 ∇ m|c 2 b 2 γ pc 2 ́1́jqb 2 ̄́c 4 ́1 à g"0 δ ́gω 2 ∆ ω|c 4 ϖ 2 δ ́pc 4 ́1́gqϖ 2 ̄γν 1 2 δ τ 1
Due to distributivity holds for the following operators, γ∇ m|b γδ∆ ω|ϖ δ " δ∆ ω|ϖ δγ∇ m|b γ (Prop. 75), (C.18) is written as

γ ν 1 δ τ 1 ́c1 ́1 à i"0 γ im 1 ∇ c 1 m 1 |m γ pc 1 ́1́iqb 1 ̄γν 1 1 ̀ν2 ́c2 ́1 à j"0 γ jm 2 ∇ m|c 2 b 2 γ pc 2 ́1́jqb 2 ̄b ́c3 ́1 à l"0 δ ́lω 1 ∆ c 3 ω 1 |ω δ ́pc 3 ́1́lqϖ 1 ̄δτ 1 1 ̀τ2
́c4 ́1 à g"0 δ ́gω 2 ∆ ω|c 4 ϖ 2 δ ́pc 4 ́1́gqϖ 2 ̄γν 1 2 δ τ 1 2

Recall that ∇ m|b γ b " γ m ∇ m|b (resp. ∆ ω|ϖ δ ϖ " δ ω ∆ ω|ϖ ) (5.11) and for 0 ď n ă i, ∇ m|i γ n ∇ i|b " ∇ m|b (resp. ́i ă τ ď 0, ∆ ω|i δ τ ∆ i|ϖ " ∆ ω|ϖ ) (Remark 30). Therefore the expression above is rephrased as, Again, because distributivity holds for δ τ γ ν ∇ m|b " γ ν ∇ m|b δ τ and γ ν δ τ ∆ ω|ϖ " δ τ ∆ ω|ϖ γ ν (Prop. 75) the product (C.18) is written as which is in the required form.

C.3.1. Proof of Prop. 78

Proof. Because of Lemma 9 all elementary operators introduced in Prop. 73 can be represented by basic elements in ET . Moreover the product of two basic elements is a finite sum of basic elements, see Lemma 10. Therefore, any element s P ET can be written as a finite (resp. infinite) sum of basic elements, i.e., s " À i γ ν i δ τ i ∇ m i |b i ∆ ω i |ϖ i γ n 1 i δ τ 1 i . Recall (5.14) and (5.15), then by choosing m " lcmpm i q and ω " lcmpω i q, s can be rephrased as s " À j γ νj δ τj ∇ m| bj ∆ ω| πj γ In the following it is shown that Q1 if again a matrix in with entries in M ax in vγ, δw. Since, ∆ 1|nω δ 1́nω " ∆ 1|n ∆ 1|ω δ ́ωpń1q δ 1́ω " ∆ 1|n δ 1́n ∆ 1|ω δ 1́ω , then The extended core is a matrix with entries in M ax in vγ, δw, since ∆ 1|n δ τ ∆ n|1 " δ rτ{nsn Remark 30. The extended core Q1 is a greatest core. Consider Q2 " E Q1 E, then

[ 65 ]A

 65 Johannes Trunk, Bertrand Cottenceau, Laurent Hardouin, and Jörg Raisch. Model Decomposition of Weight-Balanced Timed Event Graphs in Dioids: Application to Control Synthesis. In 20th IFAC World Congress 2017, pages 13995-14002, Toulouse, 2017 [66] Johannes Trunk, Bertrand Cottenceau, Laurent Hardouin, and Jörg Raisch. Output reference control for weight-balanced timed event graphs. pages 4839-4846. IEEE, December 2017. ISBN 978-1-5090-2873-3. doi: 10.1109/CDC.2017.8264374 [67] Johannes Trunk, Bertrand Cottenceau, Laurent Hardouin, and Jörg Raisch. Model Decomposition of Timed Event Graphs under Partial Synchronization. In Preprints of the 14th Workshop on Discrete Event Systems pages 209-216, Sorrento Coast, Italy, May 2018. [68] Johannes Trunk, Bertrand Cottenceau, Laurent Hardouin, and Jörg Raisch. Modelling and control of periodic time-variant event graphs in dioids. Discrete Event Dynamic Systems, submitted. [69] Johannes Trunk, Bertrand Cottenceau, Laurent Hardouin, and Jörg Raisch. Output reference control of timed event graphs under partial synchronization. Discrete Event Dynamic Systems, submitted. xi Acronyms TEG Timed Event Graph WTEG Weighted Timed Event Graph WBTEG Weight-Balanced Timed Event Graph TEGPS Timed Event Graph under Partial Synchronization TEGsPS Timed Event Graphs under Partial Synchronization PS partial synchronization CWTEG Cyco-Weighted Timed Event Graph SISO single-input and single-output MIMO multiple-input and multiple-output DES Discrete Event System SDF Synchronous Data-Flow CSDF Cyclo-Static Synchronous Data-Flow PTEG Periodic Time-variant Event Graph FIFO first-in-first-out xvii matrices and vectors are bold pAq i,j pi, jq th entry of matrix A pAq i,: i th row of matrix A pAq :,j j th column of matrix A A T transposed of matrix A pD, ', bq dioid over the set D ' addition in a dioid b multiplication in a dioid ε zero element in a dioid e identity element in a dioid J top element in a complete dioid ^greatest lower bound in a complete dioid z (resp. ) left division in a complete dioid { (resp. ) right division in a complete dioid

x 2 Figure 1 . 1 .

 211 Figure 1.1. -Manipulation of the counter function x 1 by the δ 2 and γ 3 operators. The δ 2 and γ 3operators model the earliest behavior between input transition t 1 and output transition t 2 in the TEGs above. The holding time of two time units is modeled by the δ 2 operator and the three initial tokens by the γ 3 operator.

1 .

 1 Introduction namely µ m (event duplication) and β b (event division). These operators are given by, for m, b P N, ̀µm pxq ̆ptq " m ̂xptq and ̀βb pxq ̆ptq " Y xptq b

Figure 1 . 2 .

 12 Figure 1.2. -Manipulation of the counter function x 1 by the µ 2 and β 3 operators. The µ 2 and β 3 operators model the earliest behavior between input transition t 1 and output transition t 2 in the WTEGs above.

x 1 pkq x 2 Figure 1 . 3 .

 213 Figure 1.3. -Manipulation of the dater function x 1 by the ∆ 3|3 operator.

1 Figure 1

 11 Figure 1.4. -Simple TEG under PS.

Figure 2 . 1 .

 21 Figure 2.1. -Graphical illustration of s " γ 1 δ 1 ' γ 2 δ 3 ' γ 3 δ 4 P B vγ, δw.

Figure 2 . 2 .

 22 Figure 2.2. -Graphical illustration of the equivalence class represented by s 1 " γ 1 δ 1 ' γ 2 δ 3 P M ax in vγ, δw.The series s 2 " γ 1 δ 1 ' γ 2 δ 3 ' γ 3 δ 1 belongs to the same equivalence class, since both series s 1 , s 2 cover the same area in the Z 2 -plane.

Figure 2 . 3 .

 23 Figure 2.3.-Ultimately cyclic series s " pe ' γ 1 δ 1 ' γ 2 δ 3 q ' pγ 4 δ 4 ' γ 5 δ 6 qpγ 2 δ 3 q ̊in M ax

  s " γ ́3δ ́4 ' γ ́2δ 1 ' γ 3 δ 4 Pr ̀psq " γ 0 δ 1 ' γ 3 δ 4

Figure 2 . 4 .

 24 Figure 2.4. -Illustration of the causal projection Pr ̀pγ ́3δ ́4 ' γ ́2δ 1 ' γ 3 δ 4 q.

Figure 3 . 2 .

 32 Figure 3.2. -(2,3)-periodic (C/C)-function F γ 2 β 3 γ 1 µ 2 .

Figure 3 . 3 .

 33 Figure 3.3. -(2,3)-periodic (C/C)-functionsF γ 3 µ 2 β 3 and F γ 2 µ 2 β 3 γ 2 .One hasminpF γ 3 µ 2 β 3 , F γ 2 µ 2 β 3 γ 2 q " F γ 3 µ 2 β 3 'γ 2 µ 2 β 3 γ 2 .Or in other words, the intersection of the area beneathF γ 3 µ 2 β 3 and F γ 2 µ 2 β 3 γ 2 is equal to the area beneath F γ 2 β 3 γ 1 µ 2 " F γ 3 µ 2 β 3 'γ 2 µ 2 β 3 γ 2 .

F µ 3 β 6 γ 4 F γ 1 µ 3 β 6 γ 2 F γ 2 µ 3 β 6 IFigure 3 . 4

 42634 Figure 3.4. -F µ 1 β 2 is equal to minpF µ 3 β 6 γ 4 , F γ 1 µ 3 β 6 γ 2 , F γ 2 µ 3 β 6 q.

Figure 3 . 5 .

 35 Figure 3.5. -Quasi (2,3)-periodic (C/C)-functionsF γ 3 µ 2 β 3 γ 1 and quasi (2,2)-periodic (C/C)-function F γ 2 µ 2 β 3 γ 1 . For k ă 0: F γ 2 µ 2 β 2 γ 1 pkq ă F γ 3 µ 2 β 3 γ 1 pkq and for k ą 3: F γ 2 µ 2 β 2 γ 1 pkq ą F γ 3 µ 2 β 3 γ 1 pkq.

2 F µ 3 β 3 γ 2 t " 3 I

 223 (I/O-count)-plane for t ď (I/O-count)-plane for t " 3

Figure 3 . 7 . 2 ' γ 1 µ 3 β 3 γ 1 qδ 2 ' µ 3 β 3 γ 2 δ 3

 37223 Figure 3.7. -Slices of the coefficients in the (I/O-count)-plane of the polynomial pµ 3 β 3 γ 2 ' γ 1 µ 3 β 3 γ 1 qδ 2 ' µ 3 β 3 γ 2 δ 3

Figure 3 . 8 .

 38 Figure 3.8. -Graphical representation of series s " γ 1 µ 3 β 2 γ 1 δ 2 ' pγ 3 µ 3 β 2 γ 1 ' γ 5 µ 3 β 2 qδ 3 pγ 1 δ 1 q

  Graphical representation of s P M ax in vγ, δw.

  Graphical representation of Injpsq P Errδss.

Figure 3 . 9 .

 39 Figure 3.9. -Illustration of the canonical injection Inj : M ax in vγ, δw Ñ Errδss.

F

  w pkq ě F γ ν pkq ô F w pkq ě ν ̀k ô ν ď F w pkq ́k.(3.29) By considering that F w is a quasi pb, bq-periodic function the greatest ν such that (3.29) (resp. (3.28)) holds is ν "

  Graphical illustration of Inj 5 pγ 1

  (I/O-count)-plane for t ď 2

3 Figure 3 .

 33 Figure 3.13. -Graphical illustration of Inj 7 ppq " γ 1 δ 2 ' γ 2 δ 3 .

Figure 3 .

 3 Figure 3.14. -Illustration of the zero-slice mapping Ψ 3|2 psq.

Figure 3 .

 3 15. -For a comparison of the series s and Ψ 7 3|2 ̀Ψ3|2 psq ̆we examine the slices in the (O-count/t-shift)planes for all I-count values k P Z min of the graphical representation of s and Ψ 7 3|2 ̀Ψ3|2 psq ̆.

and b b is a column vector defined by b b :" " β b γ b́1 ̈̈̈β b γ 1

 1

Figure 3 . 5 3|2

 35 Figure 3.16. -For a comparison of the series s and Ψ 5 3|2 ̀Ψ3|2 psq ̆we examine the slices in the (O-count/t-shift)planes for all I-count values k P Z min of the graphical representation of s and Ψ 5 3|2 ̀Ψ3|2 psq ̆.

. 48 )

 48 Since the scalar product m i b i " e (3.43) and E " E ̊(3.48), under some conditions the left product and right product of matrices with entries in Errδss by m m and b b are invertible, see the following proposition. Proposition 28. For D P Errδss 1̂n and P P Errδss n̂1 , we have m m zD " b m b D, P{b b " P b m b . (3.49) For O P Errδss n̂m , N P Errδss b̂n , we have ̀OE ̆{m m " OE b b m , b b z̀E N ̆" m b b EN. (3.50) Proof. See Section C.1.1 in the appendix. Corollary 2. For D P Errδss m̂b , E zpEDq " ED and pDEq{E " DE.

Proposition 30 .

 30 A series s " m m Qb b P E m|b rrδss can be expressed with a multiple period pnm, nbq by extending the core matrix Q, i.e., s " m m Qb b " m nm Q1 b nb , where Q1 P M ax in vγ, δw nm̂nb and is given by

Corollary 4 .

 4 Let s " m m Qb b P E m|b rrδss and s 1 " m b Q1 b b 1 P E b|b 1 rrδss be two ultimately cyclic series, with Q, Q1 are greatest cores, the product s b s 1 " m m Q2 b b 1 P E m|b 1 rrδss is an ultimately cyclic series, where Q2 " Q Q1 is again a greatest core.

Proposition 33 .

 33 Let s " m b Qb b P E b|b rrδss. Then, s ̊" m b pQEq ̊bb P E b|b rrδss is an ultimately cyclic series. Proof. In this case, Γ psq " b{b " 1 means that Q is a square matrix. Moreover, recall that Eb b " b b (3.44) and therefore s " m b QEb b . s ̊" e ' m b QEb b ' m b QEb b m b QEb b ' ̈̈S ince, e " m b b b (3.43), E " b b m b (3.44) and E " E ̊" EE (3.48) , s ̊" m b b b ' m b QEb b ' m b QEEQEb b ' ̈̈" m b pI ' QE ' pQEq 2 ' ̈̈̈qb b " m b pQEq ̊bb .

Proposition 34 .

 34 Let s " m m Qb b , s 1 " m m Q1 b b P E m|b rrδss be two ultimately cyclic series, then s ^s1 " m m Q2 b b P E m|b rrδss is an ultimately cyclic series, where Q2 " p Q ^Q 1

1 zp

 1 6) and Corollary 2 . Second, ́mm Q1 b b 1 ̄z ̀mm Qb b ̆" ́Q 1 b b 1 ̄z ̀mm zpm m Qb b q ̆, because of (A.5), " ́Q 1 b b 1 ̄z ̀bm m m Qb b ̆, because of (3.49), " ́Q 1 b b 1 ̄z ̀Qb b ̆, as b m m m Q " Q Remark 11, " ́Q 1 b b 1 ̄z ̀Q{m b ̆, from (3.50) and Remark 11, " b b 1 z ́Q Q{m b q ̄, because of (A.5), " b b 1 z ́p Q1 z Qq{m b ̄, because of (A.6), " m b 1 p Q1 z Qqb b , because of (3.50) and (3.52).

Remark 15 .

 15 The causal projection Pr m|b : E m|b rrδss Ñ E m|b rrδss, is given by, for s " m m Qb b P E m|b rrδss Pr m|b psq " Pr m|b ̀mm Qb b ̆" m m Pr ̀̀Q ̆bb ,

m m 1 Q11 b b 1 ̈̈̈m m 1

 11 Q1g b bg . . . . . . m mp Qp1 b b 1 ̈̈̈m mp Qpg b bg

Proposition 41 .

 41 A consistent matrix A " M w QB w 1 P E m|b rrδss p̂g can be expressed with multiple periods by extending the core matrix Q, i.e.,

Figure 4 .

 4 1a illustrates the release-time function R δ 2 ∆ 4|4 δ -1 associated to the Toperator δ 2 ∆ 4|4 δ ́1 P T . The gray area shaped by R δ 2 ∆ 4|4 δ -1 corresponds to the domain of release-time functions (resp. T-operators) less than or equal to

Corollary 9 .

 9 The 1-periodic identity operator e " ∆ 1|1 can be represented in the specific form e " ώ1 à i"0 δ ́i∆ ω|ω δ 1̀íω .

Figure 4

 4 Figure 4.3. -R e ptq is equal to maxpR∆ 3|3 δ ́2 ptq, R δ ́1 ∆ 3|3 δ ́1 ptq, R δ ́2 ∆ 3|3 ptqq.

Figure 4 . 4 .

 44 The slices in the (I/O-time)-plane for the event-shift values k " 0, 1 are illustrated in Figure4.5a. These slices correspond to the release-time functionR δ 1 ∆ 4|4 δ ́1'δ ́2∆ 4|4 of the coefficient δ 1 ∆ 4|4 δ ́1 ' δ ́2∆ 4|4 for γ 0 (resp.γ 1 ) in p. The slices for k " 2, 3 and k ě 4 are shown in Figure 4.5b and Figure 4.5c. To improve readability, the graphical representation for elements s P T rrγss has been truncated to non-negative values in Figure 4.4 and Figure 4.5.

Figure 4 .

 4 Figure 4.4. -3D representation of polynomial p " pδ 1∆ 4|4 δ ́1 ' δ ́2∆ 4|4 qγ 0 ' pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 2 ' pδ 5 ∆ 4|4 ' δ 6 ∆ 4|4 δ ́1qγ 4 .

14 (c) k ě 4

 144

Figure 4 . 5 .

 45 Figure 4.5. -Slices of the coefficients of p in the (I/O-time)-plane.(a) R δ 1 ∆ 4|4 δ ́1 'δ ́2 ∆ 4|4 , (b) R δ 5 ∆ 4|4 δ ́1 'δ 2 ∆ 4|4 and (c) R δ 5 ∆ 4|4 'δ 6 ∆ 4|4 δ

Proposition 60 .Corollary 11 .Proposition 61 .

 601161 For the N matrix the following relations holdN b N " N,(4.21)N b p ω " p ω ,(4.22)d ω b N " d ω .(4.23)Proof.N b N " p ω b d ω b p ω b d ω " p ω b e b d ω " N, N b p ω " p ω b d ω b p ω " p ω b e " p ω , d ω b N " d ω b p ω b d ω " e b d ω " d ω . Observe that I ' N " N and N b N " N, hence N " I ' N ' NN ' ̈̈" N ̊. (4.24)Due to the scalar product d ω p ω " e (3.43) and N " N ̊(4.24), under some conditions the left and right product of elements in T rrγss by d ω and p ω are invertible, see the following proposition. For A P T rrγss 1̂ω and G P T rrγss ω̂1 , we haved ω zA " p ω b A, G{p ω " G b d ω . (4.25) For O P T rrγss ω̂ω we have pONq{d ω " pONq b p ω , p ω zpNOq " d ω b pNOq. (4.26) Proof. See Section C.2.2 in the appendix. Proposition 62. For D P T rrγss ω̂ω , N zpNDq " ND and pDNq{N " DN.

d ω n, 1

 1 Qn,1 p ω n,1 ̈̈̈d ωn,m Qn,m p ωn,m fi ffi ffi fl .

Figure 4 . 6 .

 46 Figure 4.6. -Subdioid structure of pT per rrγss, ', bq.

  Graphical representation of s P M ax in vγ, δw.

  n ts h i f t k i n p u t -t i m e t (b) Graphical representation of Injpsq P Tperrrγss.

Figure 4 . 7 .

 47 Figure 4.7.-Illustration of the canonical injection Inj : M ax in vγ, δw Ñ T per rrγss of the series s " γ 1 δ 2 ' ̀γ3 δ 3 ' γ 5 δ 4 ̆pγ 3 δ 2 q ̊P M ax in vγ, δw.

δ 6 R δ 5 ∆ 14 (c) k ě 4

 65144 4|4 'δ 6 ∆ 4|4 δ

Figure 4 . 9 .

 49 Figure 4.9. -Graphical illustration of Inj 7 ppq " γ 0 δ 1 ' γ 2 δ 5 ' γ 4 δ 6 .

Figure 4 .

 4 10a and Figure 4.10b.

  The (event-shift/output-time)-plane for the imput-time value 0

Figure 4 .

 4 Figure 4.10. -Illustration of the Projection Ψ 4 ppq.

  n ts h i f t k i n p u t -t i m e t (b) 3D representation of Ψ 74 pΨ 4 ppqq " pδ 1 γ 0 ' δ 5 γ 2 ' δ 6 γ 4 q∆ 4|4 .

Figure 4 .

 4 Figure 4.11. -Comparison of the polynomial p " pδ 1∆ 4|4 δ ́1 ' δ ́2∆ 4|4 qγ 0 ' pδ 5 ∆ 4|4 δ ́1 ' δ 2 ∆ 4|4 qγ 4 '

14 (c) k ě 4

 144

Figure 4 .

 4 Figure 4.12. -Graphical illustration of Ψ 74 pΨ 4 ppqq " pδ 1 γ 0 ' δ 5 γ 2 ' δ 6 γ 4 q∆ 4|4 .

  to (3.4), " à xPX ́∆ω|ϖ pxq ̄ptq due to (5.2).

. 24 )

 24 Hence, a series s is represented by s " m m,ω Qb b,ω . The entries of the matrix Q are ultimately cyclic series in the dioid pM ax in vγ, δw , ', bq.Properties of m m,ω and b b,ω .Recall the definition of b b,ω -and m m,ω -vector, b b,ω :"" ∆ 1|ω δ 1́ω b T b ̈̈̈∆ 1|ω b T b ı T , m m,ω :" " ∆ ω|1 m m ̈̈̈δ 1́ω ∆ ω|1 m m ı .Now let us consider a m i,ω -vector and a b i,ω -vector with equal indices, i.e., this implies that the m i,ω -vector and the b i,ω -vector have the same length. Then since, m m b m " e (3.43) and (5.14) the scalar product,

Proposition 80 .

 80 For D P ET 1̂mω and P P ET bω̂1 one has, m m,ω zD " b m,ω b D, P{b b,ω " P b m b,ω . (5.30) For O P ET n̂mω and G P ET bω̂n one has pOEq{m m,ω " OE b b m,ω b b,ω zpEGq " m b,ω b pEGq. (5.31) Proof. By definition, the residuated mapping m m,ω zD is the greatest solution of the inequality m m,ω b X ĺ D. (5.32) Clearly since m m,ω b m,ω " e, b m,ω D satisfies (5.32) with equality. It remains to be shown that b m,ω D is the greatest solution of (5.32). Next, assume that exists X 1 ľ b m,ω D solving (5.32), i.e., m m,ω b X 1 ĺ D. Multiplication is order preserving hence multiplication by b m,ω results in b

X 1 " 5 .

 15 P b m b,ω and P b m b,ω is indeed the greatest solution. To prove pOEq{m m,ω " OE b b m,ω , because of b m,ω m m,ω " E " EE and Pm m,ω " P{b m,ω (5.30) one has ̀OE ̆{m m,ω " pOEb m,ω m m,ω q{m m,ω " ppOEb m,ω q{b m,ω q{m m,ω . Since, px{aq{b " x{pbaq (A.1) and m m,ω b m,ω " e (see 5.38), ppOEb m,ω q{b m,ω q{m m,ω " pOEb m,ω q{pm m,ω b m,ω q " pOEb m,ω q{e " OEb m,ω . The proof of b b,ω zpEGq " m b,ω b pEGq is analogous. Proposition 81. Let s " m m,ω Qb b,ω P ET per , core equation s " m m,ω Xb b,ω has a unique greatest solution, denoted Q and given by Q " E m,ω QE b,ω . (5.33) 123 Dioid pET, ', bq Proof. Consider the inequality m m,ω Xb b,ω ĺ m m,ω Qb b,ω " s. Recall Prop. 80, therefore the greatest solution for X is X ĺ m m,ω zm m,ω Qb b,ω {b b,ω " b m,ω m m,ω Qb b,ω d b,ω " E m,ω QE b,ω " Q. Furthermore, Q solves (5.22) with equality, recall that, m m,ω " m m,ω E m,ω , b b,ω " E b,ω b b,ω (Prop. 79), therefore,

  product, p ω,i b d ω,i " This matrix is denoted by N. Then similar to Prop. 81 the greatest solution of (5.36) is NUN, which is denoted by Û. Proposition 82. For matrices D P ET 1̂mω , P P ET bω̂1 , O P ET n̂mω and G P ET bω̂n one obtains the following results for left and right division by the d ω,m -and p ω,b -vector. d ω,m zD " p ω,m b D, P{p ω,b " P b p ω,b pONq{d ω,m " ON b d ω,m , p ω,b zpNGq " p ω,b b pNGq.

Figure 6 . 1 .

 61 Figure 6.1. -Simple Petri net with a T-semiflow ξ " r1 2 1s T .

  The WBTEG which is not consistent.

Figure 6 . 2 .

 62 Figure 6.2. -Examples for consistent and non-consistent WBTEG.

Figure 6 . 4 .

 64 Figure 6.4. -Transformed TEG corresponding to the consistent WTEG shown in Figure 6.3.

  t

2 Figure 6 . 6 .

 266 Figure 6.6. -Traffic light model with a PS.

3 Figure 6 . 8 .Figure 6 . 9 .

 36869 Figure 6.8. -PTEG with holding-time functions of places p 1 , p 2 , p 3 expressed in the short form at each place.

  TEG under periodic PS by the signals S 0 , S 2 , S 3 .

Figure 6 .

 6 Figure 6.14. -In (a) PTEG and in (b) TEG under periodic PS, both models have the same input/output behavior when operating under the earliest functioning rule.

Example 55 .

 55 Consider the TEG of Figure6.15, by assigning the counter function ũ1 ptq (resp. ũ2 ptq) to the input transition t 1 (resp. t 2 ), x1 ptq (resp. x2 ptq) to internal transition t 3 (resp. t 4 ) and ỹptq to the output transition t 5 , the earliest functioning of the TEG can be described by x1 ptq " minp x2 ptq ̀2, ũ1 pt ́1q, ũ2 pt ́3q ̀1q, ỹptq " x2 ptq " x1 pt ́2q.(6.13)Then in the (min,+)-algebra, the system given in (6.13) is expressed as x1 ptq " 2 x2 ptq ' ũ1 pt ́1q ' 1 ũ2 pt ́3q, ỹptq " x2 ptq " x1 pt ́2q.(6.14)

Figure 6 .

 6 Figure 6.17. -Realization of the core-representation of the transfer function (6.17).

Figure 6 .

 6 Figure 6.18. -(a) 3D representation of the transfer function (6.17) of the consistent WTEG given in Figure 6.3. (b) the gray slice at (I-count) value 0 in the (O-count{t-shift)-plane corresponds to the impulse response phIqptq of the system.

Figure 6 .

 6 Figure 6.22. -(a) transfer function h of Example 65. (b) the gray slice at input time 1 (resp. time 5) (event-shift{ output-time)-plane correspond to the response to an impulse at time 1: δ 1 I (resp. time 5: δ 5 I) of the system.

Figure 6 .

 6 Figure 6.24. -Simple TEGPS with a periodic PS of t 2 .gives the response x2 induced by the counter function x1 under the assumption that the TEGPS is operating under the earliest functioning rule. Recall that the value xptq of a counter function

5 Figure 6 .

 56 Figure 6.25. -A simple WTEG with a periodic PS of transition t 2 .

Figure 7 . 1 .

 71 Figure 7.1. -Comparison between the reference output z and the system response ỹ to the optimal input ũ. As required, the condition z ľ ỹ is satisfied.

Figure 7 . 2 .

 72 Figure 7.2. -Comparison between the reference output z and the system response ȳ to the optimal input ū. As required, the condition z ľ ȳ is satisfied.

ỹFigure 7 . 3 .

 73 Figure 7.3. -Open-loop control structure with a prefilter P and plant model H.

Figure 7 . 5 .

 75 Figure 7.4. -Overall system with a prefilter.

Figure 7 . 6 .

 76 Figure 7.6. -Overall system with a prefilter and a feedback.

4 FeedbackFigure 7 . 7 .

 477 Figure 7.7. -Closed loop system.

  with a (C/C) function F w{β b . Clearly, if we consider the interval 0 ď k ă b we have F u ̀0̆ě F w ̀k̆t hus in general for n P Z min the (C/C)-function F u must satisfy, for nb ď k ă pn ̀1qb, F u pnq ě F w ̀k̆.

  w{β b pkq " F w ̀pk ̀1qb ́1̆" F w ̀kb ̀pb ́1q ̆" F w ̀Fµ b γ b́1 pkq ̆" F wµ b γ b́1 pkq. This corresponds to an operator representation w{β b " wµ b γ b́1 . Proposition 105 ([16]). Let s be a series in Errδss, then γ i zs " γ ́is, s{γ i " sγ ́i, (C.5) δ τ zs " δ ́τs, s{δ τ " sδ ́τ, (C.6) β b zs " µ b s, s{µ m " sβ m , (C.7) µ m zs " β m γ ḿ1 s, s{β b " sγ b́1 µ b . (C.8)

7 )

 7 the right product by µ m and the left product by β b are invertible, since β m µ m " e. For the proof of (C.8), recall Lemma 7 µ m zw " β m γ ḿ1 w with w P E and Prop. 6. Thus for a series s " À i w i δ τ i P Errδss one hasµ m zs " µ m δ 0 źà i w i δ τ i ̄" à i ́µm zw i ̄δτ i ́0 " à i β m γ ḿ1 w i δ τ i , " β m γ ḿ1 s.The proof for s{β b " sγ b́1 β m is analogous.C.1.1. Proof of Prop. 28 Proof. For the proof of the left product by m m (3.49), by definition of the residual mapping m m zD is the greatest solution of the following inequality m m b X ĺ D, can be transformed into a set of n inequalities, µ m x 1,1 ' γ 1 µ m x 2,1 ' ̈̈̈' γ ḿ1 µ m x m,1 ĺ d 1 , µ m x 1,2 ' γ 1 µ m x 2,2 ' ̈̈̈' γ ḿ1 µ m x m,2 ĺ d 2 , . . . µ m x 1,n ' γ 1 µ m x 2,n ' ̈̈̈' γ ḿ1 µ m x mn ĺ d n .

X ĺ m m zD "D

  " b m b D. Moreover, b m D satisfies (C.9) with equality, since m m b m " e. For the inequality X b b b ĺ P We have, Xb b ĺ P ô X ĺ P{b b ,

1 b.Finally

 1 x i,n ĺ p i {β b " p i γ b́1 µ b .207C. ProofsThis can be rewritten in matrix form X ĺ P{b b " P " µ b γ 1 µ b ̈̈̈γ b́1 µ b ı " P b m b . Again, Pb b satisfies (C.1.1) with equality, since m b b b " e. To prove (3.50), since b m m m " E " EE and due to (3.49) Pm m " P{b m we can write ̀OE ̆{m m " pOEb m m m q{m m " ppOEb m q{b m q{m m . Since px{aq{b " x{pbaq (A.1) and m m b m " e (see 3.43), ppOEb m q{b m q{m m " pOEb m q{pm m b m q " pOEb m q{e " OEb m . The proof of b b zpENq " m b b EN is analogous.C.1.2. Proof of Prop. 30Proof. We can extend a core matrix Q of a series, i.e.,s " m m Qb b " m nm b nm m m Qb b m nb looooooooomooooooooon Q nb .Since,β nm γ mń1 " β n β m γ mpń1q γ ḿ1 " β n γ ń1 β m γ ḿ1 then b nm " Respectively b b m nb isgiven by b b m nb " " Eµ n Eµ n γ 1 ̈̈̈Eγ ń1 µ n ı . ń1 Qµ n β n γ ń1 Qγ 1 µ n ̈̈̈β n γ ń1 Qγ ń1 µ n β n γ ń2 Qµ n β n γ ń2 Qγ 1 µ n ̈̈̈β n γ ń2 Qγ ń1 µ n . . . . . . . . . β n Qµ n β n Qγ 1 µ n ̈̈̈β n Qγ ń1 µ n

DC. 2 . 1 C. 2 . 1 .δ

 2121 nm m m Qb b m nb E, " b nm m nm b nm loooomoooon e m m Qb b m nb b nb looomooon e m nb , " b nm m m Qb b m nb " Q1 . C.1.3. Proof of Prop. 38Proof.(3.53) for the left product, by definition of the residual mapping M zD is the greatest solution of the following inequality:M w X ĺ D (C.11) » ---m m 1 x 11 ̈̈̈m m 1 x 1g . . . . . . m mp x p1 ̈̈̈m mp x pgFor every row i P t1, ̈̈̈, pu we obtain the following inequality" m m i x i1 ̈̈̈m m i x ipDue to (3.49) the greatest solution for this inequality is given by" x i1 ̈̈̈x ip ı ĺ " b m i d i1 ̈̈̈b m i d ip ıTherefore the greatest solution for the matrix inequality in (C.11) isX ĺ M w zD " " B w D.Note that B w D satisfies (C.11) with equality since M b B w " I. The proof of O{B w 1 " OM w 1 is analogous. To prove (3.54), since E w " B w M w " E w B w M w and due to (3.53) OM w " O{B w we can write pNE w q{M w " pNE w B w M w q{M w " ppNE w B w q{B w q{M w . Since px{aq{b " x{pbaq (A.1) and M w B w " I, ppNE w B w q{B w q{M w " pNE w B w q{pM w B w q " pNE w B w q{I " NE w B w . The proof of B w 1 zpE w 1 Gq " M w 1 b E w 1 G is analogous. Proofs of Section 4.Proof of Prop. 56 Proof. Let us recall that the release-time function of the ∆ ω|ω operator is given by R ∆ ω|ω ptq " rt{ωsω. Due to Remark 23 the pnω, nωq-periodic representation of this operator is ∆ ω|ω " rṕiώjq{ωsω ∆ nω|nω δ iὼj́nὼ1 , with t " iω ̀j, "

1 1

 11 ̀ν1 ̀tppc 1 ́1́iqb 1 ̀ν 1 1 ̀ν2 ̀jm 2 q{mum ∇ c 1 m 1 |c 2 b 2 γ pc 2 ́1́jqb 2 ̀ν 1 2 b ́lω 1 ̀rṕpc 3 ́ĺ1qϖ 1 ̀τ 1 1 ̀τ2 ́gω 2 q{ωsω ∆ c 3 ω 1 |c 4 ϖ 2 δ ́pc 4 ́1́gqϖ 2 ̀τ2

γ im 1 ̀ν1 ̀tppc 1 ́1́iqb 1 ̀ν 1 1 ̀ν2 ̀jm 2 q{mum b δ τ 1 ́lω 1 ̀rṕpc 3 ́ĺ1qϖ 1 ̀τ 1 1

 1 ̀τ2 ́gω 2 q{ωsω b ∇ c 1 m 1 |c 2 b 2 ∆ c 3 ω 1 |c 4 ϖ 2 γ pc 2 ́1́jqb 2 ̀ν 1 2 δ ́pc 4 ́1́gqϖ 2 ̀τ2

n 1 j δ τ 1 j 1 b

 11 , which is the required from.C.3.2. Proof of Prop. 83Proof. Since, m m,nω b b,nω " e an ultimately cyclic series s P ET per can be expressed as,s " m m,ω Qb b,ω " m m,nω b b,nω m m,ω Qb b,ω m b,nω loooooooooooooomoooooooooooooon Q b,nω .

  1́n ∆ 1|ω δ 1́ω b m . . . ∆ 1|n δ 1́n ∆ 1|ω b m 1́n b m,ω . . . ∆ 1|n b m,ω fi ffi ffi fl Hence, for b b,nω m m,ω we obtain, b m,nω m m,ω " Respectively b b,ω m b,nω is given by, b b,ω m b,nω " " E∆ n|1 ̈̈̈Eδ 1́n ∆ n|1 γ 1́nı . 1́n Q∆ n|1 ̈̈̈∆ 1|n γ 1́n Qδ 1́n ∆ n|1 . . . . . . ∆ 1|n Q∆ n|1 ̈̈̈∆ 1|n Qδ 1́n ∆ n|1 fi ffi ffi fl .

Q2" 1 p 1 bQ1 " b m,ω d ω,m loooomoooon T UQ 1 U p ω,b m b,ω loooomoooon T UQ 2 is∇∇δ

 1112 Eb m,nω m m,ω Qb b,ω m b,nω E, " b m,nω m m,nω b m,nω loooooomoooooon e m m,ω Qb b,ω m b,nω b b,nω looooomooooon e m b,nω , " b m,nω m m,ω Qb b,ω m b,nω " Q1 .Transformation between the core matrices Q and UClearly, an ultimately cyclic series s " m m,ω Qb b,ω " d ω,m Up ω,b P ET per can be expressed in the alternative core representation (resp. core representation) as follows,s " d ω,m p ω,m m m,ω Qb b,ω d ω,b loooooooooooomoooooooooooon Û ω,b , s " m m,ω b m,ω d ω,m Up ω,b m b,ω loooooooooooomoooooooooooon Q b,ω .Then the matrixÛ1 " p ω,m m m,ω looooomooooon T QU 1 Q b b,ω d ω,blooomooonT QU 2is the greatest solutions of the alternative core equations " d ω,m Xp ω,b(5.36). For this consider the solution Û2 " N Û1 N, then Û2 " Np ω,nm m m,ω Qb b,ω d ω,nb N, " p ω,nm d ω,nm p ω,nm loooooomoooooon e m m,ω Qb b,ω d ω,nb p ω,nb looooomooooon e d ω,nb , " p nmω d ω,m Qb b,ω d ω,nb " Û1 .Respectively, the greatest solutions of the core equation s " m m,ω Xb b,ω(5.22).The matrices T QU 1 , T QU 2 , T UQ 1 and T UQ 2 are matrices with entries in M ax in vγ, δw given by, 1|m γ ḿ1 p ω ∆ ω|1 m m ̈̈̈∇ 1|m γ ḿ1 p ω δ 1́ω ∆ ω|1 m m . . . . . .∇ 1|m p ω ∆ ω|1 m m ̈̈̈∇ 1|m p ω δ 1́ω ∆ ω|1 m m 1́ω b b µ b ∆ ω|1 ̈̈̈∆ 1|ω δ 1́ω b b γ b́1 µ b ∆ ω|1 . . . . . . ∆ 1|ω b b µ b ∆ ω|1 ̈̈̈∆ 1|ω b b γ b́1 µ b ∆ ω|1 1́ω b m ∇ m|1 ∆ ω|1 ̈̈̈∆ 1|ω δ 1́ω b m γ ḿ1 ∇ m|1 ∆ ω|1 . . . . . . ∆ 1|ω b m ∇ m|1 ∆ ω|1 ̈̈̈∆ 1|ω b m γ ḿ1 ∇ m|1 ∆ ω|1 1|b γ b́1 p ω ∆ ω|1 m b ̈̈̈∇ 1|b γ b́1 p ω δ 1́ω ∆ ω|1 m b . . . . . .∇ 1|b p ω ∆ ω|1 m b ̈̈̈∇ 1|b p ω δ 1́ω ∆ ω|1 m b fi ffi ffi fl and for 0 ď a ă ω and 0 ď c ă i ∇ 1|i γ c p ω δ ́a∆ ω|1 m i " ́1 ̈̈̈δ ́1 γδ ́1 ̈̈̈γδ ́1 ́ab i γ c ∇ i|1 d ω "

  which proves that β b is lower semi-continuous. For the proof of γ ν , since ν P Z is an integer then @t P Z, ̀γν pεq ̆ptq " ν ̀εptq " ν ̀8 " 8, thus ̀γν pεq ̆ptq " εptq. To prove lower semi-continuity of γ ν we have for all finite and infinite subsets X Ď Σ,

	́γν ̀à xPX	x ̆̄ptq " ν	̀́à xPX	x ̄ptq " ν ̀min xPX	̀xptq ̆" min xPX	̀ν ̀xptq	"
				́à xPX	γ ν pxq ̄ptq.
	Proposition 10 ([16]
			̀à xPX	x ̆̄ptq " m	̂̀à xPX	xPX x ̆ptq " m ̂min	̀xptq ̆,
						" min xPX	̀m ̂xptq ̆" min xPX	́̀µ m pxq ̆ptq ̄,
						"	́à xPX	µ m pxq ̄ptq,
	which shows that µ m is lower semi-continuous. For the mapping ̀βb pxq ̆ptq, again b P N
	is a finite positive integer, therefore @t P Z, ̀βb pεq ̆ptq " tεptq{bu " t8{bu " 8, thus
	̀βb pεq ̆ptq " εptq. Moreover, for all finite and infinite subsets X Ď Σ,
		́βb	̀à xPX	x ̆̄ptq "	Y	̀ÀxPX x ̆ptq b	]	"	] Y min xPX ̀xptq b ,
						" min xPX	́Y xptq b	]̄"	min xPX	́̀β b pxq ̆ptq ̄,
						"	́à xPX	β b pxq ̄ptq,

  .14) Note that operators in E only manipulate values of the mapping x P Σ, therefore an Eoperator can be equally described by a function F : Z min Ñ Z min . The value xptq is called counter-value. And the function associated with an operator w P E is called C/C (countervalue to counter-value) function, see the following definition.

  Due to the quotient structure(3.20) of the dioid pErrδss, ', bq the variable δ in Errδss matches with the operator δ P O defined in(3.19). Moreover, the zero and unit element in Errδss are given by the zero and unit element of O, i.e., @x P Σ, εpxq " ε and epxq " x, see(3.6).Monomial, Polynomial and ultimately cyclic Series in E m|b rrδssThe subset of Errδss obtained by restricting the coefficients spτq to E m|b , i.e. the set of pm, bq-periodic operators, is denoted by E m|b rrδss. For instance, µ 2 β 3 γ 1 δ 2 P E 2|3 rrδss, since the µ 2 β 3 γ 1 E-operator is p2, 3q-periodic. A monomial in E m|b rrδss is defined as wδ τ where

	A series s P Errδss is expressed as s " for the definition of addition and multiplication in dioids of formal power series. Therefore, À τPZ spτqδ τ , with spτq P E. Recall (2.13) and (2.14)
	given two series s 1 , s 2 P Errδss,
	s 1 ' s 2 "	à	̀s1 pτq ' s 2 pτq ̆δτ ,
		τPZ
	s 1 b s 2 "	à τ ́à t̀t 1 "τ	s 1 ptq b s 2 pt 1 q ̄δτ .
	pwxqpt ́τq, due to (3.19),
	" pwδ τ xqptq, again due to (3.19).
	Definition 34 ([16]). We denote by pErrδss, ', bq the quotient dioid in the set of formal power
	series in one variable δ with exponents in Z and coefficients in the non-commutative complete
	dioid pE, ', bq induced by the equivalence relation @s P Errδss,
	s " pδ ́1q ̊s " spδ ́1q ̊.	(3.20)
	̊" s 2 pδ ́1q ̊. It is helpful to think of spδ ́1q ̊as the representative of the equivalence class of s. Hence we identify two series s 1 , s 2 P Errδss with the same equivalence class if s 1 pδ ́1q

  Qγ 1 µ n ̈̈̈β n γ ń2 Qγ ń1 µ n

				fi
				ffi
	. . .	. . .	. . .	ffi fl ffi ffi	.
	β n Qµ n	β n Qγ 1 µ n	̈̈̈β n Qγ ń1 µ n
	Proof. See Section C.1.2 in the appendix.

  This product B w M w is denoted by E w . As in the scalar case, one has E w E w " E w ; M w E w " M w and E w 1 B w 1 " B w 1 .Proposition 38. For M w (resp. B w 1 ) we haveM w zD " B w D, O{B w 1 " OM w 1 ,(3.53)pNE w q{M w " pNE w qB w , B w 1 zpE w 1 Gq " M w 1 pE w 1 Gq,(3.54)where D, O, N and G are matrices of appropriate size and with entries in Errδss.

							fi		»		fi
			. . .	. . .	. . . . .	. . . ε	ffi ffi ffi ffi ffi fl	"	------	E m 1 ε ̈̈̈ε ε . . . . . . . . . . . . . . . ε . . .	ffi ffi ffi fl ffi ffi	.
			ε	̈̈̈ε b mp m mp				ε ̈̈̈ε E mp
		»
	A "	---	a 11 ̈̈̈a 1n . . . . . .
			a p1 ̈̈̈a pn

Proof. See Section C.1.3 in the Appendix. Proposition 39. Let A P E m|b rrδss p̂g be a consistent matrix, then A can be decomposed in the following form:

  Proposition 64. An ultimately cyclic series s " d ω Qp ω P T per rrγss can be expressed with a multiple period nω by extending the core matrix Q, i.e., s " d ω Qp ω " m nω Q1 b nω , where Q1 P M ax in vγ, δw nω̂nω and is given by Q∆ n|1 ∆ 1|n δ 1́n Qδ ́1∆ n|1 ̈̈̈∆ 1|n δ 1́n Qδ 1́n ∆ n|1 ∆ 1|n δ 2́n Q∆ n|1 ∆ 1|n δ 2́n Qδ ́1∆ n|1 ̈̈̈∆ 1|n δ 2́n Qδ 1́n ∆ n|1

	Q1 "	» -----	∆ 1|n δ 1́n . . .	. . .	. . .	fi ffi ffi ffi ffi fl
			∆ 1|n Q∆ n|1	∆ 1|n Qδ ́1∆ n|1	̈̈̈∆ 1|n Qδ 1́n ∆ n|1

.

Proof. See Section C.2.3. Proposition 65. Let s " d ω Qp ω , s 1 " d ω Q 1 p ω be two ultimately cyclic series in T per rrγss, the sum s ' s 1

  1 are ultimately cyclic series in M ax in vγ, δw. Because of Theorem 2.6, the sum and product of ultimately cyclic series in M ax in vγ, δw are again ultimately cyclic series in M ax in vγ, δw. Therefore, entries of Q 2 are ultimately cyclic series in M ax in vγ, δw and the product s b s 1 " d ω Q 2 p ω is an ultimately cyclic series in T per rrγss. Corollary 13. Let s " d ω Qp ω , s 1 " d ω Q1 p ω be two ultimately cyclic series, with Q, Q1 are greatest cores, the product s b s 1 " d ω Q2 p ω P T per rrγss is an ultimately cyclic series,

	where	Q2	" Q Q1 is again a greatest core.

Proof. Because of NN " N (Prop. 60),

Q Q1

" NQNNQ 1 N " Q2 . Proposition 67. Let s " d ω Qp ω P T per rrγss be an ultimately cyclic series in T per rrγss. Then, s ̊" d ω pQNq ̊pω , (4.28) is an ultimately cyclic series in T per rrγss. Proof. Clearly, QN is a core of s P T per rrγss, since e " d ω p ω and N " p ω d ω then d ω Qp ω e " d ω Qp ω d ω p ω " d ω QNp ω . The Kleene star of series s can be written as s ̊" e ' d ω QNp ω ' d ω QNp ω d ω QNp ω ' ̈̈R ecall that Q is a square matrix, e " d ω p ω (4.19), N " p ω d ω (4.20) and N " N ̊" NN (4.24), therefore

z

  Q̄N ,since: pabq zx " b z pa zxq (A.5) and Q " QN Due to Theorem 2.6, the quotient Q z Q1 is a matrix composed of ultimately cyclic series in M ax in vγ, δw and therefore the quotient s 1 zs " d ω p Q1 z Qqp ω is an ultimately cyclic series in T per rrγss. The proof of s{s 1 " d ω p Q{ Q1 qp ω is analogous.

		" ́́Q 1	z ̀Q{N ̆̄N ̄{N, since: Prop. 62 twice
		"	́́́Q	1	z	̄N̄{ N Q̄{ N, since: pa zxq{b " a zpx{bq (A.6)
		" ́Q	1	z	Q̄{ N, because ppx{aqaq{a " x{a (A.4)
		"	Q1	z ̀Q{N ̆" Q1
	Second,
	́dω	Q1 p ω ̄z ̀dω Qp ω	̆" ́Q

z Q, since: pa zxq{b " a zpx{bq (A.6) and Prop. 62 . 1 p ω ̄z ̀dω zpd ω Qp ω q ̆, because of (A.5),

" ́Q 1 p ω ̄z ̀pω d ω Qp ω ̆,

because of (4.25) " ́Q 1 p ω ̄z ̀Qp ω ̆, as p ω d ω Q " Q Remark 26, " ́Q 1 p ω ̄z ̀Q{d ω ̆, from (4.26) and Remark 26, " p ω z ́Q 1 zp Q{d ω q ̄, because of (A.5), " p ω z ́p Q1 z Qq{d ω ̄, because of (A.6), " d ω p Q1 z Qqp ω , because of (4.26) and (4.29).

  The size of Q is then ωn ̂ωn. Note that in contrast to the decomposition of matrices with entries s P E m|b rrδss, see Section 3.4.1, the decomposition of matrices with entries in T per rrγss is simpler. Unlike to(3.55) in Prop. 39 the matrices D w and P w 1 are block diagonal matrices with same entries d ω and p ω . Moreover, note that the core representation in (4.30) is clearly not the most efficient one in terms of expressing A with a core Q P M ax in vγ, δw of minimal dimensions.

				fi
	. . .	. . .	. . . . ε . . .	ffi ffi ffi fl ffi ffi	.	(4.30)
	ε ̈̈̈ε p ω
	loooooooooooomoooooooooooon
		P w 1

  Example 37. Figure 4.6 illustrates the subdioid structure of pT per rrγss, ', bq. It is shown that pM ax in vγ, δw , ', bq, pT 3 rrγss, ', bq and pT 4 rrγss, ', bq are subdioids of pT per rrγss, ', bq. Moreover, pM ax in vγ, δw , ', bq is a subdioid of pT 3 rrγss, ', bq and pT 4 rrγss, ', bq.

		T per rrγss
	T 3 rrγss	M ax in vγ, δw	T 4 rrγss

  Since vγ n ľ γ n δ τ ô v ľ δ τ , it remains to find the greatest τ such that (4.34) holds. By considering the isomorphism between T-operators and release-time functions, see (4.12), this is equivalent to R v ptq ě R δ τ ptq, @t P Z max . By using R δ τ ptq " τ ̀t, see (4.8), one obtains R v ptq ě τ ̀t ô τ ď R v ptq ́t, @t P Z max .

					(4.35)
	Since R v is a quasi ω-periodic function it is sufficient to evaluate the function for @t P t0, ̈̈̈, ω ́1u. Therefore the greatest τ such that (4.35) (resp. (4.34)) holds is
	τ "	ώ1 min t"0	̀Rv ptq	́t̆.
				see, (2.28).	(4.34)

  ∆ 1|ω δ ́ω " δ ́1∆ 1|ω and ∆ 1|ω δ j ∆ ω|1 " e for ́ω ă j ď 0 see Remark 30, hence

				fi
		1 ̈̈̈γ 1
	. . . . . .	. . .	. . . . . . γ 1 . . .	ffi ffi ffi fl ffi ffi	,
	e ̈̈̈̈̈̈e
	(3.44) in Section 3.3,

  1 1

	t 3 1
	t 3 2
		t 4 1
	t 3 3
	t 3 4	t 4 2
	t 1 2
	t 3 5
		t 4 3
	t 3 6
	t 2 1

  Figure 6.5. -(a) standard TEG. (b) PS of t 2 by ta, triggered every ω time units. (c) equivalent PS expressed by a signal Sω.Definition 63. A periodic signal S : Z Ñ t0, 1u is defined by a string tn 0 , n 1 , ̈̈̈, n I u P N 0 and a period ω P N, such that ̈̈̈, n I u is strictly ordered, i.e., @i P t1, ̈̈̈, Iu, n í1 ă n i , and n I ă ω.

					ω	p 3
						ta			Sω
	p 1		p 2		p 1	p 2		p 1	p 2
	1	t 2	t 3	t 1	t 2	t 3	t 1	t 2	t 3
	(a) standard TEG.	(b) TEG with PS.		(c) PS by signal Sω.
		$
	Sptq "	& 1 % 0	if t P tn 0 ̀ωj, n 1 ̀ωj, ̈̈̈, n I ̀ωj |j P Zu, otherwise,
	where the string tn 0 , n 1 , Example 48. The signal
			$
	S 1 ptq "	& 1 if t P ẗ̈̈, ́3, 0, 1, 4, 5, 8, 9, ̈̈̈u, % 0 otherwise,
	is a periodic signal with a period ω " 4 and a string t0, 1u. Therefore,
			$
	S 1 ptq "	% 0 otherwise. & 1 if t P t0 ̀4j, 1 ̀4j |j P Zu,				(6.2)

  )where x (resp. ũ, ỹ) refers to the vector of counter functions of internal (resp. input, output) transitions and A, B and C are matrices with entries in E m|b rrδss of appropriate size. Clearly, A ̊B is the least solution of the implicit equation in (6.16), Theorem 2.1. Therefore the transfer function matrix of a consistent WTEG is obtained by H " CA ̊B. Moreover, this matrix is a consistent matrix with entries in E m|b rrδss, see the following propositions.Proposition 94 ([START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]). For a g inputs and p outputs WBTEG, the entries of the transfer matrix H " CA ̊B are ultimately cyclic series in E m|b rrδss.Proposition 95. Let (N , M 0 , ϕ) be a consistent WTEG with g input and p output transitions, then its transfer matrix H P E m|b rrδss p̂g is consistent.Proof. Since consistent WTEGs are a strict subclass of WBTEGs (Prop. 92) the transfer function matrix H is composed of ultimately cyclic series in E m|b rrδss, see Prop. 94. It remains to show that H P E m|b rrδss p̂g is consistent. Recall, Remark 35 the gain of a path is equivalent to the gain of its operational representation. Moreover, N admits a T-semiflow ξ, with subvectors ξ t i " rξ i 1 ̈̈̈ξ ig s associated with input transitions and ξ to " rξ o 1 ̈̈̈ξ op s associated with output transitions. Due to Prop. 91, the relation between gain and T-semiflow must hold for all paths in N . Therefore, the gain matrix Γ pHq is of rank 1 and is given by

	Γ pHq "	" ξ o 1 ̈̈̈ξ op	ı T "	1 ξ i 1	̈̈̈1 ξ ig	ı	,
	"	» ---	ξo 1 ξ i 1 . . .	i 1 ̈̈̈ξ ξ ig . . .	fi ffi ffi ffi	.
		-	ξo p ξ ig	ξ ig ̈̈̈ξ op	fl
	Example 56.

  i P M ax in vγ, δw and h P E m|b rrδss be the transfer function of a consistent WTEG, then A more convenient way to obtain the output of a consistent WTEG is to represent the input counter function ũ and the output counter function ỹ as series u, y P M ax in vγ, δw. Proposition 96. For a consistent SISO WTEG with an pm, bq-periodic transfer function h P E m|b rrδss and an input u P M ax in vγ, δw, the output y P M ax in vγ, δw is obtained byy " Ψ m|b ̀h b Injpuq ̆.Proof. First, let us recall the canonical injection from M ax in vγ, δw into Errδss, see Section 3.2, thus we can represent the input u P M ax in vγ, δw as an element in E m|b rrδss. Then, Due to Remark 36, this is equivalent to y " Ψ m|b ̀h b Injpuq ̆.Clearly, Prop. 96 can be extended to a consistent MIMO WTEG with a transfer function matrix H P E m|b rrδss p̂g .

		ỹptq " ̀h	ũ̆p tq " ̀hpuIq ̆ptq " ̀hpInjpuqIq ̆ptq
				" ̀ph b InjpuqqI ̆ptq.
				̃h́à
	ỹptq " ̀h	ũ̆p tq "	i	γ ν i δ t i I ̧̄ptq,
	as h is lower semi-continuous,
	ỹptq "	́à i	h ̀γν i δ t i I ̆̄ptq.

  .[START_REF] David-Henriet | Discrete event systems with standard and partial synchronizations[END_REF] To show equality, R v is evaluated for intervals defined in (6.6). E.g., for the interval pn I Figure 6.19. -Simple TEG with a periodic PS of t 2 .

					S 1
		t 1	p 1		t 2
			S 2
	t 1	p 1	t 2	p 3	t 3	p 4	t 4
				1		1
				p 2
	Figure 6.20. -Example of a TEG under periodic PS.

  1 ∆ 4|4 δ ́3 and δ ́3∆ 4|4 ' ∆ 4|4 δ ́1 are the T-operators corresponding to H 3 " x1 3 2 1y and H 1 " x0 0 2 1y, see Example 62. The holding-time functions in PTEGs correspond to causal periodic T-operators, see Prop. 98. As every monomial/polynomial in T per rrγss is a specific ultimately cyclic series, the entries of the A, B and C matrices are ultimately cyclic series in T per rrγss. The sum (resp. product, Kleene star) of ultimately cyclic series in T per rrγss are again ultimately cyclic series in T per rrγss, see Prop. 65 (resp. Prop. 66, Prop. 67). Thus, the transfer matrix CA ̊B is also composed of ultimately cyclic series in T per rrγss.Corollary 16. For a g-input p-output TEG under periodic PS, see Definition 63, the transfer function matrix is given by H " CA ̊B P T per rrγss p̂g . Moreover, the entries of the transfer function matrix H are ultimately cyclic series in T per rrγss.

	Example 64. Let us recall the TEG under periodic PS given in Example 61, the transfer function
	for this system is obtained by
	h

Transfer Functions Matrices for TEGs under periodic PS and PTEGs Theorem 6.1 (Transfer function matrix of PTEG). The input-output behavior of a g-input and p-output PTEG can be described by a transfer function matrix H P T per rrγss p̂g of ultimately cyclic series in T per rrγss. This transfer function matrix is obtained by H " CA ̊B.

Proof.

  Moreover, x1 and x2 are counter functions associated to the transitions t 1 and t 2 . Table6.1

					, with a periodic PS of transition
	t 2 by,
		$
	S 2 "	% &	0 otherwise. 1 if t P t0 ̀3ju,		(6.24)
					S 1
			t 1	p 1	t 2

Table 6 .

 6 1. -Response x2 induced by the counter function x1 .gives the accumulated number of firings strictly before time t. Therefore, the counter function x1 is interpreted as, no firing of transition t 1 before time t " 1. Exactly one firing at time t " 1 and one additional firing at time t " 3 (resp. time t " 6). The counter function x2 is interpreted as, no firing of transition t 2 before time t " 3. Two firings at time t " 3 and one additional Proposition 99. A periodic partial synchronization of a transition by signal S, see Definition 63, has an operator representation in ET , given by v S "δ n 0 ∆ ω|ω δ ́nI ' δ n 1 ́ω∆ ω|ω δ ́n0 ' ̈̈̈' δ n I ́ω∆ ω|ω δ ́npÍ1q . (6.27)Proof. This proof is similar to the proof of Prop. 97. There a periodic PS is modeled by an operator in T .̀vS x̆p tq " ̀pδ n 0 ∆ ω|ω δ ́nI ' δ n 1 ́ω∆ ω|ω δ ́n0 ' ̈̈̈' δ n I ́ω∆ ω|ω δ ́npÍ1q q x̆p tq E.g. for the interval n 0 ̀ωj ă t ď n 1 ̀ωj observe that,

	Because of (3.4) and (3.1),
	̀vS	x̆p tq " ̀δn 0 ∆ ω|ω δ ́nI	x̆p tq ' ̀δn 1 ́ω∆ ω|ω δ ́n0	x̆p	tq ' ̈̈'
			̀δn I ́ω∆ ω|ω δ ́npÍ1q	x̆p	tq,
		" min ́̀δ n 0 ∆ ω|ω δ ́nI	x̆p tq, ̀δn 1 ́ω∆ ω|ω δ ́n0	x̆p	tq, ̈̈,
											̄.
			̀δn I ́ω∆ ω|ω δ ́npÍ1q	x̆p	tq
	Recall (5.2) and (5.4), therefore
	̀vS	x̆p tq " min ́x ́ωY t ́n0 ω	́1	]	̀nI	̀1̄, x́ωY t ́n1 ̀ω ω	́1	]	̀n0	̀1̄,
			̈̈̈,	x́ωY t ́nI ̀ω ω	́1	]	̀nÍ1	̀1̄"
			x́m in	́ωY t ́n0 ω	́1	]	̀nI ̀1, ω Y t ́n1 ̀ω ω	́1	]	̀n0 ̀1, ̈̈,
				ω Y t ́nI ̀ω ω	́1	]	̀nÍ1	̀1̄"
			x́m in	́ωY t ́n0 ω	́1	]	̀nI ̀1, ω Y t ́n1 ω	́1	]	̀n0 ̀ω ̀1, ̈̈,
				ω Y t ́nI ω	́1	]	̀nÍ1 ̀ω	̀1̄̄.
	Recall (6.26), it remains to show that ̀vS	x̆p tq " xpK S ptq̀1q. For this ̀vS	x̆p	tqq is evaluated
	for intervals defined in (6.25). Y t ́ni ́1 ω ] " Q t ́ni ω	́ω	U		because of tn{ωu " rpn ́ω ̀1q{ωs.
				$
			"	& j for i " 0
				% j ́1 for i " 1, ̈̈̈, I
	hence,
	̀vS	x̆p tq "	x̀m in ̀ωj ̀nI ̀1, ωj ̀n0 ̀1, ̈̈̈, ωj ̀nÍ1	̀1̆"
			xpn 0 ̀ωj ̀1q.

  Theorem 6.2. For a consistent g-input p-output WTEG under periodic PSs, see Definition 63, the transfer function matrix is given by H " CA ̊B P ET per p̂g . Moreover, the entries of the transfer function matrix H are ultimately cyclic series in ET per .Proof. First, periodic PS of a transition by a periodic signal refers to a periodic ET -operator, see Prop. 99. Then, as every basic sum in ET per is a specific ultimately cyclic series, the entries of the A, B and C matrices are ultimately cyclic series in ET per . The sum (resp. product, Kleene star) of ultimately cyclic series in ET per are again ultimately cyclic series in ET per , see Prop. 85 (resp. Prop. 86, Prop. 87). Hence, the entries of the transfer matrix CA ̊B are ultimately cyclic series in ET per .Example 70. Consider the consistent WTEG shown in Figure6.26 with a PS of transition t 2 by the signal

		$
	S 2 ptq "	& 1 if t P t1 ̀2ju, % 0 otherwise.
				S 2
		t 1	p 1	t 2	p 3	t 3	p 4	t 4
				2
					1		1
					p 2
				2
		Figure 6.26. -Example of a WTEG under periodic PS.

  Proposition 102. Let H P E m|b rrδss p̂g and G P E m|b rrδss p̂g be to consistent matrices, then the open loop transfer matrix HP opt , with P opt " H zG, is a consistent matrix with entries in E

m|b rrδss, if and only if, Dc P Q, c ą 0 such that, cΓ pGq k,1 " Γ pHq k,1 , @k P 1, ̈̈̈, p. (7.7)

  4 the place p c3 contains 4 initial tokens. Moreover, the holding-time-function of p c3 isH p c3 ptq " max ́Rδ 1 ∆ 4|4 δ ́1 ptq, R δ 4 ∆ 4|4 δ ́2 ptq ̄́t,

	" max ̂1	̀R t	́1 4	V	4, 4	̀R t	́2 4	V	4 ̇́t,
	" x4 3 3 5y

  ∆ 1|n δ 1́n Q∆ n|1 ∆ 1|n δ 1́n Qδ ́1∆ n|1 ̈̈̈∆ 1|n δ 1́n Qδ 1́n ∆ n|1 ∆ 1|n δ 2́n Q∆ n|1 ∆ 1|n δ 2́n Qδ ́1∆ n|1 ̈̈̈∆ 1|n δ 2́n Qδ 1́n ∆ n|1The extended core is a matrix with entries in M ax in vγ, δw, since ∆ 1|n δ τ ∆ n|1 " δ rτ{nsn (Remark 21). Furthermore, the extended core Q1 is a greatest core. For this one has to show that Np nω d ω Qp ω d nω N, " p nω d nω p nω looomooon Qp ω d nω p nω looomooon " p nω d ω Qp ω d nω " Q1 . Lemma 9. All elementary operators introduced in Prop. 73 can be represented as basic elements in ET .

	Q2	" N	Q1 N "	Q1 .
	--" C.3. Proofs of Chapter 5 ∆ 1|n δ 1́n ∆ 1|ω δ 1́ω ∆ 1|n δ 1́n ∆ 1|ω δ 2́ω . . . ∆ 1|n δ 1́n ∆ 1|ω . . . » -----∆ 1|n ∆ 1|ω δ 1́ω ∆ 1|n ∆ 1|ω δ 2́ω ffi ffi fl . . . fi ffi ffi Q2	fi ffi ffi ffi ffi fl	ffi ffi fl fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi	"	» -----	∆ 1|n δ 1́n p ω ∆ 1|n δ 2́n p ω . . . ∆ 1|n p ω	fi ffi ffi ffi ffi fl	.
								∆ 1|n ∆ 1|ω
			This leads to
							»	∆ 1|n δ 1́n N fi
				p nω d ω "	-----	ffi ∆ 1|n δ 2́n N ffi fl . . . ffi ffi	.
								∆ 1|n N
			Respectively, p ω d nω is given by
							"		ı
				p ω d nω "	N∆ n|1 Nδ ́1∆ n|1 ̈̈̈Nδ 1́n ∆ n|1	.
			Finally, we obtain
						» ∆ 1|n δ 1́n N fi
				Q1	"	-----∆ 1|n δ 2́n N ffi fl . . . ffi ffi ffi	Q	" N∆ n|1 Nδ ́1∆ n|1 ̈̈̈Nδ 1́n ∆ n|1	ı	,
							∆ 1|n N
						»			fi
						-			ffi
					"	----		. . .	. . .	. . .	ffi fl ffi ffi	.
							∆ 1|n Q∆ n|1	∆ 1|n Qδ ́1∆ n|1	̈̈̈∆ 1|n Qδ 1́n ∆ n|1

e d ω e d nω ,

Acknowledgments

vielen Dank to the members of the Control Systems Group at TU Berlin and merci beaucoup to the members of the

Proof. The canonical form is obtained by applying (5.11).

The ordering of two canonical basic elements m 1 " γ ν 1 δ τ 1 ∇ m|b 1 ∆ ω|ϖ 1 γ ν 1 1 δ τ 1 1 , m 2 " γ ν 2 δ τ 2 ∇ m|b 2 ∆ ω|ϖ 2 γ ν 1 2 δ τ 1 2 P ET with equal indices m, ω can be checked by

b 1 " b 2 and ϖ 1 " ϖ 2 and ́γν 1 δ τ 1 ľ γ ν 2 δ τ 2 and γ ν 1

or γ ν 1 δ τ 1 ́ω ľ γ ν 2 δ τ 2 and γ ν 1 1 δ τ 1 1 ̀ϖ1 ľ γ ν 1 2 δ τ 1 2 or γ ν 1 ̀mδ τ 1 ́ω ľ γ ν 2 δ τ 2 and γ ν 1

Hence, @j P Z Hptq "

n 0 if t " 0 ̀ωj, n 1 if t " 1 ̀ωj, . . . n ώ1 if t " pω ́1q ̀ωj, (6.3) where for i P t0, ̈̈̈, ω ́1u, n i P Z are the holding times in each period.

The short form of a holding-time function is defined as a string xn 0 n 1 ̈̈̈n ώ1 y. The period ω is implicitly given by the number of elements in the string. For the modeling process of TEGs in the (max,+)-algebra, it is necessary that tokens must enter and leave each place in the same order [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF]. In other words, a place must respect FIFO behavior. This property leads to the following constraint on holding-time functions @t P Z, Hpt ̀1q ̀1 ě Hptq.

(6.4)

A holding-time function which respects (6.4) is called FIFO holding-time function. Moreover, a holding-time function is called causal if all holding times are nonnegative, i.e., @i P t0, ̈̈̈, ω ́1u, n i P N 0 .

Definition 65 (Periodic Time-variant Event Graph). A PTEG is a TEG where the holding times of places are given by causal FIFO holding-time functions.

Example 50. Consider the PTEG in Figure 6.7a where the holding time of p 1 is changing according to, @j P Z

This holding-time function satisfies (6.4) hence the holding time is such that tokens enter and leave place p 1 in the same order. In contrast, let us consider the TEG in Figure 6.7b, where the holding time of place p 2 is changing according to H 2 ptq " x3 0 2 1y. In this case, tokens which enter the place p 2 at time instant t " 0 enable the firing of transition t 4 at time instant 0 ̀H2 p0q " 3. Tokens which enter the place p 2 at time instant t " 1 immediately enable the firing of t 4 , since H 2 p1q " 0. The function H 2 violates the FIFO condition of p 2 , and therefore the TEG in Figure 6.7b is not in the class of PTEGs. As Hpt ̀1q ̀1 ě Hptq, it follows that Rpt ̀1q " Hpt ̀1q ̀t ̀1 ě Hptq ̀t " Rptq, i.e. R is isotone. The release-time function can be seen as an alternative representation of the time-variant behavior of a place in a PTEG. This function describes the time when a token in a place is available to contribute to the firing of the downstream transition of the place. The argument of this function is the time t when the token enters the place and its value is the time when the token is available to leave the place. By defining n i " ni ̀i, we can express a release-time function as

Clearly, nonnegative holding-times n i (causal holding-time functions) lead to causality of R.

Example 51 (PTEG). Figure 6.8 shows a PTEG with holding-time functions of places p 1 , p 2 , p 3 given by

The corresponding release-time functions are, @j P Z max

Therefore, if a car arrives at times 2, 6, ̈̈̈it has to wait for 2 time instants, if it arrives at times 3, 7, ̈̈̈, it has to wait for 1 time instant.

Remark 33. The behavior of a TEG under periodic PS operating under the earliest functioning rule can be modeled by an "equivalent" PTEG. For this, the time-variant delays caused by periodic PSs of the transitions are shifted to the upstream places of the transitions. For instance consider the simple TEG shown in Figure 6. [START_REF] Cofer | A generalized max-algebra model for performance analysis of timed and untimed discrete event systems[END_REF] is associated, defined by, @j P Z max ,

The value of R S 2 can be interpreted as the next time when the signal S 2 enables the firing of the corresponding transition. Clearly, an ω-periodic signal S 2 leads to a corresponding function R S 2 which satisfies @t P Z max , R S 2 pt ̀ωq " ω ̀RS 2 ptq. First, this function is partitioned into a constant offset τ and a remaining causal release-time function,

where @i P t0, ̈̈̈, ω ́1u, n 1 i " n i ́τ and τ " minpn i ́iq. Then, the PTEG shown in Figure 6.12 can be modeled by the TEG under periodic PS shown in Figure 6.13, where the periodic signals S 0 , S 1 , ̈̈̈, S ώ1 are given by, @j P Z pF N vγw , ', bq is discussed. This dioid pF N vγw , ', bq is an alternative to the dioid pT per rrγss, ', bq to model TEG under PS.

Example 53. Consider the simple PTEG shown in Figure 6.14a with a holding time function x1 0 2 2y of place p 1 . The release-time function to x1 0 2 2y is

For this example, τ " 0, since n 1 ́1 " 1 ́1 " 0, and therefore R 1 p 1 ptq " R p 1 ptq. Then the periodic signals S 0 , S 1 , S 2 and S 3 are

SDF Graphs, called Cyclo-Static Synchronous Data-Flow (CSDF) Graphs. This system class was studied, e.g. in [START_REF] Bilsen | Cycle-static dataflow[END_REF][START_REF] De Groote | Multi-rate equivalents of cyclo-static synchronous dataflow graphs[END_REF][START_REF] Parks | A comparison of synchronous and cycle-static dataflow[END_REF].

Dioid Model of Timed Event Graphs

Dater and Counter

In analogy with [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF], in the following dater and counter functions are briefly introduced. For a more exhaustive representation, the reader is invited to consult [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][Chap. 5]. An event can be seen as an instantaneous action, such as the push of a button, the start of a production process or the successive firings of a transition in a Petri net. For timed DESs the occurrences of an event can be described by a sequence generated by an increasing counting mechanism over time. For instance, the successive firings of a transition in a Petri net can be described by a time sequence, e.g. pk 0 , t 0 qpk 0 ̀1, t 1 qpk 0 ̀2, t 2 q ̈̈̈, where the firings are enumerated starting from an arbitrary value k 0 P Z. Then the pair pk i , t i q is interpreted as: The firing numbered by k i has taken place at time t i . For instance, the sequence p0, 2q, p1, 3q, p2, 3q, where k 0 is chosen to 0, means the first firing of a transition, numbered by 0, has taken place at time instant 2, the second and third firings numbered by 1 and 2 have taken place at time instant 3. This kind of sequences can either be represented by a dater function k Þ Ñ dpkq in the "event-domain" or equivalently as a counter function t Þ Ñ cptq in the "time-domain". The following section introduces dater and counter functions for the purpose of modeling WTEGs (resp. PTEGs) in dioids.

Dater

A dater is defined as a mapping d : Z Ñ Z max , k Þ Ñ dpkq, where the index k P Z numbers the consecutive firings of a transition starting from an initial value k 0 " 0 and dpkq is the time when the firing numbered by k has taken place. It is important to mention that by convention the first firing of a transition is numbered by 0. Therefore, dpkq is the time when the pk ̀1q st firing of the transition has taken place. More precisely, Therefore, the series in M ax in vγ, δw corresponding to an impulse Ipkq, see (6.7), is the unit element e " γ 0 δ 0 in the dioid pM ax in vγ, δw , ', bq. For a more detailed description of the transformation, see e.g. [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF][START_REF] Cohen | Second order theory of minlinear systems and its application to discrete event systems[END_REF].

Counter

A counter is defined as a mapping c : Z Ñ Z min , t Ñ cptq, where the time t P Z is given by a discrete value and cptq is the accumulated number of firings strictly before time t.

where the latter equality holds as δ 0 " e. This operator describes the firing relation between t 1 and t 2 , i.e. x2 " pδ ́3∆ 4|4 ' ∆ 4|4 δ ́1qx 1 . Therefore, x2 pkq " maxṕ3 ̀rx 1 pkq{4s4, rpx 1 pkq 1q{4s4q.

Remark 37. Due to the influence of the PS, this firing relation between t 1 and t 2 is timevariant. Note again that, x1 pkq indicates the pk ̀1q st firing of t 1 . Then for instance, if the pk ̀1q st firing of t 1 is at time instant x1 pkq " 1, then the pk ̀1q st firing of t 2 is at x2 pkq " 1, i.e., we have no delay. In contrast, if the pk ̀1q st firing of t 1 is at time instant x1 pkq " 2, then the pk ̀1q st firing of t 2 is at x2 pkq " 4, and the delay is 2.

Hence we have shown that,

Corollary 15. Since Hptq " Rptq ́t, the T-operator associated with a holding-time function xn 0 n 1 ̈̈̈n ώ1 y can be obtained by

Note that the operator representation of a causal release-time function R, i.e. Rptq ě t, leads to a periodic and causal T-operator.

Example 62. Consider H 1 ptq " x0 0 2 1y given in Example 51. This holding-time function corresponds to an operator given by

because of (4.10): δ ́1 ' δ ́2 ' δ ́3 " δ ́1. Respectively, H 3 ptq " x1 3 2 1y corresponds to the operator ∆ 4|4 ' δ 1 ∆ 4|4 δ ́3.

We can use T-operators and the event shift operator γ to describe the transfer behavior of PTEGs. The firing-relation between the two transitions t i , t i in Figure 6.21 is represented by xi " v i γ pM 0 q i xi , where pM 0 q i is the initial marking in place p i , v i is the T-operator associated with the holding-time function H i of place p i and xi , xi are the dater functions M ax in vγ, δw. Since, pM ax in vγ, δw , ', bq is a subdioid of pT per rrγss, ', bq and by using the canonical injection Inj, the input can be represented as a series Injpuq P T per rrγss. The output y P M ax in vγ, δw of the system is then computed as follows y " Ψ ω ̀h b Injpuq ̆. (6.23)

Example 67. Recall the transfer function h " δ 3 pγ 2 δ 2 q ̊∆2|2 δ ́1 of the TEGPS shown in Figure 6.20. Moreover, consider the input dater function,

́8 for k ă 0; 0 for k " 0;

2 for k " 1, 2;

3 for k " 3, 4, 5, 6;

8 for k ě 7.

The series u P M ax in vγ, δw to this dater function is u " γ 0 δ 0 ' γ 1 δ 2 ' γ 3 δ 3 ' γ 7 δ ̊. The output y P M ax in vγ, δw of the system is then

Ψ ω ̀δ3 ∆ 2|2 δ ́1pγ 2 δ 2 q ̊b pγ 0 δ 0 ' γ 1 δ 2 ' γ 3 δ 3 ' γ 7 δ ̊q"

pδ 3 ' δ 5 γ 1 ' δ 6 γ 3 qpγ 2 δ 2 q ̊' δ 3 γ 7 δ " pδ 3 ' δ 5 γ 1 qpγ 2 δ 2 q ̊' δ 3 δ ̊γ7 " pδ 3 ' δ 5 γ 1 ' δ 7 γ 3 ' δ 9 γ 5 ' δ 11 γ 7 ' ̈̈̈q ' δ 3 δ ̊γ7 " δ 3 ' δ 5 γ 1 ' δ 7 γ 3 ' δ 9 γ 5 ' δ ̊γ7 .

Moreover, y is the series in M ax in vγ, δw associated with the dater function,

́8 for k ă 0;

3 for k " 0;

5 for k " 1, 2;

7 for k " 3, 4;

9 for k " 5, 6;

Dioid Model of Weighted Timed Event Graphs under periodic Partial Synchronization

In analogy to the modeling process of consistent WTEGs in the dioid pErrδss, ', bq and Timed Event Graphs under Partial Synchronization (TEGsPS) in the dioid pT per rrγss, ', bq, the earliest functioning of consistent WTEGs under periodic PS can be modeled in the dioid pET , ', bq. For this, a counter function is associated with each transition. Then the influence of transitions on each other are coded as operators in ET , see Chapter 5 for the definition of the dioid pET , ', bq.

PS and Counters

Section 6.2.4 describes how the time-variant behavior of a periodic PS is expressed in the "event-domain" based on dater functions. In the following, a periodic PS is expressed in the "time-domain" based on counter functions. For this the ∆ ω|ϖ is redefined as a mapping from the set Σ into itself, see (5.2). Moreover, recall that Σ is the set of antitone mappings firing at time t " 6. Observe that, a firing of a transition at time t is represented in the counterfunction ta time t ̀1. Or differently, xpt ́1q gives the accumulated number of firings up to (including) time t. Hence, the firing relation between transition t 1 and t 2 is described by,

To describe the time-variant behavior of a PS caused by an arbitrary periodic signal S, a function K S ptq : Z Ñ Z is associated to this periodic signal S. This function is defined by, @j P Z,

Again, if the signal S is ω-periodic then the corresponding function K S ptq satisfies @t P Z, K S pt ̀ωq " ω ̀KS ptq. The value of K S ptq can be interpreted as the last time when the signal S enabled the firing of the corresponding transition. Then the firing relation between t 1 and t 2 is described by

where K S 2 is associated to the signal S 2 .

Example 69. Recall Example 68 with the signal S 2 given in (6.24). The function K S 2 ptq associated with S 2 is then, since ω " 3 and n I " 0,

Therefore, x2 ptq " x1 ptpt ́1q{3u ̂3 ̀1q.

To prove that a periodic PS of a transition admits an operator representation in the dioid pET , ', bq we must show that an operator v P ET exists such that, v x1 ptq " x1 ̀KS 2 ptq̀1 ̆. For this recall the definition of the ∆ ω|ϖ operator and the δ τ operator in ET , see Prop. 73,

τ P Z δ τ : @ x P Σ, t P Z ̀δτ p xq ̆ptq " xpt ́τq.

We have to show that the behavior of a periodic PS can be expressed by sum and composition of the δ τ and ∆ ω|ω operators.

Second, for n 1 ̀ωj ă t ď n 2 ̀ωj, By going through the remaining intervals it is shown that,

where K S ptq is given by, @j P Z

Modeling of consistent WTEGs under periodic PS in ET

Let us consider a basic path t i Ñ p i Ñ t i in a consistent WTEG with a periodic PS of transition t i by a signal S i . The influence of transition t i on transition t i is described by the following operator, xi " v t i ∇ 1|wpp i ,t i q δ pϕq i γ pM 0 q i ∇ wpt i ,p i q|1 xi , where xi and xi refer to the counter functions of transition t i and t i , v t i is the operator representation of the signal S i corresponding to the PS of t i , wpt i , p i q and wpp i , t i q are weights of the arcs pt i , p i q and pp i , t i q, pϕq i is the holding time of place p i and pM 0 q i is the initial marking of p i . For instance, consider the basic path given in Figure 6.25, with a PS of transition t 2 by the periodic signal

where z is a counter function describing the desired output schedule -a priori known signal -and ũ is the unknown input -a counter function describing the input schedule -that we want to optimize under the "just-in-time" criterion. Let us recall the calculation of a system output in Prop. 96 and the relation between counter functions and M ax in vγ, δw series. Hence, (7.1) can be written as,

where z, u are series in M ax in vγ, δw corresponding to the counter functions z and ũ. Note that for h P E m|b rrδss and u P M ax in vγ, δw , Injpuq P E 1|1 rrδss and thus the product h b Injpuq P E m|b rrδss, see Prop. [START_REF] Cottenceau | Weight-balanced timed event graphs to model periodic phenomena in manufacturing systems[END_REF]. In other words, the periodicity of h and h b Injpuq are the same. Finding the optimal input in (7.2), according to the "just-in-time" criterion, amounts to compute the following sum

Proposition 100. The greatest solution of z ľ Ψ m|b ph b Injpuqq, (7.2), is given by u opt " Inj 7 ph zΨ 7 m|b pzqq.

Proof. Since h P E m|b rrδss and Ψ 7 m|b pzq P E m|b rrδss, i.e., they have the same period, u " h zΨ 7 m|b pzq P E b|b rrδss is pb, bq-periodic, see Prop. 20, which is the required form for a potential non zero solution of Inj 7 puq, see Prop. 22.

Example 71. Let us consider the consistent WTEG of Example 56 with a transfer function h P E 3|2 rrδss given by h "µ 3 β 2 δ 2 ' pγ 2 µ 3 β 2 γ 1 ' γ 3 µ 3 β 2 qδ 3 ' γ 3 µ 3 β 2 δ 4 ' pγ 4 µ 3 β 2 γ 1 ' γ 6 µ 3 β 2 qδ 5 ' pγ 5 µ 3 β 2 γ 1 ' γ 6 µ 3 β 2 qδ 6 ' pγ 1 δ 1 q ̊̀pγ 6 µ 3 β 2 γ 1 ' γ 8 µ 3 β 2 qδ 7 ̆.

Moreover, consider the following reference counter function,

4 ̀j for 7 ̀2j ď t ď 8 ̀2j with j P N 0 .

This counter function corresponds to the series z " δ 3 ' γ 3 δ 6 pγ 1 δ 2 q ̊P M ax in vγ, δw. Then Ψ 7

3|2 pzq " µ 3 β 2 δ 3 ' pγ 1 δ 2 q ̊pγ 3 µ 3 β 2 δ 6 q and u opt " Inj 7 ph zΨ 7 3|2 pzqq " e ' γ 1 δ 1 ' γ 2 δ 4 pγ 2 δ 6 q ̊.

Optimal Control

The response y of the consistent WTEG to the optimal input u opt is y " Ψ 3|2 ph b Injpu opt qq " δ 3 ' ̀γ3 δ 6 ' γ 5 δ 7 ̆pγ 3 δ 6 q ̊.

This series corresponds to the counter function,

where the series z, u P M ax in vγ, δw correspond to the dater functions z and ū.

Proposition 101. The greatest solution of z ľ Ψ ω ph b Injpuqq, (7.4), is given by u opt " Inj 7 ph zΨ 7 ω pzqq.

Proof. The proof is similar to the proof of Prop. 100.

Example 72. Let us consider the TEG under periodic PS of Example 64 with a transfer function h P T per rrγss given by h " δ 3 pγ 2 δ 2 q ̊∆2|2 δ ́1.

Moreover, consider the following reference dater function,

This dater function corresponds to the series z " δ 3 ' γ 2 δ 6 pγ 1 δ 2 q ̊P M ax in vγ, δw. Then Ψ 7 2 pzq " δ 3 ∆ 2|2 ' pγ 1 δ 2 q ̊pγ 2 δ 6 ∆ 2|2 q and u opt " Inj 7 ̀h zΨ 7 2 pzq ̆" δ 1 ' γ 2 δ 3 pγ 1 δ 2 q ̊.

The response y of the TEG under periodic PS to the optimal input u opt is

This series corresponds to the dater function,

for k ă 0, 3 for k " 0, 1, 5 ̀2j for k " 2 ̀j with j P N 0 .

Figure 7.2 illustrates the reference output z and the system output ȳ resulting from the optimal input ūopt , clearly z ľ ȳ.

A

Formula for Residuation

The following list provides some basic relations of left and right division, for the proofs and a more detailed list the reader is invited to consult [START_REF] Baccelli | Synchronisation and Linearity: An Algebra for Discrete Event Systems[END_REF] Proof. This proof is taken from [START_REF] Cottenceau | Modeling and control of weight-balanced timed event graphs in dioids[END_REF]. To prove the left equation of (C.1), by definition of the residuated mapping the greatest solution for x of the following inequality where X is of size n ̂m and G is of size n ̂1. Then,

We obtain for each i P t1, ̈̈̈, nu the following inequalities

x i,1 ĺ g i {p∆ 1|ω δ 1́ω q " g i ∆ ω|1 , . . .

x i,n ĺ g i {∆ 1|ω " g i δ 1́ω ∆ ω|1 .

This can be expressed in matrix form

Again Gd ω satisfies (C.17) with equality, since d ω p ω " e.

Declaration

I declare that this thesis has been composed by myself, that the work contained herein is my own except where explicitly stated otherwise, and that this work has not been submitted for any other degree or professional qualification except as specified.

Berlin, July 2019

Johannes Trunk