This thesis concerns delay-dependent robust stability analysis and stabilization for time-delay system with uncertainties. By choosing new Lyapunov-Krasovskii functional and using methods which can reduce conservativeness of stability condition in the literature, new delay-dependent sufficient stability conditions are obtained in terms of linear matrix inequality. It is shown that the new stability conditions can provide less conservative results than some existing ones. Furthermore, the new stability conditions are also used to design the state feedback controllers. Finally, numerical examples are given to show the derived results and compared with results in the literature.

With the help of rapidly advancing communication technology, control systems are increasingly integrated via communication networks. Networked control systems (NCSs) bring significant advantages such as flexible and scalable structures, easy implementation and maintenance, and efficient resources distribution and allocation. NCSs empowers to finish some complicated tasks using some relatively simple systems in a collaborated manner. However, there are also some challenges and constraints subject to the imperfection of communication channels. In this thesis, the stabilization problems and the performance limitation problems of control systems subject to networked-induced constraints are studied. Overall, the thesis mainly includes two parts: 1) Consensus and consensusability of multi-agent systems (MASs); 2) Delay margins of NCSs. Chapter 2 and Chapter 3 deal with the consensus problems of MASs, which aim to properly design the control protocols to ensure the state convergence of all the agents. Chapter 4 and Chapter 5 focus on the consensusability analysis, exploring how the dynamics of the agents and the networked induced constraints impact the overall systems for achieving consensus. Chapter 6 pays attention to the delay margins of discrete-time linear time-invariant (LTI) systems, studying how the dynamics of the plants limit the time delays that can be tolerated by LTI controllers. In Chapter 2, the leader-following consensus problem of MASs with general linear dynamics and arbitrary switching topologies is considered. The MAS with arbitrary switching topologies is formulated as a switched system. Then the leaderfollowing consensus problem is transformed to the stability problem of the corresponding switched system. A necessary and sufficient consensus condition is derived. The condition is also extended to MASs with time-varying delays. In Chapter 3, the consensus problem of MASs with general linear dynamics is studied. Motivated by the multiple-input multiple-output (MIMO) communication technique, a general framework is considered in which different state variables are exchanged in different independent communication topologies. This novel framework could improve the control system design flexibility and potentially improve the system performance. Fully distributed consensus protocols are proposed and analyzed for the settings of fixed and switching multiple topologies. The protocols can be applied using only local information. And the control gains can be designed depending on the dynamics of the individual agent. By transforming the overall MASs into cascade systems, necessary and sufficient conditions are provided to guarantee the consensus under fixed and switching state-variables-dependent topologies, respectively. Chapter 4 investigates the consensusability problem for MASs with time-varying delays. The bounded delays can be arbitrarily fast time-varying. The communication topology is assumed to be undirected and fixed. Considering general linear dynamics under average state protocols, the consensus problem is then transformed into a robust control problem. Sufficient frequency domain criteria are established in terms of small-gain theorem by analyzing the delay dependent gains for both continuoustime and discrete-time systems. The controller synthesis problems can be solved by applying the frequency domain design methods. The consensusablity problem of general linear MASs considering directed topologies are explored from a frequency domain perspective in Chapter 5. By investigating the properties of Laplacian spectra, a consensus criterion is established based on the stability of several complex weighted closed-loop systems. Furthermore, for singleinput MASs, frequency domain consensusability criteria are proposed on the basis of the stability margins, which depend on the H∞ norm of the complementary sensitivity function determined by the agents’ unstable poles. The corresponding design procedure is also developed. Chapter 6 studies the delay margin problem of discrete-time LTI systems. For general LTI plants with multiple unstable poles and nonminimum phase zeros, we employ analytic function interpolation and rational approximation techniques to derive bounds on delay margins. Readily computable and explicit lower bounds are found by computing the real eigenvalues of a constant matrix, and LTI controllers can be synthesized based on the H∞ control theory to achieve the bounds. The results can be also consistently extended to the case of systems with time-varying delays. For first-order unstable plants, we also obtain bounds achievable by proportionalintergral- derivative (PID) controllers, which are of interest to PID control design and implementation. It is worth noting that unlike its continuous-time counterpart, the discrete-time delay margin problem being considered herein constitutes a simultaneous stabilization problem, which is known to be rather difficult. While previous work on the discrete-time delay margin led to negative results, the bounds developed in this chapter provide instead a guaranteed range of delays within which the delayed plants can be robustly stabilized, and in turn solve the special class of simultaneous stabilization problems in question. Finally, in Chapter 7, the thesis is summarized and some future research topics are also presented. / Graduate

L'industrie sidérurgique rencontre depuis des années un développement important. Les caractéristiques des produits destinés aux différents secteurs industriels deviennent de plus en plus exigeantes. Actuellement des régulateurs avancés de type PID sont utilisés avec une structure multi-boucles locales. Afin de traiter les cahiers des charges de plus en plus contraints, il devient intéressant de revoir cette stratégie de contrôle. Ce sujet de thèse CIFRE entre ArcelorMittal et le CRAN a pour objectif de proposer de nouvelles stratégies de contrôle avancé pour le laminage à froid des bandes d'acier de type tôle fine, notamment des stratégies centralisées de contrôle remplaçant des boucles de régulation locales et indépendantes, afin d'élargir le champ des matériaux traitables, d'augmenter et d'optimiser la productivité et les capabilités du procédé ainsi que de minimiser les coûts de production. Dans une première partie, une étude bibliographique sera effectuée afin de mieux cerner les différents modèles métallurgiques et dynamiques du système à piloter. Dans une seconde partie, le cahier des charges sera traduit en termes d'un problème d'optimisation dont la solution est un contrôleur de type retour d'état. Enfin dans une troisième et dernière partie, une synthèse de retour de sortie dynamique sera proposée pour résoudre le problème des mesures manquantes / The steel industry has been developing for years. The characteristics of the products intended for the various industrial sectors are becoming more and more demanded. Currently advanced PID controllers are used with a local multi-loop structure. In order to deal with the increasingly constrained specifications, it is interesting to review this control strategy. This CIFRE PhD thesis between ArcelorMittal and CRAN aims to propose new advanced control strategies for the cold rolling of thin sheet steel strips, including centralized control strategies replacing local and independent control loops, to broaden the scope of treatable materials, increase and optimize process productivity and capabilities, and minimize production costs. In a first part, a bibliographic study will be carried out in order to better define the various metallurgical and dynamic models of the system to be controlled. In a second part, the specifications will be translated in terms of an optimization problem whose solution is the state feedback controller. Finally, in a third and final part, a dynamic output feedback synthesis will be proposed to solve the problem of missing measurements

This thesis investigates Quasilinear Control (QLC) of time-delay systems with nonlinear actuators and sensors and analyzes the accuracy of stochastic linearization for these systems. QLC leverages the method of stochastic linearization to replace each nonlinearity with an equivalent gain, which is obtained by solving a transcendental equation. The idea of QLC is to stochastically linearize the system in order to analyze and design controllers using classical linear control theory. In this thesis, the existence of the equivalent gain for a closed-loop time-delay system is discussed. To compute the equivalent gain, two methods are explored. The first method uses an explicit but complex algorithm based on delay Lyapunov equation to study the time-delay, while the second method uses Pade approximant. It is shown that, under a suitable criterion, Pade approximant can be effectively applied for QLC of time-delay systems. Furthermore, the method of Saturated-Root Locus (S-RL) is extended to nonlinear time-delay systems. It turns out that, in a time-delay system, S-RL always terminates prematurely as opposed to a delay-free system, which may or may not terminate prematurely. Statistical experiments are performed to investigate the accuracy of stochastic linearization compared to a system without time-delay. The impact of increasing the time-delay in the approach of stochastic linearization is also investigated. Results show that stochastic linearization effectively linearizes a nonlinear time-delay system, even though delays generally degrade accuracy. Overall, the accuracy remains relatively high over the selected parameters. Finally, this approach is applied to pitch control in a wind turbine system as a practical example of a nonlinear time-delay system, and its performance is analyzed to demonstrate the efficacy of the approach.

A distributed delay is a linear input-output operators and appears in many control problems. We investigate distributed delay and its applications. After introducing the definition and the main properties of the distributed delay, the numerical implementation problem of distributed delays is analyzed and a general method for its approximation is given. Then three applications are focused on where distributed delay appears. The first application is the stable inversion and model matching. A new class of stable inversion and model matching problem for finite dimensional linear time-invariant systems is defined. The stable inversion (resp. model matching) is an approximation of the inverse of a given model (resp. model matching), where exact inversion (resp. exact matching) is reached after a time $t=h$, which is a parameter of our procedure. The second application is concerned with stabilization and finite spectrum assignment for a class of infinite dimensional systems. The last application concerns observer synthesis for estimation or output control. For linear finite dimensional systems. A closed-loop memoryless observer by input injection is introduced. Asymptotic convergence as well as finite time convergence of the estimation are analyzed by output injection and input information via distributed delay. At last, we introduce a new class for approximation of distributed parameter systems. We work on the graph topology, and show that under some weak assumptions, such an approximation can be realized using distributed delay.

Nous considérons deux classes de systèmes fractionnaires linéaires invariants dans le temps avec des ordres commensurables et des retards discrets. La première est composée de systèmes fractionnaires à entrées multiples et à une sortie avec des retards en entrées ou en sortie. La seconde se compose de systèmes fractionnaires de type neutre avec retards commensurables. Nous étudions la stabilisation de la première classe de systèmes à l'aide de l'approche de factorisation. Nous obtenons des factorisations copremières à gauche et à droite et les facteurs de Bézout associés: ils permettent de constituer l'ensemble des contrôleurs stabilisants. Pour la deuxième classe de systèmes, nous nous sommes intéressés au cas critique où certaines chaînes de pôles sont asymptotiques à l'axe imaginaire. Tout d'abord, nous réalisons une approximation des pôles asymptotiques afin de déterminer leur emplacement par rapport à l'axe. Le cas échéant, des conditions nécessaires et suffisantes de stabilité H-infini sont données. Cette analyse de stabilité est ensuite étendue aux systèmes à retard classiques ayant la même forme. Enfin, nous proposons une approche unifiée pour les deux classes de systèmes à retards commensurables de type neutre (standards et fractionnaires). Ensuite, la stabilisation d'une sous-classe de systèmes neutres fractionnaires est étudiée. Premièrement, l'ensemble de tous les contrôleurs stabilisants est obtenu. Deuxièmement, nous prouvons que pour une grande classe de contrôleurs fractionnaires à retards il est impossible d'éliminer dans la boucle fermée les chaînes de pôles asymptotiques à l'axe imaginaire si de telles chaînes sont présentes dans les systèmes à contrôler. / We consider two classes of linear time-invariant fractional systems with commensurate orders and discrete delays. The first one consists of multi-input single-output fractional systems with output or input delays. The second one consists of single-input single-output fractional neutral systems with commensurate delays. We study the stabilization of the first class of systems using the factorization approach. We derive left and right coprime factorizations and Bézout factors, which are the elements to constitute the set of all stabilizing controllers. For the second class of systems, we are interested in the critical case where some chains of poles are asymptotic to the imaginary axis. First, we approximate asymptotic poles in order to determine their location relative to the axis. Then, when appropriate, necessary and sufficient conditions for H-infinity-stability are derived. This stability analysis is then extended to classical delay systems of the same form and finally a unified approach for both classes of neutral delay systems with commensurate delays (standard and fractional) is proposed. Next, the stabilization of a subclass of fractional neutral systems is studied. First, the set of all stabilizing controllers is derived. Second, we prove that a large class of fractional controllers with delays cannot eliminate in the closed loop chains of poles asymptotic to the imaginary axis if such chains are present in the controlled systems.

In recent years, there has been increasing interest in studying time-delayed coupled networks of oscillators since these occur in many real life applications. In many cases symmetry, patterns can emerge in these networks; as a consequence, a part of the system might repeat itself, and properties of this symmetric subsystem represent the whole dynamics. In this thesis, an analysis of a second order N-node time-delay fully connected network is made. This study is carried out using symmetry groups. The existence of multiple eigenvalues forced by symmetry is shown, as well as the possibility of uncoupling the linearization at equilibria, into irreducible representations due to the symmetry. The existence of steady-state and Hopf bifurcations in each irreducible representation is also proved. Three different models are used to analyze the network dynamics, namely, the full-phase, the phase, and the phase-difference model. A finite set of frequencies ω is also determined, which might correspond to Hopf bifurcations in each case for critical values of the delay. Although we restrict our attention to second order nodes, the results could be extended to higher order networks provided the time-delay in the connections between nodes remains equal. / Nos últimos anos, tem havido um crescente interesse em estudar redes de osciladores acopladas com retardo de tempo uma vez que estes ocorrem em muitas aplicações da vida real. Em muitos casos, simetria e padrões podem surgir nessas redes; em consequência, uma parte do sistema pode repetir-se, e as propriedades deste subsistema simétrico representam a dinâmica da rede toda. Nesta tese é feita uma análise de uma rede de N nós de segunda ordem totalmente conectada com atraso de tempo. Este estudo é realizado utilizando grupos de simetria. É mostrada a existência de múltiplos valores próprios forçados por simetria, bem como a possibilidade de desacoplamento da linearização no equilíbrio, em representações irredutíveis. É também provada a existência de bifurcações de estado estacionário e Hopf em cada representação irredutível. São usados três modelos diferentes para analisar a dinâmica da rede: o modelo de fase completa, o modelo de fase, e o modelo de diferença de fase. É também determinado um conjunto finito de frequências ω, que pode corresponder a bifurcações de Hopf em cada caso, para valores críticos do atraso. Apesar de restringir a nossa atenção para nós de segunda ordem, os resultados podem ser estendido para redes de ordem superior, desde que o tempo de atraso nas conexões entre nós permanece igual.

Dans ce manuscrit, nous étudions le retard distribué et ses applications. Nous présentons la définition du retard distribué et l'étude de ses propriétés. Nous expliquons aussi le problème de la mise en œuvre du retard distribué et donnons une méthode générale pour son approximation. Ensuite nous présentons trois applications à l'aide de retard distribué, qui sont différentes avec les applications apparaissent dans la littérature. Le premier est l'inversion stable et le modèle appariement. Nous introduisons une nouvelle classe d'inversion stable et modèle d'appariement pour les systèmes linéaires de dimension finie invariables dans le temps. L'inversion stable (resp. modèle d'appariement) est une sorte d'inversion (resp. modèle d'appariement) de rapprochement. En fait, nous obtenons l'inversion exacte (resp. exacte modèle d'appariement) après un temps t = h, où le temps t = h peut être choisi arbitrairement. La deuxième application est le contrôle de la stabilité et du pôle placement finie pour une classe de système de dimension infinie. La dernière application du retard distribué est la synthèse de l'observateur pour l'estimation ou la commande de sortie. Nous craignons seulement avec les systèmes linéaires de dimension finie. Nous introduisons une boucle fermée observateurs sans mémoire par injection d'entrée. Convergence asymptotique ainsi que la convergence en temps fini de l'estimation sont analysés par injection de sortie et des informations d'entrée via retard distribué. Enfin, nous introduisons une nouvelle classe de l’approximation des systèmes à paramètres distribués. Nous travaillons sur la topologie du graphe, et montrons que sous certaines hypothèses faibles, une telle approximation peut être réalisé en utilisant retard distribué. / A distributed delay is a linear input-output operators and appears in many control problems. We investigate distributed delay and its applications. After introducing the definition and the main properties of the distributed delay, the numerical implementation problem of distributed delays is analyzed and a general method for its approximation is given. Then three applications are focused on where distributed delay appears. The first application is the stable inversion and model matching. A new class of stable inversion and model matching problem for finite dimensional linear time-invariant systems is defined. The stable inversion (resp. model matching) is an approximation of the inverse of a given model (resp. model matching), where exact inversion (resp. exact matching) is reached after a time $t=h$, which is a parameter of our procedure. The second application is concerned with stabilization and finite spectrum assignment for a class of infinite dimensional systems. The last application concerns observer synthesis for estimation or output control. For linear finite dimensional systems. A closed-loop memoryless observer by input injection is introduced. Asymptotic convergence as well as finite time convergence of the estimation are analyzed by output injection and input information via distributed delay. At last, we introduce a new class for approximation of distributed parameter systems. We work on the graph topology, and show that under some weak assumptions, such an approximation can be realized using distributed delay.

In recent years, there has been increasing interest in studying time-delayed coupled networks of oscillators since these occur in many real life applications. In many cases symmetry, patterns can emerge in these networks; as a consequence, a part of the system might repeat itself, and properties of this symmetric subsystem represent the whole dynamics. In this thesis, an analysis of a second order N-node time-delay fully connected network is made. This study is carried out using symmetry groups. The existence of multiple eigenvalues forced by symmetry is shown, as well as the possibility of uncoupling the linearization at equilibria, into irreducible representations due to the symmetry. The existence of steady-state and Hopf bifurcations in each irreducible representation is also proved. Three different models are used to analyze the network dynamics, namely, the full-phase, the phase, and the phase-difference model. A finite set of frequencies ω is also determined, which might correspond to Hopf bifurcations in each case for critical values of the delay. Although we restrict our attention to second order nodes, the results could be extended to higher order networks provided the time-delay in the connections between nodes remains equal. / Nos últimos anos, tem havido um crescente interesse em estudar redes de osciladores acopladas com retardo de tempo uma vez que estes ocorrem em muitas aplicações da vida real. Em muitos casos, simetria e padrões podem surgir nessas redes; em consequência, uma parte do sistema pode repetir-se, e as propriedades deste subsistema simétrico representam a dinâmica da rede toda. Nesta tese é feita uma análise de uma rede de N nós de segunda ordem totalmente conectada com atraso de tempo. Este estudo é realizado utilizando grupos de simetria. É mostrada a existência de múltiplos valores próprios forçados por simetria, bem como a possibilidade de desacoplamento da linearização no equilíbrio, em representações irredutíveis. É também provada a existência de bifurcações de estado estacionário e Hopf em cada representação irredutível. São usados três modelos diferentes para analisar a dinâmica da rede: o modelo de fase completa, o modelo de fase, e o modelo de diferença de fase. É também determinado um conjunto finito de frequências ω, que pode corresponder a bifurcações de Hopf em cada caso, para valores críticos do atraso. Apesar de restringir a nossa atenção para nós de segunda ordem, os resultados podem ser estendido para redes de ordem superior, desde que o tempo de atraso nas conexões entre nós permanece igual.

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