There is often freedom in choosing the location of actuators on systems governed by partial differential equations. The actuator locations should be selected in order to optimize the performance criterion of interest. The main focus of this thesis is to consider H-∞-performance with state-feedback. That is, both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output of a full-information plant. Optimal H-∞-disturbance attenuation as a function of actuator location is used as the cost function. It is shown that the corresponding actuator location problem is well-posed. In practice, approximations are used to determine the optimal actuator location. Conditions for the convergence of optimal performance and the corresponding actuator location to the exact performance and location are provided. Examples are provided to illustrate that convergence may fail when these conditions are not satisfied. Systems of large model order arise in a number of situations; including approximation of partial differential equation models and power systems. The system descriptions are sparse when given in descriptor form but not when converted to standard first-order form. Numerical calculation of H-∞-attenuation involves iteratively solving large H-∞-algebraic Riccati equations (H-∞-AREs) given in the descriptor form. An iterative algorithm that preserves the sparsity of the system description to calculate the solutions of large H-∞-AREs is proposed. It is shown that the performance of our proposed algorithm is similar to a Schur method in many cases. However, on several examples, our algorithm is both faster and more accurate than other methods. The calculation of H-∞-optimal actuator locations is an additional layer of optimization over the calculation of optimal attenuation. An optimization algorithm to calculate H-∞-optimal actuator locations using a derivative-free method is proposed. The results are illustrated using several examples motivated by partial differential equation models that arise in control of vibration and diffusion.

There is often freedom in choosing the location of actuators on systems governed by partial differential equations. The actuator locations should be selected in order to optimize the performance criterion of interest. The main focus of this thesis is to consider H-∞-performance with state-feedback. That is, both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output of a full-information plant. Optimal H-∞-disturbance attenuation as a function of actuator location is used as the cost function. It is shown that the corresponding actuator location problem is well-posed. In practice, approximations are used to determine the optimal actuator location. Conditions for the convergence of optimal performance and the corresponding actuator location to the exact performance and location are provided. Examples are provided to illustrate that convergence may fail when these conditions are not satisfied. Systems of large model order arise in a number of situations; including approximation of partial differential equation models and power systems. The system descriptions are sparse when given in descriptor form but not when converted to standard first-order form. Numerical calculation of H-∞-attenuation involves iteratively solving large H-∞-algebraic Riccati equations (H-∞-AREs) given in the descriptor form. An iterative algorithm that preserves the sparsity of the system description to calculate the solutions of large H-∞-AREs is proposed. It is shown that the performance of our proposed algorithm is similar to a Schur method in many cases. However, on several examples, our algorithm is both faster and more accurate than other methods. The calculation of H-∞-optimal actuator locations is an additional layer of optimization over the calculation of optimal attenuation. An optimization algorithm to calculate H-∞-optimal actuator locations using a derivative-free method is proposed. The results are illustrated using several examples motivated by partial differential equation models that arise in control of vibration and diffusion.

Tout phénomène, qu’il soit biologique, géologique ou mécanique peut être décrit à l’aide de lois de la physique en termes d’équations différentielles, algébriques ou intégrales, mettant en relation différentes variables physiques. Les objectifs de la thèse sont de montrer comment les systèmes à paramètres distribués peuvent être modélisés par un modèle bond graph, qui est par nature un modèle à paramètres localisés. Deux approches sont possibles : - utiliser une technique d’approximation qui discrétise le modèle initialement sous forme d’équations aux dérivées partielles (EDP) dans le domaine spatial, en supposant que les phénomènes physiques distribués peuvent être considérés comme homogènes dans certaines parties de l’espace, donc localisés. - déterminer la solution des EDP qui dépend du temps et de l’espace, puis à approximer cette solution avec différents outils numériques. Le premier chapitre rappelle quelques méthodes classiques utilisées pour l’approximation des EDP et les modèles bond graphs correspondants.Dans le deuxième chapitre, l’approche port-Hamiltonienne est présentée et son extension aux systèmes à paramètres distribués est proposée. Dans le troisième chapitre, les principaux modèles utilisés pour la représentation des flux de trafic routier sont rappelés et mis en œuvre en simulation. Ceci conduit à des comparaisons, d’une part entre différentes méthodes de résolution numérique et d’autre part entre différents modèles. Dans le quatrième chapitre, une approche originale propose d’étendre la représentation bond graph issue de la méthodologie Computational Fluid Dynamics au flux de trafic, en utilisant un modèle EDP à deux équations proposé par Jiang / Virtually every phenomenon in nature, whether biological, geological, or mechanical, can be described with the aid of the laws of physics, in terms of algebraic, differential, or integral equations relating various quantities of interest. The objectives of the thesis were to show how distributed parameter systems can be modeled using a bond graph model, which is by its nature itself a lumped parameter model. Two ways are possible :- using an approximation technique to discretize the model in the space domain, assuming that physical distributed phenomena can be considered as homogenous in some parts of space, and thus lumped. Different bond graph models can be obtained depending on the technique used.- determining a solution of the PDE depending on space and time, and thus to approximate this solution by means of different kinds of tools.In chapter 1, some classical methods used for approximation of partial differential equations are recalled and the corresponding bond graph model is designed. For each of them advantages and drawbacks are presented.In the second chapter, the port-Hamiltonian approach for distributed parameter system is presented, and a new result is proposed for telegrapher’s equation solving.In the third chapter, the main models used for traffic flow representation are presented and some of them are implemented in simulation. A comparison is done on one hand on different numerical methods applied on the first class of models (1-eq. model) and on the other hand between 1-equation and 2- equation models.In chapter 4, we have proposed an original approach extending Computational Fluid Dynamics bond graph representation to traffic flow, using Jiang’s model

Dans cette thèse, on s'intéresse au contrôle du profil de facteur de sécurité dans un plasma tokamak. Cette variable physique est liée à plusieurs phénomènes dans le plasma, en particulier des instabilités magnétohydrodynamiques (MHD). Un profil de facteur de sécurité adéquat est particulièrement important pour avoir des modes d'opération avancés dans le tokamak, avec haut confinement et stabilité MHD. Pour cela faire, on se focalise sur la commande du gradient du profil de flux magnétique poloidal dans le tokamak. L'évolution de cette variable est donnée par une équation de diffusion avec des coefficients distribuées et temps-variants. En utilisant des techniques de type Lyapunov et les propriétés de stabilité entrée-état du système on propose une loi de commande robuste qui prend en compte des contraintes non-linéaires dans l'action imposées par la physique des actionneurs. / In this thesis, we are interested in the control of the safety factor profile or q-profile in a tokamak plasma. This physical quantity has been found to be related to several phenomena in the plasma, in particular magnetohydrodynamic (MHD) instabilities. Having an adequate safety factor profile is particularly important to achieve advanced tokamak operation, providing high confinement and MHD stability. To achieve this, we focus in controlling the gradient of the poloidal magnetic flux profile. The evolution of this variable is given by a diffusion equation with distributed time-varying coefficients. Based on Lyapunov techniques and the Input-to-State stability properties of the system we propose a robust control law that takes into account nonlinear constraints on the control action imposed by the physical actuators.

Distributed parameter systems (DPS) comprise an important class of engineering systems ranging from "traditional" such as tubular reactors, to cutting edge processes such as nano-scale coatings. DPS have been studied extensively and significant advances have been noted, enabling their accurate simulation. To this end a variety of tools have been developed. However, extending these advances for systems design is not a trivial task . Rigorous design and operation policies entail systematic procedures for optimisation and control. These tasks are "upper-level" and utilize existing models and simulators. The higher the accuracy of the underlying models, the more the design procedure benefits. However, employing such models in the context of conventional algorithms may lead to inefficient formulations. The optimisation and control of DPS is a challenging task. These systems are typically discretised over a computational mesh, leading to large-scale problems. Handling the resulting large-scale systems may prove to be an intimidating task and requires special methodologies. Furthermore, it is often the case that the underlying physical phenomena span various temporal and spatial scales, thus complicating the analysis. Stiffness may also potentially be exhibited in the (nonlinear) models of such phenomena. The objective of this work is to design reliable and practical procedures for the optimisation and control of DPS. It has been observed in many systems of engineering interest that although they are described by infinite-dimensional Partial Differential Equations (PDEs) resulting in large discretisation problems, their behaviour has a finite number of significant components , as a result of their dissipative nature. This property has been exploited in various systematic model reduction techniques. Of key importance in this work is the identification of a low-dimensional dominant subspace for the system. This subspace is heuristically found to correspond to part of the eigenspectrum of the system and can therefore be identified efficiently using iterative matrix-free techniques. In this light, only low-dimensional Jacobians and Hessian matrices are involved in the formulation of the proposed algorithms, which are projections of the original matrices onto appropriate low-dimensional subspaces, computed efficiently with directional perturbations.The optimisation algorithm presented employs a 2-step projection scheme, firstly onto the dominant subspace of the system (corresponding to the right-most eigenvalues of the linearised system) and secondly onto the subspace of decision variables. This algorithm is inspired by reduced Hessian Sequential Quadratic Programming methods and therefore locates a local optimum of the nonlinear programming problem given by solving a sequence of reduced quadratic programming (QP) subproblems . This optimisation algorithm is appropriate for systems with a relatively small number of decision variables. Inequality constraints can be accommodated following a penalty-based strategy which aggregates all constraints using an appropriate function , or by employing a partial reduction technique in which only equality constraints are considered for the reduction and the inequalities are linearised and passed on to the QP subproblem . The control algorithm presented is based on the online adaptive construction of low-order linear models used in the context of a linear Model Predictive Control (MPC) algorithm , in which the discrete-time state-space model is recomputed at every sampling time in a receding horizon fashion. Successive linearisation around the current state on the closed-loop trajectory is combined with model reduction, resulting in an efficient procedure for the computation of reduced linearised models, projected onto the dominant subspace of the system. In this case, this subspace corresponds to the eigenvalues of largest magnitude of the discretised dynamical system. Control actions are computed from low-order QP problems solved efficiently online.The optimisation and control algorithms presented may employ input/output simulators (such as commercial packages) extending their use to upper-level tasks. They are also suitable for systems governed by microscopic rules, the equations of which do not exist in closed form. Illustrative case studies are presented, based on tubular reactor models, which exhibit rich parametric behaviour.

Cette thèse s’intéresse à la gestion décentralisée à l’aide des systèmes multi-agents de l’énergie, notamment de sources renouvelables, dans le contexte des réseaux électriques intelligents (smart grids). Nos travaux de recherche visent à minimiser la facture énergétique des consommateurs en se focalisant sur deux verrous essentiels que nous nous proposons de lever : (1) résoudre le problème de l’intermittence des énergies renouvelables; (2) minimiser les pertes d’énergie. Pour pallier le problème d’intermittence des énergies renouvelables et dans le but de maintenir un coût énergétique peu onéreux même lors des pics d’utilisation, nous avons intégré un système de stockage intelligent. Nous avons, en effet, proposé des algorithmes permettant d’utiliser les systèmes de stockage intelligents et la négociation multi-agents pour réduire la facture énergétique tout en conservant un taux de décharge minimal de la batterie et une perte énergétique minimale. La validation par simulation de nos contributions a montré que celles-ci répondent aux enjeux identifiés, notamment en réduisant le coût de l’énergie pour les consommateurs en comparaison aux travaux de l’état de l’art. / This thesis focuses on the decentralized management using multi-agent systems of energy, including renewable energy sources, in the smart grid context. Our research aims to minimize consumers’ energy bills by answering two key challenges: (1) handle the problem of intermittency of renewable energy sources; (2) reduce energy losses. To overcome the problem of renewable resources intermittency and in order to minimize energy costs even during peak hours, we integrated an intelligent storage system. To this end, we propose many algorithms in order to use intelligent storage systems and multi-agent negotiation algorithm to reduce energy cost while maintaining a minimal discharge rate of the battery and minimal energy loss. The validation of our contributions has shown that our proposals respond to the identified challenges, including reducing the cost of energy for consumers, in comparison to the state of the art.

L’objet de cette thèse est l’étude des propriétés de stabilité et contrôle pour des systèmes linéaires écrits à l’aide d’équations aux dérivées partielles (EDP) ou d’équations à retard. Nous souhaitons exploiter dans cette thèse les liens qui existent entre ces deux classes de systèmes de dimension infinie afin de développer une nouvelle approche permettant leur analyse. En effet dans plusieurs applications, il est possible de choisir l’un ou l’autre de ces deux types de systèmes pour modéliser la dynamique considérée. Par exemple, les phénomènes de congestion dans un réseau routier peuvent être modélisés à l’aide d’EDP de type transport [JKC], mais aussi par un modèle à retard distribué [MMN] ou encore à retard discret [SN]. On peut également renvoyer aux travaux de Krstic [K] sur la formulation d’un système à retard comme un système EDP. Ces deux classes de systèmes sont des cas particuliers de systèmes de dimension infinie, et contrairement aux cas de systèmes de dimension finie, on parle de fonctions d’état plutôt que vecteur d’état. Cela implique que l’analyse associée est plus délicate et fait appel à des outils dédiés. Dans le cadre de la thèse, l’étudiant se focalisera sur les approches basées sur une extension du théorème de Lyapunov pour les systèmes de dimension infinie utilisant des fonctionnelles spécifiques. Comme pour la modélisation, l’analyse de stabilité des systèmes à retard ou de type EDP peut être menée à l’aide de fonctionnelles de Lyapunov très similaires. Nous souhaitons que cette thèse tire parti des travaux existants dans les deux communautés sur les systèmes à retards et de type EDP pour développer une approche novatrice et unifiée pour l’analyse et le contrôle de systèmes de dimension infinie. Pour cela, le candidat s’appuiera sur ses acquis en automatique et en mathématiques ainsi que sur l’expertise des deux encadrants. Plusieurs contributions sont attendues durant la thèse. Dans un premier temps, il sera question d’étendre des résultats récents [SG1,2] développés pour l’analyse de stabilité des systèmes à retards au cas de systèmes régis par des EDP. Ces premiers résultats auront vocation à servir de base pour l’étude de la synthèse de commandes robustes dans le cadre d’applications telles que le contrôle de trafic routier [MMN], le contrôle de vibration [RBPA], etc… Cette thèse en automatique requiert plusieurs compétences parmi lesquelles des connaissances sur la théorie de Lyapunov pour les systèmes avec ou sans retard, sur les inégalités matricielles linéaires tout en s’appuyant sur les outils de mathématiques appliquées pour l’étude des équations aux dérivées partielles (algèbre linéaire, analyse fonctionnelle, espaces de Hilbert, de Sobolev). / The purpose of this thesis is the study of stability and control properties for linear systems described by partial differential equations (PDE) or delay differential equations. We wish to use in this thesis the relationship between these two classes of infinite-dimensional systems in view of developing a new paradigm for their analysis. Indeed, in many applications, it is possible to choose one or the other of these two classes of systems to model the dynamics of the system under consideration. For example, traffic flow can be modeled using PDE type of transportation [JKC], but also by a distributed delay model [SMP] or discrete delay [SN]. We may also refer to the work of Krstic [K] on the formulation of a delay system as an PDE system. These two classes of systems are special cases of infinite dimensional systems, unlike the case of finite-dimensional systems, we better called state functions rather than the state vector. This implies that the analysis is more delicate and refers to the use of dedicated tools. As part of the thesis, the student will focus on approaches based on an extension of Lyapunov theorem for infinite dimensional systems using specific functional. As for the modeling process, the stability analysis of delayed or PDE type systems can be conducted using very similar Lyapunov functionals. We hope that this thesis builds on existing work in the two communities on delay systems and PDE to develop an innovative and unified approach to the analysis and control of infinite dimensional systems. To do so, the candidate will build on its skills in automatic and mathematics as well as the on from expertise of both supervisors. Several contributions are expected during the thesis . Initially, we aim at extedning recent results [SG13,14] developed in the context of the stability analysis of delay systems to the case of systems governed by PDE. These first results will provide the basis for the design of robust control laws for various applications including traffic control, vibration control, etc ... Cette thèse portera sur l’étude des propriétés de stabilité et de contrôle des systèmes linéaires de dimension infinie, plus particulièrement écrits à l’aide d’EDP ou d’équations à retard. L’intérêt naturel pour l’étude de cette classe de systèmes à la frontière entre mathématiques appliquées et automatique connaît un succès grandissant de part la large gamme d’applications en contrôle pouvant être décrites par ces modèles : en ingénierie, biologie, informatique… L’émulation scientifique entre systèmes à retard et systèmes de type EDP permettra en outre à cette thèse de tirer parti des méthodes et outils propres à chacun des ces domaines. This PhD proposal in automatic control requires several skills including knowledge on Lyapunov theory for systems with or without delay , on linear matrix inequalities while relying on mathematical tools applied in the study of partial differential equations ( linear algebra functional analysis , Hilbert spaces , Sobolev) .

The main objective of this work is to demonstrate some new computational methods for estimation, optimization and modeling of dynamical systems that use automatic differentiation. Particular focus will be upon dynamical systems arising in Aerospace Engineering. Automatic differentiation is a recursive computational algorithm, which enables computation of analytically rigorous partial derivatives of any user-specified function. All associated computations occur, in the background without user intervention, as the name implies. The computational methods of this dissertation are enabled by a new automatic differentiation tool, OCEA (Object oriented Coordinate Embedding Method). OCEA has been recently developed and makes possible efficient computation and evaluation of partial derivatives with minimal user coding. The key results in this dissertation details the use of OCEA through a number of computational studies in estimation and dynamical modeling. Several prototype problems are studied in order to evaluate judicious ways to use OCEA. Additionally, new solution methods are introduced in order to ascertain the extended capability of this new computational tool. Computational tradeoffs are studied in detail by looking at a number of different applications in the areas of estimation, dynamical system modeling, and validation of solution accuracy for complex dynamical systems. The results of these computational studies provide new insights and indicate the future potential of OCEA in its further development.

Le retard est un phénomène naturel présent dans la majorité des systèmes physiques et dans les applications d’ingénierie, ainsi, les systèmes à retard ont été un domaine de recherche très actif en automatique durant les 60 dernières années. La conception d’observateur est un des sujets les plus importants qui a été étudié, ceci est dû à l’importance des observateurs en automatique et dans les systèmes de commande en absence de capteur pour mesurer une variable. Dans ce travail, l’objectif principal est de concevoir des observateurs pour différentes classes de systèmes à retard avec un retard arbitrairement large, et ce en utilisant différentes approches. Dans la première partie de cette thèse, la conception d’un observateur a été réalisée pour une classe de systèmes non linéaires triangulaires avec une sortie échantillonnée et un retard arbitraire. Une l’autre difficulté majeure avec cette classe de systèmes est le fait que la matrice d’état dépend du signal de sortie non-retardé qui est immesurable. Un nouvel observateur en chaine, composé de sous-observateurs en série est conçu pour compenser les retards arbitrairement larges. Dans la seconde partie de ce travail, un nouvel observateur a été conçu pour un autre type de systèmes non linéaires triangulaires, où le retard a été considéré, cette fois-ci, comme une équation aux dérivées partielles de type hyperbolique du premier ordre. La transformation inverse en backstepping et le concept de l’observateur en chaine ont été utilisés lors de la conception de cet observateur afin d’assurer son efficacité en cas de grands retards. Dans la dernière partie de cette thèse, la conception d’un nouvel observateur a été réalisée pour un type de système modélisé par des équations paraboliques non linéaires où les mesures sont issues d’un nombre fini de points du domaine spatial. Cet observateur est constitué d’une série de sous-observateurs en chaine. Chaque sous-observateur compense une fraction du retard global. L'analyse de la stabilité des systèmes d’erreur a été fondée sur différentes fonctionnelles Lyapunov-Krasovskii. Par ailleurs, différents instruments mathématiques ont été employés au cours des différentes preuves présentées. Les résultats de simulation ont été présentés dans le but de confirmer l'exactitude des résultats théoriques / Time-delay is a natural phenomenon that is present in most physical systems and engineering applications, thus, delay systems have been an active area of research in control engineering for more than 60 years. Observer design is one of the most important subject that has been dealt with, this is due to the importance of observers in control engineering systems not only when sensing is not sufficient but also when a sensing reliability is needed. In this work, the main goal was to design observers for different classes of nonlinear delayed systems with an arbitrary large delay, using different approaches. In the first part, the problem of observer design is addressed for a class of triangular nonlinear systems with not necessarily small delay and sampled output measurements. Another major difficulty with this class of systems is the fact that the state matrix is dependent on the un-delayed output signal which is not accessible to measurement. A new chain observer, composed of sub-observers in series, is designed to compensate for output sampling and arbitrary large delays.In the second part of this work, another kind of triangular nonlinear delayed systems was considered, where this time the delay was considered as a first order hyperbolic partial differential equation. The inverse backstepping transformation was invoked and a chain observer was developed to ensure its effectiveness in case of large delays. Finally, a new observer was designed for a class of nonlinear parabolic partial differential equations under point measurements, in the case of large delays. The observer was composed of several chained sub-observers. Each sub-observer compensates a fraction of the global delay. The stability analyses of the error systems were based on different Lyapunov-Krasovskii functionals. Also different mathematical tools have been used in order to prove the results. Simulation results were presented to confirm the accuracy of the theoretical results

<p>Predmet istraživanja je identifikaciji i adaptivna estimacija<br />parametara široke klase linearnih sistema. Predloženi algoritmi<br />za adaptivnu estimaciju parametara su primenjivi na sisteme koji se<br />opisuju funkcijama prenosa proizvoljnog oblika, što uključuje sisteme<br />sa kašnjenjem, distribuiranim parametrima, frakcione sisteme i<br />druge sisteme opisane iracionalnim funkcijama prenosa. Na<br />posletku, dat je algoritam za identifikaciju CNG sistema koji se ne<br />izvršava u realnom vremenu i pretpostavlja da struktura modela nije<br />poznata unapred.</p> / <p>The subject of this research is the system identification and adaptive<br />parameter estimation in wide class of linear processes. Proposed<br />approaches for adaptive parameter estimation can be applied to systems<br />described by transfer functions of arbitrary form, including systems with<br />delay, distributed-paratemeter systems, fractional order systems, and other<br />system described by irrational transfer functions. In the final part, an offline<br />algorithm for identification of CNG system which does not assume any a<br />priori known model structure is proposed.</p>