Les méthodologies de l'automatique ont joué au cours des dernières décennies un ´r^ole essentiel au sein de nombreux secteurs technologiques avancées. Cependant, de nombreuse questions restent ouvertes. Parmi celles-ci, celles concernant la stabilité et la stabilisation de systèmes non linéaires sont d'intérêt primordial. Afin de stabilizer un système (physique ou non), il est nécessaire de capter et interpreter en temps réel les informations hétérogènes caractérisant son fonctionnement afin intervenir efficacement. Actuellement ces informations ne sont pas captées en temps continu, mais de façon synchrone ou asynchrone et ceci est valable aussi pour les actuateurs. De façon très naturelle, on définit donc un système hybride, caractérisé par des dynamiques à la fois discrètes et continues. Dans ce contexte, cette thèse est orientée au développement de nouvelles méthodologies pour la stabilisation de systèmes échantillonnés non linéaires en se focalisant sur la stabilisation de formes cascades qui se retrouvent dans de nombreuse situations concretes. Pour cela, on étudiera l'effet de l'échantillonnage sur les propriétés de la dynamique continue et l'on proposera des méthodologies pour la conception de lois de commande qui ne requièrent pas d'assumptions supplémentaires au cas continu.Enfin, on étudiera l'effet de l'échantillonnage sur des systèmes présentant de retards sur les entrées. On développera des lois de commande stabilisantes exploitant la structure en cascade induite par l'échantillonnage. Des exemples académiques illustreront les calcules des solutions et leur performances tout au long du manuscript. / Over the last decades the methodologies of dynamical systems and control theory have been playing an increasingly relevant role in a lot of situations of practical interest. Though, a lot of theoretical problem still remain unsolved. Among all, the ones concerning stability and stabilization are of paramount importance. In order to stabilize a physical (or not) system, it is necessary to acquire and interpret heterogeneous information on its behavior in order to correctly intervene on it. In general, those information are not available through a continuous flow but are provided in a synchronous or asynchronous way. This issue has to be unavoidably taken into account for the design of the control action. In a very natural way, all those heterogeneities define an hybrid system characterized by both continuous and discrete dynamics. This thesis is contextualized in this framework and aimed at proposing new methodologies for the stabilization of sampled-data nonlinear systems with focus toward the stabilization of cascade dynamics. In doing so, we shall propose a small number of tools for constructing sampled-data feedback laws stabilizing the origin of sampled-data nonlinear systems admitting cascade interconnection representations. To this end, we shall investigate on the effect of sampling on the properties of the continuous-time system while enhancing design procedures requiring no extra assumptions over the sampled-data equivalent model.Finally, we shall show the way sampling positively affects nonlinear retarded dynamics affected by a fixed and known time-delay over the input signal by enforcing on the implicit cascade representation the sampling process induces onto the retarded system. Academic examples will illustrate the computational aspects together with their performances throughout the whole manuscript.

The work in this dissertation concerns the construction of a robust iterative learning control (ILC) algorithm for a class of systems characterized by measurement delays, parametric uncertainty, and linear parameter varying (LPV) dynamics. One example of such a system is the twin roll strip casting process, which provides a practical motivation for this research. I propose three ILC algorithms in this dissertation that advance the state of the art. The first algorithm compensates for measurement delays that are longer than a single iteration of a periodic process. I divide the delay into an iterative and residual component and show how each component effects the asymptotic stability properties of the ILC algorithm. The second algorithm is a coupled delay estimation and ILC algorithm that compensates for time-varying measurement delays. I use an adaptive delay estimation algorithm to force the delay estimate to converge to the true delay and provide stability conditions for the coupled delay estimation and ILC algorithm. The final algorithm is a norm optimal ILC algorithm that compensates for LPV dynamics as well as parametric uncertainty and time delay estimation error. I provide a tuning method for the cost function weight matrices based on a sufficient condition for robust convergence and an upper bound on the norm of the error signal. The functionality of all three algorithms is demonstrated through simulated case studies based on an identified system model of the the twin roll strip casting process. The simulation testing is also augmented with experimental testing of select algorithms through collaboration with an industrial sponsor.

Cette thèse est dédiée à l’analyse du retard (de calcul ou induit par la communication), qui représente un des paramètres sensibles, et qui doit être pris en compte, pour la mise en œuvre de la Commande Prédictive en temps réel d’un processus dynamique. Dans la première partie, nous avons abordé le problème d’existence des ensembles D-invariants et avons fourni par la suite des conditions nécessaires et/ou suffisantes pour l’existence de ces ensembles. En outre, nous avons détaillé quelques nouvelles idées sur la construction des ensembles D-invariants en utilisant des algorithmes itératifs et d’autres algorithmes basés sur des techniques d’optimisation à deux niveaux. La seconde partie a été consacrée à l’étude du problème de robustesse des systèmes linéaires discrets affectés par un retard variable en boucle fermée avec un contrôleur affine par morceaux défini sur une partition polyédrale de l’espace d’état. L’étude a porté sur l’analyse de la fragilité d’une telle loi commande en présence du retard dans la boucle. Nous avons décrit les marges d’invariance robustes définies comme étant le plus grand sous-ensemble de l’incertitude paramétrique pour lequel l’invariance positive est garantie par rapport à la dynamique en boucle fermée en présence du retard. La dernière partie de cette thèse s’est articulée autour de la conception des lois de commande prédictives avec un attention particulière aux modèles linéaires discrets décrivant des dynamiques affectées par des contraintes en présence du retard. Nous avons proposé plusieurs méthodes offrant différentes solutions au problème de stabilisation locale sans contrainte. Afin d’assurer la stabilité et de garantir la satisfaction des contraintes, nous avons exploité le concept d’invariance et à l’aide du formalisme "ensemble terminal-coût terminal", un problème d’optimisation a été formulé où les états sont forcés d’atteindre l’ensemble maximal admissible d’états retardés/D-invariant à la fin de l’horizon de prédiction. Enfin, nous avons étudié le problème de stabilisation des systèmes continus commandés en réseau soumis à des retards incertains et éventuellement variant dans le temps. Nous avons montré que les ensembles λ-D-contractifs peuvent être utilisés comme ensembles cibles où la stratégie de commande consiste en un simple problème de programmation linéaire ’LP’ qui peut être résolu en ligne. / The research conducted in this thesis has been focusing on Model Predictive Control (MPC) and the implication of network induced time-varying delays. We have addressed, in the first part of this manuscript, the existence problem and the algorithmic computation of positive invariant sets in the state space of the original discrete delay difference equation. The second part of these thesis has been devoted to the study of the robustness problem for a specific class of dynamical systems, namely the piecewise affine systems, defined over a polyhedral partition of the state space in the presence of variable input delay. The starting point was the construction of a predictive control law which guarantees the existence of a non-empty robust positive invariant set with respect to the closed-loop dynamic. The variable delay inducing in fact a model uncertainty, the objective was to describe the robust invariance margins defined as the largest subset of the parametric uncertainty for which the positive invariance is guaranteed with respect to the closed-loop dynamics in the presence of small and large delays. The last part has been dedicated to Model Predictive Control design with a specific attention to linear discrete time-delay models affected by input/state constraints. The starting point in the analysis was the design of a local stabilizing control law using different feedback structures. We proposed several design methods offering different solutions to the local unconstrained stabilization problem. In order to ensure stability and guarantee input and state constraints satisfaction of the moving horizon controller, the concept of positive invariance related to time-delay systems was exploited. Using the "terminal setterminal cost" design, the states were forced to attain the maximal delayed-state admissible set at the end of the prediction horizon. Finally, we have investigated the stabilization problem of Networked Control Systems ’NCSs’ subject to uncertain, possibly time-varying, network-induced delays. We showed that λ-D-contractive sets can be used as a target sets in a set induced Lyapunov function control fashion where a simple Linear Programming ’LP’ problem is required to be solved at each sampling instance.

No description available.

The high-dimensional property of delay differential equations makes them useful for various purposes. The applications of systems modelled with delay differential equations demand different degrees of complexity. One solution to tune this property is to make the dynamics of the current state dependent on more delayed states. How the system responds to more delayed states depends on the system under study, as both decreases and increases in the complexity were observed in different nonlinear systems. However, it is also known that when there is an infinite number of delays that follow a continuous distribution, simpler dynamics usually expected due to the averaging over previous states that the delay kernel provides. The present thesis investigates the role of multiple delays in nonlinear time delay systems, as well as methods for evaluating their complexity. Through the use of pseudospectral differentiation, we first compute the Lyapunov exponents of such multi-delay systems. In systems with a large number of delays, chaos is found to be less likely to occur. However, in systems with oscillatory feedback functions, the entropy can increase just by adding a few delays. Our study also demonstrates that the transition to simpler dynamics in nonlinear delay systems can be either monotonous or abrupt. This is particularly true in first-order nonlinear systems, where increasing the width of the distribution of delays results in complexity collapse, even in the presence of a few discrete delays. The roots of the characteristic equation around a fixed point can be used to approximate the degree of complexity of the dynamics of such time-delay systems, as they can be linked to other dynamical invariants such as the Kolmogorov-Sinai entropy. The phenomenon of complexity collapse uncovered in our work was further studied in an 80/20 ratio excitatory-inhibitory neural network. We found that the smaller the time delay, the higher the likelihood of chaotic dynamics, and this also promotes asynchronous spiking activity. But for larger values of the delay, the neurons show synchronized oscillatory spiking activity. A global inhibition at a longer delay results in a novel dynamical pattern of randomly occurring epochs of synchrony within the chaotic dynamics. The final part of the thesis examines the behavior of time delay reservoir computing when there are multiple time delays. It is shown that the choice of spacing between time delays is crucial, and depends on the task at hand. The system was studied for a prediction task with one chaotic input as well as for a mixture of two chaotic inputs. It was found that, similar to the single delay case, there is a resonance when the difference between delays is equal to the clock cycle. Together, our research provides valuable insights into the dynamics and complexity of nonlinear multi-delay systems and the importance of the spacing between delays.

Transit operators are increasingly interested in improving efficiency, reliability, and performance of commuter trains while reducing their operating costs. In this context, the application of optimal control theory to the problem of train control can help towards achieving some of these objectives. However, the traction and braking systems of commuter trains often exhibit significant time delays, making the control of commuter trains highly challenging. Previous literature on optimal train control ignores delays in actuation due to the inherent difficulty present in the optimal control, and in general, the control, of input-delay systems. In this thesis, optimal control of a commuter train is presented under two cases: (i) equal, and (ii) unequal time delays in the train traction and braking commands. The solution approach uses the economic model predictive control framework, which involves formulating and solving numerical optimization problems to achieve minimum mixed energy-time optimal control in discretized spatial and time domains. The optimization problems are re-solved repeatedly along the track for the remainder of the trip, using the latest sensor measurements. This would essentially establish a feedback mechanism in the control to improve robustness to modelling errors. A key feature of the proposed methods is that they are model-based controllers, they explicitly incorporate model information, including time delays, in controller synthesis hence avoiding performance degradation and potential instability. To address the issue of input-delays, the well-established predictor approach is used to compensate for input-delays. The case of equal traction-braking delays is treated in discretized spatial domain, which uses an already developed convex approximation to the optimization problem. The use of the convex approximation allows for robust and rapid computation of the optimal control solution. The non-equal traction-braking delays scenario is formulated in time domain, leading to a nonconvex optimization problem. An alternative formulation for minimum-time optimal control problems is presented for delay-free systems that simplifies the solution of minimum-time optimal control problems compared to conventional minimum-time optimal control formulations. This formulation along with the predictor approach is used to help solve the train optimal control problem in the case of non-equal traction-braking delays. The non-equal traction-braking delay controller is compared with the equal traction-braking delay controller by insertion of an artificial delay to make the shorter delays equal to the longer delay. Results of numerical simulations demonstrate the validity and effectiveness of the proposed controllers. / Thesis / Master of Applied Science (MASc)

<p>Nonlinear systems occur in industrial processes, economical systems, biotechnology and in many other areas. The thesis treats methods for system identification and control of such nonlinear systems, and applies the proposed methods to a solar heating/cooling plant. </p><p>Two applications, an anaerobic digestion process and a domestic solar heating system are first used to illustrate properties of an existing nonlinear recursive prediction error identification algorithm. In both cases, the accuracy of the obtained nonlinear black-box models are comparable to the results of application specific grey-box models. Next a convergence analysis is performed, where conditions for convergence are formulated. The results, together with the examples, indicate the need of a method for providing initial parameters for the nonlinear prediction error algorithm. Such a method is then suggested and shown to increase the usefulness of the prediction error algorithm, significantly decreasing the risk for convergence to suboptimal minimum points. </p><p>Next, the thesis treats model based control of systems with input signal dependent time delays. The approach taken is to develop a controller for systems with constant time delays, and embed it by input signal dependent resampling; the resampling acting as an interface between the system and the controller.</p><p>Finally a solar collector field for combined cooling and heating of office buildings is used to illustrate the system identification and control strategies discussed earlier in the thesis, the control objective being to control the solar collector output temperature. The system has nonlinear dynamic behavior and large flow dependent time delays. The simulated evaluation using measured disturbances confirm that the controller works as intended. A significant reduction of the impact of variations in solar radiation on the collector outlet temperature is achieved, though the limited control range of the system itself prevents full exploitation of the proposed feedforward control. The methods and results contribute to a better utilization of solar power.</p>

Os controladores PID são largamente utilizados em processos industriais. Recentemente, novos resultados de projeto de controladores PID foram obtidos para sistemas lineares invariantes no tempo sem atraso. Entretanto, o comportamento de diversas plantas industriais pode ser descrito matematicamente por sistemas lineares invariantes no tempo com atraso. O problema de estabilidade de sistemas lineares invariantes no tempo com atraso envolve encontrar a locação de raízes de funções transcedentais. Neste trabalho, o teorema de Hermite-Biehler é usado para estabelecer resultados para o projeto de controladores PID para uma classe de sistemas lineares com atraso. Usando a propriedade de entrelaçamento em altas freqüências da classe estudada e programação linear obtém-se o conjunto de todos os controladores PID. Até onde se sabe resultados anteriores de síntese de controladores PID envolvem a solução de equações transcendentais. / The PID controller is widely used in industrial processes. Recently, new results on the design of PID controllers for invariant time linear systems without time delays have appeared. However, the dynamic behavior of many industrial plants may be mathematically described by linear time invariant systems with time delays. The problem of stability of linear time invariant systems with time delays involves finding the location of roots of transcendental functions. In this work, the Hermite-Biehler theorem is used to establish results on the design of proportional plus integral plus derivative (PID) controllers for a class of time delay systems. Using the property of interlacing at high frequencies of the class of systems considered and linear programming we obtain the set of all stabilizing PID controllers. As far as we know, previous results on the synthesis of PID controllers rely on the solution of transcendental equations.

Le travail développé dans ce manuscrit porte sur la synchronisation des systèmes chaotiques. Il est articulé autour de deux axes principaux: la synthèse d'observateur et la détection de mode. Dans un premier temps, quelques rappels sur le chaos et les principales architectures de systèmes de chiffrement chaotiques sont effectués. Ensuite, nous montrons comment les systèmes chaotiques à non linéarité polynomiale ou affines à commutation peuvent se réécrire sous forme LPV polytopique. Une revue des principaux résultats sur la synthèse d'observateurs LPV polytopiques reposant sur l'utilisation des LMI est faite. Une extension des résultats aux observateurs polytopiques à entrées inconnues, à la fois dans le cas déterministe, bruité ou incertain est proposée. Ces observateurs assurent la synchronisation du chaos et donc le déchiffrement dans les systèmes de chiffrement "modulation paramétrique", "commutation chaotique", "transmission à deux canaux" et "chiffrement par inclusion". Pour les systèmes affines à commutation utilisés en tant que générateur du chaos, le cas où l'état discret n'est pas accessible est considéré. Une présentation unifiée des méthodes fondées sur les espaces de parité, proposées dans la littérature pour les systèmes linéaires et affines à commutation à temps discret, est réalisée. Le problème de discernabilité fait l'objet d'une étude approfondie. Une approche pour estimer les retards variables des systèmes affines et affines à commutation à temps discret, formulée en termes de détection de mode, est proposée en tant que solution à l'estimation de retard pour le chiffrement par injection de retard / The work developed in this manuscript addresses the synchronization of chaotic systems. It is organized around two main axes: the observer synthesis and the mode detection. In a first step, we recall the main architectures of chaotic encryption systems and show how chaotic systems with polynomial nonlinearities or switched affine dynamics can be rewritten in a polytopic LPV form. A review of the main LMI based results for polytopic LPV observers synthesis is made. An extension to polytopic unknown input observers, both in the deterministic case and noisy or uncertain case, is proposed. These observers ensure chaos synchronization and information recovering in the framework of the following encryption systems: "parametric modulation", "chaotic switching", "two channels transmission" and "inclusion encryption". For affine switched systems used as a generator of chaos, the case where the discrete state is not available is considered. A unified presentation of mode detection methods based on parity spaces proposed in the literature for linear and affine switched discrete time systems is proposed. The problem of discernibility is the subject of a complete study. An approach to estimate time varying delays for affine switched discrete time systems, formulated in terms of mode detection, is proposed as a solution for delay injection encryption

Many dynamical systems in nature and technology involve delays in the interaction of variables forming the system. Furthermore, many of such systems involve external inputs or perturbations which might force the system to have arbitrary initial function. The conventional way to model these systems is using delay differential equations (DDE). However, DDEs with arbitrary initial functions has serious problems for finding analytical and computational solutions. This fact is a strong motivation for considering abstractions and approximations for dynamical systems involving delay. In this thesis, the piecewise linear systems with delay on piecewise constant part which is a useful subclass of hybrid dynamical systems is studied. We introduced various representations of these systems and studied the state transition conditions. We showed that there exists fixed point and periodic stable solutions. We modelled the genomic regulation of fission yeast cell cycle. We discussed various potential uses including approximating the DDEs and finally we concluded.