Contributions à la stabilisation des systèmes à commutation affine

Cette thèse porte sur la stabilisation des systèmes à commutation dont la commande, le signal de commutation, est échantillonné de manière périodique. Les difficultés liées à cette classe de systèmes non linéaires sont d'abord dues au fait que l'action de contrôle est effectuée aux instants de calcul en sélectionnant le mode de commutation à activer et, ensuite, au problème de fournir une caractérisation précise de l'ensemble vers lequel convergent les solutions du système, c'est-à-dire l'attracteur. Dans cette thèse, les contributions ont pour fil conducteur la réduction du conservatisme fait pendant la définition d'attracteurs, ce qui a mené à garantir la stabilisation du système à un cycle limite. Après une introduction générale où sont présentés le contexte et les principaux résultats de la littérature, le premier chapitre contributif introduit une nouvelle méthode basée sur une nouvelle classe de fonctions de Lyapunov contrôlées qui fournit une caractérisation plus précise des ensembles invariants pour les systèmes en boucle fermée. La contribution présentée comme un problème d'optimisation non convexe et faisant référence à une condition de Lyapunov-Metzler apparaît comme un résultat préliminaire et une étape clé pour les chapitres à suivre. La deuxième partie traite de la stabilisation des systèmes affines commutés vers des cycles limites. Après avoir présenté quelques préliminaires sur les cycles limites hybrides et les notions dérivées telles que les cycles au Chapitre 3, les lois de commutation stabilisantes sont introduites dans le Chapitre 4. Une approche par fonctions de Lyapunov contrôlées et une stratégie de min-switching sont utilisées pour garantir que les solutions du système nominal en boucle fermée convergent vers un cycle limite. Les conditions du théorème sont exprimées en termes d'Inégalités Matricielles Linéaires (dont l'abréviation anglaise est LMI) simples, dont les conditions nécessaires sous-jacentes relâchent les conditions habituelles dans cette littérature. Cette méthode est étendue au cas des systèmes incertains dans le Chapitre 5, pour lesquels la notion de cycles limites doit être adaptée. Enfin, le cas des systèmes dynamiques hybrides est exploré au Chapitre 6 et les attracteurs ne sont plus caractérisés par des régions éventuellement disjointes mais par des trajectoires fermées et isolées en temps continu. Tout au long de la thèse, les résultats théoriques sont évalués sur des exemples académiques et démontrent le potentiel de la méthode par rapport à la littérature récente sur le sujet.This thesis deals with the stabilization of switched affine systems with a periodic sampled-data switching control. The particularities of this class of nonlinear systems are first related to the fact that the control action is performed at the computation times by selecting the switching mode to be activated and, second, to the problem of providing an accurate characterization of the set where the solutions to the system converge to, i.e. the attractors. The contributions reported in this thesis have as common thread to reduce the conservatism made in the characterization of attractors, leading to guarantee the stabilization of the system at a limit cycle. After a brief introduction presenting the context and some main results, the first contributive chapter introduced a new method based on a new class of control Lyapunov functions that provides a more accurate characterization of the invariant set for a closed-loop system. The contribution presented as a nonconvex optimization problem and referring to a Lyapunov-Metzler condition appears to be a preliminary result and the milestone of the forthcoming chapters. The second part deals with the stabilization of switched affine systems to limit cycles. After presenting some preliminaries on hybrid limit cycles and derived notions such as cycles in Chapter 3, stabilizing switching control laws are developed in Chapter 4. A control Lyapunov approach and a min-switching strategy are used to guarantee that the solutions to a nominal closed-loop system converge to a limit cycle. The conditions of the theorem are expressed in terms of simple linear matrix inequalities (LMI), whose underlying necessary conditions relax the usual one in this literature. This method is then extended to the case of uncertain systems in Chapter 5, for which the notion of limit cycle needs to be adapted. Finally, the hybrid dynamical system framework is explored in Chapter 6 and the attractors are no longer characterized by possibly disjoint regions but as continuous-time closed and isolated trajectory. All along the dissertation, the theoretical results are evaluated on academic examples and demonstrate the potential of the method over the recent literature on this subject

Contraction analysis of switched Filippov systems via regularization

We study incremental stability and convergence of switched (bimodal) Filippov systems via contraction analysis. In particular, by using results on regularization of switched dynamical systems, we derive sufficient conditions for convergence of any two trajectories of the Filippov system between each other within some region of interest. We then apply these conditions to the study of different classes of Filippov systems including piecewise smooth (PWS) systems, piecewise affine (PWA) systems and relay feedback systems. We show that contrary to previous approaches, our conditions allow the system to be studied in metrics other than the Euclidean norm. The theoretical results are illustrated by numerical simulations on a set of representative examples that confirm their effectiveness and ease of application.Comment: Preprint submitted to Automatic

Switching Quantum Dynamics for Fast Stabilization

Control strategies for dissipative preparation of target quantum states, both pure and mixed, and subspaces are obtained by switching between a set of available semigroup generators. We show that the class of problems of interest can be recast, from a control--theoretic perspective, into a switched-stabilization problem for linear dynamics. This is attained by a suitable affine transformation of the coherence-vector representation. In particular, we propose and compare stabilizing time-based and state-based switching rules for entangled state preparation, showing that the latter not only ensure faster convergence with respect to non-switching methods, but can designed so that they retain robustness with respect to initialization, as long as the target is a pure state or a subspace.Comment: 15 pages, 4 figure

Contributions to Control of Electronic Power Converters

This thesis deals with the control of electronic power converters. In its development two main parts have been differentiated. On the one hand, the problem of the voltage balance in the capacitors of the dc-link in a three-level NPC converter is addressed. On the other hand, given that the techniques used in the first part to model the converters need to make certain assumptions and, with the intention of avoiding averaged models, in the second part, switched affine models have been developed to design the control of the output voltage in DC-DC boost type converters. In this way, in the first part several control laws have been developed using an averaged model formulated by duty cycles for each level in each phase. This formulation allows to consider, in the controllers design stage, the degree of freedom associated with the homopolar voltage injection. Therefore, the controllers are designed as well as a part of the modulation, so that control and modulation are integrated in the same stage. In this way, three controllers have been designed where, apart from the objective of the voltage balance of the capacitors, other objectives such as the number of commutations or the quality of the output signal have also been improved. In the second part of the thesis, four methods have been developed for the design of control laws taking advantage of the modeling of converters as switched affine systems given their hybrid behaviour. Thus, the first two laws take advantage of this modeling using the delta operator to avoid numerical problems when using systems where the sampling time is very low. The first of these controllers is based on Lyapunov’s function while the second is independent of this function, thus obtaining less conservative results. The other two laws developed for switched affine systems use an alternative model to that performed in the first two controllers, so certain existing disadvantages are avoided using again a design not based on Lyapunov’s function. Thus, the first law presents a basic control but, even so, improves the results of other existing laws in the literature. Finally, a design method to deal with systems with variations in their parameters has been presented.La presente tesis trata sobre el control de convertidores electrónicos de potencia. En su desarrollo se han diferenciado dos partes principales. Por un lado, se trata el problema del balance de tensiones en los condensadores que forman el dc-link en un convertidor NPC de tres niveles. Por otro lado, dado que las técnicas utilizadas en la primera parte para modelar los convertidores necesitan realizar determinadas suposiciones y, con la intención de evitar modelos promediados, en la segunda parte se han desarrollado modelos afines conmutados para diseñar el control de la tensión de salida en convertidores DC-DC tipo boost. De esta forma, en la primera parte se han desarrollado varias leyes de control utilizando un modelo promediado formulado mediante ciclos de trabajo para cada nivel en cada fase. Esta formulación permite considerar en la fase de diseño de los controladores, un grado de libertad asociado a la inyección de tensión homopolar. Por lo tanto, se diseñan los controladores a la vez que una parte de la modulación, de forma que se integra control y modulación en una misma fase. De esta forma, se han diseñado tres controladores donde, a parte del objetivo de balancear la tensión de los condensadores, se ha ido buscando mejorar también otros objetivos como el número de conmutaciones o la calidad de la señal de salida. En la segunda parte de la tesis, se han desarrollado cuatro leyes de control aprovechando el modelado de convertidores como sistemas afines conmutados dada su naturaleza híbrida. De esta forma, las dos primeras leyes, aprovechan dicho modelado usando el operador delta para evitar problemas numéricos al utilizar sistemas donde el tiempo de muestreo es muy bajo. El primero de dichos controladores está basado en la función de Lyapunov mientras que el segundo es independiente de dicha función obteniendo así resultados menos conservadores. Las otras dos leyes desarrolladas para sistemas afines conmutados utilizan un modelado alternativo al realizado en las dos primeras, de forma que se evitan ciertas desventajas existentes y mantienen un diseño no basado en la función de Lyapunov. Así, la primera ley presenta un control más básico pero que, aun así, mejora los resultados de otras leyes existentes en la literatura. Por último, se ha presentado un procedimiento de diseño que hace frente a sistemas con variaciones en sus parámetros

Hybrid modeling and control of mechatronic systems using a piecewise affine dynamics approach

This thesis investigates the topic of modeling and control of PWA systems based on two experimental cases of an electrical and hydraulic nature with varying complexity that were also built, instrumented and evaluated. A full-order model has been created for both systems, including all dominant system dynamics and non-linearities. The unknown parameters and characteristics have been identi ed via an extensive parameter identi cation. In the following, the non-linear characteristics are linearized at several points, resulting in PWA models for each respective setup. Regarding the closed loop control of the generated models and corresponding experimental setups, a linear control structure comprised of integral error, feed-forward and state-feedback control has been used. Additionally, the hydraulic setup has been controlled in an autonomous hybrid position/force control mode, resulting in a switched system with each mode's dynamics being de ned by the previously derived PWA-based model in combination with the control structure and respective mode-dependent controller gains. The autonomous switch between control modes has been de ned by a switching event capable of consistently switching between modes in a deterministic manner despite the noise-a icted measurements. Several methods were used to obtain suitable controller gains, including optimization routines and pole placement. Validation of the system's fast and accurate response was obtained through simulations and experimental evaluation. The controlled system's local stability was proven for regions in state-space associated with operational points by using pole-zero analysis. The stability of the hybrid control approach was proven by using multiple Lyapunov functions for the investigated test scenarios.publishedVersio