La classe des systèmes non-linéaires dont la dynamique est définie par un champ de vecteurs polynomial est étudié. Des modèles polynomiaux peuvent représenter différents systèmes réels ou bien definir des approximations plus riches que des modèles linéaires pour des systèmes non-linéaires différentiables. Des techniques de programmation semi-définie développées récemment ont rendu possible l'étude de cette classe de systèmes avec des outils numériques. Le problème d'analyse en stabilité locale est résolu via des conditions basées sur la positivité de polynomes. Dans le cadre de la synthèse de lois de commande nous proposons un changement de variables linéaire pour traiter la synthèse de lois de commande non-linéaire qui garantissent la stabilité locale. Les ensembles définissant des estimations de la région d'attraction, définis par des courbes de niveau de la fonction de Lyapunov pour le système, sont également donnés par des fonctions polynomiales.

Este trabalho aborda o problema de seguimento e rejeição de sinais periódicos em sistemas lineares sujeitos a saturação nos atuadores. Para garantir o seguimento/rejeição, dois controladores baseados no princípio do modelo interno são considerados: o primeiro baseia-se no modelo interno em sua formulação clássica, isto é, um controlador dinâmico contendo um número finito de modos (marginalmente) instáveis da referência/perturbação é introduzido na malha de controle, em uma abordagem chamada de controladores ressonantes; a segunda abordagem considera o controlador repetitivo, onde um elemento de atraso é inserido namalha de controle emumlaço de realimentação positiva, fazendo o papel do modelo interno de ordem infinita. Nos dois casos, o objetivo principal é a obtenção de condições na forma de inequações matriciais lineares (do inglês, Linear Matrix Inequalities - LMIs) para a síntese simultânea de uma realimentação de estados estabilizante e do ganho do laço estático de anti-windup. Partindo do pressuposto que as referências e perturbações pertencem a um certo conjunto admissível, estes ganhos garantem que as trajetórias do sistema em malha fechada inciadas em um certo conjunto elipsoidal convergem para outro conjunto elipsoidal invariante contido na região de operação linear do sistema. Nesta região, a presença do modelo interno na malha de controle garante o seguimento e a rejeição dos sinais de interesse. Nas duas abordagens são propostos problemas de otimização visando a maximização do conjunto invariante de estados admissíveis e/ou a maximização do conjunto de referências/perturbações admissíveis. Extensões da metodologia para sistemas de tempo discreto também são apresentadas. / This work addresses the tracking/rejection problem of periodic signals for linear systems subject to control saturation. To ensure the tracking/rejection, two internal model based controllers are considered: the first one considers the internal model in a classical framework, i.e. a dynamic controller containing a finite number of (marginally) unstable modes of the reference/disturbance signal is introduced in the control loop. In this work, this approach is called resonant controller. The second approach considers the repetitive controller, where a delay element is introduced in the control loop in a positive feedback loop, playing the role of an infinite order internal model. In both cases, the main objective is to obtain conditions in the form of LMIs to simultaneously compute a stabilizing state feedback gain and an anti-windup gain. Assuming that the references and disturbances signals belong to a certain admissible set, these gains guarantee that the trajectories of the closed-loop system starting in a certain ellipsoidal set contract to another invariant ellipsoidal set inside the linearity region of the closed-loop system. In this region the presence of the internal model ensures tracking/rejection of the considered periodic signals. In both frameworks, optimization problems aiming at the maximization of the invariant set of admissible states and/or the maximization of the set of admissible references/disturbances are proposed. Extensions of the proposed framework to discrete-time systems are also presented.

Este trabalho aborda o problema de seguimento e rejeição de sinais periódicos em sistemas lineares sujeitos a saturação nos atuadores. Para garantir o seguimento/rejeição, dois controladores baseados no princípio do modelo interno são considerados: o primeiro baseia-se no modelo interno em sua formulação clássica, isto é, um controlador dinâmico contendo um número finito de modos (marginalmente) instáveis da referência/perturbação é introduzido na malha de controle, em uma abordagem chamada de controladores ressonantes; a segunda abordagem considera o controlador repetitivo, onde um elemento de atraso é inserido namalha de controle emumlaço de realimentação positiva, fazendo o papel do modelo interno de ordem infinita. Nos dois casos, o objetivo principal é a obtenção de condições na forma de inequações matriciais lineares (do inglês, Linear Matrix Inequalities - LMIs) para a síntese simultânea de uma realimentação de estados estabilizante e do ganho do laço estático de anti-windup. Partindo do pressuposto que as referências e perturbações pertencem a um certo conjunto admissível, estes ganhos garantem que as trajetórias do sistema em malha fechada inciadas em um certo conjunto elipsoidal convergem para outro conjunto elipsoidal invariante contido na região de operação linear do sistema. Nesta região, a presença do modelo interno na malha de controle garante o seguimento e a rejeição dos sinais de interesse. Nas duas abordagens são propostos problemas de otimização visando a maximização do conjunto invariante de estados admissíveis e/ou a maximização do conjunto de referências/perturbações admissíveis. Extensões da metodologia para sistemas de tempo discreto também são apresentadas. / This work addresses the tracking/rejection problem of periodic signals for linear systems subject to control saturation. To ensure the tracking/rejection, two internal model based controllers are considered: the first one considers the internal model in a classical framework, i.e. a dynamic controller containing a finite number of (marginally) unstable modes of the reference/disturbance signal is introduced in the control loop. In this work, this approach is called resonant controller. The second approach considers the repetitive controller, where a delay element is introduced in the control loop in a positive feedback loop, playing the role of an infinite order internal model. In both cases, the main objective is to obtain conditions in the form of LMIs to simultaneously compute a stabilizing state feedback gain and an anti-windup gain. Assuming that the references and disturbances signals belong to a certain admissible set, these gains guarantee that the trajectories of the closed-loop system starting in a certain ellipsoidal set contract to another invariant ellipsoidal set inside the linearity region of the closed-loop system. In this region the presence of the internal model ensures tracking/rejection of the considered periodic signals. In both frameworks, optimization problems aiming at the maximization of the invariant set of admissible states and/or the maximization of the set of admissible references/disturbances are proposed. Extensions of the proposed framework to discrete-time systems are also presented.

Este trabalho aborda o problema de seguimento e rejeição de sinais periódicos em sistemas lineares sujeitos a saturação nos atuadores. Para garantir o seguimento/rejeição, dois controladores baseados no princípio do modelo interno são considerados: o primeiro baseia-se no modelo interno em sua formulação clássica, isto é, um controlador dinâmico contendo um número finito de modos (marginalmente) instáveis da referência/perturbação é introduzido na malha de controle, em uma abordagem chamada de controladores ressonantes; a segunda abordagem considera o controlador repetitivo, onde um elemento de atraso é inserido namalha de controle emumlaço de realimentação positiva, fazendo o papel do modelo interno de ordem infinita. Nos dois casos, o objetivo principal é a obtenção de condições na forma de inequações matriciais lineares (do inglês, Linear Matrix Inequalities - LMIs) para a síntese simultânea de uma realimentação de estados estabilizante e do ganho do laço estático de anti-windup. Partindo do pressuposto que as referências e perturbações pertencem a um certo conjunto admissível, estes ganhos garantem que as trajetórias do sistema em malha fechada inciadas em um certo conjunto elipsoidal convergem para outro conjunto elipsoidal invariante contido na região de operação linear do sistema. Nesta região, a presença do modelo interno na malha de controle garante o seguimento e a rejeição dos sinais de interesse. Nas duas abordagens são propostos problemas de otimização visando a maximização do conjunto invariante de estados admissíveis e/ou a maximização do conjunto de referências/perturbações admissíveis. Extensões da metodologia para sistemas de tempo discreto também são apresentadas. / This work addresses the tracking/rejection problem of periodic signals for linear systems subject to control saturation. To ensure the tracking/rejection, two internal model based controllers are considered: the first one considers the internal model in a classical framework, i.e. a dynamic controller containing a finite number of (marginally) unstable modes of the reference/disturbance signal is introduced in the control loop. In this work, this approach is called resonant controller. The second approach considers the repetitive controller, where a delay element is introduced in the control loop in a positive feedback loop, playing the role of an infinite order internal model. In both cases, the main objective is to obtain conditions in the form of LMIs to simultaneously compute a stabilizing state feedback gain and an anti-windup gain. Assuming that the references and disturbances signals belong to a certain admissible set, these gains guarantee that the trajectories of the closed-loop system starting in a certain ellipsoidal set contract to another invariant ellipsoidal set inside the linearity region of the closed-loop system. In this region the presence of the internal model ensures tracking/rejection of the considered periodic signals. In both frameworks, optimization problems aiming at the maximization of the invariant set of admissible states and/or the maximization of the set of admissible references/disturbances are proposed. Extensions of the proposed framework to discrete-time systems are also presented.

Ce travail de thèse est consacré à l'extension de l'Inversion Dynamique non-linéaire (NDI-Nonlinear Dynamic Inversion) pour un ensemble plus grand de systèmes non-linéaires, tout en garantissant des conditions de stabilité suffisantes. La NDI a été étudiée dans le cas de diverses applications, y compris en aéronautique et en aérospatiale. Elle permet de calculer des lois de contrôle capables de linéariser et de découpler un modèle non-linéaire à tout point de fonctionnement de son enveloppe d'état. Cependant cette méthode est intrinsèquement non-robuste aux erreurs de modélisation et aux saturations en entrée. En outre, dans un contexte non-linéaire, l'obtention d'une garantie quantifiable du domaine de stabilité atteint reste à l'heure actuelle complexe. Contrairement aux approches classiques de la NDI, notre méthodologie peut être considérée comme un cadre de compensation non-linéaire généralisé qui permet d'intégrer les incertitudes et les saturations en entrée dans le processus de conception. En utilisant des stratégies de contrôle antiwindup, la loi de pilotage peut être calculée grâce à un simple processus en deux phases. Dans ce cadre de travail généralisé des transformations linéaires fractionnaires (LFT - Linear Fractional Transformations) de la boucle fermée non-linéaire peuvent être facilement déduites pour l'analyse de la stabilité robuste en utilisant des outils standards pour de systèmes linéaires. La méthode proposée est testée pour le pilotage d'un véhicule de rentrée atmosphérique de type aile delta lors de ses phases hypersonique, transsonique et subsonique. Pour cette thèse, un simulateur du vol incluant divers facteurs externes ainsi que des erreurs de modélisation a été développé dans Simulink. / This thesis work is devoted to extending Nonlinear Dynamic Inversion (NDI) for a large scale of nonlinear systems while guaranteeing sufficient stability conditions. NDI has been studied in a wide range of applications, including aeronautics and aerospace. It allows to compute nonlinear control laws able to decouple and linearize a model at any operating point of its state envelope. However, this method is inherently non-robust to modelling errors and input saturations. Moreover, obtaining a quantifiable guarantee of the attained stability domain in a nonlinear control context is not a very straightforward task. Unlike standard NDI approaches, our methodology can be viewed as a generalized nonlinear compensation framework which allows to incorporate uncertainties and input saturations in the design process. Paralleling anti-windup strategies, the controller can be computed through a single multichannel optimization problem or through a simple two-step process. Within this framework, linear fractional transformations of the nonlinear closed-loop can be easily derived for robust stability analysis using standard tools for linear systems. The proposed method is tested for the flight control of a delta wing type reentry vehicle at hypersonic, transonic and subsonic phases of the atmospheric reentry. For this thesis work, a Flight Mechanics simulator including diverse external factors and modelling errors was developed in Simulink.

Au cours du siècle dernier, la communauté scientifique a traité le contrôle des véhicules aériens principalement par l'élaboration de stratégies ad hoc, mais aucune approche unifiée n'a été développé jusqu'à présent. Cette thèse participe à l'élaboration d'une approche unifiée pour le contrôle des véhicules aériens en prenant en compte les forces aérodynamiques dans la conception de la commande. Nous supposons les effets aérodynamiques de rotation et les effets non stationnaires négligeables. Les actionneurs du véhicule sont supposés être composés d'une force de poussé fixée au corps pour le mouvement en translation, et d'un couple de contrôle pour la régulation d'attitude. Cette thèse se concentre ensuite sur la boucle de guidage, traitant du contrôle de la vitesse linéaire. L'un des principaux objectifs a été de déterminer la façon de réguler la force de poussée et l'orientation du véhicule pour compenser les forces extérieures. Tout d'abord nous abordons la modélisation, l'analyse et le contrôle de la dynamique longitudinale de l'avion. Ensuite nous étendons certaines de ces études aux mouvements tridimensionnels d'avions au corps symétrique, tels que les missiles. Un résultat original de cette thèse est de préciser les conditions sur la force aérodynamique permettant de reformuler le problème du contrôle dans celui de la commande d'un corps sphérique, pour lequel des résultats de stabilité peuvent être démontrés. Les lois de commande proposées intègrent des termes intégraux et anti-wind up sans reposer sur une politique de commutation entre plusieurs lois de commande. / Over the last century, the scientific community has dealt with the control of flying machines by mainly developing different strategies in relation to different classes of aircraft, and no unified control approach has been developed so far. The present thesis contributes towards the development of a unified control approach for aerial vehicles by maintaining aerodynamic forces in the control design. It is assumed, however, that the aerodynamic effects of rotational and unsteady motions are negligible, and that the means of actuation for an aerial vehicle consist of a body-fixed thrust force for translational motion and a control torque for attitude monitoring. This thesis then focuses on the guidance loop of the control problem. One of the main objectives has been to determine how to regulate the thrust intensity and the vehicle orientation to compensate for the orientation-dependent external forces. In particular, the modeling, analysis, and control of the longitudinal aircraft dynamics is first addressed. Then, some of these studies are extended to three-dimensional motions of symmetric aircraft, such as missile-like bodies. An original outcome of this thesis is to state conditions on the aerodynamic force that allow the control problem to be recasted into that of controlling a spherical body. In this case, strong stability results can be shown. The proposed control laws incorporate integral and anti-wind up terms and do not rely on a switching policy between several control laws.

Le contrôle automatique de véhicules sous-actionnés suscite depuis de nombreuses années un grand intérêt pour des applications diverses et variées. Cette thèse est consacrée au problème général du contrôle automatique de véhicules aériens, en particulier des véhicules à décollage et atterrissage vertical. Ce travail présente deux contributions théoriques. La première contribution concerne le développement d'une approche de commande générique pour une large classe de véhicules sous-actionnés. Cette approche exploite la structure d'actionnement commune à la plupart de véhicules conçus par l'homme, à savoir une seule commande en poussée dans une direction privilégiée du véhicule et un actionnement complet de la dynamique de rotation. La méthode de synthèse est conçue de façon incrémentale afin de traiter différents modes opérationnels: stabilisation de la direction de poussée, de la vitesse, ou de la position du véhicule. Une nouvelle technique d'intégrateur non-linéaire est proposée afin de garantir un comportement robuste vis-à-vis de perturbations extérieures ou d'erreurs de modèle. La seconde contribution concerne deux nouvelles méthodes d'estimation d'attitude du véhicule à partir de mesures fournies par une centrale inertielle et de mesures GPS. Les solutions proposées utilisent la mesure de vitesse linéaire pour estimer l'accélération du véhicule, et améliorent significativement la précision de l'attitude estimée, notamment en cas d'accélérations importantes du système.

This thesis is broadly concerned with online-optimizing anti-windup control. These are control structures that implement some online-optimization routines to compensate for the windup effects in constrained control systems. The first part of this thesis examines a general framework for analyzing robust preservation in anti-windup control systems. This framework - the robust Kalman conjecture - is defined for the robust Lur’e problem. This part of the thesis verifies this conjecture for first-order plants perturbed by various norm-bounded unstructured uncertainties. Integral quadratic constraint theory is exploited to classify the appropriate stability multipliers required for verification in these cases. The remaining part of the thesis focusses on accelerated gradient methods. In particular, tight complexity-certificates can be obtained for the Nesterov gradient method, which makes it attractive for implementation of online-optimizing anti-windup control. This part of the thesis presents a proposed algorithm that extends the classical Nesterov gradient method by using available secant information. Numerical results demonstrating the efficiency of the proposed algorithm are analysed with the aid of performance profiles. As the objective function becomes more ill-conditioned, the proposed algorithm becomes significantly more efficient than the classical Nesterov gradient method. The improved performance bodes well for online-optimization anti-windup control since ill-conditioning is common place in constrained control systems. In addition, this thesis explores another subcategory of accelerated gradient methods known as Barzilai-Borwein gradient methods. Here, two algorithms that modify the Barzilai-Borwein gradient method are proposed. Global convergence of the proposed algorithms for all convex functions is established by using discrete Lyapunov theorems.

This master‘s thesis deals with the design of modern process control algorithms suitable for SISO industrial control equipments. In this thesis the analysis of the controller design methodology using Mixed-sensitivity function is described. This method consists in shaping frequency characteristics of the sensitivity function and complementary sensitivity fiction using the weighting functions. The proposed H controllers are compared with the classical structure of a discrete PID controller with filtered derivative part. The proposed controllers are compared in terms of robustness, performance, complexity of the design and requirements necessary for their practical application (anti-windup, smooth switchover). Robustness of the controller is evaluated on the basis of modulus stability margin. Estimation of model´s parameters is solved least-square method. The proposed control system consists of an industrial PC and Automation Panel from the company B&R, decentralized system I/O and the real plant. Visualization for control workplace was created in the program Automation studio. This visualization is used to the easy transfer of information from a controlled process.

Cette thèse est consacrée au développement de techniques de commande avancées pour des classes de systèmes non linéaires en général et pour des applications automobiles en particulier.Pour répondre au besoin du contrôle moteur, la première partie propose des nouveaux résultats théoriques sur la technique de commande non linéaire à base de modèles de type Takagi-Sugeno soumis à la saturation de la commande. La saturation de la commande est traitée en utilisant sa représentation polytopique ou une stratégie anti-windup.La deuxième partie porte sur la commande du système d'air d'un moteur turbocompressé à allumage commandé. Deux approches originales sont proposées. Dans la première, l'outil théorique concernant les modèles Takagi-Sugeno à commutation développé dans la première partie est directement appliqué. La seconde approche est basée sur une commande linéarisante robuste. L'originalité de ces approches multivariables consiste dans sa simplicité de mise en œuvre et son efficacité par rapport à celles qui existent dans la littérature.La dernière partie vise à développer des stratégies pour la gestion énergétique des systèmes électriques d'un véhicule obtenues en se basant sur le Principe du Minimum de Pontryagin. À cet effet, deux approches sont considérées : l'approche hors ligne d'optimisation utilisant les informations du futur concernant les conditions de roulage et l'approche en ligne qui est adaptée de la précédente. Ensuite, ces deux approches sont implémentées et évaluées dans un simulateur avancé. / This thesis addresses the development of some advanced control design tools for a class of nonlinear systems in general and for automotive systems in particular.Motivated by automotive applications, Part I proposes some novel theoretical results on control design for nonlinear systems under Takagi-Sugeno form subject to the control input saturation. The input saturation is dealt with by using its polytopic representation or an anti-windup strategy.Part II deals with our automotive application concerning the control of a turbocharged air system of a spark ignition engine. To this end, two novel control approaches are proposed in this part. For the first one, the theoretical design tool on switching Takagi-Sugeno controller developed in Part I is directly applied. The second one is based on a robust feedback linearization control technique. The originality of these MIMO approaches consist in their simplicity and effectiveness compared to other ones existing in the literature.Part III aims at developing the strategies, which are based on the Pontryagin's Minimum Principle in optimal control theory, for the energy management of the vehicular electric power systems in a hybrid engine configuration. To this end, both offline optimization approach using the future information of driving conditions and online implementable one have been developed and evaluated in an advanced simulator.