The H_infinity control problem is studied for linear constant coefficient descriptor systems. Necessary and sufficient optimality conditions as well as controller formulas are derived in terms of deflating subspaces of even matrix pencils for problems of arbitrary index. A structure preserving method for computing these subspaces is introduced. In combination these results allow the derivation of a numerical algorithm with advantages over the classical methods.

This thesis presents strict LMI conditions for the bounded real lemma of discrete descriptor systems. Compared with existing nonstrict LMI conditions, the proposed new conditions are more tractable and reliable in numerical computations, in the sense that they can be tested easily by using the LMI Control Toolbox of Matlab. Based on the strict LMI conditions, the state feedback design for H-infinite control problem is also addressed. A sufficient LMI condition is derived so that the constructed feedback gain matrix from its solution will meet the design criteria of the closed-loop systems. Furthermore, we can probe into the problems of robust H-infinite control and pole-clustering in a disk for uncertain discrete descriptor systems subject to time-invariant norm-bounded uncertainty and convex polytopic uncertainty in the state matrix, respectively. Some sufficient LMI conditions are derived for analysis and design of these problems as well. Numerical examples are included to illustrate the results.

We study linear descriptor systems with rectangular variable coefficient matrices. Using local and global equivalence transformations we introduce normal and condensed forms and get sets of characteristic quantities. These quantities allow us to decide whether a linear descriptor system with variable coefficients is regularizable by derivative and/or proportional state feedback or not. Regularizable by feedback means for us that their exist a feedback which makes the closed loop system uniquely solvable for every consistent initial vector.

The H_infinity control problem is studied for linear constant coefficient descriptor systems. Necessary and sufficient optimality conditions as well as controller formulas are derived in terms of deflating subspaces of even matrix pencils for problems of arbitrary index. A structure preserving method for computing these subspaces is introduced. In combination these results allow the derivation of a numerical algorithm with advantages over the classical methods.

Cette thèse est consacrée au développement des techniques d’estimation et de commande pour systèmes descripteurs non linéaires. Les développements sont centrés sur une famille particulière de systèmes descripteurs non linéaires avec une matrice descripteur de rang plein. Toutes les approches présentées utilisent un formalisme de modélisation du type Takagi-Sugeno (TS) pour représenter les modèles descripteurs non linéaires. Un objectif très important est de développer des conditions sous la forme d’inégalités matricielles linéaires (LMI, en anglais). Dans la littérature, les conditions pour l’estimation des modèles TS descripteurs s’écrivent sous forme d’inégalités matricielles bilinéaires (BMI, en anglais). En plus, à notre connaissance, il n’y pas de résultats dans la littérature concernant la commande/estimation pour les modèles TS descripteurs en temps discret (avec une matrice descripteur régulière non linéaire).Trois problèmes ont été examinés : commande par retour d’état, estimation de l’état et commande statique par retour de la sortie. Dans le cas continu, des conditions moins conservatives ont été développées pour la commande par retour d’état. Pour l’estimation d’état, des conditions LMI ont été obtenues (au lieu des usuelles BMI) en utilisant un différent vecteur d’erreur augmenté. Pour la commande statique par retour de la sortie, des conditions LMI sont proposées si une matrice auxiliaire est fixée. Pour le temps discret, des nouveaux résultats sous la forme LMI ont été développées pour la commande/estimation, comblant ainsi certains manques de la littérature. Des exemples ont été inclus pour montrer l’applicabilité de tous les résultats que nous avons obtenus et ainsi l’importance de garder la structure originale des descripteurs. / This thesis addresses the estimation and control for nonlinear descriptor systems. The developments are focused on a family of nonlinear descriptor models with a full-rank descriptor matrix. The proposed approaches are based on a Takagi-Sugeno (TS) descriptor representation of a given nonlinear descriptor model. This type of TS models is a generalization of the standard TS ones. One of the mains goals is to obtain conditions in terms of linear matrix inequalities (LMIs). In the existing literature, the observer design for TS descriptor models has led to bilinear matrix inequality (BMI) conditions. In addition, to the best of our knowledge, there are no results in the literature on controller/observer design for discrete-time TS descriptor models (with a non-constant and invertible descriptor matrix).Three problems have been addressed: state feedback controller design, observer design, and static output feedback controller design. LMI conditions have been obtained for both continuous and discrete-time TS descriptor models. In the continuous-time case, relaxed LMI conditions for the state feedback controller design have been achieved via parameterdependent LMI conditions. For the observer design, pure LMI conditions have been developed by using a different extended estimation error. For the static output feedback controller, LMI constraints can be obtained once an auxiliary matrix is fixed. In the discretetime case, results in the LMI form are provided for state/output feedback controller design and observer design; thus filling the gap in the literature. Several examples have been included to illustrate the applicability of the obtained results and the importance of keeping the original descriptor structure instead of computing a standard state-space.

Ce mémoire de thèse est consacré à la conception de méthodes de diagnostic à base de modèles fondées sur les observateurs pour les systèmes non linéaires modélisés comme des systèmes singuliers (D) linéaires à paramètres variants (LPV), notés D-LPV (Descriptor-Linear Parameter Varying). Les systèmes D-LPV constituent une classe particulière de systèmes approximant avec un certain degré de précision la dynamique des systèmes complexes non linéaires à partir d’une combinaison de modèles linéaires locaux pondérés par des fonctions convexes d'ordonnancement. Dans le contexte de l’apparition de défauts capteurs ou actionneurs, ce travail de thèse s’attache aux systèmes pour lesquels ces fonctions d'ordonnancement sont non mesurables mais dépendent de l'état du système. Afin de détecter et isoler des défauts, ce travail de thèse développe des synthèses d'observateurs appropriés en développant des nouvelles conditions suffisantes en termes d’inégalités matricielles linéaires (LMI) pour garantir la synthèse de résidus sensibles aux défauts et robustes aux erreurs d’estimation inhérentes aux fonctions d'ordonnancement non mesurables. - Étendant des méthodes H∞ afin d’effectuer l'estimation d'état, la détection de pannes, la localisation et la reconstruction de défaut sur les capteurs ; - Garantissant une sensibilité “optimale” aux pannes vis-À-Vis du rejet de perturbations au travers le développement d’observateurs de type H_/H∞. À cette fin, le mémoire de thèse est organisé en cinq chapitres : Le Chapitre 1 est consacré à l’introduction générale, aux objectifs et contributions de ce travail. Le Chapitre 2 présente les éléments nécessaires pour décrire la représentation, la modélisation, les propriétés, l'analyse et la conception d'observateur pour les systèmes D-LPV ainsi qu’un état de l’art détaillé des travaux associés à ce thème de recherche. Le Chapitre 3 est dédié au développement de trois méthodes différentes fondées sur la théorie H∞ pour concevoir des observateurs de détection de défaut pour les systèmes D-LPV. Les méthodes proposées sont appliquées à un exemple dans le cadre de la détection de défaut capteurs. L’isolation de ces défauts est mise en œuvre au travers un banc d’observateurs et les performances de chacune des trois méthodes sont comparées. Le Chapitre 4 propose une méthode de détection de défauts sur la base d’observateurs établis sur le principe H_/H∞, tenant compte ainsi d'un meilleur compromis entre la sensibilité aux pannes et la robustesse aux perturbations. De nouvelles conditions suffisantes à l’aide de LMI sont proposées afin de résoudre le problème de synthèse du gain des observateurs. Le dernier chapitre est dédié à la conclusion générale et à l’analyse de problèmes ouverts pouvant être abordés dans des travaux futurs / This work is dedicated to the synthesis of model-Based fault detection and isolation (FDI) techniques based on observers for nonlinear systems modeled as Descriptor-Linear Parameter Varying (D-LPV) systems. D-LPV systems are a particular class of systems that can represent (or approximate in some degree of accuracy), complex nonlinear systems by a set of linear local models blended through convex parameter-Dependent scheduling functions. The global D-LPV System can describe both time-Varying and nonlinear behavior. Nevertheless, in many applications the time-Varying parameters in the scheduling functions could be unmeasurable. Models which depend on unmeasurable scheduling functions cover a wide class of nonlinear systems compared to models with measurable scheduling functions, but the design of control schemes for D-LPV systems with unmeasurable scheduling functions are more difficult than those with a measurable one, because the design of such control schemes involve the estimation of the scheduling vector. This topic is addressed in this work by considering the following main targets: • to design FDI in D-LPV systems based on -H∞ observers in order to guarantee robustness against disturbances and errors due the unmeasurable gain scheduling functions • to extend the proposed -H∞ methods to perform state estimation and fault detection, isolation and fault magnitude estimation in the case of sensor faults • to guarantee the best trade-Off between fault sensitivity and disturbance rejection by developing H_/H∞ fault detection observers for D-LPV systems. The thesis is organized as follows Chapter 1 is dedicated to provide a general introduction, the objectives and contribution of this work.Chapter 2 is organized in order to provide the minimum necessary elements to describe the representation, modeling, properties, analysis, and observer design of D-LPV systems. Chapter 2 is also dedicated to a detailed review of the state of the art. Chapter 3 is dedicated to the development of three different methods to design fault detection observers for D-LPV systems based on H∞ theory. Finally, the proposed methods are applied to an example, for sensor fault detection and isolation by means of an observer bank, in order to compare the performance of each method. Chapter 4 is dedicated to the design of a FDI method based on observers with H_/H∞ performance. Based on the H_/H∞ approach, which considers the best trade-Off between fault sensitivity and robustness to disturbance, adequate LMIs are obtained to guarantee sufficient conditions for the design problem. In order to illustrate the effectiveness of the proposed techniques, an example is considered

Esta tese apresenta novos algoritmos que resolvem problemas de estimativas filtrada, suavizadora e preditora para sistemas singulares no tempo discreto usando apenas argumentos determinísticos. Cada capítulo aborda inicialmente as estimativas para o sistema nominal e em seguida, as versões robustas para o sistema com incertezas limitadas. Os resultados encontrados podem ser aplicados tanto em sistemas invariantes como variantes no tempo discreto, utilizando a mesma estrutura do filtro de Kalman. Nos últimos anos, uma quantidade significativa de trabalhos envolvendo estimativas singulares foi publicada enfocando apenas a estimativa filtrada sob a justificativa de que a estimativa preditora era de significativa complexidade quando modelada pelo método dos mínimos quadrados. Por este motivo, poucos trabalhos, como NIKOUKHAH et al. (1992) e ZHANG et al. (1998), deduziram a estimativa preditora. Este último artigo apresentou também um algoritmo para a estimativa suavizadora, mas usando o modelo de inovação ARMA. No entanto, até onde foi possível identificar, nenhum trabalho até agora resolveu o problema de estimativa robusta, considerando incertezas nos parâmetros, para sistemas singulares. Para a dedução das estimativas singulares robustas, esta tese tomou como base SAYED (2001), que deduz o filtro de Kalman robusto com incertezas limitadas utilizando uma abordagem determinística, o chamado filtro BDU. Os filtros robustos para sistemas singulares apresentados nesta tese, são mais abrangentes que os apresentados em SAYED (2001). Quando particularizados para o espaço de estados sem incertezas, todos os filtros se assemelham ao filtro de Kalman. / New algorithms to optimal recursive filtering, smoothed and prediction for general time-invariant or time-variant descriptor systems are proposed in this thesis. The estimation problem is addressed as an optimal deterministic trajectory fitting. This problem is solved using exclusively deterministic arguments for systems with or without uncertainties. Kalman type recursive algorithms for robust filtered, predicted and smoothed estimations are derived. In the last years, many papers have paid attention to the estimation problems of linear singular systems. Unfortunately, all those works were concentrated only on the study of filtering problems, for nominal systems. The predicted and smoothed filters are more involved and were considered only by few works : NIKOUKHAH et al. (1992) and ZHANG et al. (1998) had proposed a unified approach for filtering, prediction and smoothing problems which were derived by using the projection formula and were calculated based on the ARMA innovation model, but they had not considered the uncertainties. In this thesis its applied for descriptor systems a robust procedure for usual state space systems developed by SAYED (2001), called BDU filter. It is obtained a robust descriptor Kalman type recursions for filtered, predicted and smoothed estimates. Considering the nominal state space, all descriptor filters developed in this work collapse to the Kalman filter.

This dissertation concerns the model reduction of linear periodic descriptor systems both in continuous and discrete-time case. In this dissertation, mainly the projection based approaches are considered for model order reduction of linear periodic time varying descriptor systems. Krylov based projection method is used for large continuous-time periodic descriptor systems and balancing based projection technique is applied to large sparse discrete-time periodic descriptor systems to generate the reduce systems. For very large dimensional state space systems, both the techniques produce large dimensional solutions. Hence, a recycling technique is used in Krylov based projection methods which helps to compute low rank solutions of the state space systems and also accelerate the computational convergence. The outline of the proposed model order reduction procedure is given with more details. The accuracy and suitability of the proposed method is demonstrated through different examples of different orders. Model reduction techniques based on balance truncation require to solve matrix equations. For periodic time-varying descriptor systems, these matrix equations are projected generalized periodic Lyapunov equations and the solutions are also time-varying. The cyclic lifted representation of the periodic time-varying descriptor systems is considered in this dissertation and the resulting lifted projected Lyapunov equations are solved to achieve the periodic reachability and observability Gramians of the original periodic systems. The main advantage of this solution technique is that the cyclic structures of projected Lyapunov equations can handle the time-varying dimensions as well as the singularity of the period matrix pairs very easily. One can also exploit the theory of time-invariant systems for the control of periodic ones, provided that the results achieved can be easily re-interpreted in the periodic framework. Since the dimension of cyclic lifted system becomes very high for large dimensional periodic systems, one needs to solve the very large scale periodic Lyapunov equations which also generate very large dimensional solutions. Hence iterative techniques, which are the generalization and modification of alternating directions implicit (ADI) method and generalized Smith method, are implemented to obtain low rank Cholesky factors of the solutions of the periodic Lyapunov equations. Also the application of the solvers in balancing-based model reduction of discrete-time periodic descriptor systems is discussed. Numerical results are given to illustrate the effciency and accuracy of the proposed methods.

Esta tese apresenta novos algoritmos que resolvem problemas de estimativas filtrada, suavizadora e preditora para sistemas singulares no tempo discreto usando apenas argumentos determinísticos. Cada capítulo aborda inicialmente as estimativas para o sistema nominal e em seguida, as versões robustas para o sistema com incertezas limitadas. Os resultados encontrados podem ser aplicados tanto em sistemas invariantes como variantes no tempo discreto, utilizando a mesma estrutura do filtro de Kalman. Nos últimos anos, uma quantidade significativa de trabalhos envolvendo estimativas singulares foi publicada enfocando apenas a estimativa filtrada sob a justificativa de que a estimativa preditora era de significativa complexidade quando modelada pelo método dos mínimos quadrados. Por este motivo, poucos trabalhos, como NIKOUKHAH et al. (1992) e ZHANG et al. (1998), deduziram a estimativa preditora. Este último artigo apresentou também um algoritmo para a estimativa suavizadora, mas usando o modelo de inovação ARMA. No entanto, até onde foi possível identificar, nenhum trabalho até agora resolveu o problema de estimativa robusta, considerando incertezas nos parâmetros, para sistemas singulares. Para a dedução das estimativas singulares robustas, esta tese tomou como base SAYED (2001), que deduz o filtro de Kalman robusto com incertezas limitadas utilizando uma abordagem determinística, o chamado filtro BDU. Os filtros robustos para sistemas singulares apresentados nesta tese, são mais abrangentes que os apresentados em SAYED (2001). Quando particularizados para o espaço de estados sem incertezas, todos os filtros se assemelham ao filtro de Kalman. / New algorithms to optimal recursive filtering, smoothed and prediction for general time-invariant or time-variant descriptor systems are proposed in this thesis. The estimation problem is addressed as an optimal deterministic trajectory fitting. This problem is solved using exclusively deterministic arguments for systems with or without uncertainties. Kalman type recursive algorithms for robust filtered, predicted and smoothed estimations are derived. In the last years, many papers have paid attention to the estimation problems of linear singular systems. Unfortunately, all those works were concentrated only on the study of filtering problems, for nominal systems. The predicted and smoothed filters are more involved and were considered only by few works : NIKOUKHAH et al. (1992) and ZHANG et al. (1998) had proposed a unified approach for filtering, prediction and smoothing problems which were derived by using the projection formula and were calculated based on the ARMA innovation model, but they had not considered the uncertainties. In this thesis its applied for descriptor systems a robust procedure for usual state space systems developed by SAYED (2001), called BDU filter. It is obtained a robust descriptor Kalman type recursions for filtered, predicted and smoothed estimates. Considering the nominal state space, all descriptor filters developed in this work collapse to the Kalman filter.

Cette thèse porte sur la commande adaptative directe robuste et son application au contrôle d’attitude des satellites de la filière Myriade du CNES. Après avoir présenté les différents types de commande variant dans le temps, nous rappelons les caractéristiques d’un contrôleur adaptatif direct, en particulier le fait que la seule connaissance d’un retour de sortie stabilisant le système à contrôler suffit pour concevoir un contrôleur adaptatif direct. Parallèlement, nous présentons la théorie des systèmes descripteurs. Modéliser un système sous forme descripteur est non conventionnel mais présente de nombreux avantages dans le contexte de la commande adaptative directe robuste. A l’aide des résultats existants sur la commande adaptative directe d’une part, et de la théorie des systèmes descripteurs d’autre part, nous fournissons une méthode permettant de calculer, connaissant un retour de sortie constant, les paramètres d’un contrôleur adaptatif direct robuste stabilisant. Cette méthode repose sur la résolution d’inégalités matricielles linéaires. Le contrôleur adpatatif est plus robuste que le contrôleur constant, mais on ne peut prouver que la stabilité globale que vers un voisinage du point d’équilibre. Nous présentons ensuite une méthode, également basée sur la résolution d’inégalités matricielles linéaires, permettant de concevoir un contrôleur adaptatif direct robuste de meilleur niveau de rejet des perturbations extérieures que le contrôleur constant à partir duquel il est construit. L’ensemble de ces résultats théoriques est ensuite appliqué au contrôle d’attitude des satellites de la filière Myriade du CNES. En particulier, nous concevons un contrôleur d’attitude stabilisant le satellite quelle que soit la valeur de son inertie. Ce contrôleur d’attitude est également capable d’éviter aux roues à réaction du satellite de saturer. Nous concevons ensuite un contrôleur d’attitude adaptatif, robuste, et qui rejette mieux les perturbations extérieures que le contrôleur constant à partir duquel il est construit. Ce contrôleur constant est d’ailleurs actuellement implémenté à bord des satellites de la filière Myriade du CNES. Enfin, nous validons l’ensemble des résultats de cette thèse à l’aide d’un simulateur SCAO du CNES, où nous simulons le déploiement des mâts d’un satellite, ainsi que des scénarii de sauts de guidage. / This manuscript deals with robust direct adaptive control, and its application to CNES microsatellites attitude control. After listing the different types of time-varying controllers, we recall the characteristics of direct adaptive control. In particular, we recall that the knowledge of a stabilizing static output feedback is sufficient to design a direct adaptive controller. In parallel, we introduce the descriptor system theory. Modelizing a system into descriptor form is not usual but fits well with robust direct adaptive control. Starting from existing results about adaptive control and descriptor system theory, we provide an LMI based method which allows to compute, with the knowledge of a stabilizing static output feedback, the parameters of a stabilizing direct adaptive controller. A first result proves that the adaptive controller is at least as robust as the static output feedback. The second result allows to prove improved robustness at the expense of relaxing stability of the equilibrium point to practical stability, that is convergence to a neighborhood of the equilibrium. Then, we provide a method, LMI based as well, which allows to design a robust direct adaptive controller which has a better level of rejection of the perturbations than the static output feedback from which it is designed. All these theoretical results are applied to the attitude control of CNES microsatellites. We design a controller which stabilizes the attitude of the satellite whatever the value of its inertia. This attitude controller can also avoid the satellite reaction wheels to saturate. We design another robust adaptive attitude controller which has a better level of rejection of the perturbations than the static controller which is currently implemented aboard CNES satellites. Finally, we validate all the results of this manuscript by simulating on a AOCS CNES simulator the deployment of the satellite masts and some guiding jumps.