Orientador: Pedro Luis Dias Peres / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-06T10:51:24Z (GMT). No. of bitstreams: 1 Oliveira_RicardoCoracaodeLeaoFontourade_D.pdf: 881205 bytes, checksum: 053263f18afcf3085a0fc073e1594d2d (MD5) Previous issue date: 2006 / Resumo: A principal contribuição desta tese é a proposta de uma metodologia para solução de desigualdades matriciais lineares dependentes de parâmetros que freqüentemente aparecem em problemas de análise e controle robusto de sistema lineares com incertezas na forma politópica. O método consiste na parametrização das soluções em termos de polinômios homogêneos com coeficientes matriciais de grau arbitrário. Para a construção dessas soluções, um procedimento baseado em resoluções de problemas de otimiza¸c¿ao na forma de um número finito de desigualdades matriciais lineares 'e proposto, resultando em seqüências de relaxações que convergem para uma solução polinomial homogênea sempre que uma solução existe. Problemas de análise robusta e custo garantido s¿ao analisados em detalhes tanto para sistemas a tempo contínuo quanto para sistemas discretos no tempo. Vários exemplos numéricos são apresentados ilustrando a eficiência dos métodos propostos em termos da acurácia dos resultados e do esforço computacional quando comparados com outros métodos da literatura / Abstract: This thesis proposes, as main contribution, a new methodology to solve parameterdependent linear matrix inequalities which frequently appear in robust analysis and control problems of linear system with polytopic uncertainties. The proposed method relies on the parametrization of the solutions in terms of homogeneous polynomials of arbitrary degree with matrix valued coefficients. For constructing such solutions, a procedure based on optimization problems formulated in terms of a finite number of linear matrix inequalities is proposed, yielding sequences of relaxations which converge to a homogeneous polynomial solution whenever a solution exists. Problems of robust analysis and guaranteed costs are analyzed in details for continuous and discrete-time uncertain systems. Several numerical examples are presented illustrating the efficiency of the proposed methods in terms of accuracy and computational burden when compared to other methods from the literature / Doutorado / Automação / Doutor em Engenharia Elétrica

O presente trabalho propõe, como inovação, o limite superior para a energia da saída do sistema em malha fechada como índice de desempenho a ser usado no projeto de controladores robustos para amortecer oscilações eletromecânicas de baixa freqüência em sistemas de potência. A saída do sistema em malha fechada é especificada de forma que a energia da saída corresponda ao valor acumulado do desvio da energia cinética do sistema. O índice de desempenho proposto mostrou-se adequado ao problema de oscilações eletromecânicas. Tal índice de desempenho é utilizado na formulação de uma metodologia sistemática de projeto. O problema de controle é estruturado na forma de desigualdades matriciais lineares, permitindo a obtenção de uma solução numérica para o problema. A obtenção da solução do problema de controle proposto (energia da saída como índice de desempenho) exige menos dispêndio computacional, quando comparado com o tradicional fator de amortecimento mínimo na forma de posicionamento regional de pólos. Tal característica pode ser significativa para o tempo computacional requerido pelo projeto de controladores envolvendo modelos sistemas de potência de ordem elevada. O projeto de controladores de amortecimento robustos, baseados em modelos multimáquinas que dispensem a hipótese de existência de um barramento infinito, constitui a segunda parte da pesquisa proposta. Os problemas inerentes à hipótese do barramento infinito são resolvidos pelo uso de duas abordagens que não adotam tal hipótese. A primeira abordagem adotada refere-se ao uso de um modelo multimáquinas com o ângulo de uma das máquinas do sistema como referência angular. A segunda alternativa proposta é a incorporação do sistema primário de controle de velocidade no modelo multimáquinas referente à primeira alternativa. Além de resolver o problema referente ao uso do barramento infinito, o presente trabalho também propõe análises da influência da hipótese de tal barramento no projeto de controladores de amortecimento. A influência da hipótese do barramento infinito no projeto dos controladores é delineada por meio dos fundamentos expostos e dos resultados obtidos com as diferentes alternativas utilizadas / The present work proposes, as innovation, an upper bound for the output energy of the closed loop system to be used as a performance index in the design of robust controllers to damp low-frequency electromechanical oscillations in electric power systems. The output of the closed loop system is specified so that the output energy corresponds to the accumulated value of the kinetic energy deviation of the closed loop system. The performance index used in the procedure has shown to be suitable to the oscillation problem. The proposed performance index is used in the formulation of a systematic design methodology. The control problem is structured in the form of linear matrix inequalities, allowing a numerical solution to the control problem. The adopted performance index is less costly in terms of computational effort when compared with the traditional minimum damping ratio (performance index usually accepted in power system as small signal stability margin) via regional pole placement in the LMI formulation. This characteristic may be significant to the computational time required for the controller design involving large power system models. The robust damping controller design, based on multimachine models without the infinite-bus assumption, constitutes the second part of this research proposal. The problems inherent to the infinte-bus assumption are solved by means of two approaches which do not use such assumption. The first proposed approach refers to the use of a multimachine model adopting one machine angle of the system as angular reference. The second adopted approach is the incorporation of the primary speed control in the multimachine model regarding to the first approach. Besides solving the problem regarding to the use of infinite-bus assumption, the present work also proposes analyses of such assumption influence in the design of damping controllers. The influence of such assumption, in the design of the controllers, is outlined by means of exposed fundamentals and results obtained with the proposed approaches

Orientador: Marcelo Carvalho Minhoto Teixeira / Resumo: Neste trabalho apresentam-se condições descritas por desigualdades matriciais lineares, LMIs (do inglês: Linear Matrix Inequalities), para o projeto de controladores robustos para sistemas dinâmicos de ordem α ∈ [0,1). Os controladores propostos utilizam a realimentação da derivada de ordem α ∈ [0,1) do vetor de estado, a chamada realimentação α-derivativa, e também a realimentação do vetor de estado. A literatura clássica apresenta resultados que utilizam o método direto de Lyapunov e a estabilização quadrática no projeto de controladores para sistemas de ordem inteira. Os teoremas propostos neste trabalho para sistemas fracionários são condições suficientes análogas a estes resultados. Esta analogia é possível através da extensão fracionária, recentemente disponível na literatura, do método direto de Lyapunov e de um limitante superior para a derivada de ordem α ∈ [0,1) da função de Lyapunov do tipo quadrática, Dα V(x(t)). Nesse sentido, as LMIs propostas para estabilização quadrática são análogas aos casos clássicos, pois não dependem da ordem α ∈ [0,1) do sistema. Em particular, o foco deste trabalho recai no controle do tipo chaveado, que trata da minimização do limitante superior de Dα V(x(t)). O controle chaveado dispensa o conhecimento das funções de pertinência quando da utilização de modelos fuzzy Takagi-Sugeno, permitindo trabalhar com plantas lineares e não lineares, ambas incluindo parâmetros incertos. Dessa forma, a estabilização quadrática possibilitou a obtenç... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: This work proposes linear matrix inequalities (LMIs) conditions for the design of robust controllers for dynamic systems of order α ∈ [0,1). The proposed controllers use the feedback of the state vector derivative of of order α ∈ [0,1), the so-called α -derivative feedback, and also the feedback of the state vector. The classical literature presents results that use the Lyapunov direct method and the quadratic stabilization in the design of the controllers for integer order systems. The theorems proposed in this work for fractional systems are sufficient conditions analogous to these results. This analogy is possible through the fractional extension, recently available in the literature, of the direct Lyapunov method and an upper bound for the a α ∈ [0,1) order derivative of the quadratic Lyapunov function, Dα V(x(t)). In this sense, the proposed LMIs for quadratic stabilization are analogous to the classical ones, since they do not depend on the order α ∈ [0,1) of the system. In particular, the focus of this work lies in the switched control, which deals with the minimization of the upper bound of Dα V(x(t)). The switched control dispenses the knowledge of the membership functions when using the Takagi-Sugeno fuzzy models, allowing to work with linear and nonlinear plants, both of them with uncertain parameters. Therefore, the quadratic stabilization allowed to obtain new results for the robust control problem of α ∈ [0,1) order systems, considering the main analogous resu... (Complete abstract click electronic access below) / Mestre

Le but de cette thèse est de présenter de nouveaux résultats sur l'analyse et la synthèse de systèmes à retard. Dans la première partie, nous étendons l'utilisation du système invariant d'ordre fini, appelé "système de comparaison", à la conception d'un contrôleur qui dépend non seulement de la sortie à l'heure actuelle et du délai maximum, mais également d'un nombre arbitraire de valeurs entre celles-ci. Cette approche nous permet d'augmenter le délai maximal stable sans exiger d'informations supplémentaires. Les méthodes présentées ici concernent la conception de systèmes de contrôle avec des retards en utilisant des routines numériques classiques basées sur la théorie Hoo. La deuxième partie de ce travail traite d'une nouvelle approche pour développer une enveloppe englobant tous les pôles d'un système à retard. Grâce aux LMIs, nous sommes en mesure de déterminer les enveloppes pour les systèmes à retard du type retardé et du type neutre. Les enveloppes proposées sont non seulement plus étroites que celles de la littérature, mais, avec notre procédure, elles peuvent également être appliquées pour vérifier la stabilité du système et pour projeter contrôleurs de retour d'état qui répondent aux exigences de conception relatives à alpha-stabilité et sont robustes face aux incertitudes paramétriques. Les systèmes fractionnaires sont également discutés dans les deux chapitres mentionnés ci-dessus. La troisième et dernière partie étudie les systèmes stochastiques avec des retards. Nous discutons d'abord des systèmes à temps continu soumis à des sauts de Markov. Nous définissons la stabilité et obtenons des LMIs pour le contrôle par retour d'état de telle sorte que la relation entre les taux de transition entre les modes soit affine, ce qui permet donc de traiter le cas dans lequel les taux sont incertains. Nous discutons ensuite des systèmes positifs avec retards, tant pour le cas continu que pour le cas discret. Des systèmes équivalents sont obtenus et la stabilité dépendante du retard est abordée. De nombreux exemples sont illustrés tout au long de la thèse. / The aim of this dissertation is to present new results on analysis and control design of time-delay systems. On the first part, we extend the use of a finite order LTI system, called 'comparison system', to design a controller which depends not only on the output at the present time and maximum delay, but also on an arbitrary number of values between those. This approach allows us to increase the maximum stable delay without requiring any additional information. The methods presented here consider time-delay systems control design with classical numeric routines based on Hoo theory. The second part of this work deals with a new approach to develop an envelope that engulfs all poles of a time-delay system. Through LMIs, we are able to determine envelopes for retarded and neutral time-delay systems. The envelopes proposed are not only tighter than the ones in the literature but, with our procedure, they can also be applied to verify the stability of the system and design state-feedback controllers which cope with design requirements regarding alpha-stability and are robust in face of parametric uncertainties. Fractional systems are also discussed for both chapters mentioned above. The third and last part studies stochastic time-delay systems.First we discuss continuous-time systems that are subjected to Markov jumps. We define stability and obtain LMIs for the state-feedback control in such a way that the relation with the transition rates between the modes is affine, allowing, therefore, to treat the case in which the rates are uncertain. We then discuss positive systems with delays, both for the continuous case as for the discrete case. Equivalent systems are obtained and delay dependent stability is addressed. A fair amount of examples are presented throughout the dissertation.

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

Dans cette thèse, nous étudions la rétroaction de systèmes linéaires invariants dans le temps reliés entre eux par trois blocs non linéaires spécifiques : un opérateur de lecture/arrêt, un mécanisme de réinitialisation de commutation et une zone morte adaptative. Cette configuration ressemble au problème de Lure étudié dans le cadre de stabilité absolue, mais les types de non-linéarités considérés ici ne satisfont pas (en général) une condition sectorielle. Ces blocs non linéaires donnent lieu à toute une série de phénomènes intéressants, tels que des ensembles compacts d’équilibres, des ensembles hybrides oméga-limites et des contraintes d’état. Tout au long de la thèse, nous utilisons le formalisme des systèmes hybrides pour décrire ces phénomènes et analyser ces boucles. Nous obtenons des conditions de stabilité très précises qui peuvent être formulées sous forme d’inégalités matricielles linéaires, donc vérifiables avec des solveurs numériques efficaces. Enfin, nous appliquons les résultats théoriques à deux applications automobiles. / In this thesis we study linear time-invariant systems feedback interconnected with three specific nonlinear blocks; a play/stop operator, a switching-reset mechanism, and an adaptive dead-zone. This setup resembles the Lure problem studied in the absolute stability framework, but the types of nonlinearities considered here do not satisfy (in general) a sector condition. These nonlinear blocks give rise to a whole range of interesting phenomena, such as compact sets of equilibria, hybrid omega-limit sets, and state constraints. Throughout the thesis, we use the hybrid systems formalism to describe these phenomena and to analyze these loops. We obtain sharp stability conditions that can be formulated as linear matrix inequalities, thus verifiable with numerically efficient solvers. Finally, we apply the theoretical findings to two automotive applications.

Les principales contributions de cette thèse concernent le développement de méthodes pour la synthèse de contrôleurs et pour l'analyse de la stabilité des systèmes linéaires, soit variant ou invariant dans le temps. Concernant les systèmes invariant dans le temps, le but est la synthèse de contrôleurs robustes d'ordre réduit pour les systèmes en temps continu qui présentent des paramètres incertains. La méthode présentée pour la synthèse est basée sur une technique en deux étapes, où un gain de retour d'état est construit dans la première étape, et appliqué à la deuxième, fournissant le contrôleur robuste souhaité. Chaque étape consiste à la résolution de conditions sous la forme d'inégalités matricielles linéaires.Dans le cas des systèmes variant dans le temps, en général, en fonction des informations disponibles, deux modèles mathématiques peuvent être utilisés. D'un côté, pour des systèmes dont les éléments variant dans le temps sont bornés mais pas complètement connus, on peut utiliser des modèles dépendant de paramètres variants, ce qui donne une représentation polytopique. Dans ce cas là, la technique de stabilisation proposée est basée sur la méthode en deux étapes, pour générer des contrôleurs dépendants des paramètres. On suppose que les paramètres sont mesurables en ligne, et les contrôleurs sont synthétisés pour qu'ils soient robustes à des bruits de mesures. De l'autre côté, si les dynamiques variantes dans le temps sont connues, on peut traiter directement le système sans utiliser aucune paramétrisation. Deux techniques de synthèse sont proposées pour ce cas: la construction des gains stabilisants en utilisant directement la matrice de transition d'état, et une technique de synthèse conçue à partir d'un nouveau critère de vérification de la stabilité du système. La validité des méthodes proposées est illustrée par plusieurs exemples numériques, qui montrent la qualité des résultats qui peuvent être obtenus / The main contributions of this thesis concern the development of methods for the stability analysis and the synthesis of controllers for linear systems, either time-varying or time-invariant. Concerning time-invariant systems, the objective is the synthesis of reduced-order robust controllers for continuous-time systems presenting uncertain parameters. The method presented for the synthesis is based on a two-stages technique, in which a stabilizing state-feedback gain is constructed in the first stage and then applied on the second stage to search for the desired controller. Each stage consists in the resolution of conditions based on linear matrix inequalities. In the case of time-varying systems, depending on the amount of available information, twomathematical models may be used. On one hand, if the time-varying elements of the system are not entirely known, one can model the system as function of time-varying parameters, resulting on a polytopic representation. In this case, the stabilization method proposed is based on the two-stages technique, which yields parameter-dependent controllers. The parameters are supposed to be real-time measurable, and the controllers are robust with respect to noises and uncertainties on the measures. On the other hand, if the time-varying dynamics are known, the system may be directly handled without using any parameterization. Two synthesis techniques are proposed in this case: the construction of stabilizing gains by using the state transition matrix, and a synthesis technique derived from a new stability criterion for time-varying systems. The validity of the proposed methods is illustrated through numerical examples, that show the efficiency of the results that can be obtained

Dans cette thèse, nous nous intéressons à l'étude des structures déterminantielles apparaissent dans l'optimisation semi-définie (SDP), le prolongement naturel de la programmation linéaire au cône des matrices symétrique semi-définie positives. Si l'approximation d'une solution d'un programme semi-défini peut être calculé efficacement à l'aide des algorithmes de points intérieurs, ni des algorithmes exacts efficaces pour la SDP sont disponibles, ni une compréhension complète de sa complexité théorique a été atteinte. Afin de contribuer à cette question centrale en optimisation convexe, nous concevons un algorithme exact pour décider la faisabilité d'une inégalité matricielle linéaire (LMI) $A(x)\succeq 0$. Quand le spectraèdre associé (le lieu $\spec$ des $x \in \RR^n$ ou $A(x)\succeq 0$) n'est pas vide, la sortie de cet algorithme est une représentation algébrique d'un ensemble fini qui contient au moins un point $x \in \spec$: dans ce cas, le point $x$ minimise le rang de $A(x)$ sur $\spec$. La complexité est essentiellement quadratique en le degré de la représentation en sortie, qui coïncide, expérimentalement, avec le degré algébrique de l'optimisation semi-définie. C'est un garantie d'optimalité de cette approche dans le contexte des algorithmes exacts pour les LMI et la SDP. Remarquablement, l'algorithme ne suppose pas la présence d'un point intérieur dans $\spec$, et il profite de l'existence de solutions de rang faible de l'LMI $A(x)\succeq 0$. Afin d'atteindre cet objectif principal, nous développons une approche systématique pour les variétés déterminantielles associées aux matrices linéaires. Nous prouvons que décider la faisabilité d'une LMI $A(x)\succeq 0$ se réduit à calculer des points témoins dans les variétés déterminantielles définies sur $A(x)$. Nous résolvons ce problème en concevant un algorithme exact pour calculer au moins un point dans chaque composante connexe réelle du lieu des chutes de rang de $A(x)$. Cet algorithme prend aussi avantage des structures supplémentaires, et sa complexité améliore l'état de l'art en géométrie algébrique réelle. Enfin, les algorithmes développés dans cette thèse sont implantés dans une nouvelle bibliothèque Maple appelé Spectra, et les résultats des expériences mettant en évidence la meilleure complexité sont fournis. / In this thesis we focus on the study of determinantal structures arising in semidefinite programming (SDP), the natural extension of linear programming to the cone of symetric positive semidefinite matrices. While the approximation of a solution of a semidefinite program can be computed efficiently by interior-point algorithms, neither efficient exact algorithms for SDP are available, nor a complete understanding of its theoretical complexity has been achieved. In order to contribute to this central question in convex optimization, we design an exact algorithm for deciding the feasibility of a linear matrix inequality (LMI) $A(x) \succeq 0$. When the spectrahedron $\spec = \{x \in \RR^n \mymid A(x) \succeq 0\}$ is not empty, the output of this algorithm is an algebraic representation of a finite set meeting $\spec$ in at least one point $x^*$: in this case, the point $x^*$ minimizes the rank of the pencil on the spectrahedron. The complexity is essentially quadratic in the degree of the output representation, which meets, experimentally, the algebraic degree of semidefinite programs associated to $A(x)$. This is a guarantee of optimality of this approach in the context of exact algorithms for LMI and SDP. Remarkably, the algorithm does not assume the presence of an interior point in the spectrahedron, and it takes advantage of the existence of low rank solutions of the LMI. In order to reach this main goal, we develop a systematic approach to determinantal varieties associated to linear matrices. Indeed, we prove that deciding the feasibility of a LMI can be performed by computing a sample set of real solutions of determinantal polynomial systems. We solve this problem by designing an exact algorithm for computing at least one point in each real connected component of the locus of rank defects of a pencil $A(x)$. This algorithm admits as input generic linear matrices but takes also advantage of additional structures, and its complexity improves the state of the art in computational real algebraic geometry. Finally, the algorithms developed in this thesis are implemented in a new Maple library called {Spectra}, and results of experiments highlighting the complexity gain are provided.

Ce mémoire de thèse traite de la stabilité en temps fini et de la stabilisation en temps fini des systèmes dynamiques. En effet, il est souvent important de garantir que pendant le régime transitoire, les trajectoires d'état ne dépassent pas certaines limites prédéfinies afin d'éviter les saturations et l'excitation des non-linéarités du système. Un système dynamique est dit stable en temps fini FTS si, pour tout état initial appartenant à un ensemble borné prédéterminé, la trajectoire d'état reste comprise dans un autre ensemble borné prédéterminé pendant un temps fini et fixé. Lorsque le système est perturbé, on parle de bornitude en temps fini FTB. Premièrement, des nouvelles conditions suffisantes assurant la synthèse d'un correcteur FTB par retour de sortie dynamique des systèmes linéaires continus invariants perturbés ont été développées via une approche descripteur originale. Le résultat a été établi par une transformation de congruence particulière. Les conditions obtenues sont sous forme de LMIs. Deuxièmement, l'utilisation de la notion d'annulateur combinée avec le lemme de Finsler, permet d’obtenir des nouvelles conditions sous formes LMIs garantissant la stabilité et la stabilisation en temps fini des systèmes non linéaires quadratiques. Enfin, pour obtenir des conditions encore moins pessimistes dans un contexte de stabilité en temps fini, de nouveaux développements ont été proposés en utilisant des fonctions de Lyapunov polynomiales / This dissertation deals with the finite time stability and the finite time stabilization of dynamic systems. Indeed, it is often important to ensure that during the transient regime, the state trajectories do not exceed certain predefined limits in order to avoid saturations and excitations of the nonlinearities of the system. Hence the interest is to study the stability of the dynamic system in finite time. A dynamic system is said to be stable in finite time (FTS) if, for any initial state belonging to a predetermined bounded set, the state trajectory remains within another predetermined bounded set for a finite and fixed time. When the system is disturbed, it is called finite time boundedness (FTB). In this manuscript, the goal is to improve the results of finite time stability used in the literature. First, new sufficient conditions expressed in terms of LMIs for the synthesis of an FTB controller by dynamic output feedback have been developed via an original descriptor approach. An original method has been proposed which consists in using a particular congruence transformation. Second, new LMI conditions for the study of finite time stability and finite time stabilization have been proposed for disturbed and undisturbed nonlinear quadratic systems. Third, to obtain even less conservative conditions, new developments have been proposed using polynomial Lyapunov functions

Orientador: Edvaldo Assunção / Resumo: Neste trabalho é proposta a resolução do problema do regulador linear quadrático (Linear Quadratic Regulator - LQR) via desigualdades matriciais lineares (Linear Matrix Inequalities - LMIs) para sistemas lineares e invariantes no tempo sujeitos a incertezas politópicas, bem como para sistemas lineares sujeitos a parâmetros variantes no tempo (Linear Parameter Varying - LPV). O projeto dos controladores é baseado na realimentação derivativa. A escolha da realimentação derivativa se dá devido à sua fácil implementação em certas aplicações como, por exemplo, no controle de vibrações. Os sinais usados na realimentação são aceleração e velocidade, sendo obtidos por meio de acelerômetros. Por meio do método proposto é possível obter condições LMIs para a síntese de controladores que garantam a estabilização do sistema em malha fechada, sendo que os controladores possuem desempenho otimizado. Para a formulação das condições LMIs, uma função de Lyapunov do tipo quadrática é utilizada. Exemplos teóricos e simulações são utilizados como forma de validação dos métodos propostos, além de mostrar que os novos resultados apresentam condições menos conservadoras. Além disso, ao final é apresentada uma implementação prática em um sistema de suspensão ativa, produzida pela Quanser®. / Abstract: The resolution of linear quadratic regulator (LQR) problem via linear matrix inequalities (LMIs) for linear time-invariant systems subject to polytopic uncertainties, as linear systems subjects to linear parameter varying (LPV), is proposed in this work. The controllers' designs are based on the state derivative feedback. The aim to the choice of the state derivative feedback is your easy implementation in a class of mechanical systems, such as in vibration control, for example. The signals used for feedback are acceleration and velocity, it is obtained by means of accelerometers. Through the proposed method it is possible to obtain LMIs conditions for the synthesis of controllers that guarantee the stabilisation of the closed-loop system, being that the controllers have optimised performance. For the LMIs conditions formulations, a Lyapunov function of type quadratic is used. As a form of validation, theoretical examples and simulations are performed, besides to show that the new results are less conservative. Furthermore, a practical implementation in an active suspension system, produced by Quanser®, is performed. / Mestre