Les travaux de cette thèse portent sur la stabilisation des systèmes non linéaires à grandes dimensions. Il s’agit de décomposer le système de grandes dimensions en un ensemble de n sous-systèmes interconnectés. Ensuite, en tenant compte de la nature des sous-systèmes, un ensemble de lois de commande décentralisées, composée de n lois de commande locale, peut être synthétisé. Ainsi, chaque loi de commande locale est associée à un sous-système et permet de le stabiliser indépendamment des autres sous-systèmes. Trois classes de systèmes sont ainsi traitées dans le cadre de cette thèse : les systèmes non linéaires interconnectés et les systèmes hybrides interconnectés (linéaires ou non linéaires).Premièrement, on s’est intéressé à l’étude de la stabilisation des systèmes non linéaires interconnectés représentées par des multi-modèles de type Takagi-Sugeno (TS). En effet, un multi-modèle TS permet de représenter un système non linéaire affine par un ensemble de modèles linéaires interconnectés par des fonctions non linéaires. Afin de réduire le conservatisme des approches quadratiques de Lyapunov existantes, de nouvelles conditions écrites sous forme d’inégalités linéaires matricielles (LMI) ont été proposées sur la base d’une fonction non quadratique de Lyapunov candidate. Ces conditions permettent ainsi la synthèse des contrôleurs TS décentralisés proposés. Dans le but d’étendre les résultats obtenus au cas des systèmes non linéaires et hybrides de grandes dimensions, on s’est ensuite intéressé à l’étude de la stabilisation des systèmes linéaires à commutations interconnectés via l’emploi de fonctions multiples de Lyapunov. Enfin, des conditions de stabilisation pour les systèmes non linéaires interconnectés à commutations, composés de sous-systèmes TS à commutations, ont été proposées sous forme LMI. / This thesis focuses on the stabilization of nonlinear large scale systems. Hence, a global large scale systemcan be described by a set of n interconnected subsystems. Then, based on the subsystem’s nature, adecentralized control law, composed of a set of n local control law is proposed. Each local control law isable to stabilize the subsystem for which it is synthesized. In this thesis, three classes of dynamical systemsare considered : nonlinear interconnected systems, switched linear interconnected systems and switchednonlinear interconnected systems.First, based on Takagi Sugeno (TS) modeling, the decentralized stabilization of nonlinear interconnectedsystems has been studied. In order to relax quadratic Lyapunov approaches, one has considered nonquadratic Lyapunov functions. Then, based on multiple Lyapunov switched functions, the stabilization oflarge scale linear switched systems is proposed. Hence, a decentralized switched control law designmethodology has been developed to ensure the stability of the proposed class of hybrid systems. Finally,the stabilization of nonlinear switched interconnected systems as been considered such that each nonlinearsubsystem is described by a TS model.

<p>In order to design robust controllers, H-infinity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the plant, the problem is no longer convex and it is then relatively hard to solve. These problems become very complex, even when the order of the system to be controlled is low.</p><p>The approach used in the thesis is based on formulating the constraint on the maximum order of the plant as a polynomial equation. By using the fact that the polynomial is non-negative on the feasible set, the problem is reformulated as an optimization problem where the nonconvex polynomial function is to be minimized over a convex set defined by linear matrix inequalities.</p><p>To solve this optimization problem, two methods have been proposed. The first method is a barrier method and the second one is a method based on a primal-dual framework. These methods have been evaluated on several problems and compared with a well-known method found in the literature. To motivate this choice of method, we have made a brief survey of available methods available for solving the same or related problems.</p><p>The proposed methods emerged as the best methods among the three for finding lower order controllers with the same or similar performance as the full order controller. When the aim is to find the lowest order controller with no worse than +50% increase in the closed loop H-infinity norm, then the three compared methods perform equally well.</p>

In order to design robust controllers, H-infinity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the plant, the problem is no longer convex and it is then relatively hard to solve. These problems become very complex, even when the order of the system to be controlled is low. The approach used in the thesis is based on formulating the constraint on the maximum order of the plant as a polynomial equation. By using the fact that the polynomial is non-negative on the feasible set, the problem is reformulated as an optimization problem where the nonconvex polynomial function is to be minimized over a convex set defined by linear matrix inequalities. To solve this optimization problem, two methods have been proposed. The first method is a barrier method and the second one is a method based on a primal-dual framework. These methods have been evaluated on several problems and compared with a well-known method found in the literature. To motivate this choice of method, we have made a brief survey of available methods available for solving the same or related problems. The proposed methods emerged as the best methods among the three for finding lower order controllers with the same or similar performance as the full order controller. When the aim is to find the lowest order controller with no worse than +50% increase in the closed loop H-infinity norm, then the three compared methods perform equally well.

Made available in DSpace on 2014-06-11T19:22:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-05-27Bitstream added on 2014-06-13T20:47:51Z : No. of bitstreams: 1 esteves_tt_me_ilha.pdf: 642482 bytes, checksum: 6d87652f7c370af770e00c245690a225 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho é realizada a análise da estabilidade de sistemas fuzzy Takagi-Sugeno (TS) contínuos no tempo, através de Funções de Lyapunov Fuzzy (FLF), Funções de Lyapunov Metzler (FLM) e de Funções de Lyapunov Fuzzy-Metzler (FLFM) introduzida nesta disser- tação. Novas propostas são feitas a partir destas análises, sendo apresentadas condições su- ficientes para a estabilidade assintótica destes sistemas no sentido de Lyapunov. As soluções obtidas são baseadas em desigualdades lineares matriciais (LMIs, do inglês Linear Matrix Ine- qualities) e dependem da solução de um conjunto de desigualdades de Lyapunov-Metzler, que podem ser de difícil solução. Então, foram apresentadas condições de estabilidade baseadas em uma subclasse de matrizes de Metzler que, quando factíveis, podem ser resolvidas através de LMIs com a necessidade de uma busca unidimensional. Foram propostos métodos que genera- lizam os já existentes na literatura, baseados em FLF, para a estabilidade assintótica dos sistemas fuzzy TS / This work addresses the stability analysis of Takagi-Sugeno (TS) fuzzy systems via Fuzzy Lyapunov Functions (FLF), Metzler Lyapunov Functions (MLF) and Fuzzy-Metzler Lyapunov Functions (FMLF) that was proposed in this dissertation. New proposals are made from these analyses, and sufficient conditions for asymptotic stability of these systems in the sense of Lyapunov are presented. The results obtained are based on LMIs (Linear Matrix Inequalities) and depend on the solutions of a set of Lyapunov-Metzler inequalities, that are usually difficult to solve. Then, conditions for stability based on a subclass of Metzler matrices that, when feasible, can be described by a set of LMIs with an unidimensional search, are presented. The proposed methods generalize the similar methods available in the literature, based on FLF, for the asymptotic stability of TS fuzzy systems

Orientador: Vicente Lopes Júnior / Banca: Gilberto Pechoto de Melo / Banca: Carlos Alberto Bavastri / Resumo: Neste trabalho é discutida a aplicação do controle ativo para atenuação de vibrações mecânicas em estruturas flexíveis. Como atuadores são utilizados os materiais piezelétricos, também conhecidos como materiais inteligentes. O projeto do controlador por realimentação de estados é desenvolvido utilizando a otimização H com restrições de projeto e, a solução matemática do problema é feita através de otimização convexa envolvendo Desigualdades Matriciais Lineares (LMIs). Os estados utilizados para realimentar o sistema são estimados por um observador. Para modelar a estrutura a ser controlada, duas técnicas de identificação experimental são aplicadas: o Algoritmo de Realização de Autosistemas e a Exponencial Complexa. A aquisição dos dados para o processo de identificação é feita para uma limitada faixa de freqüência, apenas para os primeiros modos de vibrar. Os exemplos de aplicação experimental são feitos em uma estrutura do tipo placa e em uma estrutura do tipo treliça. A robustez do projeto do controlador é garantida pelos diferentes métodos e procedimentos de identificação do sistema, obtendo-se mais de um modelo. / Abstract: In this work is discussed the application of active control to attenuation of mechanical vibrations in flexible structures. Piezoelectric materials are used as actuators, also known as smart materials. The controller design by state feedback is developed using the H optimization with project restrictions, where the mathematical solution of the problem is done by convex optimization involving Linear Matrix Inequalities (LMIs). The states used to feedback the system are estimated by an observer. Two experimental identification approaches are applied, in order to find the mathematical model of the structure: the Eigensysten Realization Algorithm and Complex Exponential. The data acquisition for the identification process is done in a limited frequency band, only including the first vibration modes. The examples of experimental application are done in two kinds of structures: plate and truss structures. The robustness of the controller design is guaranteed by different methods and procedures of identification of the system obtaining more than one model. / Mestre

Die vorliegende Arbeit beschäftigt sich mit den finiten matriziellen Momentenproblemen von Stieltjes-Typ und beschreibt unter Verwendung der Methode der Fundamentalen Matrixungleichungen die Lösungsmenge durch gebrochen lineare Transformationen.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: This research focuses on the development of new efficient algorithms for multi-path planning and multi-rigid body constrained attitude control. The work is motivated by current and future applications of these algorithms in: intelligent control of multiple autonomous aircraft and spacecraft systems; control of multiple mobile and industrial robot systems; control of intelligent highway vehicles and traffic; and air and sea traffic control. We shall collectively refer to the class of mobile autonomous systems as “agents”. One of the challenges in developing and applying such algorithms is that of complexity resulting from the nontrivial agent dynamics as agents interact with other agents, and their environment. In this work, some of the current approaches are studied with the intent of exposing the complexity issues associated them, and new algorithms with reduced computational complexity are developed, which can cope with interaction constraints and yet maintain stability and efficiency. To this end, this thesis contributes the following new developments to the field of multipath planning and multi-body constrained attitude control: • The introduction of a new LMI-based approach to collision avoidance in 2D and 3D spaces. • The introduction of a consensus theory of quaternions by applying quaternions directly with the consensus protocol for the first time. • A consensus and optimization based path planning algorithm for multiple autonomous vehicle systems navigating in 2D and 3D spaces. • A proof of the consensus protocol as a dynamic system with a stochastic plant matrix. • A consensus and optimization based algorithm for constrained attitude synchronization of multiple rigid bodies. • A consensus and optimization based algorithm for collective motion on a sphere. / AFRIKAANSE OPSOMMING: Hierdie navorsing fokus op die ontwikkeling van nuwe koste-effektiewe algoritmes, vir multipad-beplanning en veelvuldige starre-liggaam beperkte standbeheer. Die werk is gemotiveer deur huidige en toekomstige toepassing van hierdie algoritmes in: intelligente beheer van veelvuldige outonome vliegtuig- en ruimtevaartuigstelsels; beheer van veelvuldige mobiele en industrile robotstelsels; beheer van intelligente hoofwegvoertuie en verkeer; en in lug- en see-verkeersbeheer. Ons sal hier “agente” gebruik om gesamentlik te verwys na die klas van mobiele outonome stelsels. Een van die uitdagings in die ontwikkeling en toepassing van sulke algoritmes is die kompleksiteit wat spruit uit die nie-triviale agentdinamika as gevolg van die interaksie tussen agente onderling, en tussen agente en hul omgewing. In hierdie werk word sommige huidige benaderings bestudeer met die doel om die kompleksiteitskwessies wat met hulle geassosieer word, bloot te l^e. Verder word nuwe algoritmes met verminderde berekeningskompleksiteit ontwikkel. Hierdie algoritmes kan interaksie-beperkings hanteer, en tog stabiliteit en doeltreffendheid behou. Vir hierdie doel dra die proefskrif die volgende nuwe ontwikkelings by tot die gebied van multipad-beplanning van multi-liggaam beperkte standbeheer: • Die voorstel van ’n nuwe LMI-gebasseerde benadering tot botsingsvermyding in 2D en 3D ruimtes. • Die voorstel van ’n konsensus-teorie van “quaternions” deur “quaternions” vir die eerste keer met die konsensusprotokol toe te pas. • ’n Konsensus- en optimeringsgebaseerde padbeplanningsalgoritme vir veelvoudige outonome voertuigstelsels wat in 2D en 3D ruimtes navigeer. • Die bewys van ’n konsensusprotokol as ’n dinamiese stelsel met ’n stochastiese aanlegmatriks. • ’n Konsensus- en optimeringsgebaseerde algoritme vir beperkte stand sinchronisasie van veelvoudige starre liggame. • ’n Konsensus- en optimeringsgebaseerde algoritme vir kollektiewe beweging op ’n sfeer.

A análise de estabilidade de sistemas não-lineares surge em vários campos da engenharia. Geralmente, esta análise consiste na determinação de conjuntos atratores estáveis e suas respectivas áreas de atração. Os métodos baseados no método de Lyapunov fornecem estimativas destes conjuntos. Entretanto, estes métodos envolvem uma busca não sistemática por funções auxiliares chamadas funções de Lyapunov. Este trabalho apresenta um procedimento sistemático, baseado no método de Lyapunov, para estimar conjuntos atratores e as respectivas áreas de atração para uma classe de sistemas não-lineares, aqui chamado de problema de Lure estendido. Este problema consiste de sistemas não-lineares que podem ser escritos na forma do problema de Lure, cuja função não-linear pode violar a condição de setor em torno da origem. O procedimento desenvolvido é baseado na extensão do princípio de invariância de LaSalle e usa as funções de Lyapunov genéricas do problema de Lure para estimar o conjunto atrator e sua respectiva área de atração. Os parâmetros das funções de Lyapunov são obtidos resolvendo um problema de otimização que pode ser colocado na forma de desigualdades matriciais lineares (LMIs). / The stability analysis of nonlinear systems is present in several engineering fields. Usually, the concern is the determination of stable attractor sets and their associated attraction areas. Methods based on the Lyapunov method provide estimates of these sets. However, these methods involve a nonsystematic search for auxiliary functions called Lyapunov functions. This work presents a systematic procedure, based on Lyapunov method, to estimate attractor sets and their associated attraction areas of a class of nonlinear systems, called in this work extended Lure problem. The extended Lure problem consists of nonlinear systems like those of Lure problem where the nonlinear functions can violate the sector conditions around the origin. The developed procedure is based on the extension of invariance LaSalle principle and uses the general Lyapunov functions of Lure problem to estimate the attractor set and their associated attraction area. The parameters of the Lyapunov functions are obtained solving an optimization problem write like a linear matrix inequality (LMI).

Resumo: Este trabalho tem como propósito utilizar técnicas de controle robusto para atenuação ativa de vibração mecânica em estruturas acopladas a atuadores e sensores piezelétricos. Os controladores são projetados segundo o enfoque de otimização convexa, com os requisitos envolvendo desigualdades matriciais lineares (LMIs). A proposta é ilustrar duas sínteses diferentes de realimentação via LMIs. A primeira é o projeto de controladores por realimentação de estados, estimados por um observador, considerando incertezas paramétricas do tipo politópicas. A segunda metodologia é baseada no controle H8 via realimentação do sinal de saída, considerando incertezas dinâmicas limitadas por norma. Os sensores/atuadores são posicionados em pontos ótimos utilizando-se a norma H8 como índice de desempenho. Os modelos matemáticos utilizados na síntese dos controladores foram obtidos a partir do método dos elementos finitos considerando o acoplamento eletromecânico entre os atuadores/sensores e a estrutura base ou a partir de métodos de identificação. Neste contexto, este trabalho também discute e exemplifica o algoritmo de realização de autosistemas (ERA). Três exemplos são solucionados para exemplificar a metodologia implementada: uma estrutura tipo placa, uma viga engastada-livre e a supressão ativa de flutter em um aerofólio 2-D, problema de grande interesse na indústria aeronáutica. Os resultados mostraram uma significante atenuação da vibração estrutural na faixa de freqüência de interesse e o atendimento dos requisitos impostos na fase de projeto. / Abstract: The proposal of this work is to use robust control techniques in order to suppress mechanical vibration in structures with pieozoelectric sensors and actuators coupled. The controllers are designed by convex optimization and the constraints are dealt through linear matrix inequalities (LMIs) frameworks. Two different methodologies to feedback the system by using LMIs are explained. The first one is the observer-based state-feedback considering polytopic uncertainties. The second one is the H output feedback control considering norm-bound uncertainties. The sensors/actuators are located in optimal placements by using H norm as performance index. The mathematical models used in the controller design were obtained by finite element methods considering eletromechanical effects between the host structure and piezoelectric sensors/actuators patches or by using identification methods. In this sense, it is also discussed the eigensystem realization algorithm (ERA). Three different applications are proposed and solved in order to illustrate the applicability of the methodology: a cantilever plate; a cantilever beam; and an active flutter suppression in a 2-D airfoil, a problem of considered interest in the aeronautic industry. The results showed the vibration suppression in the bandwidth of interest when submited to the requirements imposed by practical situations. / Orientador: Vicente Lopes Junior / Coorientador: Edvaldo Assunção / Banca: Vicente Lopes Junior / Banca: Marcelo Carvalho Minhoto Teixeira / Banca: Edilson Hiroshi Tamai / Mestre

La dérégulation du marché de l'électricité, qui concerne progressivement tous les pays d'Europe, modifie profondément l'approche technico-économique dans l'exploitation et l'optimisation des réseaux électriques. C'est dans ce nouveau contexte que les spécialistes des .réseaux électriques se voient de plus en 'plus confrontés à de nombreux défis. Le développement des dispositifs FACTS (Flexible AC Transmission System) ouvre de nouvelles perspectives pour une meilleure exploitation des réseaux par leur action continue et rapide _sur les différents paramètres du réseau. Toutefois, ces dispositifs, ne sont pas à l'abri d'éventuelles interactions entre eux mais aussi avec d'autres éléments du réseau tels que les charges dynamiques. Cette thèse, qui s'inscrit dans le cadre de l'analyse et l'amélioration de la sécurité dynamique des réseaux électriques, s'est focalisé sur l'impact de l'interaction des dispositifs FACTS avec les charges dynamiques. Dans ce travail, il a été démontré que les boucles d'amortissement des oscillations de puissance dimensionnées par les techniques classiques de sensibilité ne sont pas robustes vis-à-vis des variations des modèles de charge. Pour remédier ,à ces problèmes, une première méthode, basée sur la sensibilité des valeurs propres et tenant compte. des variations des modèles de charge, a été proposée. Deux autres méthodes plus robustes, basées sur les techniques LMI, ont été mises en oeuvre. La première, basée Sllr une représentation polytopique des systèmes incertains, convient au cas d'incertitudes non structurées. La deuxième, basée sur une représentation LPV (système linéaire à paramètres variants), permet de traiter le cas d'incertitudes structurées.