In mathematical epidemiology, disease transmission is commonly assumed to behave in accordance with the law of mass action; however, other disease incidence terms also exist in the literature. A homogeneous Susceptible-Infectious-Removed (SIR) model with a generalized incidence term is presented along with analytic and numerical results concerning effects of the generalization on the global disease dynamics. The spatial heterogeneity of the metapopulation with nonrandom directed movement between populations is incorporated into a heterogeneous SIR model with nonlinear incidence. The analysis of the combined effects of the spatial heterogeneity and nonlinear incidence on the disease dynamics of our model is presented along with supporting simulations. New global stability results are established for the heterogeneous model utilizing a graph-theoretic approach and Lyapunov functions. Numerical simulations confirm nonlinear incidence gives raise to rich dynamics such as synchronization and phase-lock oscillations.

Les travaux de cette thèse portent sur les problèmes d'analyse de stabilité et de synthèse de commande de systèmes non-linéaires à commutations en temps discret. Nos résultats obtenus sont fondés sur une nouvelle fonction de Lyapunov-Lur'e adaptée au temps discret. Nous reprenons le problème classique d'analyse de stabilité globale de systèmes linéaires connectés à une non-linéarité du type secteur borné. Notre fonction permet de traiter une classe de non-linéarités plus générale que celle des approches fondées sur la fonction de Lur'e classique. Ensuite, la stabilité locale et la synthèse de commande de ces systèmes avec une loi de commande non-linéaire saturée sont résolues en considérant les lignes de niveau de notre fonction de Lyapunov comme estimation du bassin d'attraction de l'origine. Notre estimation est composée par des ensembles non-connexes et non-convexes qui s'adaptent bien à l'allure du bassin d'attraction et donc est moins conservative que les ensembles ellipsoïdaux. Nous étendons nos résultats pour étudier les systèmes à commutations lorsque chacun des modes présente une non-linéarité du type secteur et la saturation. D'une part, en supposant que la loi de commutation est arbitraire, nous obtenons des conditions suffisantes pour assurer la propriété de stabilité pour toute loi de commutation. Dans ce cadre, notre fonction s'avère intéressante afin de fournir une estimation bien adaptée au bassin d'attraction. D'autre part, en considérant la loi de commutation comme une variable de commande, nous proposons une stratégie de commutation sur le minimum des fonctions de Lyapunov modales. Cette stratégie définit des partitions de l'espace d'état relatives à l'activation des modes qui ne sont pas uniquement des régions coniques, normalement exhibées par des approches fondées sur les fonctions quadratiques commutées / In this PhD thesis, several problems of stability analysis and control design of discrete-time switched nonlinear systems are addressed. As main contribution, a new class of Lyapunov functions which takes the nonlinearity into account has been proposed. We show that these functions are suitable to solve the classical stability analysis problem of linear systems connected to a cone bounded nonlinearity. Instead of the original Lyapunov Lur'e function, the assumptions about the nonlinearity variation are not required. Furthermore, the local stability analysis and control synthesis problems of Lur'e systems subject to control saturation are tackled by considering the level set of our function as an estimate of the basin of attraction. We expose that this estimate, which is given by non-convex and disconnected sets, is less conservative than ellipsoidal sets. We extend these results in order to deal with the problems of stability analysis and stabilization of discrete-time switched nonlinear systems. On one hand, we consider the case of arbitrary switching such that our sufficient conditions assure the properties of stability for all possible switching rules. In this framework, we highlight that our function is able to provide a suitable estimate of the basin of attraction. On the other hand, we tackle the problem of switching rule design aiming at the stabilization of discrete-time switched systems with nonlinear modes. We propose a switching strategy depending on the minimum of our switched Lyapunov Lur'e function. Hence, our framework leads to state space partitions, related to the mode activation, which are not restricted to conic sets, commonly exhibited by the switched quadratic functions approaches

Ce travail de thèse concerne la synthèse des observateurs et des lois de commande des systèmes d’ordre fractionnaire. Le document présenté est constitué de 4 chapitres : Le premier chapitre du manuscrit de thèse contient une introduction, traitant les notions mathématiques de base et de stabilités des systèmes d’ordre fractionnaire ainsi qu’une présentation des différentes définitions. Les conditions de stabilités de ces systèmes et quelques exemples de systèmes modélisés par des équations différentielles fractionnaires sont présentés. Dans le deuxième chapitre, nous nous sommes intéressés à la conception de plusieurs types d’observateurs dits d’ordre réduit, d’ordre plein et des observateurs fonctionnels pour les systèmes d’ordre fractionnaire avec et sans retards. Dans le cas où il n’y a pas de retards dans la dynamique du système, des observateurs d’ordres plein et réduit ont été synthétisé afin d’assurer l’estimation des pseudo-états. Dans un deuxième temps, un observateur fonctionnel a été synthétisé dans le cas où le retard est présent dans la dynamique du système. Dans le chapitre 3, nous avons travaillé sur la synthèse d’observateur pour les systèmes d’ordre fractionnaire incertains. Nos contributions sont classées en trois grandes lignes : premièrement, quand le système considéré est affecté par des entrées inconnues, un observateur fonctionnel a été proposé. En deuxième partie, des observateurs H∞ pour les systèmes d’ordre fractionnaire avec et sans retards ont été synthétisés afin d’assurer la stabilité de l’erreur d’observation. Il s’agit en fait de garantir une borne du gain L2 entre l’erreur d’observation et les perturbations non mesurables affectant la dynamique du système : ce gain L2 est aussi appelé norme H∞. Ce chapitre présente aussi la synthèse d’un observateur robuste vis-à-vis des incertitudes de modélisation pour cette classe de systèmes. Les conditions suffisantes de convergence des erreurs d’estimations des pseudo-états obtenues sont établies sous la forme d’un ensemble d’inégalités matricielles LMIs. Le dernier chapitre du manuscrit est consacré à la commande basée sur les différents observateurs obtenus. Nous nous sommes intéressés à la commande basée sur un observateur pour les systèmes d’ordre fractionnaire. Cette commande est basée sur les observateurs proposés dans les chapitres précédents. Des conditions de stabilité et des procédures de synthèse sont présentées / This work focuses on the synthesis of observers and the controller laws for fractional order systems. The presented document consists of 4 chapters: The first chapter of the theses manuscript contains an introduction dealing with the basic mathematical notions and the stability analysis of fractional systems as well as a presentation of the different definitions. The stability conditions of these systems and some examples of systems modeled by fractional differential equations are presented. In the second chapter, we were interested in the design of several types of observers of reduced order, full order, and functional observers for fractional systems with and without delays. In the case where there are no delays in the dynamics of the system, observers of full and reduced orders have been synthesized in order to ensure the estimation of the pseudo-states. In a second step, a functional observer was synthesized in the case where the delay is present in the dynamics of the system. In Chapter 3, we worked on observer synthesis for uncertain fractional order systems. Our contributions are classified into three main lines: first, when the system under consideration is affected by unknown inputs, a functional observer has been proposed. In the second part, H∞ observers for fractional order systems with and without delays have been synthesized to ensure the stability of the estimation error. It is a question of guaranteeing a bound of the L2 gain between the observation error and the non-measurable perturbations affecting the dynamics of the system: this gain L2 is also called H∞ norm. In last part of this chapter, the synthesis of a robust observer with respect to modeling uncertainties for this class of systems is presented. The sufficient conditions of convergence of the estimation errors of the pseudo-states obtained are established in the form of a set of matrix inequalities LMIs. The last chapter of the manuscript is devoted to the command based on the different observers obtained. We were interested in observer-based control for fractional order systems. This command is based on the observers proposed in the previous chapters. Stability conditions and synthesis procedures are presented

Made available in DSpace on 2015-03-04T18:51:12Z (GMT). No. of bitstreams: 1 Dissertacao Daniela Paula.pdf: 457309 bytes, checksum: 2cece1f133cd1224c92821fe3bd36e8b (MD5) Previous issue date: 2009-07-27 / Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior / Due, in part, to the nowadays considerable body of theoretical results for Markov Jump Linear Systems (MJLS), there has been recently an intense interplay between the classical switch systems and MJLS theory. Although the development of these theories came up independently, in a broad way MJLS can be seen as a class of switch systems with a stochastic switching mecanism. Motivated by the diversity of methods of these theories and its potentiality in the treatment of systems with requires tolerance to failure (the so-called safety-critical and highintegrity systems), it is our intention in this dissertation to make up a synthesis of the most relevant methods, setting against the two theories. In view of the huge amount of results of these theories, we focus here just on the stability problem. We begin presenting well known tools such as common Lyapunov functions and others which are related to involving classes of linear subsistems with certain particularities such as commutativity and solubility of Lie algebra. Rigth after, we present the concept of average dwell time, part Lyapunov functions and results about design of switch. Using the average dwell time at the linear systems with stable and unstable systems with the rules already demonstrated we claim some results about stability that applied at linear systems with markovian switch. / Devido em parte, ao considerável corpo de resultados teóricos para Sistemas Lineares com Saltos Markovianos (SLMS), tem havido recentemente uma intensa interação entre a teoria clássica de sistemas com chaveamento (switched systems) e a teoria de SLSM. Apesar do desenvolvimento dessas teorias terem acontecido essencialmente de maneira independentes, num sentido amplo SLMS pode ser visto como um sistema com chaveamento cujo mecanismo de chaveamento é estocástico. Motivados pela diversidade de métodos dessas teorias e sua enorme potencialidade no tratamento de sistemas que exigem comportamentos tolerantes a falhas (faz parte do que se denomina na literatura especializada como safety-critical and high integrity systems) é nossa intenção nesta dissertaçãoo fazer uma síntese dos métodos mais relevantes, contrapondo as duas teorias. Tendo em vista a enorme quantidade de resultados, focaremos apenas o problema de estabilidade. Começaremos o estudo com critérios já conhecidos como a construção de uma função comum de Lyapunov para os sistemas e outros que dizem respeito à estabilidade em classes de subsistemas lineares que possuem certas particularidades como comutatividade e solubilidade da álgebra de Lie gerada pela coleção de matrizes. Em seguida, apresentaremos os conceitos de tempo médio de habitação, funções de Lyapunov por partes e os resultados sobre design de switch. Através do estudo do tempo médio de habitação em sistemas lineares com matrizes estáveis e instáveis, juntamente com os critérios já estudados referentes às classes de subsistemas para as quais é possível a construção de uma função comum de Lyapunov, chegamos a alguns resultados para estabilidade, que aplicamos ao caso de chaveamento Markoviano.

The present research study is comprised of two main parts. The first part aims at the development of a systematic system-theoretic framework that allows the derivation of optimal chemotherapy protocols for HIV patients. The proposed framework is conceptually aligned with a notion of continuous-time model predictive control of nonlinear dynamical systems, and results in an optimal way to control viral replication, while maintaining low antiretroviral drug toxicity levels. This study is particularly important because it naturally integrates powerful system-theoretic techniques into a clinically challenging problem with worldwide implications, namely the one of developing chemotherapy patterns for HIV patients that are effective and do not induce adverse side-effects. The second part introduces a new digital controller design methodology for nonlinear (bio)chemical processes, that reflects contemporary necessities in the practical implementation of advanced process control strategies via digital computer-based algorithms. The proposed methodology relies on the derivation of an accurate sampled-data representation of the process, and the subsequent formulation and solution to a nonlinear digital controller synthesis problem. In particular, for the latter two distinct approaches are followed that are both based on the methodological principles of Lyapunov design and rely on a short-horizon model-based prediction and optimization of the rate of“energy dissipation" of the system, as it is realized through the time derivative of an appropriately selected Lyapunov function. First, the Lyapunov function is computed by solving the discrete Lyapunov matrix equation. In the second approach however, it is computed by solving a Zubov-like functional equation based on the system's drift vector field. Finally, two examples of a chemical and a biological reactor that both exhibit nonlinear behavior illustrate the main features of the proposed digital controller design method.

Orientador: Pedro Luis Dias Peres / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-06T06:53:10Z (GMT). No. of bitstreams: 1 Valmorbida_Giorgio_M.pdf: 1105118 bytes, checksum: 1fec640e2bfaeb6dac37afed313255ef (MD5) Previous issue date: 2006 / Resumo: Este trabalho apresenta resultados no contexto de estabilidade de sistemas com atraso. A estabilidade de sistemas incertos com atraso é estudada utilizando o Teorema do Pequeno Ganho Escalonado a partir de um sistema de comparação. Aplicando resultados do Lema de Finsler e empregando matrizes de Lyapunov dependentes de parâmetro nas desigualdades matriciais lineares do Teorema do Pequeno Ganho, são obtidas condições independentes e condições dependentes do atraso para sistemas incertos. Sistemas com atraso que apresentam entrada com saturação em posição são estudados visando obter condições para cômputo de ganhos de realimentação de estados e visando obter uma estimativa para a região de atração do sistema em malha fechada. É considerada uma lei de controle com realimentação do estado atual e do estado atrasado. Funcionais de Lyapunov-Krasovskii são utilizados na obtenção das condições de estabilizabilidade. A maximização das estimativas das regiões de atração é feita a partir da solução de problemas de otimizaçã.o com restrições na forma de desigualdades matriciais lineares / Abstract: This work presents results in the context of time-delay system stability. Uncertain time-delay systems are studied by means of the Scaled Small-Gain Theorem. By applying results from Finsler's Lemma and using parameter-dependent Lyapunov matrices, delay-dependent and delay-independent conditions for uncertain systems are obtained in terms of linear matrix inequalities. Time-delay system presenting amplitude-saturating inputs are analyzed aiming to establish conditions to compute state-feedback gains and to obtain an estimate of the bassin of attraction of the system. A control law composed by a current state-feedback and a delayed state-feedback is considered. Lyapunov-Krasovskii functionals are the starting point to obtain the stabilizability conditions. The maximization the estimates of the bassin of attraction is carried out by solving an optimization problem whose constraints are linear matrix inequalities / Mestrado / Automação / Mestre em Engenharia Elétrica

Orientadores: Pedro Luis Dias Peres, Ricardo Coração de Leão Fontoura de Oliveira / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-14T06:55:42Z (GMT). No. of bitstreams: 1 Moreira_HeberRocha_M.pdf: 2003528 bytes, checksum: 9b091e7706d78e1298bd1d0603504e7f (MD5) Previous issue date: 2009 / Resumo: Esta dissertação trata de um dos mais importantes problemas em aberto na teoria de controle, o projeto de controladores por realimentação estática de saída. A principal contribuição é propor um método para computar controladores robustos estáticos por realimentação de saída para sistemas lineares incertos discretos no tempo, usando um controlador por realimentação de estados como parâmetro de entrada do método. Além disso, os resultados são estendidos para tratar do projeto de controle H2 robusto por realimentação de saída. As condições de síntese são formuladas em termos de um procedimento convexo de otimização, baseado em um conjunto finito de desigualdades matriciais lineares. Exemplos numéricos são apresentados para ilustrar a eficiência dos metodos propostos quando comparados com outros métodos existentes na literatura. / Abstract: This thesis deals with one of the most important open problems in control theory, the design of static output feedback controllers. The main contribution is to propose a method to compute robust static output feedback controllers for uncertain linear discrete-time systems, using a state feedback controller as an input parameter for the method. Additionally, the results are extended to cope with H2 static output feedback control design. The synthesis conditions are formulated in terms of a convex optimization procedure, based on a finite set of linear matrix inequalities. Numerical examples are presented to illustrate the effectiveness of the proposed approach compared to other methods available in the literature. / Mestrado / Automação / Mestre em Engenharia Elétrica

Orientadores: Pedro Luis Dias Peres, Ricardo Coração de Leão Fontoura de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-25T15:07:40Z (GMT). No. of bitstreams: 1 Lacerda_MarcioJunior_D.pdf: 1206868 bytes, checksum: c5abb83323581461eb40c95f27c95097 (MD5) Previous issue date: 2014 / Resumo: Este trabalho apresenta novas condições na forma de desigualdades matriciais lineares para o projeto de filtros H-infinito de ordem completa em três diferentes contextos: i) sistemas lineares incertos discretos com um atraso variante no tempo afetando os estados; ii) sistemas lineares com parâmetros variantes no tempo, contínuos e discretos, sujeitos a incertezas nas medições dos parâmetros; iii) sistemas não lineares quadráticos contínuos e discretos no tempo. Para cada contexto, o objetivo é projetar filtros: i) com termos atrasados nos estados; ii) dependentes dos parâmetros incertos medidos; iii) com termos quadráticos. Em cada um dos casos, o ponto de partida é a existência de uma função de Lyapunov que assegure estabilidade e um limitante para a norma H-infinito do sistema aumentado, ou seja, o sistema original conectado com o filtro de ordem completa. As condições de projeto são obtidas impondo-se uma determinada estrutura para as variáveis de folga, resultando em desigualdades matriciais com parâmetros escalares. A eficácia das condições apresentadas é ilustrada por meio de comparações numéricas utilizando exemplos da literatura / Abstract: This work presents new conditions in the form of linear matrix inequalities for full order H-infinity filter design in three different contexts: i) uncertain linear discrete-time systems with a time-varying delay affecting the states ii) linear parameter-varying systems, continuous and discrete-time, subject to inexactly measured parameters; iii) continuous and discrete-time nonlinear quadratic systems. For each context, the aim is to design filters: i) with state-delayed terms; ii) dependent upon the inexactly measured parameters; iii) with quadratic terms. In each case, the starting point is the existence of a Lyapunov function that assures stability and a bound to the H-infinity norm of the augmented system, that is, the original system conected with the full order filter. The design conditions are obtained by imposing a given structure to the slack variables, resulting in matrix inequalities with scalar parameters. The effectiveness of the proposed conditions is illustrated by means of numerical comparisons and benchmark examples from the literature / Doutorado / Automação / Doutor em Engenharia Elétrica

Orientador: Pedro Luis Dias Peres, Ricardo Coração de Leão Fontoura de Oliveira / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-19T07:25:52Z (GMT). No. of bitstreams: 1 Tognetti_EduardoStockler_D.pdf: 1309075 bytes, checksum: 3848aea254594b5081e40f2de57efe3f (MD5) Previous issue date: 2011 / Resumo: A principal contribuição dessa tese é a proposta de novas condições, em termos de desigualdades matriciais lineares, para a análise de estabilidade e síntese de controladores de sistemas nebulosos Takagi-Sugeno, casos discretos e contínuos no tempo. A lei de controle considerada é por realimentação de estados, sendo que para sistemas contínuos no tempo são também projetados controladores de realimentação de saída estática e dinâmica, de ordem arbitrária. São adotados como critérios de desempenho os custos garantidos H2 e H?. As funções de Lyapunov propostas, mais gerais que as existentes na literatura, são construídas como a combinação nebulosa de funções quadráticas nos estados, polinomiais homogêneas de grau arbitrário em termos das funções de pertinência do sistema nebuloso. As funções de pertinência são modeladas em um espaço definido pelo produto Cartesiano de simplexos no mesmo instante de tempo, no caso contínuo, ou em múltiplos instantes de tempo, permitindo o projeto de controladores com memória finita de parâmetros presentes e passados, no caso discreto. Em sistemas contínuos, as variáveis premissas usadas na lei de controle podem ser escolhidas pelo projetista, de acordo com a disponibilidade ou não para implementação em tempo real, dando um caráter seletivo ao controlador. Nas condições de síntese de controladores, duas situações distintas para as taxas de variação das funções de pertinência são consideradas. Um conjunto convexo modela o espaço ao qual pertencem as derivadas temporais das funções de pertinência, no caso em que algumas taxas máximas de variação são conhecidas. Para taxas de variações arbitrárias, quando as variáveis premissas são os estados, utilizam-se funções de Lyapunov do tipo integral de linha associadas a uma estratégia de síntese em dois estágios. No primeiro estágio um ganho de realimentação de estados é projetado e utilizado como entrada no segundo estágio, que retorna, se possível, um controlador de realimentação de estados que atende especificações de desempenho mais exigentes, ou ganhos de realimentação estática ou dinâmica de saída, de ordem completa ou reduzida. Exemplos numéricos ilustram os resultados, mostrando que os métodos propostos podem reduzir o conservadorismo nos problemas estudados, sendo mais eficientes quando comparados com outras abordagens da literatura / Abstract: This thesis proposes, as main contribution, new linear matrix inequalities conditions for stability analysis and control synthesis of continuous and discrete-time Takagi-Sugeno fuzzy systems. A state feedback control law is considered and, for continuous-time systems, static and dynamic output feedback controllers of arbitrary order are also determined. The H2 and H? guaranteed costs are adopted as performance criteria. The proposed Lyapunov functions, more general than the others in the literature, are constructed as fuzzy combinations of quadratic in the state functions that are homogeneous polynomials of arbitrary degree depending on the membership functions of the fuzzy systems. The membership functions are modeled in a space defined by the Cartesian product of simplexes in the same instant of time, in the continuous-time case, or in multiple instants of time, yielding finite memory controllers depending on the present and past parameters, in the discrete-time case. In the continuous-time synthesis conditions, the premise variables used in the control law can be defined by the designer, accordingly to real-time availability, as a selective feature of the controller. Two different situations are considered in terms of the rates of variation of the membership functions. A convex set models the space of the timederivatives of the membership functions if some upper bounds of the variation rates are known. For arbitrary rates, when the premise variables are the states of the system, a line-integral Lyapunov function is considered associated with a two steps procedure. At the first step, a state feedback is computed and used as an input for the second step, that provides, if possible, a state feedback controller that take into account more stringent performance specifications, or static or dynamic controllers, of full and reduced orders. Numerical examples illustrate the results, showing that the proposed approaches can be less conservative and more efficient when compared with other methods available in the literature / Doutorado / Automação / Doutor em Engenharia Elétrica

Orientador: Juan Francisco Camino dos Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-22T01:28:47Z (GMT). No. of bitstreams: 1 Rodrigues_LuisAntonio_M.pdf: 1882720 bytes, checksum: efb662d3473168965c88de41357ef029 (MD5) Previous issue date: 2012 / Resumo: O presente trabalho investiga os problemas de estabilidade assintótica e desempenho H'INFINITO' de sistemas lineares a parâmetros variantes discretos no tempo. São fornecidas condições suficientes para análise de estabilidade, análise de desempenho H'INFINITO' e síntese de controladores estáticos de realimentação de saída robustos e por ganho escalonado. Além disso, é proposto um método de parametrização polinomial homogênea de sistemas LPV afins. Assume-se que a matriz do sistema tem dependência polinomial homogênea de grau arbitrário sobre os parâmetros que variam dentro de um politopo com conhecidos limitantes sobre suas taxas de variação. As propriedades geométricas do domínio politópico são exploradas para se obter um conjunto finito de desigualdades matriciais lineares que levam em consideração os limitantes sobre as taxas de variação dos parâmetros. As condições LMIs são obtidas usando uma função de Lyapunov quadrática nos estados com dependência polinomial homogênea dos parâmetros variantes em instantes sucessivos de tempo. As condições fornecidas são aplicadas no modelo LPV de um sistema vibroacústico. Comparações com resultados numéricos encontrados na literatura mostram os benefícios das técnicas propostas / Abstract: This work investigates stability and H'INFINITE' performance of discrete-time linear parameter varying systems. Sufficient conditions for stability analysis, H'INFINITE' performance analysis and synthesis of both robust and gain-scheduled static output feedback controller are provided. It is assumed that the system matrices have a homogeneous polynomial dependence of arbitrary degree on the time-varying parameters. Thus, a homogeneous-polynomially parametrization method for affine LPV systems is proposed. The parameters are assumed to vary inside a polytope and to have known bounds on their rates of variation. The geometric properties of the polytopic domain are exploited to derive a finite set of LMIs that take into account the bounds on the rates of variation of the scheduling parameters. The LMI conditions are obtained using a quadratic in the state Lyapunov function with a homogeneous polynomial dependence on the scheduling parameters at successive instants of time. The proposed techniques are applied to an LPV model of a vibroacoustic setup. Comparisons with numerical results found in literature show the benefits of the proposed approach / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica