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MASTER's DISSERTATION No 1200 ROBUST STABILITY OF MASTER’S DISSERTATION No 1200 ROBUST STABILITY OF POLYTOPIC LPV SYSTEMS: GEOMETRIC ISSUES AND THE COMPUTATIONAL COMPLEXITY by Igor Pereira Vieira School of Engineering Graduate Program in Electrical Engineering Belo Horizonte - MG July 2020 UNIVERSIDADE FEDERAL DE MINAS GERAIS SCHOOL OF ENGINEERING GRADUATE PROGRAM IN ELECTRICAL ENGINEERING ROBUST STABILITY OF POLYTOPIC LPV SYSTEMS: GEOMETRIC ISSUES AND THE COMPUTATIONAL COMPLEXITY Igor Pereira Vieira Dissertation presented to the Graduate Program in Electrical Engineering at the Universidade Federal de Minas Gerais in partial fulfillment of the requirements for obtaining a M.Sc. degree in Electrical En- gineering. Concentration area: Signals and Systems Research line: Modeling, Analysis and Control of Nonlinear Systems (MACSNL) Advisors: Prof. Leonardo Amaral Mozelli, Ph.D. Prof. Fernando de Oliveira Souza, Ph.D. Belo Horizonte - MG July 2020 UNIVERSIDADE FEDERAL DE MINAS GERAIS ESCOLA DE ENGENHARIA PROGRAMA DE PÓS-GRADUAÇÃO EM ENGENHARIA ELÉTRICA ESTABILIDADE ROBUSTA DE SISTEMAS LPV POLITÓPICOS: ASPECTOS GEOMÉTRICOS E COMPLEXIDADE COMPUTACIONAL Igor Pereira Vieira Dissertação apresentada ao Programa de Pós-Graduação em Engenharia Elétrica da Universidade Federal de Minas Gerais como requisito parcial para a obtenção do título de Mestre em Ciências, na área de Engenharia Elétrica. Área de Concentração: Sinais e Sistemas Linha de Pesquisa: Modelagem, Análise e Controle de Sistemas Não Lineares Orientadores: Prof. Leonardo Amaral Mozelli, Dr. Prof. Fernando de Oliveira Souza, Dr. Belo Horizonte - MG Julho de 2020 Vieira, Igor Pereira. V658r Robust stability of polytopic LPV systems: [recurso eletrônico] : geometric issues and the computational complexity / Igor Pereira Vieira. – 2020. 1 recurso online (100 f. : il., color.) : pdf. Orientador: Leonardo Amaral Mozelli. Coorientador: Fernando de Oliveira Souza. Dissertação (mestrado) Universidade Federal de Minas Gerais, Escola de Engenharia. Inclui bibliografia e índice. Exigências do sistema: Adobe Acrobat Reader. 1. Engenharia elétrica - Teses. 2. Liapunov, Funções de - Teses. 3. Complexidade computacional - Teses. 4. Controle robusto - Teses. I. Mozelli, Leonardo Amaral. II. Souza, Fernando de Oliveira. III. Universidade Federal de Minas Gerais. Escola de Engenharia. IV. Título. CDU: 621.3(043) Ficha catalográfica: Biblioteca Profº Mário Werneck, Escola de Engenharia da UFMG "Robust Stability of Polytopic LPV Systems: Geometric Issues and the Computational Complexity" Igor Pereira Vieira Dissertação de Mestrado submetida à Banca Examinadora designada pelo Colegiado do Programa de Pós-Graduação em Engenharia Elétrica da Escola de Engenharia da Universidade Federal de Minas Gerais, como requisito para obtenção do grau de Mestre em Engenharia Elétrica. Aprovada em 28 de julho de 2020. Por: ______________________________________ Prof. Dr. Leonardo Amaral Mozelli DELT (UFMG) - Orientador ______________________________________ Prof. Dr. Fernando de Oliveira Souza DELT (UFMG) - Co-orientador ______________________________________ Prof. Dr. Leonardo Antônio Borges Tôrres DELT (UFMG) ______________________________________ Prof. Dr. Víctor Costa da Silva Campos DELT (UFMG) ACKNOWLEDGEMENTS My most sincere thanks to my advisors Prof. Leonardo A. Mozelli and Prof. Fernando O. Souza. I am especially grateful for their con- fidence, patience, time and support in all stages of this work. Their dedication and continued commitment to excellence are truly inspiring. It is a pleasure to express my gratitude to the members of the exa- mination committe, Prof. Víctor C. S. Campos and Prof. Leonardo A. B. Tôrres, for their careful reading and their many insightful comments and suggestions. I also thank Prof. Leonardo Tôrres for the opportu- nity to participate in his magnificent course on Fundamentals of Nonlin- ear Control. To all the professors of the Department of Electronics Engineering (DELT/EE-UFMG) and of the Department of Electrical Engineering (DEE/EE-UFMG) who indirectly contributed to the preparation of this work. Special acknowledgments are made to: Prof. Reinaldo M. Pal- hares, Prof. Luis A. Aguirre, Prof. Rodney R. Saldanha, Prof. Eduardo M. A. M. Mendes and Prof. Bruno O. S. Teixeira. I am grateful to the Universidade Federal de Minas Gerais and the Graduate Program in Electrical Engineering for the opportunity. I also thank the National Council for Scientific and Technological Develop- ment (CNPq) for the partial support. I am grateful as well to my family, for the constant support and encouragement. [...] As mais soberbas pontes e edifícios, o que nas oficinas se elabora, o que pensado foi e logo atinge distância superior ao pensamento, os recursos da terra dominados, e as paixões e os impulsos e os tormentos e tudo que define o ser terrestre ou se prolonga até nos animais e chega às plantas para se embeber no sono rancoroso dos minérios, dá volta ao mundo e torna a se engolfar, na estranha ordem geométrica de tudo, e o absurdo original e seus enigmas, suas verdades altas mais que todos monumentos erguidos à verdade: e a memória dos deuses, e o solene sentimento de morte, que floresce no caule da existência mais gloriosa, tudo se apresentou nesse relance e me chamou para seu reino augusto, afinal submetido à vista humana. [...] A Máquina do Mundo Carlos Drummond de Andrade VIEIRA, I.P. Robust Stability of Polytopic LPV Systems: Geometric Issues and the Computational Complexity. Master Dissertation (Electrical Engineering) – School of Engineering, Universidade Federal de Minas Gerais. Belo Horizonte, 2020. ABSTRACT This work addresses the problem of computational complexity in robust stability certification for continuous-time Linear Parameter Varying (LPV) systems based on Parameter-Dependent Lyapunov Functions (PDLFs). New strategies are presented for the inclusion of the uncertain parameters time derivatives, designed to balance the computational cost, resulting from the reduction of the number of Linear Ma- trix Inequalities (LMIs) to be evaluated, and the conservatism. The methodology is grounded in the exploration of geometric aspects of the LPV polytopic representation. In a first approach, the gradual transition between the regular simplex – the convex object with least possible number of vertices in each dimension – and the minimum- hypervolume polytope – with factorial vertex growth as function of the number of time-varying uncertain parameters – is outlined by exchanging elements between these sets. Analytical relations that make possible evaluations about the impact of suppressing one or more simplex vertices, for the case of symmetrical uncertainties parametrically described, are provided, leading to a bendable trade-off solution. It is also shown that conservatism can occasionally be improved through simple proce- dures involving the rotation of the simplex vertices in the time-derivative parameter space, without any intervention in the geometric structure of the polytope. Subse- quently, procedures that enable the employment of cubic and orthoplectic convex hulls, in detriment of the original one, are proposed. Along with the regular sim- plexes, already covered in the literature, these polytopic families are the only ones to remain regular as the dimensionality grows. In the numerical analysis conducted, such geometries provided better results, with regard to conservatism, as the system scale was increased. Finally, using well-consolidated tools of the control theory, the previous approaches, conceived within the scope of the stability analysis, are gener- alized to the context of robust state feedback control synthesis. Keywords: Linear Parameter-Varying Systems, Parameter-Dependent Lyapunov Func- tions, Linear Matrix Inequalities, Computational Complexity, Robust Control, Poly- topic Geometry, Regular Polytopic Families VIEIRA, I.P. Estabilidade Robusta de Sistemas LPV Politópicos: Aspectos Geométri- cos e Complexidade Computacional. Dissertação de Mestrado (Engenharia Elétrica) – Escola de Engenharia, Universidade Federal de Minas Gerais. Belo Horizonte, 2020. RESUMO Este trabalho investiga o problema da complexidade computacional no âmbito da geração de certificados de estabilidade robusta para sistemas lineares a parâmetros variantes (LPV), de tempo contínuo, empregando funções de Lyapunov dependen- tes de parâmetros (PDLFs). São apresentadas novas estratégias para a inclusão das derivadas temporais dos parâmetros incertos destinadas a balancear o custo computa- cional, a partir da redução da quantidade de desigualdades matriciais lineares (LMIs) a serem avaliadas, e o conservadorismo. A metodologia é fundamentada na mani- pulação dos aspectos geométricos da representação politópica destes sistemas. Em uma primeira abordagem, é delineada a transição gradual entre o simplex regular – o objeto convexo com o menor número possível de vértices em cada dimensão – e o politopo de menor hipervolume – com crescimento fatorial de vértices em função do número de parâmetros incertos variantes no tempo –, intercambiando elementos entre esses conjuntos. São fornecidas relações analíticas que possibilitam a avaliação acerca do impacto da supressão de um ou mais vértices do simplex para o caso de incertezas simétricas descritas parametricamente, culminando em uma solução de compromisso flexível. Além disso, é mostrado que o conservadorismo pode, ocasionalmente, ser reduzido por procedimentos simples envolvendo a rotação dos vértices do simplex no espaço de parâmetros das derivadas temporais, sem qualquer intervenção na es- trutura
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