In dieser Arbeit werden lineare zeitinvariante endlichdimensionale Systeme (LTI-Systeme) mit m > 1 Eingängen und p > 1 Ausgängen untersucht (MIMO-Systeme). Diese lassen sich darstellen durch lineare Gleichungen mit Matrizen, deren Einträge Polynome im Ableitungsoperator d/dt sind. Bei Nutzung der Laplace-Transformation handelt es sich um Polynome in s. Algebraisch bilden diese einen Euklidischen Ring. Durch Überführung der Matrizen in die Hermitesche Normalform werden m Basisgrößen definiert. Die Verläufe oder Trajektorien der Basisgrößen lassen sich frei vorgegeben. Damit werden die Trajektorien sämtlicher übrigen Signale, insbesondere die der erforderlichen Eingangssignale, festgelegt und können ohne Integration berechnet werden. Ein linksteilerfremdes (auch steuerbar genanntes) Modell ist dabei nicht zwingend erforderlich. Damit eignen sich die Basisgrößen besonders zur Planung von Trajektorien. Genauer untersucht wird die Planung mit Polynomen in der Zeit als Ansatzfunktionen und die Planung von Trajektorien, die ein quadratisches Kostenfunktional minimieren. In der technischen Praxis werden die Systeme stets von den geplanten Trajektorien abweichen. Insbesondere bei instabilen Regelstrecken ist deshalb ein stabilisierender Folgeregler unentbehrlich. Die Struktur der Folgeregelung wird eingeführt und es wird deutlich gemacht, dass jede Methode zum Entwurf linearer Regler angewendet werden kann. Die Nullstellenzuweisung durch dynamische Ausgangsrückführung mit Reglern vorgegebener möglichst geringer dynamischer Ordnung wird detailliert untersucht und eine neue Lösungsmöglichkeit aufgezeigt. Durch Nutzung der modifizierten z-Transformation lässt sich die Theorie auf ein hybrides System, bestehend aus einer zeitkontinuierlichen Regelstrecke und einer zeitdiskreten digitalen Steuerung und Regelung, ausdehnen. Dabei werden die Verläufe der Signale zwischen den Abtastzeitpunkten in die Planung einbezogen. Zum Schluss werden die linearen Beobachter im Licht der polynomialen Matrizendarstellung neu untersucht. Es wird gezeigt, dass die polynomiale Matrizendarstellung einen theoretischen Rahmen bietet, in dem sich sämtliche linearen Beobachter mit einer Methode entwerfen lassen. - (Die Dissertation ist veröffentlicht in der Reihe Fortschritt-Berichte VDI, Reihe 8 - Mess-, Steuerungs- und Regelungstechnik, Band 1164 im VDI Verlag GmbH, Düsseldorf, ISBN 978-3-18-516408-8) / In this thesis linear time invariant lumped systems (LTI-systems) with m>1 inputs and p > 1 outputs (MIMO-systems) are investigated. These systems can be represented by linear equations with matrices, whose entries are polynomials in the differential operator d/dt. If Laplace-transform is employed, the polynomials are in s. Algebraically polynomials form a Euclidean ring. The conversion of the matrices to the Hermite form leads to defining m basic variables. The trajectories of the basis variables may be chosen arbitrarily. With that choice the trajectories of all remaining variables and especially the input variables are determined and can be calculated without integration. A left coprime (also called controllable) model is not required. Hence basis variables are particularly useful for planning trajectories. Special attention is paid to planning trajectories with polynomials in time as basic functions and planning trajectories which minimise a quadratic functional of costs. In engineering practice the systems will always differ from the planed trajectories. Especially with unstable plants a stabilising tracking controller is compulsory. The structure of the tracking control is introduced. It becomes apparent that every linear theory for the design of closed loop controllers is suitable. Pole assignment by dynamic output feedback with low order controllers of a fixed structure is looked at in more detail. A new approach to this problem is presented. Using the modified z-transform the theory is extended to hybrid systems consisting of a digital or discrete time controller and a plant in continuous time. Thereby the course of the signals between the sampling moments is taken into account. Finally linear observers are reinvestigated using the polynomial matrix representation. It is shown that the polynomial matrix representation provides a theoretical framework in which all linear observers can be designed.

This Thesis is concerned with the approximation of high order controllers or the controller reduction problem. We firstly consider approximating high-order controllers by low order controllers based on the closed-loop system approximation. By approximating the closed-loop system transfer function, we derive a new parametrized double-sided frequency weighted model reduction problem. The formulas for the input and output weights are derived using three closed-loop system configurations: (i) by placing a controller in cascade with the plant, (ii) by placing a controller in the feedback path, and (iii) by using the linear fractional transformation (LFT) representation. One of the weights will be a function of a free parameter which can be varied in the resultant frequency weighted model reduction problem. We show that by using standard frequency weighted model reduction techniques, the approximation error can be easily reduced by varying the free parameter to give more accurate low order controllers. A method for choosing the free parameter to get optimal results is being suggested. A number of practical examples are used to show the effectiveness of the proposed controller reduction method. We have then considered the relationships between the closed-loop system con gurations which can be expressed using a classical control block diagram or a modern control block diagram (LFT). Formulas are derived to convert a closed-loop system represented by a classical control block diagram to a closed-loop system represented by a modern control block diagram and vice versa.

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

Orientador: Pedro Luis Dias Peres / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-13T19:35:14Z (GMT). No. of bitstreams: 1 Borges_RenatoAlves_D.pdf: 1110575 bytes, checksum: 60707903258309c73eb16c6b81e8e7ba (MD5) Previous issue date: 2009 / Resumo: A principal contribuição desta tese é a proposta de condições de sintese de filtros e controladores lineares, tanto robustos quanto dependentes de parametros, para sistemas discretos variantes no tempo.Os controladores, ou filtros, são obtidos solucionando problemas de otimização formulados em termos de desigualdades matriciais bililineares, por meio de um metodo que se baseia na alternancia de problemas convexos descritos por desigualdades matriciais lineares. Para obtenção das condiçoes de sintese foram utilizadas tanto funções de Lyapunov afins nos parâmetros quanto ametros, alem de variáveis multi-afins em diferentes instantes de tempo dos parâmetros, alem de variaveis extras introduzidas pelo lema de Finsler. Nesse contexto, sao tratados problemas de sintese com custo garantido H, assegurando robustez em relação a incertezas não estruturadas. Simulaçoes numéricas ilustram a eficiencia dos metodos propostos em termos de desempenho H quando comparados com outros metodos da literatura / Abstract: The main contribution of this dissertation is to propose conditions for linear filter and controller design, considering both robust and parameter dependent structures, for discrete time-varying systems. The controllers, or filters, are obtained through the solution of optimization problems, formulated in terms of bilinear matrix inequalities, using a method that alternates convex optimization problems described in terms of linear matrix inequalities. Both affine and multi-affine in different instants of time (path dependent)Lyapunov functions were usedto obtain the design conditions, as wellas extra variables introduced bythe Finsler's lemma.Design problems that take into account an H guaranteed cost were investigated, providing robustness with respect to unstructured uncertainties. Numerical simulations show the effciency of the proposed methods in terms of H performance when compared with other strategies from the literature / Doutorado / Automação / Doutor 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

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While moment matching can effectively reduce the dimension of a linear, time-invariant system, it can simultaneously fail to improve the stable time-step for the forward Euler scheme. In the context of a semi-discrete heat equation with spatially smooth forcing, the high frequency modes are virtually insignificant. Eliminating such modes dramatically improves the stable time-step without sacrificing output accuracy. This is accomplished by modal filtration, whose computational cost is relatively palatable when applied following an initial reduction stage by moment matching. A bound on the norm of the difference between the transfer functions of the moment-matched system and its modally-filtered counterpart yields an intelligent choice for the mode of truncation. The dual-stage algorithm disappoints in the context of highly nonnormal semi-discrete convection-diffusion equations. There, moment matching can be ineffective in dimension reduction, precluding a cost-effective modal filtering step.

Ce mémoire est dédié à l’étude de la synthèse de l’estimation d’état en temps fini par une approche algébrique (les techniques développées au sein de l’équipe ALIEN) pour les systèmes linéaires à paramètres invariant dans le temps (LTI) sujets à des perturbations extérieures inconnues, les systèmes linéaires à paramètres variant dans le temps (LTV) et les systèmes linéaires à commutation en temps continu (SLC). Pour les systèmes LTI et LTV, une expression formelle de l’état en fonction des intégrales itérées des sorties et de l’entrée a été donnée. Pour les systèmes linéaires à commutation, en combinant les résultats de l’estimation d’état pour les systèmes LTI et de la détection de l’instant de commutation en temps réel présentée dans le chapitre 4, nous donnons la démarche principale de l’estimation en temps réel du mode courant et l’état continu du système. Pour ce faire, on applique certains outils mathématiques : la transformation de Laplace, les outils issus du calcul opérationnel et la théorie des distributions / This PhD thesis is dedicated to the synthesis of the state estimation in a finite time by an algebraic approach (the techniques developed within the ALIEN group) for the linear time-invariant systems (LTI) subject to the external unknown disturbances, the linear time-varying systems (LTV) and the switched linear systems (SLC) in continuous time. For the LTI and LTV systems, a formal expression of state as a function of iterated integrals of the output and the input is obtained. For switched linear systems, combining the results of state estimation for LTI systems and switch instant detection presented in Chapter 4, we give the main approach of current mode estimation and the continuous state estimation in real time. To do this, one applies some mathematical tools: Laplace transforms, the operational calculus and the theory of distribution

Orientador: Pedro Luis Dias Peres / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-05T03:23:54Z (GMT). No. of bitstreams: 1 Leite_ValterJuniordeSouza_D.pdf: 4310851 bytes, checksum: bcd0414d19eb46e02496290857fdf9bc (MD5) Previous issue date: 2005 / Resumo: Este trabalho trata da aplica¸c¿ao de funcionais de Lyapunov e Lyapunov-Krasovskii dependentes de parâmetro a alguns problemas selecionados da área de controle robusto, a saber: D-estabilidade robusta de polítipo de matrizes, D-estabilidade robusta de politopos de polinômios matriciais, estabilidade robusta de sistemas neutrais com atrasos variantes no tempo e controle robusto H8 de sistemas discretos no tempo com atraso nos estados. É utilizada a representação politópica para as incertezas dos sistemas estudados. São obtidas formulações convexas, na forma de desigualdades matriciais lineares, suficientes para a solução dos problemas selecionados. Essas condições podem ser resolvidas numericamente de maneira eficiente por meio de algoritmos especializados baseados em pontos interiores. Os resultados obtidos são menos conservadores que os encontrados na literatura, baseados em geral na estabilidade quadrática, isto é, as matrizes dos funcionais são fixas e independentes da incerteza / Abstract: This work deals with the application of parameter dependent Lyapunov and Lyapunov-Krasovskii functionals to some selected problems of robust control: robust D-stability of polytopes of matrices, robust D-stability of polytopes of polynomial matrices, robust stability of uncertain neutral systems with timevarying delays and robust H8 control of uncertain discrete time delay systems. The polytopic representation is used to describe the uncertainties. Convex formulations are obtained, in terms of inear matrix inequalities, that are sufficient for the solution of the selected problems. Those conditions can be solved in a efficient way through specialized interior point algorithms. The obtained results are less conservative than those from the literature, in general based on quadratic stability, i.e., the matrices in the functionals are fixed and do not depend on the uncertainty / Doutorado / Automação / Doutor em Engenharia Elétrica