[pt] O trabalho apresenta uma teoria de fatorações coprimas para sistemas não lineares e aplicações dessa teoria em problemas de controle. A parte inicial é exatamente a teoria de fatorações coprimas, que se assemelha à versão linear. O problema da estabilização de sistemas não lineares é resolvido através de realimentação aditiva, com pré e pós compensadores dinâmicos não lineares. A solução para esse problema é dada na forma da classe de compensadores que estabilizam o sistema. São também apresentadas condições para a estabilidade na presença de ruídos aditivos. Outro problema bastante relevante do ponto de vista de controles é o da especificação da dinâmica do sistema de malha fechada. O enfoque apresenta soluções de caráter local, o que permite que a dinâmica a ser especificada seja definida apenas sobre uma restrição do espeço de entrada. Dessa forma tornou-se factível a especificação de dinâmicas dentro de uma classe relativamente ampla. São discutidas possibilidades para o problema da regulação. Também utilizando condiçòes locais é apresentada uma teoria de estabilização robusta com relação a perturbações não estruturadas. Algumas soluções explícitas e relativamente estruturadas são apresentadas. / [en] The control of nonlinear systems via coprime factorization is the subject of this dissertation. Initially, a broad theory concerning nonlinear factorizations is presented. The class of stabilizing controllers for a given nonlinear plant is derived using that theory. Then, there are derived sufficient conditions for the closed loop system are also presented. One of the major departures from the original work on nonlinear factorizations is the fact that the solutions presented need only to be locally derived, which allows a wider class of dynamics to be assigned for the closed loop input- output transference relation. The robust control of nonlinear systems is achieved through the use of locally defined solutions, allowing to control systems subject to some relatively structured perturbations.

abstract: The problem of systematically designing a control system continues to remain a subject of intense research. In this thesis, a very powerful control system design environment for Linear Time-Invariant (LTI) Multiple-Input Multiple-Output (MIMO) plants is presented. The environment has been designed to address a broad set of closed loop metrics and constraints; e.g. weighted H-infinity closed loop performance subject to closed loop frequency and/or time domain constraints (e.g. peak frequency response, peak overshoot, peak controls, etc.). The general problem considered - a generalized weighted mixed-sensitivity problem subject to constraints - permits designers to directly address and tradeoff multivariable properties at distinct loop breaking points; e.g. at plant outputs and at plant inputs. As such, the environment is particularly powerful for (poorly conditioned) multivariable plants. The Youla parameterization is used to parameterize the set of all stabilizing LTI proper controllers. This is used to convexify the general problem being addressed. Several bases are used to turn the resulting infinite-dimensional problem into a finite-dimensional problem for which there exist many efficient convex optimization algorithms. A simple cutting plane algorithm is used within the environment. Academic and physical examples are presented to illustrate the utility of the environment. / Dissertation/Thesis / M.S. Electrical Engineering 2013

Nous considérons deux classes de systèmes fractionnaires linéaires invariants dans le temps avec des ordres commensurables et des retards discrets. La première est composée de systèmes fractionnaires à entrées multiples et à une sortie avec des retards en entrées ou en sortie. La seconde se compose de systèmes fractionnaires de type neutre avec retards commensurables. Nous étudions la stabilisation de la première classe de systèmes à l'aide de l'approche de factorisation. Nous obtenons des factorisations copremières à gauche et à droite et les facteurs de Bézout associés: ils permettent de constituer l'ensemble des contrôleurs stabilisants. Pour la deuxième classe de systèmes, nous nous sommes intéressés au cas critique où certaines chaînes de pôles sont asymptotiques à l'axe imaginaire. Tout d'abord, nous réalisons une approximation des pôles asymptotiques afin de déterminer leur emplacement par rapport à l'axe. Le cas échéant, des conditions nécessaires et suffisantes de stabilité H-infini sont données. Cette analyse de stabilité est ensuite étendue aux systèmes à retard classiques ayant la même forme. Enfin, nous proposons une approche unifiée pour les deux classes de systèmes à retards commensurables de type neutre (standards et fractionnaires). Ensuite, la stabilisation d'une sous-classe de systèmes neutres fractionnaires est étudiée. Premièrement, l'ensemble de tous les contrôleurs stabilisants est obtenu. Deuxièmement, nous prouvons que pour une grande classe de contrôleurs fractionnaires à retards il est impossible d'éliminer dans la boucle fermée les chaînes de pôles asymptotiques à l'axe imaginaire si de telles chaînes sont présentes dans les systèmes à contrôler. / We consider two classes of linear time-invariant fractional systems with commensurate orders and discrete delays. The first one consists of multi-input single-output fractional systems with output or input delays. The second one consists of single-input single-output fractional neutral systems with commensurate delays. We study the stabilization of the first class of systems using the factorization approach. We derive left and right coprime factorizations and Bézout factors, which are the elements to constitute the set of all stabilizing controllers. For the second class of systems, we are interested in the critical case where some chains of poles are asymptotic to the imaginary axis. First, we approximate asymptotic poles in order to determine their location relative to the axis. Then, when appropriate, necessary and sufficient conditions for H-infinity-stability are derived. This stability analysis is then extended to classical delay systems of the same form and finally a unified approach for both classes of neutral delay systems with commensurate delays (standard and fractional) is proposed. Next, the stabilization of a subclass of fractional neutral systems is studied. First, the set of all stabilizing controllers is derived. Second, we prove that a large class of fractional controllers with delays cannot eliminate in the closed loop chains of poles asymptotic to the imaginary axis if such chains are present in the controlled systems.