Thesis: M.S., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 1977 / Includes bibliographical references. / M.S. / M.S. Massachusetts Institute of Technology, Department of Aeronautics and Astronautics

This dissertation addresses the design of controllers for multi variable finite-dimensional, linear, autonomous dynamical systems with distinct sets of slow and fast dynamics, which thus display multiple time scale behavior. It seeks, specifically, to compose overall controllers from lower-order dynamic output compensators, which are designed separately for slow and fast approximating models of the given plant. Reduction of the dynamic order of the design problem and the avoidance of numerically ill-conditioned interaction between modes of disparate orders of magnitude are among the patent advantages which pertain to such a design. As is well known, the explicit singularly perturbed systems, as a class, possess the multiple time scale property, while the broader class of implicit singularly perturbed systems and the multiple time scale systems are partially overlapping system classes. A composite state feedback controller scheme for the explicit singularly perturbed system has long been known. In connection with dynamic output controllers, however, only the case of the explicit singularly perturbed system with, restrictively, open-loop-stable fast dynamics has so far received attention in the literature. The dissertation, in providing a composite dynamic controller design suitable as well to the implicit singularly perturbed multiple time scale system, which furthermore is permitted to exhibit fast (or "parasitic") as well as slow (or "normal") open-loop instabilities, thus presents a more comprehensive dynamic controller strategy for this system than so far reported in the literature. Working with multivariable transfer functions, the dissertation applies certain fractional representation techniques of modem Algebraic System Theory to the frequency domain study of the multiple time scale system. Following the work of D. W. Luse and H. K. Khalil, we replace the transfer matrix of the multiple time scale system with two or more lower-order transfer functions, each of which has validity, in its own respective frequency range, as an approximation to the first. Following the work. of M. Vidyasagar, we write the rational transfer function of each of these approximating lower-order subsystems as a "fraction" over the Ring of proper and stable rational matrices. Parametrizations, in terms of "free" matrices belonging to this Ring, of the sets of stabilizing controllers for the lower-order subsystems :md the corresponding achievable stable closed-loop behaviors then enable the relevant design syntheses to be achieved. In this development we exploit, specifically, a Theorem proved by Luse and Khalil concerning the relation of the closed-loop poles in a feedback configuration of multiple time scale systems to the poles in corresponding lower-order closed-loop systems. The dissertation's novel contribution thus resides in (i) interpreting the Theorem of Luse and Khalil as the outline for a possible separate and-composite dynamic output controller strategy, and in (ii) adapting algebraic techniques derivative from Vidyasagar for actually realizing the putative strategy as a set of concretely implementable design procedures. Three specific design procedures are developed and formalized in the dissertation: the first for achieving mere stabilization of the multiple time scale system, the second for the placement of slow and fast poles within specified subregions of the Complex Plane, and the third for achieving entirely arbitrary pole placement. Since these procedures derive, methodologically, from Vidyasagar's fractional representations, they are intrinsically multivariable in character. Since the procedures are validated by the Theorem of Luse and Khalil, they are applicable, in principle, to the very broadest class of linear autonomous multiple time scale systems. The dissertation presents its three design procedures in high-Ievel-algorized form. For application to the explicit singularly perturbed system, the three procedures entail no further matrix operations than addition, multiplication, inversion and the determination of linear constant controller and observer gains, the most basic of operations in any available Control software. In connection with the implicitly singularly perturbed multiple time scale system, their concrete application requires some further computational development pertaining to the attainment of coprime factorizations of general rational matrices, a topic of independent active interest in the current literature on Control. Elsewhere in the literature, singularly perturbed discrete time, distributed, multidimensional, time-varying and nonlinear systems have been studied. Such systems have been studied, furthermore, in several contexts involving optimal and stochastic control, but nearly always from the time domain point of view. The dissertation tackles only the problems of robust stabilization and pole placement in the finite-dimensional, linear, autonomous case. Future work will attempt to extend its results on stabilization and pole-placement, appropriately, to some of the other general multiple time scale system classes. Further frequency domain investigations, related to the dissertation, may as well explore other problems pertaining to the multiple time scale systems which so far have received treatment only in the available time domain literature.

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The objective of this research is to mount a software package entitled Optimal Regulator Algorithms for the Control of Linear Systems (ORACLS), on a Control Data Corporation (CDC) Cyber 74 digital computer system at Florida State University so that is can be operated from the University of Central Florida. The software package contains 60 subroutines which can be used for the analysis and design of state variable feedback control alws for time-invariant linear systems. The procedure for using this package is documented. Several examples are presented to illustrate the capability ORACLS in both digital and continuous linear-quadratic-gaussian (LQG) controller design and additionally, to demonstrate the construction of typical executive programs.

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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 2018-08-27T08:37:55Z (GMT). No. of bitstreams: 1 Braga_MarcioFeliciano_D.pdf: 944285 bytes, checksum: 3305c11425207a3bbee3feae45b3b073 (MD5) Previous issue date: 2015 / Resumo: Esta tese investiga o problema de discretização de sistemas lineares incertos em dois cenários distintos. O primeiro supõe que as matrizes do sistema e a taxa de amostragem são incertas e invariantes no tempo, descritas em termos de politopos. No segundo caso supõe-se que todos os parâmetros incertos, incluindo o intervalo de amostragem, podem variar no tempo; que os parâmetros do sistema possuem limitantes conhecidos para a taxa de variação e são continuamente monitorados, de forma que uma nova amostra é coletada sempre que ocorrer uma mudança significativa nos parâmetros ou quando um intervalo máximo desde a última amostragem for atingido, caracterizando-se assim a chamada amostragem baseada em eventos. O objetivo é projetar controladores digitais para o sistema discretizado e garantir teoricamente a estabilidade robusta em malha fechada do sistema contínuo original. Considera-se também um atraso induzido pela rede no modelo discreto. Do ponto de vista da discretização, propõem-se extensões da expansão em série de Taylor até um grau arbitrário l aplicada ao sistema original como uma solução para tratar as expressões envolvendo exponenciais de matrizes incertas. O modelo discreto resultante é composto por matrizes polinomiais homogêneas de grau l com parâmetros pertencentes ao produto cartesiano de simplexos unitários, chamado multi-simplex, mais um termo aditivo limitado em norma que representa o erro residual de aproximação. Modelos mais acurados são obtidos com o aumento do grau l que, consequentemente, produz resíduos com menor norma, possibilitando resultados de síntese menos conservadores. Também são propostas condições na forma de desigualdade matriciais lineares para a síntese de controladores digitais robustos por realimentação de estados e de saída, cômputo de norma H2 e análise de estabilidade para sistemas discretos polinomiais com incertezas aditivas limitadas em norma. As condições empregam funções de Lyapunov com dependência polinomial nos parâmetros incertos para certificar a estabilidade em malha fechada do sistema controlado. Em alguns casos, as desigualdades matriciais possuem um parâmetro escalar, tornando-se lineares para valores fixos desse parâmetro. Os resultados obtidos são gradativamente menos conservadores com o aumento do grau da matriz de Lyapunov, podendo ainda ser melhorados com o auxílio de uma busca linear no parâmetro escalar. Experimentos numéricos são apresentados para ilustrar a versatilidade da metodologia proposta, aplicável a uma classe mais geral de problemas de controle por rede que os métodos existentes na literatura / Abstract: This thesis investigates the problem of discretization of uncertain linear systems in two distinct scenarios. The first one supposes that the matrices of the system and the sampling rate are uncertain and time-invariant, described in terms of polytopes. The second case considers that all uncertain parameters, including the sampling interval, can vary in time and that the parameters of the system have known bounds for the rate of variation and are continuously monitored, in such a way that a new sample is collected whenever a significant change in the parameters occurs or when a maximum time interval since the previous sample is reached, thus characterizing the so-called event-based sampling. The aim is to design digital controllers for the discretized system and to guarantee the closed-loop robust stability of the original continuous-time system. A network-induced delay is also considered in the discretetime model. From the discretization point of view, extensions of the Taylor series expansion of an arbitrary degree ℓ applied to the original system are proposed as a solution to deal with expressions involving the exponential of uncertain matrices. The resulting discrete-time model consists of homogeneous polynomial matrices of degree ℓ with parameters belonging to the Cartesian product of unit simplexes, called multi-simplex, plus an additive norm-bounded term which represents the approximation residual error. More accurate models are obtained with the increase in the degree ℓ which, consequently, yields residues with lower norms, allowing to produce less conservative synthesis results. Conditions based on linear matrix inequalities for the synthesis of robust state-feedback and output-feedback digital controllers, H2 norm computation, and stability analysis for polynomial discrete-time systems with additive norm-bounded terms are also proposed. The conditions employ Lyapunov functions with polynomial dependency on the uncertain parameters to certify the closed-loop stability of the controlled system. In some cases, the matrix inequalities have a scalar parameter, becoming linear for fixed values of the parameter. The obtained results are gradually less conservative with the increase of the degree of the Lyapunov matrix and can be further enhanced with the help of a line search in the scalar parameter. Numerical experiments are presented to illustrate the versatility of the proposed methodology, applicable to a more general class of networked control problems than the existing methods in the literature. / Doutorado / Automação / Doutor em Engenharia Elétrica