Studies in Robust Stability

Abstract

In this thesis, questions in the analysis and synthesis of stability robustness properties for linear and nonlinear control systems are considered. The first part of this work is devoted to linear systems., where the emphasis is on obtaining necessary and sufficient conditions for stability of parametrized families of systems. This class of robustness problems has recently received significant attention in the literature [1]. In the second part of the thesis, questions of stabilization of nonlinear systems by feedback are considered. Part I of this work addresses the generalized stability, i.e. stability with respect to a given domain in the complex plane, of parametrized families of linear time-invariant systems. The main contribution is the introduction and application of the new concepts of "guarding map" and "semiguarding map" for a given domain. Basically, these concepts allow one to replace the original parametrized system stability problem with a finite number of stability tests. Moreover, the tool is very powerful in that it allows the treatment of a large class of domains in the complex plane. The parametrized stability problem is completely solved for the case stability of a one- parameter family with respect to guarded and semiguarded domains. The primary interest in semiguarded domains arises in a process of reduction of a given multiparameter problem to one involving fewer parameters. For the two-parameter case, we consider stability of families of matrices relative to domains with a polynomial guarding map. The first step replaces the two- parameter problem by a one-parameter stability problem relative to a new domain. The second step employs a polynomial semiguarding map for the new domain to obtain necessary and sufficient conditions for stability of the new problem. The case of three or more parameters, which involves technical questions not encountered in the one- or two-parameter case, is also considered. In Part II, a class of nonlinear control systems for which the linear part satisfies special stabilizability conditions is considered. These conditions naturally give rise to certain nonstandard algebraic issues in linear systems. Sufficient conditions for the existence of a linear feedback control which stabilizes a given nonlinear control system within a prescribed ball of given radius (possibly infinite) are given. The feedback control is found to be robust in a certain sense against a class of modeling errors. A complete design methodology is obtained for planar systems and extended to a class of higher dimensional singularly perturbed nonlinear control systems. For these systems, nonlinear feedback laws achieving stabilization within prescribed cylindrical regions are presented.