Stability robustness of linear systems: a field of values approach

Abstract

One active area of research in stability robustness of linear time invariant systems is concerned with stability of matrix polytopes. Various structured real parametric uncertainties can be modeled by a family of matrices consisting of a convex hull of a finite number of known matrices, the matrix poly tope. An interval matrix family consisting of matrices whose entries can assume any values in given intervals are special types of matrix polytopes and it models a commonly encountered parametric uncertainty. Results that allow the inference of the stability of the whole polytope from stability of a finite number of elements of the polytope are of interest. Deriving such results is known to be difficult and few results of sufficient generality exist. In this thesis, a survey of results pertaining to robust Hurwitz and Schur stability of matrix polytopes and interval matrices are given. A seemingly new tool, the field of values, and its elementary properties are used to recover most results available in the literature and to obtain some new results. Some easily obtained facts through the field of values approach are as follows. Poly topes with normal vertex matrices turn out to be Hurwitz and Schur stable if and only if the vertex matrices are Hurwitz and Schur stable, respectively. If the polytope contains the transpose of each vertex matrix, Hurwitz stability of the symmetric part of the vertices is necessary and sufficient for the Hurwiz stability of the polytope. If the polytope is nonnegative and the symmetric part of each vertex matrix is Schur stable, then the polytope is also stable. For polytopes with spectral vertex matrices, Schur stability of vertices is necessary and sufficient for the Schur stability of the polytope.