Perturbation Theory

Abstract

The aim of this chapter is to study how the root and eigenvalue locations of polynomials and matrices change under perturbations. The chapter is quite a substantial one since we address a number of different issues. First and foremost we consider a variety of perturbation classes, ranging from highly structured perturbations which are determined via a single parameter to unstructured perturbations where all the entries of the matrix or coefficients of the polynomial are subject to independent variation. The size of the perturbations will, in the main, be measured by arbitrary operator norms. Moreover we will develop the theory for both complex and real perturbations which often require quite different approaches. The first section is concerned with polynomials. We establish some continuity and analyticity results for the roots, then describe the sets of all Hurwitz and Schur polynomials in coefficient space. We also consider the problem of determining conditions under which all polynomials with real coefficients belonging to prescribed intervals are stable and prove Kharitonov's Theorem. The effect of perturbations on the eigenvalues of matrices is considered in Section 4.2. We first state some simple continuity and analyticity results which follow directly from the results of Section 4.1. Then we assume that the matrix depends analytically on a single parameter and examine the smoothness of eigenvalues, eigenprojections and eigenvectors. Section 4.3 deals with singular values and singular value decompositions which are important tools in the quantitative perturbation analysis of linear systems. Section 4.4 is dedicated to structured perturbations and presents some elements of μ-analysis, both for complex and for real parameter perturbations. We finish the chapter in Section 4.5 with a brief introduction to some numerical issues which are important for Systems Theory, focussing on those aspects which have a relationship with the material of this and the previous chapters.