On Robust Eigenvalue Location.

Abstract

The concepts of guardian and semiguardian maps were recently introduced as tools for assessing robust generalized stability of parametrized families of matrices or polynomials. Necessary and sufficient conditions were obtained for stability of parametrized families with respect to a large class of open subsets of the complex plane, namely those with which one can associate a polynomic guardian or semiguardian map. This note focuses on a class of disconnected subsets of the complex plane, of interest in the context of dominant pole assignment and filter design. It is first observed that the robust stability conditions originally put forth are in fact necessary and aufficient for the number of eigenvalues (matrices) or zeros (polynomials) in any given connected component to the same for all the members of the given family. Polynomic semiguardian maps are then identified for a class of disconnected regions of interest. These maps are in fact "essentially guarding with respect to one-parameter families."