On root loci in several variables: Continuity in the high gain limit

Abstract

In this paper we discuss the continuity of the closed-loop poles of a linear multivariable system with respect to a multidimensional polynomial family of direct output gains K(λ1,…,λr). This is based on, and contains an exposition of, the geometric formulation for including infinite gains which was developed in the lectures [2] and extended and applied in [1] to the study of output feedback systems. This has been a basic tool in recent work on the classical problem of poleplacement by output feedback and in [1] the lack of continuity of the root-loci, in certain situations, was discussed with special emphasis on the complex case. Here, after presenting two somewhat surprising counterexamples to this continuity, we give in Theorem 1 and the ensuing discussion necessary and sufficient conditions for continuity of the root-loci at a real infinite gain. This should have significant impact on the problem of constructing graphical tests for the stability of systems subject to 2-dimensional variations in the gain parameter.