On Topological Properties of the Set of Stabilizing Feedback Gains

Abstract

This article presents a fairly complete account on various topological and metrical aspects of feedback stabilization for single-input single-output (SISO) continuous and discrete time linear time-invariant (LTI) systems. In particular, we prove that the set of stabilizing output feedback gains for a SISO system with <inline-formula><tex-math notation= LaTeX >$n$</tex-math></inline-formula> states have at most <inline-formula><tex-math notation= LaTeX >$\\lceil {\\frac{n}{2}}\\rceil$</tex-math></inline-formula> connected components. Furthermore, our analysis yields an algorithm for determining intervals of stabilizing gains for general continuous and discrete LTI systems; the proposed algorithm also computes the number of unstable roots in each unstable interval.