Regularity Properties of the Phase for Multivariable Systems

Abstract

For multivariable, input-output systems that are represented as rational, transfer-function matrices, the most frequently used measures of relative stability are gain based. However, there are a number of important physical applications where the phase of a perturbation can also have a significant effect on relative stability. Such applications led J. R. Bar-on and E. A. Jonckheere [J. R. Bar-on, {Phase and Gain Margins for Multivariable Control Systems, Ph.D. thesis, University of Southern California, 1990; J. R. Bar-on and E. A. Jonckheere, Internat. J. Control, 52 (1990), pp. 485--498] to define precisely the notions of phase, minimum-phase mapping, and phase margin for multivariable systems. The objective of this paper is to establish conditions under which the phase and minimum-phase mappings have certain desired regularity properties (e.g., continuity or differentiability). After a review of the definitions of the phase concepts under consideration, we collect a few well-known results about set-valued maps that have direct applications to parametrized families of constrained optimization problems. Using these results we show that, under very mild conditions, the minimum-phase mapping is lower semicontinuous as a function of frequency; as a consequence, the phase margin (initially defined as the infimum of the phase of all destabilizing unitary perturbations in the range of frequencies where destabilizing perturbations can occur) is achieved as the phase of a specific destabilizing unitary perturbation. We then establish sufficient conditions of gradually increasing strength for the minimum-phase mapping to be continuous and real analytic as a function of frequency. The proof of the real analyticity of the minimum-phase mapping relies on the implicit function theorem and the Lagrange multiplier theorem.