Regularity properties of the phase for multivariable systems

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 important physical applications where the phase of a perturbation can also have a significant effect on relative stability. Such applications led Bar-on and Jonckheere (1990, 1992) to formulate the notions of phase, minimum-phase mapping, and phase margin for multivariable systems. The objective of this paper is to present conditions under which the phase and minimum-phase mappings have certain desired regularity properties. After a review of the definitions of the phase concepts under consideration, we summarize some well known results about solutions of 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. We then outline sufficient conditions, of gradually increasing strength, for the minimum-phase mapping to be continuous and real analytic as a function of frequency. Two examples are presented to illustrate the theoretical results