MIMO disturbance and plant uncertainty attenuation by feedback

This paper investigates the ability of feedback to reduce plant and disturbance uncertainties in the multiple-input-multiple-output (MIMO) case, by solving two fundamental problems posed by Zames in the late seventies. The first problem is termed the MIMO extension of the optimal robust disturbance attenuation problem, the second is a filtering of plant uncertainty two-degree of freedom feedback problem. The optimal solutions of these problems are characterized using Banach space duality theory and shown to satisfy an "allpass" condition and an extremal identity. Moreover, the duality description leads to a dual pair of optimizations and the introduction of two nonstandard matrix norms. In particular, the primal-dual optimization reduces naturally to approximate finite dimensional convex optimizations within desired tolerance. Whereas the computation of the nonstandard matrix norms are shown to be equivalent to specific semi-definite programming problems, and a numerical solution based on convex programming is provided. It is also shown using Douglas' range inclusion theorem that performance is a monotonic function of uncertainty, and some qualitative implications for feedback are derived.