Sensitivity minimization in an H ∞ norm: parametrization of all suboptimal solutions

Abstract

A number of problems in the control of linear feedback systems can be reduced to the following: we are given three stable rational matrix functions K, ϕ, ψ of sizes p ₁ x q ₁, p ₁ x q ₂ and p ₂ x q ₁ respectively, and seek a stable rational q ₂ x p ₂ matrix function S so as to minimize ¦K + ϕSψ¦_∞. We assume that p ₁ ≥ q ₂, p ₂≤q ₁ and that ϕ and ψ have maximal rank (q ₂ and p ₂ respectively) on the jω-axis. Given a tolerance level μ sufficiently large, we obtain a linear fractional map G → F = [0 ₁₁ G + 0 ₁₂, 0 ₁₃][0 ₂₁ G + 0 ₂₂, 0 ₂₃]⁻¹ such that F = K + ϕSψ with S stable and ¦F _∞ ≤ μ if and only if G is a stable q ₂ x p ₂ matrix function with ¦G¦_∞ ≤ 1. The computation of 0 = [0 _ij ] (1 ≤ i ≤ 2, 1 ≤ j ≤ 3) reduces to solving a pair of symmetric Wiener–Hopf factorization problems. For the special case where ϕ = [I _q2, 0]^T, ψ = [I _p2, 0] (and K not necessarily stable) to which the general case can be reduced, we provide explicit state-space formulae for 0 in terms of a state-space realization of K and the solutions of some related Riccati equations. The approach is a natural extension of that of Ball–Helton and Ball–Ran for the case p ₁ = q ₂ and p ₂ = q ₁.