Sensitivity minimization in an <i>H</i> ^∞ norm: parametrization of all suboptimal solutions

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.