Optimal Hankel norm model reductions and Wiener-Hopf factorization II: The non-canonical case

Abstract

We consider the problem of approximating a given stable rational matrix function G(s) of McMillan degree n by a function Ĝ(s)+F(s), where Ĝ has McMillan degree ℓ<n and F is antistable. It is known that the minimum possible L^∞-norm of the error ‖G−Ĝ−F‖_L∞, or equivalently the minimum possible Hankel norm of the stable part of the error ‖G−Ĝ‖_H, is equal to the (ℓ+1)-st Hankel singular value σ_ℓ+1(G(s)) of G. We give an explicit linear fractional map parametrization for the class of all functions Ĝ(s)+F(s) as above which satisfy ‖G−Ĝ−F‖_L∞=σ_ℓ+1(G(s)). The coefficients of the linear fractional map are completely determined by the matrices A, B and C in a realization G(s)=C(sI−A)⁻¹B for G(s) and the unique solutions of a pair of Lyapunov equations involving these matrices. (Note that without loss of generality G(∞)=0.) The basic idea is to use the approach of Ball and Helton to reduce the problem to a symmetric Wiener-Hopf factorization problem, which in turn can be solved by applying a result of Kaashoek and Ran. The results obtained here are equivalent to the results of Glover, but our analysis gives an alternative more geometric approach.