Optimal Hankel Norm Model Reductions and Wiener–Hopf Factorization I: The Canonical Case

We consider the problem for discrete time systems of approximating a given stable rational matrix function $K(z) = \sum\nolimits_{j = 1}^\infty {K_j z^{ - j} } $ of McMillan degree n by a function $\hat K(z) + H(z)$, where $\hat K$ has McMillan degree $l < n$ and H is antistable; we include the case $l = 0$ in which we then take $\hat K = 0$. The minimum possible $L^\infty $-norm (on the unit circle) of the error $\| {K - \hat K - H} \|_{L^\infty } $, or equivalently the minimum possible spectral norm of the induced Hankel matrix $\| {\mathcal{H}_{K - \hat K} } \|$, is known to be equal to the $(l + 1)$st singular value $\sigma _{l + 1} (K)$ of the Hankel matrix $\mathcal{H}_K = [K_{i + j - 1} ]_{i,j} $. Assume $\sigma _{l + 1} (K) < \sigma _l (K)$ and choose the number $\sigma $ to satisfy $\sigma _{l + 1} (K) < \sigma < \sigma _l (K)$; if $l = 0$, the condition is simply $\sigma _1 (K) = \| {\mathcal{H}_K } \| < \sigma $. We give an explicit linear fractional map parametrization of the class of all functions $\hat K + H$ as above which satisfy $\| {K - \hat K - H} \|_{L^\infty } \leqq \sigma $. The coefficients of the linear fractional map are completely determined by the matrices A, B, C in a realization $K(z) = C(zI - A)^{ - 1} B$ for $K(z)$ and the observability and controllability gramians for the discrete time system $(A,B,C)$. The analogous results for continuous time systems are derived by a linear fractional change of variable; in this way we recover some recent results of Glover. The basic idea is to use the Grassmannian approach of Ball and Helton to reduce the problem to one of spectral factorization; this in turn can be solved by the geometric factorization principle of Bart, Gohberg, Kaashoek and van Dooren. Known applications include sensitivity minimization in $H^\infty $ control theory and model reduction for linear systems.