State-space algorithm for the superoptimal Hankel-norm approximation problem

It has been demonstrated by N. T. Young [NATO ASI Series F34, Springer-Verlag, Berlin, New York, 1987] that given a stable matrix-valued function G₀(s) and a nonnegative integer k, there exists a unique superoptimal approximation Φ(s) with no more than k poles in the left half plane that minimizes the sequence (s₁^∞(G₀+Φ), s₂^∞(G₀+Φ), ···), with respect to lexicographic ordering, where s_i^∞(G₀+Φ): = sup$-ω/[s_i(G₀+Φ)(jω)] and s_i(·) are the singular values in descending order of magnitude. This paper presents a constructive state-space algorithm that evaluates the superoptimal approximating matrix function. The procedure recursively minimizes each frequency-dependent singular value with the aid of all-pass transformations constructed from the kth Schmidt pairs of a sequence of Hankel operators. The algorithm may be stopped after an arbitrary number of, say, l≤min (m,p) steps. The representation formula at the lth stage will characterize all matrix functions that have ≤k poles in the left half plane and that minimize s₁^∞(G₀+Φ), ···, s_l^∞(G₀+Φ).