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

G. D. Halikias, D. J.N. Limebeer, K. Glover

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)


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 si(G0+Φ): = sup$-ω/[si(G0+Φ)(jω)] and si(·) 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), ···, sl(G0+Φ).

Original languageEnglish
Pages (from-to)960-982
Number of pages23
JournalSIAM Journal on Control and Optimization
Issue number4
Publication statusPublished - 1993
Externally publishedYes

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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