## Abstract

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 G_{0}(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 (s_{1}^{∞}(G_{0}+Φ), s_{2}^{∞}(G_{0}+Φ), ···), with respect to lexicographic ordering, where s_{i}^{∞}(G_{0}+Φ): = sup$-ω/[s_{i}(G_{0}+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 s_{1}^{∞}(G_{0}+Φ), ···, s_{l}^{∞}(G_{0}+Φ).

Original language | English |
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Pages (from-to) | 960-982 |

Number of pages | 23 |

Journal | SIAM Journal on Control and Optimization |

Volume | 31 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1993 |

Externally published | Yes |

## ASJC Scopus subject areas

- Control and Optimization
- Applied Mathematics