Discrete \(H^\infty\)-optimization. With applications in signal processing and control systems.
type: Item
instance of: scholarly article
title: Discrete \(H^\infty\)-optimization. With applications in signal processing and control systems.
zbMATH Open document ID: 0912.93002
author: Charles K. Chui
author: Guanrong Chen
published in: Springer Series in Information Sciences
publication date: 1997-08-01T00:00:00Z
review text: [For a review of the first ed. (1992) see Zbl 0824.93002.] The \(H^\infty\)-approach is one of the most fruitful directions in the theory of control; this idea is proved by the considerable amount of papers and books in this domain published in the last two decades. The model-matching problem, the tracking problem and the robust stabilization were successfully treated in the \(H^\infty\)-framework; the Nevanlinna-Pick theory or Sarason's theory were applied to solve such important problems as the determination of the optimal feedback controllers. The objective of this monograph is to present a unified frequency-domain approach to discrete-time signal processing and systems theory; the importance of such a unified approach is revealed by the existence of common problems in the fields of approximation, identification, stability and rank reduction. The book concentrates on the fundamental work of \textit{V. M. Adamyan}, \textit{D. Z. Arov} and \textit{M. G. Krejn} (AAK) [Math. USSR, Sb. 15 (1971), 31-73 (1972); translation from Mat. Sb., Nov. Ser. 86(128), 34-75 (1971; Zbl 0243.47023)] which relates the approximation of a transfer function by stable rational functions and the approximation of the corresponding Hankel operator by finite-rank bounded operators. The aim of unifying signal processing and systems theory implies the concentration of the text on discrete-time methods; this is the reason why the presentation of the theories and techniques in Hardy spaces and within the AAK approach is restricted to the unit disk. The work is generally self-contained and easily readable although some theorems are given without proof (and the reader is sent to references). The text is very synthetic and dense; it provides a considerable amount of information. There are many ideas and theorems which are illustrated by suitable examples. Each chapter ends with a long list of problems which cover the dicussed topics, many of them representing important theoretic results. A bibliogr…
reviewed by: Valeriu Prepeliţă
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 93B36
Mathematics Subject Classification ID: 93B28
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 47N70
zbMATH DE Number: 1042197
zbMATH Keywords: approximation of the Hankel operator
zbMATH Keywords: \(H^\infty\)-optimization
zbMATH Keywords: discrete-time signal processing
zbMATH Keywords: rank reduction
zbMATH Keywords: approximation of a transfer function
zbMATH Keywords: discrete-time methods
zbMATH Keywords: digital signals
zbMATH Keywords: filters
zbMATH Keywords: minimal realizations
zbMATH Keywords: system reduction
zbMATH Keywords: sensitivity minimization
zbMATH Keywords: Nevanlinna-Pick interpolations
zbMATH Keywords: Nehari theorem
zbMATH Keywords: optimal Hankel-norm approximation
zbMATH Keywords: balanced realizations
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1042197