Discrete \(H^\infty\)-optimization. With applications in signal processing and control systems.
[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 bibliography of 101 titles selects the most important and useful works in the domain. The first chapter is a nontraditional presentation of the basic properties of digital signals and filters. Chapter 2 introduces the fundamental concepts of linear systems: state-space description, controllability, observability and stability, transfer matrices and minimal realizations, system reduction, general MIMO linear feedback systems and their sensitivity minimization problems. Chapter 3 deals with approximation by polynomials and stable rational functions using Hardy space techniques which include minimum-norm and Nevanlinna-Pick interpolations. Chapter 4 contains the Nehari theorem and related results. A detailed constructive proof of the fundamental AAK theorem is given for finite rank and real Hankel matrices. System reduction and \(H^\infty\)-minimization for SISO systems are discussed. Chapter 5 is devoted to the general theory of optimal Hankel-norm approximation. A detailed proof of the AAK main theorem extended to bounded Hankel matrices is presented. In Chapter 6 the general theory of \(H^\infty\)-optimization and problems on system reduction for MIMO linear systems are studied, using the technique of balanced realizations.
scientific article; zbMATH DE number 1042197