Tracking poles and representing Hankel operators directly from data
type: Item
instance of: scholarly article
title: Tracking poles and representing Hankel operators directly from data
zbMATH Open document ID: 0726.30003
DOI: 10.1007/BF01385646
author: J. William Helton
author: N. J. Young
author: P. G. Spain
published in: Numerische Mathematik
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133522
review text: This paper addresses the estimation of the poles of a rational function near the unit circle \(\partial D\) sample values of the function on a grid of points on \(\partial D\). It is motivated by numerical difficulties in the solution of the Nehari problem in the presence of poles close to \(\partial D\), and this in turn is motivated by sundry standard problems in \(H^{\infty}\) control theory, such as robust stabilization. Suppose given a rational function G restricted to \(\partial D\) and bounded there. We seek the rational function \(\hat G,\) bounded and analytic in D which minimises \(\| G-\hat G\|_{\infty}\). Mathematically the problem is solved once we know the singular vectors of the Hankel operator with symbol G. It transpires that a straightforward numerical approximation to the matrix of \(H_ G\) (based on Fourier coefficients) is effective as long as the poles of G are not close to \(\partial D\). However, unfavourably placed poles do occur in applications, and so a refinement of the obvious representation was proposed and analysed in \textit{J. W. Helton}, and \textit{N. J. Young} [Signal processing, Pt. II: Control theory and applications, Lect., Minneapolis/MN (USA) 1988, IMA Vol. Math. Appl. 23, 115-137 (1990; Zbl 0711.93030)]. In broad terms, we use orthonormal bases for the domain and range of \(H_ G\) which are better tuned to the action of \(H_ G\) than are the standard bases \((e^{in\theta})\), and to this end we need estimates for the unfavourable poles of G. The present paper puts forward a method of obtaining such estimates based on a local best \(L^ 2\)-fit by rational functions of degree 1. A convergence theorem is proved and numerical examples are given.
Mathematics Subject Classification ID: 30-04
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 93-04
zbMATH DE Number: 4198390
zbMATH Keywords: H\({}^{\infty }\) control theory
zbMATH Keywords: Carathéodory-Féjer method
zbMATH Keywords: pole- tracking
zbMATH Keywords: Nehari problem
zbMATH Keywords: Hankel operator
describes a project that uses: NAG
describes a project that uses: nag
MaRDI profile type: Publication
cites work: Infinite Hankel matrices and generalized Caratheodory-Fejer and Riesz problems
cites work: All optimal Hankel-norm approximations of linear multivariable systems and theirL,∞-error bounds†
cites work: Worst case analysis in the frequency domain: The H&lt;sup&gt;∞&lt;/sup&gt;approach to control
cites work: Broadbanding:Gain equalization directly from data
cites work: The distance of a function to \(H^\infty\) in the Poincaré metric; electrical power transfer
cites work: Q3951096
cites work: Model reduction via balanced state space representations
cites work: Near-circularity of the error curve in complex Chebyshev approximation
cites work: The singular-value decomposition of an infinite Hankel matrix
description: scientific article; zbMATH DE number 4198390