The 2-block superoptimal AAK problem
type: Item
instance of: scholarly article
title: The 2-block superoptimal AAK problem
zbMATH Open document ID: 0916.93025
DOI: 10.1007/BF02741893
author: George Halikias
author: Imad M. Jaimoukha
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 1999-07-19T00:00:00Z
review text: This paper is a continuation of other works of some previous authors which solved the 1- and 2-block AKK problems. First, the authors outline the solution to the 2-block optimal AKK problem. Afterwards, the authors deal with the 2-block superoptimal AKK problem: Given a rational and antistable matrix function \(R(s)= [R_{11}(s), R_{12}(s)]\) and a nonnegative integer \(k\), find all superoptimal approximations \(Q(s)\), with no more than \(k\) poles in the right-half complex plane, that minimize the supremum, over the imaginary axis, of the singular values of the error function \(E(s)= [R_{11}(s), R_{12}(s)+ Q(s)]\), with respect to lexicographic ordering. The authors present a simple algorithm for the solution to both 1- and 2-block superoptimal AKK problems, namely, it is proved that, under certain conditions, it is possible to construct a sequence of 2-block functions \(R_1,\dots, R_t\), of the same form as \(R\) (but of progressively reduced dimensions and MacMillan degree) such that \(s_i(R,k)= s_1(R_i,k_i)\) and \({\mathcal S}_i(R,k)= {\mathcal T}_{i-1}[{\mathcal S}_1(R_i,k_i)]\), where \({\mathcal T}_1,{\mathcal T}_2,\dots\) is a certain sequence of operators and \(\{k_i\}\) is a nonincreasing integer sequence with \(k_0= k\). The notations used are defined by: \[ \begin{aligned} s_1(R,k) & :=\inf_{Q\in{\mathcal H}^{p\times m}_\infty(k)} s^\infty_1([R_{11}, R_{12}+ Q]),\;s^\infty_1(\cdot)= \|\cdot\|_\infty;\\ {\mathcal S}_1(R, K) & := \{Q\in{\mathcal H}^{p\times m}_\infty(k): s^\infty_1([R_{11}, R_{12}+ Q])= s_1(R,k)\};\\ s_i(R, k) & := \inf_{Q\in{\mathcal S}_{i-1}(R, k)} s^\infty_i([R_{11}, R_{12}+ Q]),\;i= 1,2,\dots;\\ {\mathcal S}_i(R, k) & := \{Q\in{\mathcal S}_{i-1}(R, k): s^\infty_i([R_{11}, R_{12}+ Q])= s_i(R, k)\},\;i= 1,2,\dots,\end{aligned} \] where \({\mathcal S}_0(R,k):= {\mathcal H}^{p\times m}_\infty(k)\) is the subset of \({\mathcal L}^{p\times m}_\infty\) consisting of the matrices with no more than \(k\) poles in \({\mathcal C}_+\). Thus, the…
Mathematics Subject Classification ID: 93B36
Mathematics Subject Classification ID: 93C62
zbMATH DE Number: 1234467
zbMATH Keywords: general distance problem
zbMATH Keywords: \(H_\infty\)-optimal control problem
zbMATH Keywords: 2-block AKK problems
zbMATH Keywords: superoptimal AKK problem
reviewed by: Marian Ivanovici
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02741893
OpenAlex ID: W2009102821
description: scientific article; zbMATH DE number 1234467