The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle
type: Item
instance of: scholarly article
title: The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle
zbMATH Open document ID: 1210.47045
published in: Integral Equations and Operator Theory
publication date: 2010-08-23T00:00:00Z
review text: A rational \(m\times m\) matrix function \(R\) with no poles on the unit circle is said to admit a right canonical factorization if \(R\) can be factored as \(R(z)=\Psi(z)\Theta(z)\), where \(\Theta\) and \(\Psi\) are regular \(m\times m\) matrix rational functions such that \(\Theta(z)\) and \(\Theta(z)^{-1}\) have no poles inside the unit disc, while \(\Psi(z)\) and \(\Psi(z)^{-1}\) have no poles outside the unit disc. The main result of the paper says that if \(R\) has a so-called stable representation, then \(R\) admits a right canonical factorization with respect to the unit circle if and only if the algebraic Riccati equation associated with the stable representation has a stabilizing solution. In this case, the right canonical factorization is explicitly given. When the values of \(R\) on the unit circle are hermitian matrices, the corresponding associated Riccati equation is a symmetric one originated from stochastic realization theory and \(R(z)\) plays the role of the so called Popov function. As an application of the main result, the case when the corresponding Toeplitz operator attached to \(R\) is tri-diagonal and the case when \(R\) has no poles on the closed unit disc are analysed. Also, to illustrate how one can use the main result of the paper, an example is given. Riccati iteration and the finite sections method is used to analyze the invertibility of the block Toeplitz operator attached to \(R\).
reviewed by: Ilie Valuşescu
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 42A99
Mathematics Subject Classification ID: 39A99
zbMATH DE Number: 5774250
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: rational matrix functions
zbMATH Keywords: canonical factorization
zbMATH Keywords: discrete algebraic Riccati equation
zbMATH Keywords: Riccati difference equation
zbMATH Keywords: Schur complements
zbMATH Keywords: finite section method
author: Arthur E. Frazho
author: Marinus Kaashoek
author: André C. M. Ran
MaRDI profile type: Publication
full work available at URL: s00020-010-1747-1
OpenAlex ID: W2124522014
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DOI: 10.1007/S00020-010-1747-1
description: scientific article; zbMATH DE number 5774250