Sensitivity analysis of the discrete-time algebraic Riccati equation
type: Item
instance of: scholarly article
title: Sensitivity analysis of the discrete-time algebraic Riccati equation
zbMATH Open document ID: 0989.93025
DOI: 10.1016/S0024-3795(97)10017-9
author: Ji-guang Sun
published in: Linear Algebra and its Applications
publication date: 2002-08-05T00:00:00Z
review text: The paper deals with the perturbation analysis of the real discrete-time algebraic Riccati equation \[ A^TXA-X- (A^TXB+S) (R+B^TXB)^{-1}(B^TXA+ S^T)+Q=0, \] where \(A\), \(Q\) and the solution \(X\) are \(n\times n\) matrices with \(Q\) and \(X\) symmetric, \(B\) and \(S\) are \(n\times m\) matrices and \(R=R^T\) is an \(m\times m\) matrix. Let the coefficient matrices be subject to perturbations \(A\to A+\Delta A\), etc., which gives rise to a perturbed solution \(X+\Delta X\). The problem is to find asymptotic as well as non-local bounds for the norm of the perturbation \(\Delta X\) as functions of the norms of the perturbations in the coefficient matrices. The analysis is done without the assumption that \(R\) is positive definite and is based on an appropriate representation of the perturbed equation. Residual bounds for an approximate solution are also given. The results obtained are illustrated by examples. Thus previous results of the reviewer, \textit{P. Hr. Petkov} and \textit{N. D. Christov} [Kybernetika 29, No. 1, 18-29 (1993; Zbl 0792.93053)] are generalized. Note that by using the technique of homogeneous first order bounds proposed in [the reviewer, \textit{M. O. Stanislavova} and \textit{P. Hr. Petkov}, Linear Algebra Appl. 285, 7-31 (1998; Zbl 0933.15025)], the asymptotic results of the type presented in Theorem 3.2 on p. 603 may be improved.
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 93C73
zbMATH DE Number: 1354734
zbMATH Keywords: residual bounds
zbMATH Keywords: perturbation analysis
zbMATH Keywords: real discrete-time algebraic Riccati equation
reviewed by: Mihail M. Konstantinov
describes a project that uses: RICPAC
MaRDI profile type: Publication
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full work available at URL: s0024-3795(97)10017-9
OpenAlex ID: W1965951227
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description: scientific article; zbMATH DE number 1354734