Necessary and sufficient condition in Lyapunov robust control: Multi- input case
type: Item
instance of: scholarly article
title: Necessary and sufficient condition in Lyapunov robust control: Multi- input case
zbMATH Open document ID: 0681.93050
DOI: 10.1007/BF00940529
author: Yanyan Li
published in: Journal of Optimization Theory and Applications
publication date: 1990-01-01T00:00:00Z
review text: We consider a linear time-invariant finite-dimensional system \(\dot x=Ax+Bu\) with multi-input u, in which the matrices A and B are in canonical controller form. We assume that the system is controllable and B has rank m. We study the Lyapunov equation \(PA+A^ TP+Q=0\), with \(Q>0\), and investigate the properties that P must satisfy in order that the canonical controller matrix A be Hurwitz. We show that, for the matrix A being Hurwitz, it is necessary and sufficient that \(B^ TPB>0\) and that the determinant of \(B^ TPW\) be Hurwitz, where \(W=block\) \(diag[w_ 1,...,w_ m]\), with element \(w_ i=[s^{k_ i-1},s^{k_ i-2},...,s,1]^ T\); here, the symbols \(k_ i\), \(i=1,2,...,m\), denote the Kronecker invariants with respect to the pair \(\{\) A,B\(\}\). This result has application in designing robust controllers for linear uncertain systems.
Mathematics Subject Classification ID: 93D05
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93C99
Mathematics Subject Classification ID: 93C35
zbMATH DE Number: 4116429
zbMATH Keywords: canonical controller
zbMATH Keywords: Lyapunov equation
zbMATH Keywords: Hurwitz
zbMATH Keywords: robust controllers
zbMATH Keywords: linear uncertain systems
zbMATH Keywords: time-invariant
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00940529
OpenAlex ID: W3013113581
description: scientific article; zbMATH DE number 4116429