Limits of generalized state space systems under proportional and derivative feedback
type: Item
instance of: scholarly article
title: Limits of generalized state space systems under proportional and derivative feedback
zbMATH Open document ID: 0935.93040
DOI: 10.1007/BF01213382
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 2000-05-09T00:00:00Z
review text: Generalized state space systems are described by time-invariant differential and algebraic equations of the form \[ E\dot x(t)= Ax(t)+ Bu(t),\quad t\geq 0,\quad E,A\in F^{q\times n},\quad B\in F^{q\times m},\tag{1} \] where \(F\) is either the field of real numbers or the field of complex numbers. Let \({\mathcal G}\) be the group of transformations of the generalized state space systems that is generated by the left multiplications (in which (1) is transformed to \(LE\dot x(t)= LAx(t)+ LBu(t)\), where \(L\in F^{q\times q}\) is invertible), the state and input changes of coordinates ((1) is transformed to \(ER\dot x(t)= ARx(t)+ BWu(t)\), where \(R\in F^{n\times n}\) and \(W\in F^{m\times m}\) are invertible), and the state feedbacks ((1) is transformed to \(E\dot x(t)= (A+ BK)x(t)+ Bu(t)\), \(K\in F^{m\times n}\)). Let \({\mathcal D}\) be the group of transformations of the generalized state space systems that is generated by \({\mathcal G}\) and the derivative feedbacks ((1) is transformed to \((E+ BK)\dot x(t)= Ax(t)+ Bu(t)\), where \(K\in F^{m\times n}\)). The canonical forms of the generalized state space systems under the actions of \({\mathcal G}\) and of \({\mathcal D}\) have been obtained, and the corresponding invariants fully described, in \textit{J. J. Loiseau}, \textit{K. Özçaldiran}, \textit{M. Malabre} and \textit{N. Karcanias} [Kybernetika 27, No. 4, 289-305 (1991; Zbl 0767.93007)]. The reviewed paper is built on these canonical forms. The high-gain feedback classification problem is studied. The problem consists of determining, in terms of the invariants, the topological closures of the orbits of the \({\mathcal G}\) (or \({\mathcal D}\)) action, where a generalized state space system (1) is identified with the ordered triple of matrices \((E,A,B)\). For regularizable systems, i.e., such that \(\text{det}(sE- (A- BK))\not\equiv 0\) (as a polynomial of \(s\)) for a suitable \(K\in F^{m\times n}\), a complete description of orbit closures under the \({…
Mathematics Subject Classification ID: 93C35
Mathematics Subject Classification ID: 93B10
Mathematics Subject Classification ID: 93B52
Mathematics Subject Classification ID: 93B17
zbMATH DE Number: 1109546
zbMATH Keywords: generalized state space systems
zbMATH Keywords: derivative feedbacks
zbMATH Keywords: canonical forms
zbMATH Keywords: high-gain feedback classification
zbMATH Keywords: invariants
zbMATH Keywords: orbits
zbMATH Keywords: regularizable systems
author: D. Hinrichsen
author: Joyce O'Halloran
reviewed by: Leiba Rodman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1109546