Model matching for linear systems with delays and 2D systems
type: Item
instance of: scholarly article
title: Model matching for linear systems with delays and 2D systems
zbMATH Open document ID: 0937.93007
DOI: 10.1016/S0005-1098(98)00177-0
author: S. H. Smith
published in: Automatica
publication date: 1998-10-27T00:00:00Z
review text: The model matching problem (MMP) is solved for linear systems with delays. A class of linear neutral delay-differential systems with commensurate delays is considered. It is shown that a delay system can be represented as the usual model for linear systems, i.e. \[ x=A(\nabla)x+B(\nabla)u\qquad y=C(\nabla)x+D(\nabla)u \] where the matrices \(A\), \(B\), \(C\), \(D\) are defined over the \(\text{PID }{\mathcal R}_u(\nabla)\) of ``realizable rational functions in \(\nabla\) with real coefficients'', \[ {\mathcal R}_u(\nabla)=\{p(\nabla)/q(\nabla)\mid p,q\in\mathbb{R}[\nabla], q(\nabla=0)\neq 0\}. \] The model matching equation is solved over the ring \({\mathcal R}_c={\mathcal R}_s\cap {\mathcal R}_u\) where \({\mathcal R}_s\) is the ring of proper rational functions in \(s\) with coefficients in \(\mathbb{R}(\nabla)\) and \({\mathcal R}_u\) is the ring of rational functions \(p/q\), where \(p,q\in\mathbb{R}[s,\nabla]\) and the leading coefficient of \(q\), considered as an element of \(\mathbb{R}[\nabla][s]\) is invertible over \(R_u(\nabla)\). It is shown that the ring \(\mathcal R\) characterizes the notion of realizability of a transfer matrix \(T(s,\nabla)\), namely \(T(s,\nabla)\) is realizable iff it is defined over \({\mathcal R}_c\). The key of the method for solving the MMP is the fact that \({\mathcal R}_c\) is the intersection of two PIDs, \({\mathcal R}_s\) and \({\mathcal R}_u\) (but \({\mathcal R}_c\) is not a PID). Necessary and sufficient conditions are given for the MMP to have a solution for the above class of linear systems with delays, firstly for the case of injective systems; then, by introducing some ``decompositions'', the problem is solved for the surjective and for the general case. These conditions are constructive and they can be used to solve some classical control problems. It is noticed that the results are also valid for 2D transfer matrices (due to the obvious correspondence of variables \(\nabla=1/z\)).
reviewed by: Valeriu Prepeliţă
Mathematics Subject Classification ID: 93A30
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 93C23
zbMATH DE Number: 1192918
zbMATH Keywords: model matching problem
zbMATH Keywords: linear systems with delays
zbMATH Keywords: realizability
MaRDI profile type: Publication
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cites work: Systems over a Principal Ideal Domain. A Polynomial Model Approach
cites work: The Disturbance Decoupling Problem for Systems over a Ring
cites work: Realization and stabilization of 2-d systems
cites work: Q5623842
cites work: Roth's theorems over commutative rings
cites work: System equivalence for AR-systems over rings -- with an application to delay-differential systems
cites work: Theory of functional differential equations. 2nd ed
cites work: Infinite structure and exact model matching problem: A geometric approach
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description: scientific article; zbMATH DE number 1192918