Key problems in the extension of module-behaviour duality
Key problems in the extension of module-behaviour duality
The duality for linear constant coefficient partial differential equations between behaviours and finitely generated modules over the operator ring is a very powerful tool linking equation structure to dynamic behaviour. This duality is critically dependent on the choice of signal space. In this paper we discuss two key algebraic problems which form an obstacle to the extension of this theory to general signal spaces. The first of these is the so-called Willems closure problem, which limits the ability of system equations to directly describe the system. The second is the elimination problem, the general solution of which depends upon an algebraic property (injectivity) of the signal space. We demonstrate the importance of these problems in the module-behaviour framework, and some of the useful consequences of a full or partial solution. The issues here are of particular relevance to the extension of the current duality theory for behaviours defined by linear partial differential equations from the case of constant to non-constant coefficients.
761-798
Wood, J
9d32c83e-9d59-4c32-8ab2-a5f64ecc26e6
2002
Wood, J
9d32c83e-9d59-4c32-8ab2-a5f64ecc26e6
Wood, J
(2002)
Key problems in the extension of module-behaviour duality.
Linear Algebra and Its Applications, 351 (2), .
Abstract
The duality for linear constant coefficient partial differential equations between behaviours and finitely generated modules over the operator ring is a very powerful tool linking equation structure to dynamic behaviour. This duality is critically dependent on the choice of signal space. In this paper we discuss two key algebraic problems which form an obstacle to the extension of this theory to general signal spaces. The first of these is the so-called Willems closure problem, which limits the ability of system equations to directly describe the system. The second is the elimination problem, the general solution of which depends upon an algebraic property (injectivity) of the signal space. We demonstrate the importance of these problems in the module-behaviour framework, and some of the useful consequences of a full or partial solution. The issues here are of particular relevance to the extension of the current duality theory for behaviours defined by linear partial differential equations from the case of constant to non-constant coefficients.
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Published date: 2002
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Electronics & Computer Science
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Local EPrints ID: 256820
URI: http://eprints.soton.ac.uk/id/eprint/256820
ISSN: 0024-3795
PURE UUID: 0ad2e246-ffac-4df3-96cb-408bd7f1b713
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Date deposited: 25 Jun 2003
Last modified: 08 Jan 2022 11:45
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J Wood
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