Loughborough University
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Equivalence transformations in linear systems theory

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posted on 2018-06-01, 10:42 authored by Paul Fretwell
There is growing interest in infinite frequency structure of linear systems, and transformations preserving this type of structure. Most work has been centred around Generalised State Space (GSS) systems. Two constant equivalence transformations for such systems are Rosenbrock's Restricted System Equivalence (RSE) and Verghese's Strong Equivalence (str.eq.). Both preserve finite and infinite frequency system structure. RSE is over restrictive in that it is constrained to act between systems of the same dimension. While overcoming this basic difficulty str.eq. on the other hand has no closed form description. In this work all these difficulties have been overcome. A constant pencil transformation termed Complete Equivalence (CE) is proposed, this preserves finite elementary divisors and non-unity infinite elementary divisors. Applied to GSS systems CE yields Complete System Equivalence (CSE) which is shown to be a closed form description of str.eq. and is more general than RSE as it relates systems of different dimensions. Equivalence can be described in terms of mappings of the solution sets of the describing differential equations together with mappings of the constrained initial conditions. This provides a conceptually pleasing definition of equivalence. The new equivalence is termed Fundamental Equivalence (FE) and CSE is shown to be a matrix characterisation of it. A polynomial system matrix transformation termed Full Equivalence (fll.e.) is proposed. This relates general matrix polynomials of different dimensions while preserving finite and infinite frequency structure. A definition of infinite zeros is also proposed along with a generalisation of the concept of infinite elementary divisors (IED) from matrix pencils to general polynomial matrices. The IED provide an additional method of dealing with infinite zeros.


Science and Engineering Research Council.



  • Science


  • Mathematical Sciences


© Paul Fretwell

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This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

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A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy at Loughborough University.


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