Sub-optimal discrete model reduction by multipoint Padé approximation
type: Item
instance of: scholarly article
title: Sub-optimal discrete model reduction by multipoint Padé approximation
zbMATH Open document ID: 0857.93018
DOI: 10.1016/0016-0032(95)00088-7
author: T. N. Lucas
published in: Journal of the Franklin Institute
publication date: 1997-02-26T00:00:00Z
review text: Let \(y(iT)\) be the impulse response of a linear discrete-time system and \(y_k(iT)\) the impulse response of a \(k\)th order reduced system. Let \(G(z)\) be the transfer function of the original system and \(G_k(z)\) the transfer function of the reduced system. If the poles of \(G_k\) are fixed to be \(p_1,\dots,p_k\), then minimizing the least squares error \(\sum^\infty_{i=0} [y(iT)-y_k(iT)]^2\) is equivalent to interpolating \(G(z)\) by \(G_k(z)\) in the reciprocals \(1/p_1,\dots,1/p_k\). Fixing the poles of the approximant has the advantage that the least squares problem is linear in the numerator coefficients and iteration can be avoided. The problem has become a multipoint Padé-type approximation problem. The problem can be solved by an efficient Routh-array type of algorithm. The method can easily adapted for the case when the \(y(iT)\) do not correspond to an impulse response but are the response to a step input or a ramp input. However, this method may give a reduced system that does not match the first Markov parameter of the given system. If one insists of matching at least this first Markov parameter, a least squares problem with a constraint should be solved. Using a Lagrange multiplier technique, this is elaborated in the paper under review and it results in a modified interpolation problem that can be solved by methods of the type given above.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 93B11
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 93B40
zbMATH DE Number: 902659
zbMATH Keywords: rational least squares approximation
zbMATH Keywords: model reduction
zbMATH Keywords: discrete-time
zbMATH Keywords: Padé-type approximation
zbMATH Keywords: first Markov parameter
zbMATH Keywords: linear
MaRDI profile type: Publication
full work available at URL: 0016-0032(95)00088-7
OpenAlex ID: W2984626375
cites work: Least squares model reduction
cites work: Optimal discrete model reduction by multipoint Padé approximation
cites work: Approximation of discrete linear systems based on a dual criterion
cites work: An iterative method for simplification of discrete systems
cites work: Reduction of high-order discrete-time systems
cites work: Optimal L//2-approximation with fixed poles
cites work: Continued fraction methods for the reduction of discrete-time dynamic systems
cites work: Order reduction of z-transfer functions via multipoint Jordan continued- fraction expansion
cites work: Reduction of discrete time systems using a modified Routh array
cites work: Low-order modelling via discrete stability equations
cites work: New matrix method for multipoint Padé approximation of transfer functions
cites work: Finding roots by deflated polynomial approximation
description: scientific article; zbMATH DE number 902659