Variance analysis of \(L_2\) model reduction when undermodeling -- the output error case.
type: Item
instance of: scholarly article
title: Variance analysis of \(L_2\) model reduction when undermodeling -- the output error case.
zbMATH Open document ID: 1064.93009
DOI: 10.1016/S0005-1098(03)00175-4
author: Fredrik Tjärnström
published in: Automatica
publication date: 2003-11-17T00:00:00Z
review text: Given a system \(y(t)=G_0(q)u(t)+e(t)\), one wants to find a model of the form \(y(t)=G(q,\theta)u(t)+e(t)\) where the rational transfer function is characterized by the parameters \(\theta\). To minimize the prediction output error (OE) \(\varepsilon(t,\theta)=y(t)-G(q,\theta)u(t)\), one can minimize \(V_N(\theta)=(1/2N)\sum_{n=1}^N \varepsilon^2(t,\theta)\). This first approach is directly based on the data. Undermodeling means that we will not get the true system \(G_0(q)\) as \(\lim_{N\to\infty}G(q,\theta)\). Another approach may be to estimate an optimal high-order model \(G(q,\theta)\) as indicated above, and then find a low-order model \(G(q,\eta)\) by minimizing the OE cost function \[ J(\eta,\theta)=(1/4\pi)\int_0^{2\pi} | G(e^{i\omega},\theta)-G(e^{i\omega},\eta)| ^2\Phi_u(\omega)d\omega \] where \(\Phi_u(\omega)\) is the covariance of the input \(u\). It was shown by \textit{F. Tjärnström} and \textit{L. Ljung} [Automatica 38, 1517--1530 (2002; Zbl 1008.93016)] that in the case of an undermodeled FIR model, the second approach is better in the sense that the covariance of \(\eta\) will be smaller. This result is generalized in this paper to general linear undermodeled OE models.
Mathematics Subject Classification ID: 93B11
Mathematics Subject Classification ID: 93E12
zbMATH DE Number: 2005468
zbMATH Keywords: model reduction
zbMATH Keywords: linear system
zbMATH Keywords: output error model
zbMATH Keywords: undermodeling
zbMATH Keywords: covariance of the input
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2005468