Variance analysis of \(L_2\) model reduction when undermodeling -- the output error case. (Q1413949)

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.