Available online at www.sciencedirect.com Automatica 39 (2003) 1721–1733 www.elsevier.com/locate/automatica Uncertainty of transfer function modelling using prior estimated noise models  R. Pintelon * , J. Schoukens, Y. Rolain Department of Electricity, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium Received 22 April 2002; received in revised form 22 January 2003; accepted 26 May 2003 Abstract Assuming “small” model errors (unmodelled dynamics and/or nonlinear distortions) and “large” signal-to-noise ratios we derive in this paper explicit expressions for the covariance matrix of a frequency domain estimator using prior estimated noise models. These analytic expressions (i) give a clear insight in the behaviour of the covariance matrix as a function of the signal-to-noise ratio, the unmodelled dynamics and the nonlinear distortions, and (ii) allow to predict accurately the order of magnitude of the actual uncertainty of the estimates. The link with the classical prediction error approach is also established. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Frequency domain; Non-parametric noise model; Model errors; Uncertainty 1. Introduction Since real life systems are mostly distributed and/or non- linear, the model errors (unmodelled dynamics and/or the nonlinear distortions) are often the limiting factor in transfer function modelling problems. The inuence of model errors on the asymptotic properties (amount of data going to in- nity) of the estimated plant model parameters ˆ (Z ) have been studied in Ljung (1999) and Pintelon and Schoukens (2001), and general closed form expressions for the asymp- totic covariance matrix Cov( ˆ (Z )) are available. The di- culty is that these expressions are not tractable in the pres- ence of model errors. Numerical methods for calculating Cov( ˆ (Z )) are described in Hjalmarsson and Ljung (1992) and Tj arnstr om and Ljung (2002), and a qualitative study of the inuence of model errors on Cov( ˆ (Z )) is available in Pintelon and Schoukens (2002a, b). The main contribution of this paper is to derive, under some suitable assumptions concerning the disturbing noise and the plant model errors, approximate, easy to use analytic expressions for the asymptotic covariance matrix Cov( ˆ (Z )) of a frequency domain estimator using prior estimated  This article is not presented at any IFAC meeting. This article was recommended for publication in revised form by Associate Editor Antonio Vicini under the direction of Editor Torsten S oderstr om. * Corresponding author. Tel.: +32-2-629-2944; fax: +32-2-629-2850. E-mail address: rik.pintelon@vub.ac.be (R. Pintelon). noise models. These analytic expressions (i) give a clear in- sight in the behaviour of Cov( ˆ (Z )) as a function of the dis- turbing noise level, the unmodelled dynamics and the non- linear distortions, (ii) allow accurate prediction of the order of magnitude of the actual uncertainty of the estimates; and (iii) establish a link between transfer function modelling us- ing prior estimated noise models and the classical prediction error framework. The theory is elaborated for identication starting from observed input/output spectra, and from mea- sured frequency response functions. Note that: (i) Except some special cases (e.g. linear least squares) it is at this moment impossible to give analytic expressions for the nite sample properties of estimators. Only asymptotic expressions are available (see, for example, Ljung, 1999), however, practise shows that these expressions predict quite well the nite sample behaviour of the estimates. (ii)Onecouldwonderwhetheritmakessensetoapplylin- ear system identication techniques to nonlinear systems. In Schoukens, Dobrowiecki, and Pintelon (1998) and Pintelon and Schoukens (2001) it has been shown that this indeed makes sense for nonlinear time-invariant systems whose out- put can be approximated arbitrarily well in least-squares sense by a Volterra series on a given input domain. For such systems (i) the inuence of the initial conditions vanishes asymptotically, and (ii) the steady-state response to a peri- odic input is a periodic signal with the same period as the 0005-1098/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0005-1098(03)00185-7