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Asymptotic properties of the least-squares method for estimating transfer functions and disturbance spectra

Part of: Inference from stochastic processes Limit theorems Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Lennart Ljung  and
Bo Wahlberg
Lennart Ljung*
Affiliation:
Linköping University
Bo Wahlberg
Affiliation:
Linköping University
*
∗ Postal address: Department of Electrical Engineering, Linköping University, S-581 83 Linköping, Sweden.
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Abstract

The problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, is considered. The set of models used consists of linear rational transfer functions and the spectral densities are estimated from a finite-order autoregressive disturbance description. The true system and disturbance spectrum are, however, not necessarily of finite order. We investigate the properties of the estimates obtained as the number of observations tends to ∞ at the same time as the model order employed tends to ∞. It is shown that the estimates are strongly consistent and asymptotically normal, and an expression for the asymptotic variances is also given. The variance of the transfer function estimate at a certain frequency is related to the signal/noise ratio at that frequency and the model orders used, as well as the number of observations. The variance of the noise spectral estimate relates in a similar way to the squared value of the true spectrum.

Keywords

ARX MODELSFREQUENCY DOMAINLINEAR SYSTEMSAUTOREGRESSIONIDENTIFICATIONALMOST SURE CONVERGENCECONVERGENCE IN LPCENTRAL LIMIT THEOREM

MSC classification

Primary: 93E12: System identification
Secondary: 93E24: Least squares and related methods 62M10: Time series, auto-correlation, regression, etc. 60F05: Central limit and other weak theorems 60F25: $L^p$-limit theorems
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 412 - 440
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

∗∗

Present address: Department of Automatic Control, Royal Institute of Technology, S-10044 Stockholm, Sweden.

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