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

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

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.

References

An, H. Z., Chen, Z. G. and Hannan, E. J. (1982) Autocorrelation, autoregression and autoregressive approximation. Ann. Statist. 10, 926–936.CrossRefGoogle Scholar
Aström, K. J. and Wittenmark, B. (1984) Computer Controlled Systems. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Baxter, G. (1963) A norm inequality for a finite section Wiener-Hopf equation. Illinois J. Math. 7, 97–103.CrossRefGoogle Scholar
Berk, K. N. (1974) Consistent autoregressive spectral estimates. Ann. Statist. 2, 489–502.CrossRefGoogle Scholar
Bhansali, R. J. (1978) Linear prediction by autoregressive model fitting in the time domain. Ann. Statist. 6, 224–231.CrossRefGoogle Scholar
Box, G. E. P. and Jenkins, G. W. (1976) Time Series Analysis: Forecasting and Control , 2nd edn. Holden-Day, San Francisco.Google Scholar
Brillinger, D. R. (1981) Time Series: Data Analysis and Theory. Holden-Day, San Francisco.Google Scholar
Gevers, M. and Ljung, L. (1986) Optimal experiment designs with respect to the intended model applications. Automatica 22.CrossRefGoogle Scholar
Hannan, E. J. and Deistler, M. (1988) The Statistical Theory of Linear Systems. Wiley, New York.Google Scholar
Hannan, E. J. and Kavalieris, L. (1984) Multivariate linear time series models. Adv. App. Prob. 16, 492–561.CrossRefGoogle Scholar
Hannan, E. J. and Kavalieris, L. (1986) Regression, autoregression models. J. Time Series Anal., 7, 27–50.CrossRefGoogle Scholar
Hannan, E. J. and Wahlberg, B. (1989) Convergence rates for inverse Toeplitz matrix forms. J. Multivariate Anal. 31, 127–135.CrossRefGoogle Scholar
Jenkins, G. M. and Watts, D. G. (1968) Spectral Analysis and its Applications. Holden-Day, San Francisco.Google Scholar
Lewis, R. and Reinsel, G. C. (1985) Prediction of multivariate autoregressive time series by autoregressive model fitting. J. Multivariate Anal. 16, 393–411.CrossRefGoogle Scholar
Ljung, L. (1984a) Analysis of stochastic gradient algorithms for linear regression problems. IEEE Trans. Inf. Theory 30, 151–160.CrossRefGoogle Scholar
Ljung, L. (1984b) Asymptotic properties of the least squares method for estimating transfer functions and disturbance spectra. Report LiTH-ISY-I-0709. Linköping University, S-581 83 Linköping, Sweden.Google Scholar
Ljung, L. (1985) Asymptotic variance expressions for identified black-box transfer function models. IEEE Trans. Autom. Control 30, 834–844.CrossRefGoogle Scholar
Ljung, L. (1987) System Identification: Theory for the User. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Ljung, L. and Yuan, Z. D. (1985) Asymptotic properties of black-box identification of transfer functions. IEEE Trans. Autom. Control. 30, 514–530.CrossRefGoogle Scholar
Shibata, R. (1981) An optimal autoregressive spectral estimate. Ann. Statist. 9, 300–306.CrossRefGoogle Scholar
Stout, W. F. (1984) Almost Sure Convergence. Academic Press, New York.Google Scholar
Söderström, T. and Stoica, P. G. (1989) System Identification. Prentice-Hall International, Hemel Hempstead, UK.Google Scholar
Söderström, T., Ljung, L. and Gustavsson, I. (1976) Identifiability conditions for linear systems operating in closed loop. IEEE Trans. Autom. Control 21, 837–840.CrossRefGoogle Scholar
Yuan, Z. D. and Ljung, L. (1984) Black-box identification of multivariable transfer functions-asymptotic properties and optimal input design. Int. J. Control 40, 233–256.CrossRefGoogle Scholar
Yuan, Z. D. and Ljung, L. (1985) Unprejudiced optimal open loop input design for identification of transfer functions. Automatica 16, 697–708.CrossRefGoogle Scholar