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Tjernström, Fredrik
Publications (4 of 4) Show all publications
Tjernström, F. (2003). Variance Analysis of L2 Model Reduction when Undermodeling: The Output Error Case. Automatica, 39(10), 1809-1815
Open this publication in new window or tab >>Variance Analysis of L2 Model Reduction when Undermodeling: The Output Error Case
2003 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 39, no 10, p. 1809-1815Article in journal (Refereed) Published
Abstract [en]

In this contribution, variance properties of L2 model reduction are studied. That is, given an estimated model of high order we study the resulting variance of an L2 reduced approximation. The main result of the paper is showing that estimating a low-order output error (OE) model via L2 model reduction of a high-order model gives a smaller variance compared to estimating a low-order model directly from data in case of undermodeling. This has previously been shown to hold for Finite Impulse Response models, but is in this paper extended to general linear OE models.

Place, publisher, year, edition, pages
Elsevier, 2003
Keywords
Linear systems, Model reduction, Reduced-order models, System identification, Variance
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-46469 (URN)10.1016/S0005-1098(03)00175-4 (DOI)
Note

© 2003 Elsevier Ltd. All rights reserved.

Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
Tjernström, F. & Ljung, L. (2003). Variance Properties of a Two-Step ARX Estimation Procedure. European Journal of Control, 9(4), 422-430
Open this publication in new window or tab >>Variance Properties of a Two-Step ARX Estimation Procedure
2003 (English)In: European Journal of Control, ISSN 0947-3580, E-ISSN 1435-5671, Vol. 9, no 4, p. 422-430Article in journal (Refereed) Published
Abstract [en]

In this contribution, the variance properties of a two-step ARX estimation scheme are discussed. An expression for the covariance of the final low-order model is calculated and it is shown how this covariance can be minimised (at least for high-model orders). The implication of the results is that identification of the dynamics of a system can very easily be performed with standard linear least squares (two times), even if the measurement noise is heavily colored. A numerical example is included, where this two-step method gives a variance which is close (but not equal) to the Cramèr-Rao lower bound. Moreover, the point estimate of the covariance is close to the one obtained through Monte Carlo simulations.

Keywords
Identification, Model reduction, Variance
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-46382 (URN)10.3166/ejc.9.422-430 (DOI)
Note

© 2003 EUCA.

Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
Tjernström, F. & Ljung, L. (2002). L2 Model Reduction and Variance Reduction. Automatica, 38(9), 1517-1530
Open this publication in new window or tab >>L2 Model Reduction and Variance Reduction
2002 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 38, no 9, p. 1517-1530Article in journal (Refereed) Published
Abstract [en]

In this contribution we examine certain variance properties of model reduction. The focus is on L2 model reduction, but some general results are also presented. These general results can be used to analyze various other model reduction schemes. The models we study are finite impulse response (FIR) and output error (OE) models. We compare the variance of two estimated models. The first one is estimated directly from data and the other one is computed by reducing a high order model, by L2 model reduction. In the FIR case we show that it is never better to estimate the model directly from data, compared to estimating it via L2 model reduction of a high order FIR model. For OE models we show that the reduced model has the same variance as the directly estimated one if the reduced model class used contains the true system.

Place, publisher, year, edition, pages
Elsevier, 2002
Keywords
Identification, Model reduction, Variance reduction
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-46912 (URN)10.1016/S0005-1098(02)00066-3 (DOI)
Note

© 2002 Elsevier Science Ltd. All rights reserved.

Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
Tjernström, F. & Ljung, L. (2002). Using the Bootstrap to Estimate the Variance in the Case of Undermodeling. IEEE Transactions on Automatic Control, 47(2), 395-398
Open this publication in new window or tab >>Using the Bootstrap to Estimate the Variance in the Case of Undermodeling
2002 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 47, no 2, p. 395-398Article in journal (Refereed) Published
Abstract [en]

This note deals with the problem of estimating the variance of an undermodeled model. Undermodeling means that the model class used is not flexible enough to describe the underlying system. The proposed solution to the problem is an algorithm that is based on the bootstrap. A simulation example shows that the variance estimates based on the proposed algorithm are in very good agreement with Monte Carlo simulations.

Keywords
Bootstrap, Identification, Model uncertainty, Simulation-based methods, Undermodeling
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-47116 (URN)10.1109/9.983387 (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
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