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Variance Expressions and Model Reduction in System Identification
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
2002 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Although system identification is now a mature research field, some problems remain unsolved. Examples of unsolved or partly unsolved problems are; accuracy of subspace identification algorithms, identification via model reduction, identification for control, and identification of non-linear systems. Some problems that fall into these categories are studied in this thesis.

This thesis discusses variance expressions in system identification. In particular, variance expressions for reduced models are analyzed.

The topic of model reduction via system identification has received little attention during the years. To understand how the variance of a high order model affects the reduced model, a general expression for the variance of the low order model as a function of the reduction method used is derived. This allows the analysis of all model reduction methods that can be written as a minimization criterion, where the function to be minimized is twice continuously differentiable. Many methods can be studied using this approach. However, the popular method of model reduction by balanced truncation of states does not immediately fit into this framework.

Many unsolved problems in system identification may be studied with the use of bootstrap methods. This statistical tool, used to assess accuracy in estimation problems, may be adopted to a series of problems in system identification and signal processing. The thesis presents how bootstrap can be adopted in the prediction error framework. In addition, we demonstrate how bootstrap can be applied to problems of constructing condence regions with a simultaneous confidence degree and calculating the variance of undermodeled models.

The thesis briefly discusses how set membership identification and prediction error identification can be combined into a more robust estimate. Finally, insights into how model validation can be performed in a more user informative way are also given.

Place, publisher, year, edition, pages
Linköping: Linköping University , 2002. , 192 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 730
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:liu:diva-98162ISBN: 91-7373-253-2 (print)OAI: oai:DiVA.org:liu-98162DiVA: diva2:652322
Public defence
2002-02-22, Visionen, Hus B, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Supervisors
Available from: 2013-10-09 Created: 2013-09-30 Last updated: 2013-10-09Bibliographically approved
List of papers
1. L2 Model Reduction and Variance Reduction
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, 1517-1530 p.Article 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
Keyword
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
2. Variance Aspects of L2 Model Reduction when Undermodeling - the Output Error Case
Open this publication in new window or tab >>Variance Aspects of L2 Model Reduction when Undermodeling - the Output Error Case
2002 (English)Report (Other academic)
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 L2reduced 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 the case of undermodeling. This has previously been shown to hold for FIR (Finite Impulse Response) models, but is in this paper extended to general linear OE models.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2002. 14 p.
Series
LiTH-ISY-R, ISSN 1400-3902 ; 2405
Keyword
Identification, Model reduction, Variance
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-55858 (URN)LiTH-ISY-R-2405 (ISRN)
Available from: 2010-04-30 Created: 2010-04-30 Last updated: 2014-10-02Bibliographically approved
3. Variance Properties of a Two-Step ARX Estimation Procedure
Open this publication in new window or tab >>Variance Properties of a Two-Step ARX Estimation Procedure
2001 (English)Report (Other academic)
Abstract [en]

In this contribution we discuss some variance properties of a two-step ARX estimation scheme. An expression for the covariance of the final low order model is calculated and it is discussed how one should minimize this covariance. The implication of the results isthat identification of the dynamics of a system could very easily be performed with standard linear least squares (two times), even if the measurement noise is heavily colored. We also show a numerical example, where this two-step estimation scheme gives a variance which is close (but not equal) to the the Cramér-Rao lower bound. Moreover, we show that the point estimate of the covariance is close to the one obtained through Monte Carlo simulations.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2001. 8 p.
Series
LiTH-ISY-R, ISSN 1400-3902 ; 2393
Keyword
Identification methods, Estimation
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-55804 (URN)Variance Properties of a Two-Step ARX Estimation Procedure (ISRN)
Available from: 2010-04-30 Created: 2010-04-30 Last updated: 2014-09-08Bibliographically approved
4. Comparison of Methods for Probabilistic Uncertainty Bounding
Open this publication in new window or tab >>Comparison of Methods for Probabilistic Uncertainty Bounding
1999 (English)In: Proceedings of the 38th IEEE Conference on Decision and Control, 1999, 522-527 vol.1 p.Conference paper, Published paper (Refereed)
Abstract [en]

The problem of computing probabilistic uncertainty regions for the frequency responses of identified models is studied. A novel method for uncertainty bounding that uses bootstrap is presented and compared to a classical method using estimated covariance information. It is shown that, with bootstrap, it is possible to compute realistic uncertainty regions that closely resemble those obtainable through Monte Carlo simulations.

Keyword
Model uncertainty, Identification, Bootstrap
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-94081 (URN)10.1109/CDC.1999.832835 (DOI)0-7803-5250-5 (ISBN)
Conference
38th IEEE Conference on Decision and Control, Phoenix, AZ, USA, December, 1999
Available from: 2013-06-16 Created: 2013-06-16 Last updated: 2013-10-09
5. Computing Uncertainty Regions with Simultaneous Confidence Degree using Bootstrap
Open this publication in new window or tab >>Computing Uncertainty Regions with Simultaneous Confidence Degree using Bootstrap
2000 (English)In: Proceedings of the 12th IFAC Symposium on System Identification, 2000, 1133-1138 p.Conference paper, Published paper (Refereed)
Abstract [en]

We discuss the importance of constructing confidence regions of simultaneous confidence degree for certain statistics, e.g., the frequency function. In this contribution we show how bootstrap can be used to obtain reliable confidence regions of simultaneous confidence degree, independently of how many confidence regions we calculate. The procedure is illustrated by comparison with classical methods and Monte Carlo simulations. We will also provide an evaluation of the quality of the obtained confidence regions.

Series
LiTH-ISY-R, ISSN 1400-3902 ; 2289
Keyword
Bootstrap, Identification, Modeling errors, Uncertainty, Simultaneous confidence degree
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-91208 (URN)978-0080435459 (ISBN)
Conference
12th IFAC Symposium on System Identification, Santa Barbara, CA, USA, June, 2000
Available from: 2013-04-17 Created: 2013-04-17 Last updated: 2014-12-18
6. Using the Bootstrap to Estimate the Variance in the Case of Undermodeling
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, 395-398 p.Article 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.

Keyword
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
7. A Novel Mixed Set Membership/Stochastic Approach to System Identification
Open this publication in new window or tab >>A Novel Mixed Set Membership/Stochastic Approach to System Identification
2001 (English)Report (Other academic)
Abstract [en]

In this paper, we propose a mixed approach to identication of linear dynamical systems corrupted by additive, amplitude bounded noise. To come up with an estimate of the system, we mix the information from a prediction error estimate and a set membership estimate. The estimate shows very good performance on a number of simulation examples, which are performed using a wide range of noise sources.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2001. 12 p.
Series
LiTH-ISY-R, ISSN 1400-3902 ; 2406
Keyword
Identification, bounded noise, stochastic noise, variance
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-55857 (URN)LiTH-ISY-R-2406 (ISRN)
Available from: 2010-04-30 Created: 2010-04-30 Last updated: 2014-10-02Bibliographically approved
8. A Nonparametric Approach to Model Error Modeling
Open this publication in new window or tab >>A Nonparametric Approach to Model Error Modeling
2000 (English)In: Proceedings of the 12th IFAC Symposium on System Identification, 2000, 157-162 p.Conference paper, Published paper (Refereed)
Abstract [en]

To validate an estimated model and evaluate its reliability is an important part of the system identification process. Recent work on model validation has shown that the use of explicit model error models provide a better way of visualizing the possible deficiencies of the nominal model. Previous contributions have mainly focused on parametric black-box models for estimating the error model. However, this requires that a correct model order for the error model has to be selected. Here we suggest an adaptive and nonparametric frequency-domain method that estimates the frequency response of the model error by an automatic procedure. A benefit with this approach is that the tuning can be done locally, i.e., that different resolutions can be used in different frequency bands. The ideas are based on local polynomial regression and utilize a statistical criterion for selecting the optimal resolution.

Series
LiTH-ISY-R, ISSN 1400-3902 ; 2230
Keyword
Model error modeling, Non-parametric regression, Frequency-response methods
National Category
Engineering and Technology Control Engineering
Identifiers
urn:nbn:se:liu:diva-91126 (URN)978-0080435459 (ISBN)
Conference
12th IFAC Symposium on System Identification, Santa Barbara, CA, USA, June, 2000
Available from: 2013-04-28 Created: 2013-04-17 Last updated: 2014-12-11

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