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Geometric analysis of stochastic model errors in system identification
KTH, School of Electrical Engineering (EES), Automatic Control.ORCID iD: 0000-0002-3672-5316
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

Models of dynamical systems are important in many disciplines of science, ranging from physics and traditional mechanical and electrical engineering to life sciences, computer science and economics. Engineers, for example, use models for development, analysis and control of complex technical systems. Dynamical models can be derived from physical insights, for example some known laws of nature, (which are models themselves), or, as considered here, by fitting unknown model parameters to measurements from an experiment. The latter approach is what we call system identification. A model is always (at best) an approximation of the true system, and for a model to be useful, we need some characterization of how large the model error is. In this thesis we consider model errors originating from stochastic (random) disturbances that the system was subject to during the experiment.

Stochastic model errors, known as variance-errors, are usually analyzed under the assumption of an infinite number of data. In this context the variance-error can be expressed as a (complicated) function of the spectra (and cross-spectra) of the disturbances and the excitation signals, a description of the true system, and the model structure (i.e., the parametrization of the model). The primary contribution of this thesis is an alternative geometric interpretation of this expression. This geometric approach consists in viewing the asymptotic variance as an orthogonal projection on a vector space that to a large extent is defined from the model structure. This approach is useful in several ways. Primarily, it facilitates structural analysis of how, for example, model structure and model order, and possible feedback mechanisms, affect the variance-error. Moreover, simple upper bounds on the variance-error can be obtained, which are independent of the employed model structure.

The accuracy of estimated poles and zeros of linear time-invariant systems can also be analyzed using results closely related to the approach described above. One fundamental conclusion is that the accuracy of estimates of unstable poles and zeros is little affected by the model order, while the accuracy deteriorates fast with the model order for stable poles and zeros. The geometric approach has also shown potential in input design, which treats how the excitation signal (input signal) should be chosen to yield informative experiments. For example, we show cases when the input signal can be chosen so that the variance-error does not depend on the model order or the model structure.

Perhaps the most important contribution of this thesis, and of the geometric approach, is the analysis method as such. Hopefully the methodology presented in this work will be useful in future research on the accuracy of identified models; in particular non-linear models and models with multiple inputs and outputs, for which there are relatively few results at present.

Place, publisher, year, edition, pages
Stockholm: KTH , 2007. , p. viii, 58, 201-208
Series
Trita-EE, ISSN 1653-5146 ; 2007:061
Keyword [en]
Automatic Control
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:kth:diva-4506ISBN: 978-91-7178-770-5 (print)OAI: oai:DiVA.org:kth-4506DiVA: diva2:12601
Public defence
2007-10-31, Hörsal F3, Lindstedtsvägen 26, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20100810Available from: 2007-10-15 Created: 2007-10-15 Last updated: 2010-08-11Bibliographically approved
List of papers
1. A Geometric Approach to Variance Analysis in System Identification: Theory and Nonlinear Systems
Open this publication in new window or tab >>A Geometric Approach to Variance Analysis in System Identification: Theory and Nonlinear Systems
2007 (English)In: PROCEEDINGS OF THE 46TH IEEE CONFERENCE ON DECISION AND CONTRO, 2007, Vol. FrA17.2, p. 5092-5097Conference paper, Published paper (Refereed)
Abstract [en]

This paper addresses the problem of quantifying the model error ("variance-error") in estimates of dynamic systems. It is shown that, under very general conditions, the asymptotic (in data length) covariance of an estimated system property (represented by a smooth function of estimated system parameters) can be interpreted in terms of an orthogonal projection of a certain function gamma, associated with the property of interest, onto a subspace determined by the model structure and experimental conditions. An explicit method to construct a suitable gamma, in such a way that the individual impacts of model structure, model order and experimental conditions become visible, is presented. The technique is used to derive asymptotic variance expressions for a Hammerstein model and a nonlinear regression problem.

Series
IEEE conference on decision and control - proceedings, ISSN 0191-2216
Keyword
covariance analysis, covariance matrices, geometry, identification, nonlinear systems, regression analysis, Hammerstein model, asymptotic covariance matrix, dynamic system, geometric approach, nonlinear regression problem, nonlinear system, system identification, variance analysis, Accuracy of identification, Asymptotic variance expressions
National Category
Control Engineering
Research subject
SRA - ICT
Identifiers
urn:nbn:se:kth:diva-7540 (URN)10.1109/CDC.2007.4434584 (DOI)000255181701278 ()2-s2.0-39549096274 (Scopus ID)978-1-4244-1497-0 (ISBN)
Conference
46th IEEE Conference on Decision and Control New Orleans, LA, DEC 12-14, 2007
Funder
Swedish Research Council, 621-2007-6271
Note
QC 20100810.Available from: 2007-10-15 Created: 2007-10-15 Last updated: 2012-01-19Bibliographically approved
2. A geometric approach to variance analysis in system identification: Linear Time-Invariant Systems
Open this publication in new window or tab >>A geometric approach to variance analysis in system identification: Linear Time-Invariant Systems
2007 (English)In: PROCEEDINGS OF THE 46TH IEEE CONFERENCE ON DECISION AND CONTROL, 2007, no ThC11.5, p. 4269-4274Conference paper, Published paper (Refereed)
Abstract [en]

This paper addresses the problem of quantifying the model error ("variance-error") in estimates of linear time invariant systems. Building on the results in H. Hjalmarsson and J. Martensson (2007), we present an explicit method to construct an expression for the asymptotic variance of system properties such as impulse response coefficients, system gain, or the performance of some (control) application where the identified model is used. The expression is such that the individual impacts of model structure, model order and experimental conditions become visible. The technique is used to derive asymptotic variance expressions for a number of system properties.

Series
IEEE conference on decision and control - proceedings, ISSN 0191-2216
Keyword
identification for control; accuracy of identification; asymptotic variance expressions; CLOSED-LOOP IDENTIFICATION; FREQUENCY FUNCTIONS; EXPRESSIONS; ERROR; QUANTIFICATION
National Category
Control Engineering
Research subject
SRA - ICT
Identifiers
urn:nbn:se:kth:diva-7541 (URN)10.1109/CDC.2007.4434586 (DOI)000255181701280 ()2-s2.0-62749150993 (Scopus ID)
Conference
PROCEEDINGS OF THE 46TH IEEE CONFERENCE ON DECISION AND CONTROL, New Orleans, DEC 12-14, 2007
Note

QC 20160601

Available from: 2007-10-15 Created: 2007-10-15 Last updated: 2016-06-01Bibliographically approved
3. Variance error quantification for identified poles and zeros
Open this publication in new window or tab >>Variance error quantification for identified poles and zeros
2009 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 45, no 11, p. 2512-2525Article in journal (Refereed) Published
Abstract [en]

This paper deals with quantification of noise induced errors in identified discrete-time models of causal linear time-invariant systems, where the model error is described by the asymptotic (in data length) variance of the estimated poles and zeros. The main conclusion is that there is a fundamental difference in the accuracy of the estimates depending on whether the zeros and poles lie inside or outside the unit circle. As the model order goes to infinity, the asymptotic variance approaches a finite limit for estimates of zeros and poles having magnitude larger than one. but for zeros and poles strictly inside the unit circle the asymptotic variance grows exponentially with the model order. We analyze how the variance of poles and zeros is affected by model order, model structure and input excitation. We treat general black-box model structures including ARMAX and Box-Jenkins models.

Keyword
Accuracy of identification, Asymptotic variance expressions
National Category
Control Engineering
Research subject
SRA - ICT
Identifiers
urn:nbn:se:kth:diva-7542 (URN)10.1016/j.automatica.2009.08.001 (DOI)000271877200005 ()2-s2.0-70349883931 (Scopus ID)
Note
QC 20101005 QC 20120119Available from: 2007-10-15 Created: 2007-10-15 Last updated: 2017-12-14Bibliographically approved
4. Variance error quantification for identified poles and zeros: Part II. Closed loop identification
Open this publication in new window or tab >>Variance error quantification for identified poles and zeros: Part II. Closed loop identification
(English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, AutomaticaArticle in journal (Refereed) Submitted
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-7543 (URN)
Note
QS 20120328Available from: 2007-10-15 Created: 2007-10-15 Last updated: 2017-12-14Bibliographically approved
5. How to Make Bias and Variance Errors Insensitive to System and Model Complexity in Identification
Open this publication in new window or tab >>How to Make Bias and Variance Errors Insensitive to System and Model Complexity in Identification
2011 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 56, no 1, p. 100-112Article in journal (Refereed) Published
Abstract [en]

Solutions to optimal input design problems for system identification are sometimes believed to be sensitive to the underlying assumptions. For example, a wide class of problems can be solved with sinusoidal inputs with the same number of excitation frequencies (over the frequency range (-pi, pi]) as the number of model parameters. The order of the true system is in many cases unknown and, hence, so is the required number of frequencies in the input. In this contribution we characterize when and how the input spectrum can be chosen so that the (asymptotic) variance error of a scalar function of the model parameters becomes independent of the order of the true system. A connection between these robust designs and the solutions of certain optimal input design problems is also made. Furthermore, we show that there are circumstances when using this type of input allows some model properties to be estimated consistently even when the model order is lower than the order of the true system. The results are derived under the assumptions of causal linear time invariant systems operating in open loop and excited by an input signal having a rational spectral factor with all poles and zeros strictly inside the unit circle.

Keyword
Asymptotic model accuracy, low-complexity models, optimal input design, robust input design, system identification
National Category
Control Engineering
Research subject
SRA - ICT
Identifiers
urn:nbn:se:kth:diva-7544 (URN)10.1109/TAC.2010.2052294 (DOI)000286108800008 ()2-s2.0-78651323084 (Scopus ID)
Note

Changed title from "Robustness issues in experiment design for system identification".

Updated from submitted to published. QC 20120411

Available from: 2007-10-15 Created: 2007-10-15 Last updated: 2017-12-14Bibliographically approved

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