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A Geometric Approach to Variance Analysis in System Identification: Theory and Nonlinear Systems
KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre. (System Identification Group)ORCID iD: 0000-0002-9368-3079
KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre. (System Identification Group)ORCID iD: 0000-0002-3672-5316
2007 (English)In: PROCEEDINGS OF THE 46TH IEEE CONFERENCE ON DECISION AND CONTRO, 2007, Vol. FrA17.2, 5092-5097 p.Conference 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.

Place, publisher, year, edition, pages
2007. Vol. FrA17.2, 5092-5097 p.
, IEEE conference on decision and control - proceedings, ISSN 0191-2216
Keyword [en]
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
URN: urn:nbn:se:kth:diva-7540DOI: 10.1109/CDC.2007.4434584ISI: 000255181701278ScopusID: 2-s2.0-39549096274ISBN: 978-1-4244-1497-0OAI: diva2:12596
46th IEEE Conference on Decision and Control New Orleans, LA, DEC 12-14, 2007
Swedish Research Council, 621-2007-6271
QC 20100810.Available from: 2007-10-15 Created: 2007-10-15 Last updated: 2012-01-19Bibliographically approved
In thesis
1. Geometric analysis of stochastic model errors in system identification
Open this publication in new window or tab >>Geometric analysis of stochastic model errors in system identification
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. viii, 58, 201-208 p.
Trita-EE, ISSN 1653-5146 ; 2007:061
Automatic Control
National Category
Control Engineering
urn:nbn:se:kth:diva-4506 (URN)978-91-7178-770-5 (ISBN)
Public defence
2007-10-31, Hörsal F3, Lindstedtsvägen 26, Stockholm, 10:00
QC 20100810Available from: 2007-10-15 Created: 2007-10-15 Last updated: 2010-08-11Bibliographically approved

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