Change search
ReferencesLink to record
Permanent link

Direct link
Variance error quantification for identified poles and zeros
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
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
2009 (English)In: Automatica, ISSN 0005-1098, Vol. 45, no 11, 2512-2525 p.Article 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.

Place, publisher, year, edition, pages
2009. Vol. 45, no 11, 2512-2525 p.
Keyword [en]
Accuracy of identification, Asymptotic variance expressions
National Category
Control Engineering
Research subject
URN: urn:nbn:se:kth:diva-7542DOI: 10.1016/j.automatica.2009.08.001ISI: 000271877200005ScopusID: 2-s2.0-70349883931OAI: diva2:12598
QC 20101005 QC 20120119Available 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

Open Access in DiVA

No full text

Other links

Publisher's full textScopus

Search in DiVA

By author/editor
Mårtensson, JonasHjalmarsson, Håkan
By organisation
Automatic ControlACCESS Linnaeus Centre
In the same journal
Control Engineering

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Altmetric score

Total: 95 hits
ReferencesLink to record
Permanent link

Direct link