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How to Make Bias and Variance Errors Insensitive to System and Model Complexity in Identification
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
2011 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 56, no 1, 100-112 p.Article 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.

Place, publisher, year, edition, pages
2011. Vol. 56, no 1, 100-112 p.
Keyword [en]
Asymptotic model accuracy, low-complexity models, optimal input design, robust input design, system identification
National Category
Control Engineering
Research subject
URN: urn:nbn:se:kth:diva-7544DOI: 10.1109/TAC.2010.2052294ISI: 000286108800008ScopusID: 2-s2.0-78651323084OAI: diva2:12600

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: 2012-04-11Bibliographically 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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