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Balanced truncation of linear time-varying systems
2004 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 49, no 2, 217-229 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, balanced truncation of linear time-varying systems is studied in discrete and continuous time. Based on relatively basic calculations with time-varying Lyapunov equations/inequalities we are able to derive both upper and lower error bounds for the truncated models. These results generalize well-known time-invariant formulas. The case of time-varying state dimension is considered. Input-output stability of all truncated balanced realizations is also proven. The method is finally successfully applied to a high-order model.

Place, publisher, year, edition, pages
2004. Vol. 49, no 2, 217-229 p.
Keyword [en]
balanced truncation, error bound, linear time-varying systems, model reduction, model-reduction, error-bounds, variable systems, periodic-systems
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:kth:diva-23182DOI: 10.1109/TAC.2003.822862ISI: 000189037200005OAI: oai:DiVA.org:kth-23182DiVA: diva2:341880
Note

QC 20100525 QC 20111110

Available from: 2010-08-10 Created: 2010-08-10 Last updated: 2017-12-12Bibliographically approved
In thesis
1. Model Reduction for Linear Time-Varying Systems
Open this publication in new window or tab >>Model Reduction for Linear Time-Varying Systems
2004 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis treats model reduction for linear time-varying systems. Time-varying models appear in many fields, including power systems, chemical engineering, aeronautics, and computational science. They can also be used for approximation of time-invariant nonlinear models. Model reduction is a topic that deals with simplification of complex models. This is important since it facilitates analysis and synthesis of controllers.

The thesis consists of two parts. The first part provides an introduction to the topics of time-varying systems and model reduction. Here, notation, standard results, examples, and some results from the second part of the thesis are presented.

The second part of the thesis consists of four papers. In the first paper, we study the balanced truncation method for linear time-varying state-space models. We derive error bounds for the simplified models. These bounds are generalizations of well-known time-invariant results, derived with other methods. In the second paper, we apply balanced truncation to a high-order model of a diesel exhaust catalyst. Furthermore, we discuss practical issues of balanced truncation and approximative discretization. In the third paper, we look at frequency-domain analysis of linear time-periodic impulse-response models. By decomposing the models into Taylor and Fourier series, we can analyze convergence properties of different truncated representations. In the fourth paper, we use the frequency-domain representation developed in the third paper, the harmonic transfer function, to generalize Bode's sensitivity integral. This result quantifies limitations for feedback control of linear time-periodic systems.

Place, publisher, year, edition, pages
Lund: Lund University, 2004. 174 p.
Series
Institutionen för reglerteknik, Lunds universitet, ISSN 0280-5316 ; 1071
Keyword
Model reduction, Linear systems, Time-varying systems, Error bounds, Frequency-domain analysis, Convergence analysis, Performance limitations
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-74698 (URN)
Public defence
2004-12-03, M:B, Maskinhuset, Lunds tekniska högskola, Lund, 10:15 (English)
Opponent
Supervisors
Note
QC 20120206Available from: 2012-02-06 Created: 2012-02-03 Last updated: 2012-02-06Bibliographically approved
2. Linear Time-Varying Systems: Modeling and Reduction
Open this publication in new window or tab >>Linear Time-Varying Systems: Modeling and Reduction
2002 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Linear time-invariant models are widely used in the control community. They often serve as approximations of nonlinear systems. For control purposes linear approximations are often good enough since feedback control systems are inherently robust to model errors. In this thesis some of the possibilities for linear time-varying modeling are studied. In the thesis it is shown that the balanced truncation procedure can be applied to reduce the order of linear time-varying systems. Many of the attractive properties of balanced truncation for time-invariant systems can be generalized into the time-varying framework. For example, it is shown that a truncated input-output stable system will be input-output stable, and computable simple worst-case error bounds are derived. The method is illustrated with model reduction of a nonlinear diesel exhaust catalyst model. It is also shown that linear time-periodic models can be used for analysis of systems with power converters. Power converters produce harmonics in the power grids and give frequency coupling that cannot be modeled with standard time-invariant linear models. With time-periodic models we can visualize the coupling and also use all the available tools for linear time-varying systems, such as balanced truncation. The method is illustrated on inverter locomotives.

Place, publisher, year, edition, pages
Lund: Lund University, 2002. 21 p.
Keyword
linear time-varying, model reduction, balanced truncation, error bounds, power converters, harmonics, periodic modeling, inverter locomotive
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-74720 (URN)
Presentation
2002-11-19, Maskinhuset, Lunds tekniska högskola, Lund, 10:15 (English)
Opponent
Supervisors
Note
QC 20120208Available from: 2012-02-08 Created: 2012-02-03 Last updated: 2012-02-08Bibliographically approved

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