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Model Reduction for Linear Time-Varying Systems
Lund University, Department of Automatic Control.ORCID iD: 0000-0003-1835-2963
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 [en]
Model reduction, Linear systems, Time-varying systems, Error bounds, Frequency-domain analysis, Convergence analysis, Performance limitations
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:kth:diva-74698OAI: oai:DiVA.org:kth-74698DiVA: diva2:489875
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
List of papers
1. A case study in model reduction of linear time-varying systems
Open this publication in new window or tab >>A case study in model reduction of linear time-varying systems
2006 (English)In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 42, no 3, 467-472 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, the balanced truncation procedure is applied to time-varying linear systems, both in continuous and in discrete time. The methods are applied to a linear approximation of a diesel exhaust catalyst model. The reduced-order systems are obtained by using certain projections instead of direct balancing. An approximate zero-order-hold discretization of continuous-time systems is described, and a new a priori approximation error bound for balanced truncation in the discrete-time case is obtained. The case study shows that there are several advantages to work in discrete time. It gives simpler implementation with fewer computations.

Keyword
balanced-truncation, periodic-systems, error-bounds
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-15437 (URN)10.1016/j.automatica.2005.10.016 (DOI)000235471200012 ()2-s2.0-31144457450 (Scopus ID)
Note

QC 20111110

Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
2. Balanced truncation of linear time-varying systems
Open this publication in new window or tab >>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.

Keyword
balanced truncation, error bound, linear time-varying systems, model reduction, model-reduction, error-bounds, variable systems, periodic-systems
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-23182 (URN)10.1109/TAC.2003.822862 (DOI)000189037200005 ()
Note

QC 20100525 QC 20111110

Available from: 2010-08-10 Created: 2010-08-10 Last updated: 2017-12-12Bibliographically approved
3. Frequency-domain analysis of linear time-periodic systems
Open this publication in new window or tab >>Frequency-domain analysis of linear time-periodic systems
2005 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 50, no 12, 1971-1983 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature.

Keyword
convergence analysis, frequency-response operators, linear time-periodic systems, series expansions, sampled-data systems, response operators, h(infinity), infinity, h(2), controllers, norm, h-2
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-15259 (URN)10.1109/TAC.2005.860294 (DOI)000234062800004 ()2-s2.0-30344466922 (Scopus ID)
Note

QC 20111110

Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved
4. A bode sensitivity integral for linear time-periodic systems
Open this publication in new window or tab >>A bode sensitivity integral for linear time-periodic systems
2005 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 50, no 12, 2034-2039 p.Article in journal (Refereed) Published
Abstract [en]

Bode's sensitivity integral is a well-known formula that quantifies some of the limitations in feedback control for linear time-invariant systems. In this note, we show that there is a similar formula for linear time-periodic systems. The harmonic transfer function is used to prove the result. We use the notion of roll-off 2, which means that the first time-varying Markov parameter is equal to zero. It then follows that the harmonic transfer function is an analytic operator and a trace class operator. These facts are used to prove the result.

Keyword
Bode sensitivity integral, linear time-periodic systems, performance limitations, frequency-response, feedback-systems, varying systems, tradeoffs, analog
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-15260 (URN)10.1109/TAC.2005.860247 (DOI)000234062800010 ()2-s2.0-30344454665 (Scopus ID)
Note

QC 20111110

Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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