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Linear Time-Varying Systems: Modeling and Reduction
Lund University, Department of Automatic Control.ORCID iD: 0000-0003-1835-2963
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 [en]
linear time-varying, model reduction, balanced truncation, error bounds, power converters, harmonics, periodic modeling, inverter locomotive
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:kth:diva-74720OAI: oai:DiVA.org:kth-74720DiVA: diva2:489904
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
List of papers
1. 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
2. Periodic Modelling of Power Systems
Open this publication in new window or tab >>Periodic Modelling of Power Systems
2001 (English)In: Proceedings of the 1st IFAC Workshop on Periodic Control Systems, 2001, 91-96 p.Conference paper, Published paper (Refereed)
Abstract [en]

This paper treats modelling of power systems with converters in a linear time-periodic framework. A power converter is a nonlinear switching device connecting an AC system to a DC system. The converter generates harmonics that might cause instabilities in systems of this kind. About a nominal periodic trajectory the power converter is well described by a periodic gain matrix, whereas the power grids often can be described by linear time-invariant models. Put together they form a linear time-periodic model. It is also shown in this paper how Integral Quadratic Constraints may be used for robustness analysis. To conclude an inverter locomotive is modeled with the described techniques.

National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-74718 (URN)
Conference
1st IFAC Workshop on Periodic Control Systems
Note
QC 20120208Available from: 2012-02-03 Created: 2012-02-03 Last updated: 2012-02-08Bibliographically approved
3. Harmonic modeling of the motor side of an inverter locomotive
Open this publication in new window or tab >>Harmonic modeling of the motor side of an inverter locomotive
2000 (English)In: Control Applications, 2000. Proceedings of the 2000 IEEE International Conference on, 2000, 918-923 p.Conference paper, Published paper (Refereed)
Abstract [en]

An AC-voltage source feeding an electric network results in a periodic excitation of the network. In steady state, all currents and voltages will be periodic with cycle time corresponding to the frequency of the voltage source. If the network is linear, all signals are sinusoidal and the network is solved using traditional methods. If the network contains components with nonlinear or switching dynamics, iterative methods based on harmonic balance are often required to obtain the periodic steady state solution. By linearization of the system around the periodic solution, a linear time periodic model is obtained. This can be used as a local description of the system in the neighborhood of the periodic solution. If only periodic signals are considered, a linearized model can be represented by a matrix, called the harmonic transfer matrix (HTM). The method is applied to the motor side of a modern inverter train. Via the HTM, the steady state response to constant or periodic disturbances or changes in reference values can be obtained

Keyword
3-phase induction motor;AC-voltage source;constant disturbances;harmonic balance;harmonic modeling;harmonic transfer matrix;inverter locomotive;iterative methods;linear network;linear time periodic model;linearized model;nonlinear dynamics;periodic disturbances;periodic excitation;steady state;steady state response;switching dynamics;voltage source frequency;induction motors;invertors;locomotives;matrix algebra;power conversion harmonics;
National Category
Control Engineering
Identifiers
urn:nbn:se:kth:diva-74697 (URN)10.1109/CCA.2000.897585 (DOI)
Conference
International Conference on Control Applications
Note
QC 20120208Available from: 2012-02-03 Created: 2012-02-03 Last updated: 2012-02-08Bibliographically approved

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