Summary
This paper presents new recursive projection techniques to compute reduced order models of time-varying linear systems. The methods produce a low rank approximation of the Gramians or of the Hankel map of the system and are mainly based on matrix operations that can exploit sparsity of the model. We show the practical relevance of our results with a few benchmark examples.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Chahlaoui, Y.: Recursive low rank Hankel approximation and model reduction. Doctoral Thesis, Université catholique de Louvain, Louvain-la-Neuve (2003)
Chahlaoui, Y. and Van Dooren, P.: Estimating Gramians of large-scale time-varying systems. In: Proc. IFAC World Congress, Barcelona, Paper 2440 (2002)
Chahlaoui, Y. and Van Dooren, P.: Recursive Gramian and Hankel map approximation of large dynamical systems. In: CD-Rom Proceedings SIAM Applied Linear Algebra Conference, Williamsburg, Paper MS14-1 (2003)
Chahlaoui, Y. and Van Dooren, P.: Recursive low rank Hankel approximation and model reduction. In: CD-Rom Proceedings ECC 2003, Cambridge, Paper 553 (2003)
Dewilde, P. and van der Veen, A.-J.: Time-varying systems and computations. Kluwer Academic Publishers, Boston (1998)
Enns, D.: Model reduction with balanced realizations: An error bound and frequency weighted generalization. In: Proc. of the IEEE Conference on Decision and Control, San Diego, 127–132 (1981)
Gallivan, K., Vandendorpe, A. and Van Dooren, P.: Sylvester equations and projection-based model reduction. J. Comp. Appl. Math., 162, 213–229 (2003)
Golub, G. and Van Loan, C.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996)
Gugercin, S., Sorenson, D. and Antoulas, A.: A modified low-rank Smith method for large-scale Lyapunov equations. Numerical Algorithms, 32(1), 27–55 (2003)
Imae, J., Perkins, J.E. and Moore, J.B.: Toward time-varying balanced realization via Riccati equations. Math. Control Signals Systems, 5, 313–326 (1992)
Lall, S., and Beck, C.: Error-bounds for balanced model-reduction of linear time-varying systems. IEEE Trans. Automat. Control, 48(6), 946–956 (2003)
Meyer, R. and Burrus, C.: A unified analysis of multirate and periodically time-varying digital filters. IEEE Trans. Circ. Systems, 22, 162–168 (1975)
Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Control, 26, 17–31 (1981)
Sandberg, H., and Rantzer, H.: Balanced model reduction of linear time-varying systems. In: Proc. IFAC02, 15th Triennial World Congress, Barcelona (2002)
Shokoohi, S., Silverman, L., and Van Dooren, P.: Linear time-variable systems: Balancing and model reduction. IEEE Trans. Automat. Control, 28, 810–822 (1983)
Tornero, J., Albertos, P., and Salt, J.: Periodic optimal control of multirate sampled data systems. In: Proc. PSYCO2001, IFAC Conf. Periodic Control Systems, Como, 199–204 (2001)
Verriest, E., and Kailath, T.: On generalized balanced realizations. IEEE Trans. Automat. Control, 28(8), 833–844 (1983)
Zhou, K., Doyle, J., and Glover, K.: Robust and optimal control. Prentice Hall, Upper Saddle River (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chahlaoui, Y., Van Dooren, P. (2005). Model Reduction of Time-Varying Systems. In: Benner, P., Sorensen, D.C., Mehrmann, V. (eds) Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27909-1_5
Download citation
DOI: https://doi.org/10.1007/3-540-27909-1_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24545-2
Online ISBN: 978-3-540-27909-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)