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AN ITERATIVE MODEL ORDER REDUCTION METHOD FOR LARGE-SCALE DYNAMICAL SYSTEMS

Published online by Cambridge University Press:  05 April 2017

K. MOHAMED ,
A. MEHDI  and
M. ABDELKADER
K. MOHAMED*
Affiliation:
Université de Tunis El Manar, École Nationale d’Ingenieurs de Tunis, Laboratoire de Recherche Analyse et Commande des Systèmes, LR-11-ES20, BP 37, Le Belvedere 1002 Tunis, Tunisia email [email protected]
A. MEHDI
Affiliation:
Université de Carthage, École Nationale d’Ingénieurs de Carthage, Laboratoire de Recherche Analyse et Commande des Systèmes, LR-11-ES20, BP 37, Le Belvedere 1002 Tunis, Tunisia email [email protected]
M. ABDELKADER
Affiliation:
Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire de Recherche Analyse et Commande des Systèmes, LR-11-ES20, BP 37, Le Belvedere 1002 Tunis, Tunisia email [email protected]
*
✉[email protected]
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Abstract

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We present a new iterative model order reduction method for large-scale linear time-invariant dynamical systems, based on a combined singular value decomposition–adaptive-order rational Arnoldi (SVD-AORA) approach. This method is an extension of the SVD-rational Krylov method. It is based on two-sided projections: the SVD side depends on the observability Gramian by the resolution of the Lyapunov equation, and the Krylov side is generated by the adaptive-order rational Arnoldi based on moment matching. The use of the SVD provides stability for the reduced system, and the use of the AORA method provides numerical efficiency and a relative lower computation complexity. The reduced model obtained is asymptotically stable and minimizes the error ($H_{2}$ and $H_{\infty }$) between the original and the reduced system. Two examples are given to study the performance of the proposed approach.

Keywords

model order reductionAORASVDGramianlarge scaleKrylovmoment matchingH∞
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 1 , July 2017 , pp. 115 - 133
Copyright
© 2017 Australian Mathematical Society 

References

Antoulas, A. C., Approximation of large-scale dynamical systems (Society for Industrial and Applied Mathematics, Philadelphia).Google Scholar
Chahlaoui, Y. and Dooren, P. V., “A collection of benchmark examples for model reduction of linear time invariant dynamical systems”, Niconet (2002)http://www.icm.tu-bs.de/NICONET/benchmodred.html.Google Scholar
Chahlaoui, Y. and Dooren, P. V., “Benchmark examples for model reduction of linear time-invariant dynamical systems”, Dimens. Reduction of Large-Scale Systems 45 (2005) 379–392;doi:10.1007/3-540-27909-1_24.Google Scholar
Chu, C. C., Lai, M. H. and Feng, W. S., “Model-order reductions for MIMO systems using global Krylov subspace methods”, Math. Comput. Simulation 79 (2008) 1153–1164;doi:10.1016/j.matcom.2007.09.007.Google Scholar
Druskin, V. and Simoncini, V., “Adaptive rational Krylov subspaces for large-scale dynamical systems”, Systems Control Lett. 60 (2011) 546–560; doi:10.1016/j.sysconle.2011.04.013.CrossRefGoogle Scholar
Frangos, M. and Jaimoukha, I. M., “Adaptive rational interpolation: Arnoldi and Lanczos-like equations”, Eur. J. Control 14 (2008) 342–354; doi:10.3166/ejc.14.342-354.Google Scholar
Gallivan, K., Grimme, E. and Dooren, P. V., “A rational Lanczos algorithm for model reduction”, Numer. Algorithms 12 (1996) 33–63; doi:10.1007/BF02141740.CrossRefGoogle Scholar
Grimme, E. J., Sorensen, D. C. and Dooren, P. V., “Model reduction of state space systems via an implicitly restarted Lanczos method”, Numer. Algorithms 12 (1995) 1–31;doi:10.1007/BF02141739.CrossRefGoogle Scholar
Gugercin, S., “An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems”, Linear Algebra Appl. 428 (2008) 1964–1986; doi:10.1016/j.laa.2007.10.041.Google Scholar
Gugercin, S., Sorensen, D. C. and Antoulas, A. C., “A modified low-rank Smith method for large-scale Lyapunov equations”, Numer. Algorithms 32 (2003) 27–55; doi:10.1023/A:1022205420182.CrossRefGoogle Scholar
Hashimoto, Y. and Nodera, T., “Inexact shift-invert Arnoldi method for evolution equations”, ANZIAM J. 58 (2016) E1–E27; doi:10.21914/anziamj.v58i0.10766.Google Scholar
Lee, H. J., Chu, C. C. and Feng, W. S., “An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems”, Linear Algebra Appl. 415 (2006) 235–261;doi:10.1016/j.laa.2004.10.011.Google Scholar
Mohamed, K., Mehdi, A. and Abdelkader, M., “Arnoldi model reduction for switched linear systems”, in: 5th International Conference Modeling, Simulation and Applied Optimization (Hammamet, Tunisia) (IEEE, Piscataway, NJ, 2013) 1–6; doi:10.1109/ICMSAO.2013.6552568.Google Scholar
Mohamed, K., Mehdi, A. and Abdelkader, M., “A survey of linear invariant time model reduction”, ICIC Express Letters, An International Journal of Research and Surveys 7 (2013) 909–916;http://www.ijicic.org/el-7(3)b.htm.Google Scholar
Mohamed, K., Mehdi, A. and Abdelkader, M., “Rational Arnoldi & adaptive order rational Arnoldi for switched linear systems”, Neural Parallel Sci. Comput. 22 (2014) 75–88;http://cat.inist.fr/?aModele=afficheN&cpsidt=28604453.Google Scholar
Mohamed, K., Mehdi, A. and Abdelkader, M., “Iterative dual rational Krylov and iterative SVD-dual rational Krylov model reduction for switched linear systems”, in: Complex system modelling and control through intelligent soft computations, Studies in Fuzziness and Soft Computing, Volume 319 (eds Zhu, Q. and Azar, A. T.), (Springer International Publishing, Switzerland, 2015) 407–435.Google Scholar
Mohamed, K., Mehdi, A. and Abdelkader, M., “SVD-AORA method for dynamic linear time invariant model order reduction”, IFAC-PapersOnLine 48(1) (2015) 695–696;doi:10.1016/j.ifacol.2015.05.175.CrossRefGoogle Scholar
Mohamed, K., Mehdi, A. and Abdelkader, M., “Arnoldi model reduction for switched linear systems”, Int. J. Oper. Res. 27 (2016) 85–112; doi:10.1504/IJOR.2016.078456.Google Scholar