Projection-based model reduction for time-varying descriptor systems: New results

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Deflation by restriction for the inverse-free preconditioned Krylov subspace method

2016, 6(1): 73-90. doi: 10.3934/naco.2016.6.73

Projection-based model reduction for time-varying descriptor systems: New results

Mohammad-Sahadet Hossain ^1,

1.	Department of Mathematics and Physics, North South University, Dhaka, Bangladesh

Received July 2015 Revised January 2016 Published January 2016

Abstract
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We have presented a Krylov-based projection method for model reduction of linear time-varying descriptor systems in [13] which was based on earlier ideas in the work of J. Philips [17] and others. This contribution continues that work by presenting more details of linear time-varying descriptor systems and new results coming from real fields of application. The idea behind the proposed procedure is based on a multipoint rational approximation of the monodromy matrix of the corresponding differential-algebraic equation. This is realized by orthogonal projection onto a rational Krylov subspace. The algorithmic realization of the method employs recycling techniques for shifted Krylov subspaces and their invariance properties. The proposed method works efficiently for macro-models, such as time varying circuit systems and models arising in network interconnection, on limited frequency ranges. Bode plots and step response are used to illustrate both the performance and accuracy of the reduced-order model.

Keywords: recycled Krylov subspaces, model-order reduction., linear time-varying systems, time-domain discretization, Krylov subspace techniques.

Mathematics Subject Classification: Primary: 37N30, 37N35, 37Mxx, 78M34, 93C83; Secondary: 93C9.

Citation: Mohammad-Sahadet Hossain. Projection-based model reduction for time-varying descriptor systems: New results. Numerical Algebra, Control and Optimization, 2016, 6 (1) : 73-90. doi: 10.3934/naco.2016.6.73

References:

[1]	Z. Bai, Krylov subspace techniques for reduced-order modeling for large-scale dynamical system, App. Numer. Math, 43(1-2) (2002), 9-44. doi: 10.1016/S0168-9274(02)00116-2.
[2]	P. Benner, Solving large-scale control problems, IEEE Control System Magazine, 24 (2004), 44-59.
[3]	P. Benner, Numerical linear algebra for model reduction in control and simulation, GAMM Mitteilungen, 23 (2006), 275-296. doi: 10.1002/gamm.201490034.
[4]	S. Bittanti and P. Colaneri, Periodic Systems: Filtering and Control, 1st edition, Springer-Verlag, London, 2009.
[5]	H. G. Brachtendorf, Theorie und Analyse von Autonomen und Quasiperiodisch Angeregten Elektrischen Netzwerken. Eine Algorithmisch Orientierte Betrachtung, Habilitation thesis, University of Bremen, Germany, 2001.
[6]	R. L. Burden and J. D. Faires, Numerical Analysis, Ninth edition, Brooks/Cole, Boston, USA, 2011.
[7]	I. Elfadel and D. D. Ling, A block Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks, in IEEE/ACM International Conference on Computer-Aided Design, IEEE, (1997), 66-71.
[8]	R. Freund, Model reduction methods based on Krylov subspaces, Acta Numerica, 12 (2003), 267-319. doi: 10.1017/S0962492902000120.
[9]	G. Golub and C. Van Loan, Matrix Computations, 3rd edition, Johns Hopkins University Press, Baltimore, 1996.
[10]	E. J. Grimme, Kryloy projection methods for model reduction, Ph.D. thesis, University of Illinois at Urbana, Champaign, 1997.
[11]	S. Gugercin, An iterative SVD-KRYLOV based method for model reduction of large-scale dynamical systems, in 44'th IEEE Conference on Decision and Control and the European control conference, IEEE, 2005, 5905-5910.
[12]	S. Gugercin, A. C. Antoulas and C. Beattie, H₂ model reduction for large-scale linear dynamical systems, SIAM J. Matrix Anal. Appl, 30 (2008), 609-638. doi: 10.1137/060666123.
[13]	M.-S. Hossain and P. Benner, Projection-based model reduction for time-varying descriptor systems using recycled Krylov subspaces, in Appllied Mathematics and Mechanics, WILEY-VCH Verlag, 2008, 10081-10084.
[14]	M. Nakhla and E. Gad, Efficient model reduction of linear time-varying systems via compressed transient system function, in Conference on Design, automation and test in Europe, IEEE, 2002, 916-922.
[15]	M. Nakhla and E. Gad, Efficient model reduction of linear periodically time-varying systems via compressed transient system function, IEEE Transactions on Circuit and Systems, 52 (2005), 1188-1204. doi: 10.1109/TCSI.2005.846661.
[16]	T. Penzl, Algorithms for model reduction of large dynamical systems, Linear Algebra Appl., 415 (2006), 322-343. doi: 10.1016/j.laa.2006.01.007.
[17]	J. Phillips, Model reduction of time-varying linear systems using multipoint Krylov-subspace projectors, in International Conference on Computer-Aided Design, ACM, 1998, 96-102.
[18]	J. Phillips, Projection-based approaches for model reduction of weakly nonlinear time-varying systems, IEEE Trans. Computer-Aided Design, 22 (2003), 171-187.
[19]	A. Rahman and M. S. Hossain, Svd-Krylov based model reduction for time-varying periodic descriptor systems, in 2nd International Conference on Electrical Engineering and Information Technology, IEEE, 2015.
[20]	J. Roychowdhury, Reduced-order modeling of time-varying systems, IEEE Control Systems Magazine, 46 (1999), 1273-1288.
[21]	J. Roychowdhury, Reduced-order modelling of linear time-varying systems, in ASP-DAC '99. Asia and South Pacific, IEEE, 1999, 53-56.
[22]	Y. Saad, Overview of Krylov subspace methods with applications to control problems, in International Symposium MTNS-89 on Signal Processing, Scattering and Operator Theory, and Numerical Methods, Birkhauser Verlag AG, 1990.
[23]	Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.
[24]	B. Salimbahrami, B. Lohmann, T. Bechtold and J. Korvink, Two-sided Arnoldi algorithm and its application in order reduction of MEMS, in 4th Fourth International Conference on Mathematical Modelling (eds. I. Troch and F. Breitenecker), Vienna, 2003, 1021-1028.
[25]	S. B. Salimbahrami, Structure Preserving Order Reduction of Large Scale Second Order Models, Ph.D. thesis, Technische Universität München, Fakultät für Maschinenwesen, Germany, 2005.
[26]	R. E. Skelton, M. Oliveira and J. Han, System modeling and model reduction,, Paper available from: , ().
[27]	T. Stykel, Low-rank iterative methods for projected generalized Lyapunov equations, Electron. Trans. Numer. Anal., 30 (2008), 187-202.
[28]	R. Telichevesky, J. White and K. Kundert, Efficient steady-state analysis based on matrix-free Krylov-subspace methods, in 32rd Design Automation Conference, IEEE, 1995, 480-484.
[29]	R. Telichevesky, J. White and K. Kundert, Efficient AC and noise analysis of two-tone RF circuits, in 33rd annual Design Automation Conference, IEEE, 1996, 292-297.
[30]	A. A. Vaidyanathan, Multirate digital filters, filters banks, polyphase networks, and applications: A tutorial, in IEEE Proceedings, IEEE, 1990, 56-93.
[31]	E. Wachspress, The ADI Model Problem,, 1995, ().
[32]	B. Yang and D. Feng, Efficient finite-difference method for quasi-periodic steady-state and small signal analyses, in IEEE/ACM International Conference on Computer-Aided Design, IEEE, 2000, 272-276.
[33]	L. Zadeh, Frequency analysis of variable networks, IEEE Transactions on Circuits and Systems, 38 (1950), 291-299.

show all references

References:

[1]	Z. Bai, Krylov subspace techniques for reduced-order modeling for large-scale dynamical system, App. Numer. Math, 43(1-2) (2002), 9-44. doi: 10.1016/S0168-9274(02)00116-2.
[2]	P. Benner, Solving large-scale control problems, IEEE Control System Magazine, 24 (2004), 44-59.
[3]	P. Benner, Numerical linear algebra for model reduction in control and simulation, GAMM Mitteilungen, 23 (2006), 275-296. doi: 10.1002/gamm.201490034.
[4]	S. Bittanti and P. Colaneri, Periodic Systems: Filtering and Control, 1st edition, Springer-Verlag, London, 2009.
[5]	H. G. Brachtendorf, Theorie und Analyse von Autonomen und Quasiperiodisch Angeregten Elektrischen Netzwerken. Eine Algorithmisch Orientierte Betrachtung, Habilitation thesis, University of Bremen, Germany, 2001.
[6]	R. L. Burden and J. D. Faires, Numerical Analysis, Ninth edition, Brooks/Cole, Boston, USA, 2011.
[7]	I. Elfadel and D. D. Ling, A block Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks, in IEEE/ACM International Conference on Computer-Aided Design, IEEE, (1997), 66-71.
[8]	R. Freund, Model reduction methods based on Krylov subspaces, Acta Numerica, 12 (2003), 267-319. doi: 10.1017/S0962492902000120.
[9]	G. Golub and C. Van Loan, Matrix Computations, 3rd edition, Johns Hopkins University Press, Baltimore, 1996.
[10]	E. J. Grimme, Kryloy projection methods for model reduction, Ph.D. thesis, University of Illinois at Urbana, Champaign, 1997.
[11]	S. Gugercin, An iterative SVD-KRYLOV based method for model reduction of large-scale dynamical systems, in 44'th IEEE Conference on Decision and Control and the European control conference, IEEE, 2005, 5905-5910.
[12]	S. Gugercin, A. C. Antoulas and C. Beattie, H₂ model reduction for large-scale linear dynamical systems, SIAM J. Matrix Anal. Appl, 30 (2008), 609-638. doi: 10.1137/060666123.
[13]	M.-S. Hossain and P. Benner, Projection-based model reduction for time-varying descriptor systems using recycled Krylov subspaces, in Appllied Mathematics and Mechanics, WILEY-VCH Verlag, 2008, 10081-10084.
[14]	M. Nakhla and E. Gad, Efficient model reduction of linear time-varying systems via compressed transient system function, in Conference on Design, automation and test in Europe, IEEE, 2002, 916-922.
[15]	M. Nakhla and E. Gad, Efficient model reduction of linear periodically time-varying systems via compressed transient system function, IEEE Transactions on Circuit and Systems, 52 (2005), 1188-1204. doi: 10.1109/TCSI.2005.846661.
[16]	T. Penzl, Algorithms for model reduction of large dynamical systems, Linear Algebra Appl., 415 (2006), 322-343. doi: 10.1016/j.laa.2006.01.007.
[17]	J. Phillips, Model reduction of time-varying linear systems using multipoint Krylov-subspace projectors, in International Conference on Computer-Aided Design, ACM, 1998, 96-102.
[18]	J. Phillips, Projection-based approaches for model reduction of weakly nonlinear time-varying systems, IEEE Trans. Computer-Aided Design, 22 (2003), 171-187.
[19]	A. Rahman and M. S. Hossain, Svd-Krylov based model reduction for time-varying periodic descriptor systems, in 2nd International Conference on Electrical Engineering and Information Technology, IEEE, 2015.
[20]	J. Roychowdhury, Reduced-order modeling of time-varying systems, IEEE Control Systems Magazine, 46 (1999), 1273-1288.
[21]	J. Roychowdhury, Reduced-order modelling of linear time-varying systems, in ASP-DAC '99. Asia and South Pacific, IEEE, 1999, 53-56.
[22]	Y. Saad, Overview of Krylov subspace methods with applications to control problems, in International Symposium MTNS-89 on Signal Processing, Scattering and Operator Theory, and Numerical Methods, Birkhauser Verlag AG, 1990.
[23]	Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.
[24]	B. Salimbahrami, B. Lohmann, T. Bechtold and J. Korvink, Two-sided Arnoldi algorithm and its application in order reduction of MEMS, in 4th Fourth International Conference on Mathematical Modelling (eds. I. Troch and F. Breitenecker), Vienna, 2003, 1021-1028.
[25]	S. B. Salimbahrami, Structure Preserving Order Reduction of Large Scale Second Order Models, Ph.D. thesis, Technische Universität München, Fakultät für Maschinenwesen, Germany, 2005.
[26]	R. E. Skelton, M. Oliveira and J. Han, System modeling and model reduction,, Paper available from: , ().
[27]	T. Stykel, Low-rank iterative methods for projected generalized Lyapunov equations, Electron. Trans. Numer. Anal., 30 (2008), 187-202.
[28]	R. Telichevesky, J. White and K. Kundert, Efficient steady-state analysis based on matrix-free Krylov-subspace methods, in 32rd Design Automation Conference, IEEE, 1995, 480-484.
[29]	R. Telichevesky, J. White and K. Kundert, Efficient AC and noise analysis of two-tone RF circuits, in 33rd annual Design Automation Conference, IEEE, 1996, 292-297.
[30]	A. A. Vaidyanathan, Multirate digital filters, filters banks, polyphase networks, and applications: A tutorial, in IEEE Proceedings, IEEE, 1990, 56-93.
[31]	E. Wachspress, The ADI Model Problem,, 1995, ().
[32]	B. Yang and D. Feng, Efficient finite-difference method for quasi-periodic steady-state and small signal analyses, in IEEE/ACM International Conference on Computer-Aided Design, IEEE, 2000, 272-276.
[33]	L. Zadeh, Frequency analysis of variable networks, IEEE Transactions on Circuits and Systems, 38 (1950), 291-299.

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