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A model reduction method in large scale dynamical systems using an extended-rational block Arnoldi method

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Abstract

In this paper, we propose a new block Krylov-type subspace method for model reduction in large scale dynamical systems. We project the initial problem onto a new subspace, generated as a combination of rational and polynomial block Krylov subspaces. Simple algebraic properties are given and expressions of the error between the original and reduced transfer functions are established. Furthermore, we present an adaptive strategy of the interpolation points that will be used in the construction of our new block Krylov subspace. We also show how this method can be used to extract an approximate low rank solution of large-scale Lyapunov equations. Numerical results are reported on some benchmark examples to confirm the performance of our method compared with other known methods.

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Notes

  1. http://slicot.org/20-site/126-benchmark-examples-for-model-reduction.

  2. https://www.mathworks.com/matlabcentral/fileexchange/59169-sssmor-toolbox.

References

  1. Abidi, O., Jbilou, K.: Balanced truncation-rational Krylov methods for model reduction in large scale dynamical systems. Comput. Appl. Math. 37(1), 525–540 (2018)

    Article  MathSciNet  Google Scholar 

  2. Bai, Z., Ye, O.: Error estimation of the Padé approximation of transfer functions via the Lanczos process. Electron. Trans. Numer. Anal. 7, 1–17 (1998)

    MathSciNet  MATH  Google Scholar 

  3. Barkouki, H., Bentbib, A.H., Jbilou, K.: An adaptive rational block Lanczos-type algorithm for model reduction of large scale dynamical systems. SIAM J. Sci. Comput. 67, 221–236 (2015)

    Article  MathSciNet  Google Scholar 

  4. Barkouki, H., Bentbib, A.H., Jbilou, K.: An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systems. Calcolo 55(1), 1–23 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bartels, R.H., Stewart, G.W.: Solution of the matrix equation \( AX+XB=C \). Commun. ACM 15, 820–826 (1972)

    Article  Google Scholar 

  6. Beattie, C.A., Gugercin, S.: Krylov-based minimization for optimal \({{\cal{H}}} _2\) model reduction. In Proceedings of the 46th IEEE Conference on Decision and Control, pp 4385–4390 (2007)

  7. El Guennouni, A., Jbilou, K., Riquet, A.J.: Krylov-subspace methods for the Sylvester equation. Linear Algebra Appl. 172, 283–313 (1992)

    Article  MathSciNet  Google Scholar 

  8. Beattie, C.A., Gugercin, S.: Krylov-based model reduction of second-order systems with proportional damping. In Proceedings of the \(44^{th}\) IEEE Conference on Decision and Control, pp. 2278–2283 (2005)

  9. Bentbib, A.H., Jbilou, K., Kaouane, Y.: A computational global tangential Krylov subspace method for model reduction of large-scale MIMO dynamical systems. J. Sci. Comput. 59, 1–19 (2017)

    MATH  Google Scholar 

  10. Benner, P., Li, J.R., Penzl, T.: Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems. Numer. Linear Algebra Appl. 15, 755–777 (2008)

    Article  MathSciNet  Google Scholar 

  11. Chahlaoui, Y., Van Dooren, P.: A collection of benchmark examples for model reduction of linear time invariant dynamical systems, Part of the Lecture Notes in imension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, (45), 379–392 (2008)

  12. Druskin, D., Lieberman, C., Zaslavsky, M.: On adaptive choice of shifts in Rational Krylov subspace reduction of evolutionary problems. SIAM J. Sci. Comput. 32(5), 2485–2496 (2010)

    Article  MathSciNet  Google Scholar 

  13. Druskin, D., Simoncini, V.: Adaptive Rational Krylov subspaces for large-scale dynamical systems. Syst. Control Lett. 60(8), 546–560 (2011)

    Article  MathSciNet  Google Scholar 

  14. Frangos, M., Jaimoukha, I.M.: Adaptive Rational Krylov algorithms for model reduction. In European Control Conference (ECC). Kos pp. 4179–4186, (2007)

  15. Frangos, M., Jaimoukha, I.M.: Adaptive Rational interpolation: Arnoldi and Lanczos-like equations. Eur. J. Control 14(4), 342–354 (2008)

    Article  MathSciNet  Google Scholar 

  16. Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L-infinity error bounds. Int. J. Control 39, 1115–1193 (1984)

    Article  Google Scholar 

  17. Glover, K., Limebeer, J.N., Doyle, J.C., Kasenally, M., Safonov, M.G.: A characterization of all solutions to the four block general distance problem. SIAM J. Control Optim. 29, 283–324 (1991)

    Article  MathSciNet  Google Scholar 

  18. Grimme, E.: Krylov projection methods for model reduction. University of Illinois at Urbana Champaign, Coordinated Science Laboratory (1997).. (Ph.D. thesis)

  19. Grimme, E., Sorensen, D., Van Dooren, P.: Model reduction of state space systems via an implicitly restarted Lanczos method. Numer. Algorithms 12, 1–32 (1996)

    Article  MathSciNet  Google Scholar 

  20. Gugercin, S., Antoulas, A.C.: A survey of model reduction by balanced truncation and some new results. Int. J. Control 77(8), 748–766 (2003)

    Article  MathSciNet  Google Scholar 

  21. Gugercin, S., Antoulas, A.C.: \({{\cal{H}}} _2\) model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30, 609–638 (2008)

    Article  MathSciNet  Google Scholar 

  22. Gugercin, S., Antoulas, A.C., Bedrossian, N.: Approximation of the international space station 1R and 12A models. In: Proceedings of the 40th IEEE Conference on Decision and Control, vol. 2, pp. 1515–1516 (2001)

  23. Heyouni, M., Jbilou, K., Messaoudi, A., Tabaa, T.: Model reduction in large scale MIMO dynamical systems via the block Lanczos method. Comput. Appl. Math. 27(2), 211–236 (2008)

    Article  MathSciNet  Google Scholar 

  24. Jbilou, K.: Low rank approximate solutions to large Sylvester matrix equations. Appl. Math. Comput. 177(1), 365–376 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Kaouane, Y.: A tangential method for the balanced truncation in model reduction. Numer. Algorithms 83, 629–652 (2020)

    Article  MathSciNet  Google Scholar 

  26. Knizhnerman, L., Druskin, D., Zaslavsky, M.: On optimal convergence rate of the rational Krylov subspace reduction for electromagnetic problems in unbounded domains. SIAM J. Numer. Anal. 47(2), 953–971 (2009)

    Article  MathSciNet  Google Scholar 

  27. Merhrmann, V., Penzl, T.: Benchmark collections in SLICOT, Technical Report SLWN 1998–5, SLICOT Working Note, ESAT, KU Leuven, K. Mercierlaan 94, Leuven-Heverlee 3100, Belgium, (1998)

  28. Penzl, T.: LYAPACK A MATLAB toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problem, and Linear-quadratic Optimal Control Problems, (2000) http://www.tu-chemnitz.de/sfb393/lyapack

  29. Ruhe, A.: A rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl. 58, 391–405 (1984)

    Article  MathSciNet  Google Scholar 

  30. Ruhe, A.: The rational Krylov algorithm for nonsymmetric eigenvalue problems III: Complex shifts for real matrices. BIT 34(1), 165–176 (1994)

    Article  MathSciNet  Google Scholar 

  31. Shamash, Y.: Stable reduced-order models using Padé type approximations. IEEE. Trans. Autom. Control AC 19, 615–616 (1974)

    Article  Google Scholar 

  32. Simocini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29(3), 1268–1288 (2007)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to express their appreciations to the Reviewers for their insightful comments and suggestions that lead to an improved paper.

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Authors and Affiliations

  1. Laboratory of Modeling Simulation and Data Analysis, Mohammed VI Polytechnic University, Hay Moulay Rachid, 43150, Ben Guerir, Morocco

    M. A. Hamadi, K. Jbilou & A. Ratnani

  2. Department of Mathematics LMPA, ULCO, 50 rue F. Buisson, Calais, 62100, France

    M. A. Hamadi & K. Jbilou

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  1. M. A. Hamadi
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  2. K. Jbilou
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  3. A. Ratnani
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Correspondence to M. A. Hamadi.

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Hamadi, M.A., Jbilou, K. & Ratnani, A. A model reduction method in large scale dynamical systems using an extended-rational block Arnoldi method. J. Appl. Math. Comput. 68, 271–293 (2022). https://doi.org/10.1007/s12190-021-01521-0

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