Adaptive rational block Arnoldi methods for model reductions in large-scale MIMO dynamical systems

Abstract

In recent years, a great interest has been shown towards Krylov subspace techniques applied to model order reduction of large-scale dynamical systems. A special interest has been devoted to single-input single-output (SISO) systems by using moment matching techniques based on Arnoldi or Lanczos algorithms. In this paper, we consider multiple-input multiple-output (MIMO) dynamical systems and introduce the rational block Arnoldi process to design low order dynamical systems that are close in some sense to the original MIMO dynamical system. Rational Krylov subspace methods are based on the choice of suitable shifts that are selected a priori or adaptively. In this paper, we propose an adaptive selection of those shifts and show the efficiency of this approach in our numerical tests. We also give some new block Arnoldi-like relations that are used to propose an upper bound for the norm of the error on the transfer function.

Keywords

• Dynamical systems,model reduction,rational block Krylov subspaces

References

1. Z. Bai, Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems, Appl. Numer. Math. 43 (2002) 944.
2. Z. Bai, Q. Ye, Error estimation of the Pad´e approximation of transfer functions via the Lanczos process, Elect. Trans. Numer. Anal. 7 (1998) 1-17.
3. P. Benner, R.C. Li, N. Truhar, On the ADI method for Sylvester equations, J. Comput. Appl. Math., 233 (2009) 1035–1045.
4. B. N. Datta, Large-Scale Matrix computations in Control, Appl. Numer. Math. 30 (1999) 53-63.
5. B. N. Datta, Krylov Subspace Methods for Large-Scale Matrix Problems in Control, Future Gener. Comput. Syst. 19(7) (2003) 1253-1263.
6. V. Druskin, V. Simoncini, Adaptive rational Krylov subspaces for large-scale dynamical systems, Systems Control Lett. 60(8)(2011) 546-560.
7. V. Druskin, C. Lieberman, M. Zaslavsky, On adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problems, SIAM J. Sci. Comput. 32(5) (2010) 2485-2496.
8. K. Gallivan, E. Grimme, P. Van Dooren, A rational Lanczos algorithm for model reduction, Numer. Alg. 12 (1996) 3363.

9. K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L-infinity error bounds. Inter. J. Cont. 39 (1984) 1115-1193.
10. K. Glover, D. J. N. Limebeer, J. C. Doyle, E. M. Kasenally, M. G. Safonov, A characterisation of all solutions to the four block general distance problem, SIAM J. Control Optim., 29:283–324, (1991).
11. E. Grimme,, Krylov projection methods for model reduction , Ph.D. Thesis, The University of Illinois at Urbana-Champaign. (1997).
12. E. Grimme, D. Sorensen and P. Van Dooren, Model reduction of state space systems via an implicitly restarted Lanczos method, Numer. Alg. 12 (1996) 1-32.
13. S. Gugercin, A.C. Antoulas, C. Beattie,H2 model reduction for large-scale linear dynamical systems, SIAMJ.Matrix Anal. Appl. 30 (2008) 609-638.
14. M. Heyouni, K. Jbilou, An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation, Elect. Trans. Num. Anal., 33(2009) 53–62.
15. M. Heyouni, K. Jbilou,Matrix Krylov subspace methods for large scale model reduction problems, App.Math. Comput., 181(2006) 1215–1228.
16. A. Bouhamidi, M. Hached and K. Jbilou, A preconditioned block Arnoldi for large Sylvester matrix equations, Numer. Lin. Alg. Appl., 2 (2013) 208-219.
17. C. Jagels, L. Reichel, The extended Krylov subspace method and orthogonal Laurent polynomials, Lin. Alg. Appl., 431(2009), 441-458.
18. M. Frangos, I.M. Jaimoukha, Adaptive rational interpolation: Arnoldi and Lanczos-like equations, Eur. J. Control., 14(4) (2008) 342-354.
19. I. M. Jaimoukha, E. M. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Matrix Anal. Appl., 31(1) (1994) 227-251.
20. L. Knizhnerman and V. Druskin and M. Zaslavsky, On optimal convergence rate of the rational Krylov subspace reduction for electromagnetic problems in unbounded domains, SIAM J. Numer. Anal., 47(2) (2009) 953–971.
21. B. C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Automatic Contr., AC-26 (1981) 17-32.
22. K. Henrik A. Olsson and A. Ruhe, Axel, Rational Krylov for eigenvalue computation and model order reduction, BIT. 46 (2006) S99-S111.
23. T. Penzl, LYAPACK A MATLAB toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problem, and Linearquadratic Optimal Control Problems, http://www.tu-chemnitz.de/sfb393/lyapack
24. T. Penzl, LYAPACK A MATLAB toolbox for Large Lyapunov and Riccati Equations, Model Reduction Problem, and Linearquadratic Optimal Control Problems, http://www.tu-hemnitz.de/sfb393/lyapack/guide.pdf
25. T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput. 21 (1999) 1064-8275.
26. A. Ruhe, Rational Krylov sequence methods for eigenvalue computation, Lin. Alg. Appl., 58 (1984) 391-405.
27. A. Ruhe, The rational Krylov algorithm for nonsymmetric eigenvalue problems. III. Complex shifts for real matrices, BIT. 34(1) (1994) 165-176.
28. Y. Saad, Iterative Methods for Sparse Linear Systems, The PWS Publishing Company. (1996).
29. Y. Shamash, Stable reduced-order models using Pad´e type approximations, IEEE. Trans. Automatic Control. AC-19 (1974) 615-616.
30. V. Simoncini, A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comp., 29(3):1268–1288, 2007.
31. E.L. Wachspress, Iterative solution of elliptic systems, and applications to the neutron diffusion equations of reactor physics, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1966) xiv+299.
32. E.L. Wachspress, The ADI minimax problem for complex spectra, Academic Press, Boston, MA in Iterative Methods for Large Linear Systems, (1990) 251-271.