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Model reduction of state space systems via an implicitly restarted Lanczos method. (English) Zbl 0870.65052

This paper uses a modified Lanczos method to derive a stable reduced order model for a single input-single output system described by the state space equation \(dx/dt =Ax +bu\), \(y=cx +du\). The basic assumption is that the matrix \(A\) is large, sparse and stable. The modification of the method lies in the fact that one uses oblique Krylov projectors to produce the model. The main interest of this approach is that the algorithm so obtained involves only inner-products and matrix-vector multiplications.

MSC:

65K10 Numerical optimization and variational techniques
93A99 General systems theory
93C15 Control/observation systems governed by ordinary differential equations
65F15 Numerical computation of eigenvalues and eigenvectors of matrices

Software:

IRAM; eigs; Algorithm 432
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Full Text: DOI

References:

[1] M.M.M. Al-Husari, B. Hendel, I.M. Jaimoukha, E.M. Kasenally, D.J.N. Limebeer and A. Portone, Vertical stabilisation of Tokamak plasmas,Proc. 30th IEEE Conf. on Decision and Control, Brighton, England (1991).
[2] G.A. Baker Jr.,Essentials of Padé Approximants (Academic Press, New York, 1975).
[3] R.H. Bartels and G.W. Stewart, Solution of the matrix equationAX+XB=C, Commun. ACM 15 (1972) 820–826. · Zbl 1372.65121
[4] D.L. Boley, Krylov space methods on state-space control models, Technical Report, Univ. of Minnesota, Minneapolis, MN 55455 (1992). · Zbl 0794.68151
[5] O.H. Bosgra, G. Schoolstra and M. Steinbuch, Robust control of a compact disc player, in:Proc. 31st IEEE Conf. on Decision and Control, Tucson, AZ (1992). · Zbl 0775.93116
[6] M.A. Brebner and J. Grad, Eigenvalues ofAx={\(\lambda\)}Bx for real symmetric matricesA andB computed by reduction to a pseudosymmetric form and theHR process, Lin. Alg. Appl. 43 (1982) 99–118. · Zbl 0485.65029
[7] A. Bunse-Gerstner, An analysis of theHR algorithm for computing the eigenvalues of a matrix, Lin. Alg. Appl. 35 (1981) 155–173. · Zbl 0462.65021
[8] A Bunse-Gerstner, On the similarity transformation to tridiagonal form, Lin. Multilin. Alg. 12 (1982) 51–56. · Zbl 0486.15005
[9] C.I. Byrnes and A. Lindquist, The stability and instability of partial realizations, Syst. Contr. Lett. 2 (1982) 99–105. · Zbl 0494.93009
[10] D. Calvetti, L. Reichel and D.C. Sorensen, An implicitly restarted Lanczos method for large symmetric eigenvalue problems, Electron. Trans. Numer. Anal. 2 (1994) 1–21. · Zbl 0809.65030
[11] J. Daniel, W.B. Gragg, L. Kaufman and G.W. Stewart, Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Math. Comp. 30 (1976) 772–795. · Zbl 0345.65021
[12] C. Davis and W.M. Kahan, The rotation of eigenvectors by a perturbation III, SIAM J. Numer. Anal. 7 (1970) 1–46. · Zbl 0198.47201
[13] R.W. Freund, G.H. Golub and N.M. Nachtigal, Iterative solution of linear systems, Acta Numer. 1 (1992) 57–100. · Zbl 0762.65019
[14] R.W. Freund and N.M. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math. 60 (1991) 315–339. · Zbl 0754.65034
[15] K. Gallivan, E. Grimme and P. Van Dooren, Asymptotic waveform evaluation via a Lanczos method, Appl. Math. Lett. (1994), to appear. · Zbl 0810.65067
[16] W.B. Gragg and A. Lindquist, On the partial realization problem, Lin. Alg. Appl. 50 (1983) 277–319. · Zbl 0519.93024
[17] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and theirL error bounds. Int. J. Contr. 39 (1984) 1115–1193. · Zbl 0543.93036
[18] G.H. Golub and C. Van Loan,Matrix Computations, 2nd ed. (Johns Hopkins University Press, Baltimore, MD, 1989). · Zbl 0733.65016
[19] E. Grimme, D. Sorensen and P. Van Dooren, Stable partial realizations via an implicitly restarted Lanczos method,Proc. Amer. Control Conf., Baltimore, MD (1994). · Zbl 0870.65052
[20] M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms, part I, SIAM J. Matrix Anal. Appl. 13 (1992) 594–639. · Zbl 0760.65039
[21] M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms, part II, SIAM J. Matrix Anal. Appl. 15 (1994) 15–58. · Zbl 0809.65028
[22] A.S. Householder,The Numerical Treatment of a Single Nonlinear Equation (McGraw-Hill, New York, 1970). · Zbl 0242.65047
[23] X. Huang, A survey of Padé approximations and their applications in model reduction, Technical Report, Carnegie Mellon Univ., Pittsburgh, PA 15213 (1990).
[24] I.M. Jaimoukha and E.M. Kasenally, Oblique projection methods for large scale model reduction, SIAM J. Matrix Anal. Appl. (1994), to appear. · Zbl 0827.65073
[25] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bureau Stand. 45 (1950) 255–282.
[26] B.C. Moore, Principal component analysis in linear systems: controllability, observability and model reduction, IEEE Trans. Auto. Contr. AC-26 (1981) 17–32. · Zbl 0464.93022
[27] C.C. Paige, The computation of eigenvalues and eigenvectors of very large sparse matrices, Ph.D. Dissertation, University of London, UK (1971).
[28] B.N. Parlett, Reduction to tridiagonal form and minimal realizations, SIAM J. Matrix Anal. Appl. 13 (1992) 567–593. · Zbl 0754.65040
[29] V. Raghavan R.A. Rohrer, L.T. Pillage, J.Y. Lee, J.E. Bracken and M.M. Alaybeyi, AWE-inspired,Proc. IEEE Custom Integrated Circuits Conf. (1993).
[30] Y. Shamash, Stable reduced-order models using Padé type approximations, IEEE Trans. Auto. Contr. AC-19 (1974) 615–616. · Zbl 0291.93019
[31] Y. Shamash, Model reduction using the Routh stability criterion and the Padé approximation technique, Int. J. Contr. 21 (1975) 475–484. · Zbl 0299.93034
[32] D.C. Sorensen, Implicit application of polynomial filters in aK-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992) 357–385. · Zbl 0763.65025
[33] T.J. Su and R.R. Craig Jr., Model reduction and control of flexible structures using Krylov vectors, J. Guidance, Contr. Dynamics 14 (1991) 260–267.
[34] T.J. Su and R.R. Craig Jr., An unsymmetric Lanczos algorithm for damped structural dynamics systems,Proc. 33rd Conf. on Structures, Structural Dynamics and Materials (1992).
[35] P. Van Dooren, Numerical linear algebra techniques for large scale matrix problems in systems and control,Proc. 31st IEEE Conf. on Decision and Control Tucson, AZ (1992).
[36] C.D. Villemagne and R.E. Skelton, Model reduction using a projection formulation, Int. J. Contr. 46 (1987) 2141–2169.
[37] J.H. Wilkinson,The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, England, 1965). · Zbl 0258.65037
[38] H. Xiheng, FF-Padé method of model reduction in frequency domain, IEEE Trans. Auto. Contr. AC-32 (1987) 243–246. · Zbl 0607.93006
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