Model reduction of state space systems via an implicitly restarted Lanczos method

The nonsymmetric Lanczos method has recently received significant attention as a model reduction technique for large-scale systems. Unfortunately, the Lanczos method may produce an unstable partial realization for a given, stable system. To remedy this situation, unexpensive implicit restarts are developed which can be employed to stabilize the Lanczos generated model.

[1]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[2]  Christopher C. Paige,et al.  The computation of eigenvalues and eigenvectors of very large sparse matrices , 1971 .

[3]  A. Portone,et al.  Vertical stabilisation of Tokamak plasmas , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[4]  M. A. Brebner,et al.  Eigenvalues of Ax = λBx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process , 1982 .

[5]  A. Bunse-Gerstner.  An analysis of the HR algorithm for computing the eigenvalues of a matrix , 1981 .

[6]  Martin H. Gutknecht,et al.  A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms, Part I , 1992, SIAM J. Matrix Anal. Appl..

[7]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[8]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[9]  Imad M. Jaimoukha,et al.  Oblique Production Methods for Large Scale Model Reduction , 1995, SIAM J. Matrix Anal. Appl..

[10]  Daniel Boley.  Krylov space methods on state-space control models , 1994 .

[11]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[12]  Roy R. Craig,et al.  An unsymmetric Lanczos algorithm for damped structural dynamics systems , 1992 .

[13]  Anders Lindquist,et al.  The stability and instability of partial realizations , 1982 .

[14]  Gene H. Golub,et al.  Matrix computations , 1983 .

[15]  W. Gragg,et al.  On the partial realization problem , 1983 .

[16]  Robert Skelton,et al.  Model reductions using a projection formulation , 1987, 26th IEEE Conference on Decision and Control.

[17]  G. A. Baker.  Essentials of Padé approximants , 1975 .

[18]  A. Householder.  The numerical treatment of a single nonlinear equation , 1970 .

[19]  P. Dooren,et al.  Asymptotic Waveform Evaluation via a Lanczos Method , 1994 .

[20]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[21]  Y. Shamash.  Stable reduced-order models using Padé-type approximations , 1974 .

[22]  Angelika Bunse-Gerstner,et al.  On the similarity transformation to tridiagonal form , 1982 .

[23]  Xiheng Hu,et al.  FF-Padé method of model reduction in frequency domain , 1987 .

[24]  M. Gutknecht.  A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms. Part II , 1994, SIAM J. Matrix Anal. Appl..

[25]  G. Golub,et al.  Iterative solution of linear systems , 1991, Acta Numerica.

[26]  E. Grimme,et al.  Stable partial realizations via an implicitly restarted Lanczos method , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[27]  N. Nachtigal.  A look-ahead variant of the Lanczos algorithm and its application to the quasi-minimal residual method for non-Hermitian linear systems. Ph.D. Thesis - Massachusetts Inst. of Technology, Aug. 1991 , 1991 .

[28]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[29]  M Maarten Steinbuch,et al.  Robust control of a compact disc player , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[30]  Beresford N. Parlett,et al.  Reduction to Tridiagonal Form and Minimal Realizations , 1992, SIAM J. Matrix Anal. Appl..

[31]  R. Craig,et al.  Model reduction and control of flexible structures using Krylov vectors , 1991 .

[32]  D. Calvetti,et al.  AN IMPLICITLY RESTARTED LANCZOS METHOD FOR LARGE SYMMETRIC EIGENVALUE PROBLEMS , 1994 .

[33]  G. Stewart,et al.  Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization , 1976 .

[34]  W. Kahan,et al.  The Rotation of Eigenvectors by a Perturbation. III , 1970 .

[35]  V. Krishnamurthy,et al.  Model reduction using the Routh stability criterion , 1978 .

[36]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[37]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[38]  Tongxing Lu,et al.  Solution of the matrix equation AX−XB=C , 2005, Computing.