Model-order reduction of large-scale second-order MIMO dynamical systems via a block second-order Arnoldi method

In this paper, we present a structure-preserving model-order reduction method for solving large-scale second-order MIMO dynamical systems. It is a projection method based on a block second-order Krylov subspace. We use the block second-order Arnoldi (BSOAR) method to generate an orthonormal basis of the projection subspace. The reduced system preserves the second-order structure of the original system. Some theoretical results are given. Numerical experiments report the effectiveness of this method.

[1]  Zhaojun Bai,et al.  A projection method for model reduction of bilinear dynamical systems , 2006 .

[2]  Thilo Penzl.  Algorithms for model reduction of large dynamical systems , 2006 .

[3]  Jing-Rebecca Li.  Model reduction of large linear systems via low rank system gramians , 2000 .

[4]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[5]  D. G. Meyer,et al.  Balancing and model reduction for second-order form linear systems , 1996, IEEE Trans. Autom. Control..

[6]  Jacob K. White,et al.  Reduction of large circuit models via low rank system gramians , 2001 .

[7]  Andras Varga,et al.  Preface Special issue on “Order reduction of large-scale systems” , 2006 .

[8]  Yunkai Zhou.  Numerical methods for large scale matrix equations with applications in LTI system model reduction , 2002 .

[9]  R. Freund.  Krylov Subspaces Associated with Higher-Order Linear Dynamical Systems , 2005, math/0501484.

[10]  R. Freund.  Krylov-subspace methods for reduced-order modeling in circuit simulation , 2000 .

[11]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

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

[13]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[14]  R. Freund.  Model reduction methods based on Krylov subspaces , 2003, Acta Numerica.

[15]  T. Stykel.  Analysis and Numerical Solution of Generalized Lyapunov Equations , 2002 .

[16]  Mark J. Balas,et al.  Trends in large space structure control theory: Fondest hopes, wildest dreams , 1982 .

[17]  D Ramaswamy,et al.  Automatic Generation of Small-Signal Dynamic Macromodels from 3-D Simulation , 2001 .

[18]  A. Preumont.  Vibration Control of Active Structures , 1997 .

[19]  R. Aldhaheri.  Model order reduction via real Schur-form decomposition , 1991 .

[20]  Zhaojun Bai,et al.  Reduced-Order Modeling , 2005 .

[21]  Liang Bao,et al.  Block second-order Krylov subspace methods for large-scale quadratic eigenvalue problems , 2006, Appl. Math. Comput..

[22]  R. D. Slone,et al.  Fast frequency sweep model order reduction of polynomial matrix equations resulting from finite element discretizations , 2002 .

[23]  William Weaver,et al.  Structural dynamics by finite elements , 1987 .

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

[25]  Rolf Schuhmann,et al.  Two-step Lanczos algorithm for model order reduction , 2002 .

[26]  Noël Tanguy,et al.  Laguerre-Gram reduced-order modeling , 2005, IEEE Transactions on Automatic Control.

[27]  Zhaojun Bai,et al.  Dimension Reduction of Large-Scale Second-Order Dynamical Systems via a Second-Order Arnoldi Method , 2005, SIAM J. Sci. Comput..

[28]  James Demmel,et al.  New Numerical Techniques and Tools in SUGAR for 3D MEMS Simulation , 2001 .

[29]  Roy R. Craig,et al.  Structural Dynamics: An Introduction to Computer Methods , 1981 .

[30]  Paul Van Dooren,et al.  A collection of benchmark examples for model reduction of linear time invariant dynamical systems. , 2002 .

[31]  R. Freund.  Padé-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems , 2004, math/0410195.

[32]  Ren-Cang Li,et al.  Structure-Preserving Model Reduction Using a Krylov Subspace Projection Formulation , 2005 .

[33]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[34]  Paul Van Dooren,et al.  Model Reduction of MIMO Systems via Tangential Interpolation , 2005, SIAM J. Matrix Anal. Appl..

[35]  D. Sorensen,et al.  A Survey of Model Reduction Methods for Large-Scale Systems , 2000 .

[36]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control) , 2005 .

[37]  Peter Benner,et al.  Dimension Reduction of Large-Scale Systems , 2005 .