International Journal of Engineering and Manufacturing (IJEM)
IJEM Vol. 2, No. 1, 29 Feb. 2012
Cover page and Table of Contents: PDF (size: 254KB)
A Parameter Free Iterative Method for Solving Projected Generalized Lyapunov Equations
Full Text (PDF, 254KB), PP.62-70
Views: 0 Downloads: 0
Author(s)
Yiqin Lin
1,*
Liping Zhou
1
Liang Bao
2
Index Terms
Projected generalized Lyapunov equation, ADI method, Parameter free method, C-stable
Abstract
This paper is devoted to the numerical solution of projected generalized continuous-time Lyapunov equations with low-rank right-hand sides. Such equations arise in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. A parameter free iterative method is proposed. This method is based upon a combination of an approximate power method and a generalized ADI method. Numerical experiments presented in this paper show the effectiveness of the proposed method.
Cite This Paper
Yiqin Lin,Liping Zhou,Liang Bao,"A Parameter Free Iterative Method for Solving Projected Generalized Lyapunov Equations", IJEM, vol.2, no.1, pp.62-70, 2012. DOI: 10.5815/ijem.2012.01.10
Reference
[1] T. Stykel, “Stability and inertia theorems for generalized Lyapunov equations,” Linear Algebra Appl., vol. 355, pp. 297–314, 2002.
[2] F. Gantmacher, Theory of Matrices. New York: Chelsea, 1959.
[3] B. Anderson and J. Moore, Optimal Control-Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice-Hall, 1990.
[4] V. Mehrmann, The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution, ser. Lecture Notes in Control and Information Sciences. Heidelberg: Springer, 1991, vol. 163
[5] T. Stykel, “Numerical solution and perturbation theory for generalized Lyapunov equations,” Linear Algebra Appl., vol. 349, pp. 155–185, 2002.
[6] T. Stykel, “Low-rank iterative methods for projected generalized Lyapunov equations,” Elect. Trans. Numer. Anal., vol. 30, pp. 187–202, 2008.
[7] T. Penzl, “A cyclic low-rank Smith method for large sparse Lyapunov equations,” SIAM J. Sci. Comput., vol. 21, pp. 1401–1418, 1999/2000.
[8] P. Benner, H. Mena, and J. Saak, “On the parameter selection problem in the Newton-ADI iteration for large-scale Riccati equations,” Elect. Trans. Numer. Anal., vol. 29, pp. 136–149, 2008.
[9] R. Nong and D. C. Sorensen, “A parameter free adi-like method for the numerical solution of large scale lyapunov equations,” 2009, preprint.
[10] A. Hodel, B. Tenison, and K. Poolla, “Numerical solution of the Lyapunov equation by approximate power iteration,” Linear Algebra Appl., vol. 236, pp. 205–230, 1996.
[11] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. Baltimore, MD: John Hopkins University Press, 1996.
[12] Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput., vol. 7, pp. 856–869, 1986.
[13] D. Vasilyev and J. White, “A more reliable reduction algorithm for behavior model extraction,” in Proc. Int. Conf. on Computer. Aided Design (ICCAD 2005), 2005, pp. 812–819.