Iterative Model Reduction of Large State-Space Systems

Huhtanen, M.

International Journal of Applied Mathematics and Computer Science, Tome 9 (1999) no. 2, pp. 245-263.

Résumé

There exist criteria for reducing the order of a large state-space model based on the accuracy of the approximate solutions to the Lyapunov matrix equations and the Hankel operator. Iterative solution techniques for the Lyapunov equations with the Arnoldi method have been proposed in a number of papers. In this paper we derive error bounds for approximations to the solutions to the Lyapunov equations as well as for the Hankel operator that indicate how to precondition while solving these equations iteratively.These bounds show that the error depends on three terms: First, on the amount of invariance of the constructed subspace for A, second, on the eigenvalues of A at least in proportion to 1/|Re l|, and third, under a certain condition on projectors P_l=W_lW_l* ,on the factor min_X in C^l x p|| B-( l I-A)W_lX|| for l on a path G surrounding the spectrum of A. Consequently, in order to compensate for those parts of the spectrum where 1/|Re l| is not small, preconditioning or an inverse iteration is needed to keep the sizes of the matrices used in construction of a reduced-order model moderate.

Détail

Keywords: model reduction, iterative methods, Lyapunov matrix equations, Hankel operator, preconditioning, Hankel singular values
Mots-clés : redukcja modeli, metoda iteracyjna, równanie macierzowe Lyapunova, kondycjonowanie, operator Hankela