Matrix Krylov subspace methods for large scale model reduction problems

The present paper considers matrix Krylov subspace methods for solving large coupled Lyapunov matrix equations of the form AP + PAT + BBT = 0 and ATQ + QA + CTC = 0 where A is a real n × n matrix, B and CT are real n × s and n × r matrices, respectively, with s ≪ n and r ≪ n. Such problems appear in many areas of control theory such as the computation of the controllability and observability Gramians of a stable Linear Time Invariant (LTI) system. The proposed methods are based on the global Arnoldi and Lanczos processes. In the second part, we show how to use matrix Krylov subspace techniques to obtain a reduced order model for LTI systems. This will be done by approximating the corresponding transfer functions. Finally, numerical experiments are reported to illustrate the behavior and the effectiveness of the global Arnoldi and Lanczos processes when applied to solve some large coupled Lyapunov equations and to approximate some transfer functions.