Large periodic Lyapunov equations: Algorithms and applications

Abstract

Two algorithms for the solution of discrete-time periodic Lyapunov equations are presented. The first one is a variant of the squared Smith iteration, which is solely based on matrix multiplications and thus attractive to parallel computing environments. The second algorithm is based on Krylov subspaces and employs a recently developed variant of the block Arnoldi algorithm. It is particularly suited for periodic Lyapunov equations with large and sparse coefficient matrices. We also demonstrate how these methods can be applied to balanced truncation model reduction of periodic discrete-time systems and the solution of periodic Riccati equations.

Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

“…, K − 1, the Gramians of the periodic standard system can also be determined by solving the periodic standard Lyapunov equations in the lifted form. The solutions of these lifted equations are block diagonal matrices with the required Gramians on the diagonal [13,29]. This result can also be extended to periodic descriptor systems.…”

“…All these methods have also been generalized to projected Lyapunov equations [27,28]. On the other hand, an extension of the Smith method and the Krylov subspace method based on a block Arnoldi algorithm to standard periodic Lyapunov equations has been presented in [13]. Unfortunately, these methods cannot be directly applied to the projected periodic Lyapunov equations.…”

Gradient Based Iterative Algorithm to Solve General Coupled Discrete-Time Periodic Matrix Equations Over Generalized Reflexive Matrices

“…Based on Smith iterations [24], iterative methods were developed for periodic standard Lyapunov matrix equations and projected generalized Lyapunov matrix equations [27,28]. Kressner introduced new variants of the squared Smith iteration and Krylov subspace based methods for the approximate solution of discrete-time periodic Lyapunov matrix equations [20]. In [17], Granat et al presented novel recursive blocked algorithms for solving various periodic triangular matrix equations.…”

Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations

“…This contribution builds upon Section-4 in (Benner et al (2011a)), where the concept for preserving the block diagonal structure of the computed solution at each iteration step is based on (Kressner (2003)). …”

“…All these methods have also been generalized to projected Lyapunov equations (Stykel (2008); Stykel and Simoncini (2012)). On the other hand, an extension of the Smith method and the Krylov subspace method based on a block Arnoldi algorithm to standard periodic Lyapunov equations has been presented in (Kressner (2003)). Unfortunately, these methods cannot be directly applied to the projected periodic Lyapunov equations.…”