Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations

Abstract

We discuss the Smith iteration for solving large-scale sparse projected discrete-time periodic Lyapunov equations which arise in periodic state feedback problems and in model reduction of periodic descriptor systems. Two algorithms are presented in this paper. The first one works with the cyclic lifted representation of the corresponding projected discrete-time periodic Lyapunov equations. In this algorithm, the block diagonal structure of the periodic solution is preserved in every iteration step by efficient permutations. The second algorithm works directly with the periodic matrix coefficients. We analyze the cyclic structure of the matrices arising in the iterative computations of the periodic solutions of the projected discretetime periodic Lyapunov equations. A low-rank version of this method is also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Iterative solvers for periodic matrix equations and model reduction for periodic control systems

“…It can be verified that each factor inside (18) preserves the block diagonal structure analogous to the solution of (7) [3]. SinceX =X ν , andX ν =R νR T ν , whereR ν = diag(R 1 , .…”

“…. , ν, in the computation of (14) where the permutation matrix P i changes at each iteration step in a cyclic manner by a forward block-row shift [3]. One nice property of this permutation matrix is that it satisfies the periodicity property, i.e., P K+k = P k , k = 0, 1, .…”

Convergence analysis of the MCGNR algorithm for the least squares solution group of discrete-time periodic coupled matrix equations

“…with periodic matrices (Chu et al, 1995; Coll et al, 2004; Varga, 2007). To solve large-scale sparse projected discrete-time periodic Lyapunov equations, Benner and Hossain discussed the Smith iteration (Benner and Hossain, 2012). Granat et al (2006) extended and applied the recursive blocking technique for solving the periodic Sylvester matrix equations.…”

Extending the CGLS method for finding the least squares solutions of general discrete-time periodic matrix equations

“…The linear periodic systems are one of important topics in engineering [27][28][29]. In the last decades of the past century the discrete-time periodic matrix equations have been used as a main tool of analysis and design problems involving periodic systems [1,5,6,[8][9][10]27].…”