Structure Preserving Iterative Solution of Periodic Projected Lyapunov Equations

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 discrete-time 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.