Recursive Smith-type iterative algorithm to solve a class of periodic Lyapunov equations arising in periodic model reduction

Periodic control systems are of interest in many engineering and mechanical research. Many important analysis of periodic control systems directly links to the periodic matrix equations, i.e., periodic Lyapunov equations or periodic Riccati equations. This paper represents the iterative methods for solving a class of periodic Lyapunov equation, known as periodic projected discrete-time Lyapunov equation. A remarkable contribution of these types of equations are seen in control problems with periodic setting, and also in dimension reduction of periodic systems in descriptor forms. We explore the Smith iterations for the iterative solution of the projected discrete-time algebraic Lyapunov equation and analyze the cyclic and repeated structure of the periodic matrices to ensure the recursive computation of the periodic solutions. We also introduce an algorithm for computing the low-rank approximations of those iterative solutions. Computational results are illustrated at the end to report the effciency of the suggested methods