An iterative algorithm for discrete periodic Lyapunov matrix equations

In this paper, a novel iterative algorithm with a tuning parameter is developed to solve the forward discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. An important feature of the proposed algorithm is that the information in the current and the last steps is used to update the iterative sequence. The convergence rate of the algorithm can be significantly improved by choosing a proper tuning parameter. It is shown that the sequence generated by this algorithm with zero initial conditions monotonically converges to the unique positive definite solution of the periodic Lyapunov matrix equation if the tuning parameter is within the interval (0,1]. In addition, a necessary and sufficient convergence condition is given for the proposed algorithm in terms of the roots of a set of polynomial equations. Also, a method to choose the optimal parameter is developed such that the algorithm has the fastest convergence rate. Finally, numerical examples are provided to illustrate the effectiveness of the proposed algorithm.