An implicit sequential algorithm for solving coupled Lyapunov equations of continuous-time Markovian jump systems

In this paper, an implicit sequential algorithm is presented for solving coupled Lyapunov matrix equations of continuous-time Markovian jump linear systems. First, some existing iterative algorithms which can be utilized to solve the coupled Lyapunov matrix equations are reviewed and discussed. Next, based on the existing parallel iterative algorithm, an implicit sequential algorithm is proposed by using the latest updated information. The proposed algorithm fills the current gap of implicit algorithms for solving continuous coupled Lyapunov matrix equations. It is shown that the proposed algorithm with zero initial conditions can monotonically converge to the unique positive definite solutions of the coupled Lyapunov matrix equations if the associated Markovian jump system is stochastically stable. Moreover, a necessary and sufficient condition is established for the proposed algorithm to be convergent. The algorithm presented in this paper has much better convergence performance than other existing iterative algorithms and requires less storage capacity. Finally, a numerical example is given to show the effectiveness of the proposed algorithm.