An SOR-type algorithm based on IO iteration for solving coupled discrete Markovian jump Lyapunov equations

Abstract

In this paper, based on the inner-outer (IO) iteration framework [17], by introducing some tunable parameters, an SOR-type IO (SIO) iteration method is proposed for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) in the discrete-time jump linear systems with Markovian transitions. Fisrtly, the SIO iteration algorithm for solving the discrete Sylvester matrix equation is developed, its convergence property is analyzed and the choices of the parameters are also discussed. Next, the SIO iteration algorithm is used to solve the CLMEs. Moreover, by using the latest estimations, a current-estimation-based SIO (CSIO)
iteration algorithms are also constructed for solving the CLMEs, respectively. The boundedness and monotonicity of the iteration sequence derived from the proposed algorithm with zero initial conditions are established. Finally, several numerical examples are implemented to illustrate the superiorities of the proposed iteration algorithms.