Convergence of gradient-based iterative solution of coupled Markovian jump Lyapunov equations

The solution of coupled discrete-time Markovian jump Lyapunov matrix equations (CDMJLMEs) is important in stability analysis and controller design for Markovian jump linear systems. This paper presents a simple and effective iterative method to produce numerical solutions to this class of matrix equations. The gradient-based algorithm is developed from an optimization point of view. A necessary and sufficient condition guaranteeing the convergence of the algorithm is established. This condition shows that the algorithm always converges provided the CDMJLMEs have unique solutions which is evidently different from the existing results that converge conditionally. A simple sufficient condition which is easy to test is also provided. The optimal step size in the algorithm such that the convergence rate of the algorithm is maximized is given explicitly. It turns out that an upper bound of the convergence rate is bounded by a function of the condition number of the augmented coefficient matrix of the CDMJLMEs. Some parameters are introduced to the algorithm that will potentially reduce the condition number and thus increase the convergence rate of the algorithm. A numerical example is used to illustrate the efficiency of the proposed approach.