Abstract
For the coupled Lyapunov problem derived from continuous Markov jump systems, a new implicit single-step splitting(ISS) iterative algorithm is proposed based on the idea of single step split iteration. After that, the accelerated implicit single-step splitting(AISS) iterative method, which is an accelerated algorithm of ISS iterative method, is proposed by using the idea of two-step alternate iteration. In this paper, the relevant convergence proofs of ISS and AISS iteration methods are given. Then, the selection range of the parameters of ISS and AISS iteration methods and the selection method of optimal parameters are given. Finally, this paper demonstrates the feasibility and advantage of the novel methods through actual calculations.
Highlights
IntroductionResearchers are focusing on coupled Lyapunov matrix equations(CLMEs)
All along, researchers are focusing on coupled Lyapunov matrix equations(CLMEs)
Inspired by the above methods, and due to the particularity of CLMEs matrix, this paper proposes an implicit single step split iteration (ISS) method to solve the non-Hermitian negative definite CLMEs related to continuous Markov jump systems, and proposes an accelerated ISS(AISS) iteration method to make ISS iterative method converge faster
Summary
Researchers are focusing on coupled Lyapunov matrix equations(CLMEs). The main reason is that by studying the coupled Lyapunov equations, for Markov jump systems, we can more conveniently conduct stability analysis and design of it. Because finding the only positive definite solution of the Lyapunov equation is equivalent to the stochastic stability analysis of the Markov jump system [8], [9], many scholars have proposed many effective solutions for coupled Lyapunov equations. In [10], an explicit direct algorithm for solving continuous CLMEs is given by applying Kronecker product and matrix vectorization. Because the algorithm uses Kronecker product, this method is not suitable for solving large problems. Recursive algorithms for continuous and discrete CLMEs are proposed in [11] and [12], respectively.
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