Overrelaxation Iterative Algorithms For Solving The Riccati Equation In Markov Jump Systems

Markov jump system is a class of stochastic control system,since in the multi-mode control system the subsystems jump with Markov property between different states,so we can describe the jump process of the system using Markov model.When the state of the control system to jumps,in order to get the minimum cost function,usually required solving a coupled algebraic matrix Riccati equation,from which we can obtained the optimal control input in each state of the control system,and then get the minimum cost function.This paper presents two iterative algorithms for solving the above Riccati equation,and the main contents are as follows:First,the Riccati equation of the coupled algebraic matrix can be transformed into a decoupled algebraic matrix Lyapunov equation,and then we can get the super relaxation Lyapunov iterative algorithm with the help of latest estimation innovation and over relaxation iterative algorithm.If the initial iteration satisfies certain conditions,the convergence of the algorithm can be theoretically given by mathematical induction.The unique stabilizability positive definite solution of the coupled equation can be obtained by several iterations.Second,the super relaxation Lyapunov iteration algorithm,this paper gives an algorithm to obtained the suitable initial value.Third,the coupled algebraic matrix Riccati equation can also be written as an iterative form of the ordinary Riccati equation,and the iterative initial value can be chosen as zero matrix.Combining the mathematical induction and the comparison theorem of the algebraic Riccati equation,the iterative algorithm can be proved convergent theoretically.If the relaxation factor is chosen properly,the convergence precision and speed are similar to the Lyapunov iterative algorithm.Fourth,according to the above two kinds of over-relaxation method,by the MATLAB simulation,if the relaxation factor is suitable,the two algorithms can be accelerate the convergence speed.By selecting different relaxation factors several times and simulation,the optimal relaxation factor can be obtained.