Accelerated Gradient Iterative Algorithms For Solving Coupled Lyapunov Matrix Equations Of Markovian Jump Systems

The Markov jump system can describe the dynamic behavior of the system due to changes in the external environment or changes in the internal structure,and therefore has a strong application background.Stability is important in such systems.The solution of the coupled Lyapunov matrix equation is closely related to the stability determination of the Markov jump system.In this paper,the gradient iterative algorithm for solving the coupled Lyapunov matrix equation of is studied.For the coupled Lyapunov matrix equation of the discrete time Markov jump system,an accelerated gradient iterative algorithm based on the latest estimation is established by introducing the latest estimated information into the iteration of the current step.The necessary and sufficient condition for the proposed algorithm to converge under arbitrary initial conditions is given.Then the Kronecker product and the auxiliary matrix are used to prove that the proposed algorithm can converge to the exact solution under arbitrary initial conditions if there is a unique solution to the matrix equation.Finally,the numerical simulation proves that the convergence speed of the gradient iterative algorithm based on the latest estimation is significantly faster than that of the direct gradient iterative algorithm under the non-zero initial condition and the zero initial condition.The optimal step length of the proposed algorithm is given by simulation.For the coupled Lyapunov matrix equation of the continuous-time Markov jump system,by minimizing the value of the quadratic objective function,,a direct gradient iterative algorithm for the continuous coupled Lyapunov matrix equation is established using the gradient search idea.Based on this,the latest estimation idea is used.An iterative algorithm based on the latest estimation is established,and the necessary and sufficient conditions for convergence of the two algorithms are given.By straightening the two algorithms and introducing the auxiliary matrix and vector forms,it is proved that the proposed algorithms can converge to the exact solution under any initial conditions if there is a unique solution to the matrix equation.Numerical simulation shows that the convergence speed of the two algorithms varies greatly under different iterative steps under non-zero and zero initial conditions,but the overall convergence speed of the accelerated gradient algorithm based on the latest estimation is significantly better than that of the direct gradient iterative algorithm.Finally,the optimal step length is obtained by simulating the spectral radius of the corresponding iterative matrix for the two algorithms.