Iterative Algorithms For Continuous Markovian Jump Lyapunov Equations And Their Convergence Analysis

If the states in a dynamic system can be represented by a countable set of models,and the relationship between the models can be represented by a Markov chain,then this system can be described by a Markov jump system model.The Markov jump systems are widely used in various fields,therefore it is of great significance to study the Markov jump systems.The solution of the Lyapunov equation plays a crucial role in the stability analysis of Markov jump system,control system design and analysis such as boundedness analysis,and optimal and robust controllers and filter design.Therefore,solving the Lyapunov equations has been a hot research direction and growing researchers attached importance to them.This thesis studies the iterative algorithms for coupled Lyapunov matrix equation corresponding to Markov jump system and the convergence conditions of these algorithms,and analyzes the applicable range and convergence performance of these algorithms.In order to study the multi-parameter forward implicit iterative algorithm,the iterative algorithm based on weighted latest estimation and the iterative algorithm with multi-step estimation are analyzed respectively.The necessary conditions and necessary and sufficient conditions for the convergence of these two algorithms under arbitrary initial conditions are given.The relationship between the parameters and the necessary and sufficient conditions in the two algorithms is described by polynomial form respectively.It is found that the convergence performance of the two algorithms depends on the tunable parameters in the two algorithms by numerical simulation.Choosing the appropriate parameters can make the convergence rate of the two algorithms faster than the convergence rate of the traditional algorithm.Moreover,numerical simulations show that the newer iterative information can improve the convergence performance of the iterative algorithm.The methods to select optimal tunable parameter in the iterative algorithm based on weighted latest estimation and the iterative algorithm with multi-step estimation are given,and the convergence performance of the two algorithms is compared.According to the necessary and sufficient conditions,the spectral radius expressions of the corresponding matrix under all eigenvalues are obtained,and the expressions of the optimal parameters such that the two algorithms have the fastest convergence rate are obtained.The consistent spectral radius expressions of the two algorithms prove that the convergence performance is theoretically consistent.Moreover,numerical simulation can be used to find the approximation of the optimal parameters.It is proved that the convergence performance between the two algorithms is consistent under certain conditions by numerical simulation.