Iterative Algorithms For Solving Lyapunov Matrix Equations In It? Stochastic Systems?

Lyapunov matrix equations play a vital role in the stability, observability and 2H analysis of stochastic systems. For example, the Lyapunov matrix equations are closely related to the mean square stability of the systems, that is to say if there exists a unique positive definite solution of the equation, the system will be mean square stable. Here, iterative algorithms for solving 3 kinds of Lyapunov matrix equations in It? stochastic systems are proposed. The main parts of this thesis are shown as below.Two algorithms for solving the coupled Lyapunov matrix equations related to continuous-time It? stochastic system with Markovian jumps are presented. Based on the latest updated information, the first algorithm introduces a series of tuning parameters to adjust the effect of the latest updated information and the estimation in the last step. This combination is used to update the estimation of the unknown matrices. The second algorithm utilizes the idea of gradient search, and iterates by minimizing a quadratic objective function.One explicit iterative algorithm is presented to solve the coupled Lyapunov matrix equations related to discrete-time It? stochastic system with Markovian jumps. Analogously based on the latest updated information, this algorithm is a kind of fixed-point iterative algorithm using the special form of the discrete-time Lyapunov matrix equation to calculate the estimation of unknown matrices in each step.One implicit round iterative algorithm for solving the Lyapunov matrix equations related to discrete-time It? stochastic systems is proposed. Utilizing the special form of the equation, this algorithm isolates a term from the accumulation item containing s items and constructs a standard Lyapunov equation in order. There are s standard discrete-time Lyapunov matrix equations needed to be solved in each step. And each equation regards the estimation updated in the last one as a known matrix. The weighting sum of all the estimation in each step is the final solution of the current step.For the algorithms with zero initial conditions, the boundedness and monotonicity of the sequences generated are given and proven. While the algorithms with non-zero initial conditions, the sufficient and necessary conditions are also provided. The numerical examples are given to show their effectiveness and difference.