Fast Iterative Algorithms For Solving Discrete-time Lyapunov Equations

Lyapunov matrix equations play a vital role in modern control theory.For example,in the stability analysis of discrete-time linear systems,we can solve the discrete-time Lyapunov matrix equation,determining the system stable whether or not according to whether it has unique positive definite solution.Iterative algorithm is an efficient method for solving the approximate solution of Lyapunov matrix equation,and the approximate solution can approach to the unique positive definite solution of the Lyapunov matrix equation.In this paper,an explicit iterative algorithm with tuning parameters is proposed for solving the discrete-time Lyapunov matrix equation related to the linear discrete time system.The main parts of this thesis are shown as follows:An explicit iterative algorithm with tuning parameters for solving the discrete-time Lyapunov matrix equations related to discrete-time linear system is presented.The algorithm introduces tuning parameters and thus introduces the updated information in the previous steps and the estimation in the last step.To some content,the convergence rate of the algorithm can be accelerated when the iterative estimation information is used more thoroughly.The boundedness and monotonicity of the sequences generated by the proposed iterative algorithm are given and proved under zero initial conditions by mathematical induction,it is proved that the iterative sequences are strictly monotone increasing and take the real solution of the equation as the upper bound,which illustrate the convergence of the iterative sequences.Numerical example is employed to show the effectiveness of the presented algorithm under zero initial conditions.According to the matrix Kronecker products and straightening operation,the matrix equation can be transformed into linear system of equations.Then an necessary and sufficient condition for the convergence of the proposed algorithm under non-zero condition is given by the important conclusions of the iteration of linear equations.Moreover,another necessary and sufficient condition is gained by the roots set of the polynomial equations.The ranges of the parameters in the proposed algorithm are given by using Jury criterion.Numerical example is employed to show the effectiveness of the presented algorithm under non-zero conditions and the numerical simulation results show that the convergence rate of the algorithm can be affected by the choice of different initial values.A mathematical technique to choose the optimal parameter for the proposed algorithm is given based on the roots of polynomial equation.Explicit expressions of the optimal parameters are given for Lyapunov matrix equations related to the unique systems,which makes the algorithm achieves to the fastest convergence rate and also the minimum spectral radius can be given by this method.Numerical examples are employed to verify that the optimal parameter can make the proposed algorithm has the fastest convergence rate.