Weighted Iterative Algorithms For The Discrete Periodic Lyapunov Matrix Equations

The discrete-time periodic Lyapunov matrix equations play a vital role in the analysis and design of the discrete-time linear periodic systems.For example,the discrete-time periodic Lyapunov matrix equations can be used to test the controllability and observability of the discrete-time linear periodic systems.The discrete-time periodic Lyapunov matrix equations are the key to calculate the minimum realization of the linear periodic systems.The asymptotic stability of the discrete-time linear periodic systems can be determined by the existence of unique positive definite solution of the corresponding discrete-time periodic Lyapunov matrix equations.Therefore,it is necessary to solve the discrete-time periodic Lyapunov matrix equations quickly,accurately and simply.In this paper,weighted iterative algorithms are proposed for the forward and backward discrete-time periodic Lyapunov matrix equations corresponding to the discrete-time linear periodic systems.When the parameter is appropriate,the proposed iterative algorithms can approach the unique positive definite solution of the equations faster.The specific research contents of this paper are as follows:A weighted iterative algorithm is proposed for the forward discrete-time periodic Lyapunov matrix equations in this paper.One of the important features of the algorithm is the constant distortion of the algorithm by adding the adjustable parameter,and the latest estimation is added to make the application of iterative information more thorough,which can significantly improve the convergence speed of the algorithm.Under the zero initial condition,the series of the solution produced by the algorithm are proved to be bounded and monotonically increasing by mathematical induction,and the upper bound is the real solution of the equations.By using vector operator and Kronecker product,the discrete-time periodic Lyapunov matrix equations are transformed into linear equations which makes the equation solving easier.The convergence of the solution sequences produced by the algorithm ais verified and the conditions of convergence are given.In order to evaluate the performance of the proposed algorithm and the existing algorithms,we compare the calculation time of different algorithms for proving that the proposed algorithm has better convergence performance.With the same idea of the forward discrete-time periodic Lyapunov matrix equations,a weighted iterative algorithm for the backward discrete-time Lyapunov matrix equations is presented.Similarly,some properties of the solution sequences produced by the algorithm are discussed,and the convergence conditions of the algorithm are given in this paper.Finally,the algorithm is validated by numerical simulation.Based on the existing results,this paper presents two kinds of multi-parameter iterative algorithms for forward and backward equations by introducing multiple adjustable parameters to optimize the two algorithms previously proposed.In the same way,this paper analyses the properties of the solution sequences produced by multi-parameter iterative algorithms and gives the conditions of convergence.Finally,numerical simulation is carried out to verify the effectiveness of multi-parameter iterative algorithms.