Iterative Algorithms For Solving Periodic Discrete-time LYAPUNOV Equations And Periodic Discrete-time RICCATI Equations

As the simplest form of time-varying linear systems,periodic linear systems have been attracting much attention during the past several decades.This is partially because this type of systems has very wide application.To investigate the stabilization problem of the periodic linear systems,it is important to achieve the solution of the periodic Lyapunov matrix equation.Similarly,the design of linear quadratic optimal state feedback controller based on the robust control is related to the stabilizing positive definite solution of Riccati equation.Based on this research background,we propose iterative algorithms for solving discrete-time periodic Lyapunov matrix equation and discrete-time periodic Riccati matrix equation.Iterative algorithms for discrete periodic Lyapunov equations are derived,respectively to the zero initial conditions and arbitrary initial conditions.And the proof of convergence is given.The effectiveness of the algorithm is verified by numerical simulation.And the latest information estimation theory is into the iterative algorithm,the proof of the convergence is also given.The validity of the algorithm is verified by numerical simulations.Finally,the simulation analysis of the two algorithms find that the convergence rate of the iterative algorithm based on the estimation of the latest information is faster than the original algorithm.It proves the superiority of the iterative algorithm adding the latest information of the estimation.Iterative algorithm for discrete periodic Riccati equations is derived,given the zero initial condition of convergence,and the effectiveness of the algorithm is verified through numerical simulation.In order to improve the algorithm with the latest estimate information,a new iterative algorithm based on the information of the latest estimation is given.The convergence of the new algorithm is proved and the validity of the algorithm is verified by numerical simulation.Through numerical simulation,the convergence curves of different weighting factors are given.It found that using the latest estimate information,the convergence speed is the fastest.Therefore,adding the latest estimation information can effectively improve the convergence speed of iterative algorithm.