An iterative algorithm for discrete periodic Lyapunov matrix equations

Abstract In this paper, a novel iterative algorithm with a tuning parameter is developed to solve the forward discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. An important feature of the proposed algorithm is that the information in the current and the last steps is used to update the iterative sequence. The convergence rate of the algorithm can be significantly improved by choosing a proper tuning parameter. It is shown that the sequence generated by this algorithm with zero initial conditions monotonically converges to the unique positive definite solution of the periodic Lyapunov matrix equation if the tuning parameter is within the interval ( 0 , 1 ] . In addition, a necessary and sufficient convergence condition is given for the proposed algorithm in terms of the roots of a set of polynomial equations. Also, a method to choose the optimal parameter is developed such that the algorithm has the fastest convergence rate. Finally, numerical examples are provided to illustrate the effectiveness of the proposed algorithm.

[1]  Musheng Wei,et al.  Iterative algorithms for solving the matrix equation AXB+CXTD=E , 2007, Appl. Math. Comput..

[2]  Feng Ding,et al.  Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations , 2005, IEEE Trans. Autom. Control..

[3]  I. Borno,et al.  Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems , 1995 .

[4]  Sergio Bittanti,et al.  The simple pendulum and the periodic LQC control problem , 1991 .

[5]  I. Borno.  Parallel computation of the solutions of coupled algebraic Lyapunov equations , 1995, Autom..

[6]  Feng Ding,et al.  Iterative least-squares solutions of coupled Sylvester matrix equations , 2005, Syst. Control. Lett..

[7]  Guang-Ren Duan,et al.  Gradient based iterative algorithm for solving coupled matrix equations , 2009, Syst. Control. Lett..

[8]  A. Varga.  Periodic Lyapunov equations: Some applications and new algorithms , 1997 .

[9]  J. Bibby.  Axiomatisations of the average and a further generalisation of monotonic sequences , 1974, Glasgow Mathematical Journal.

[10]  Ai-Guo Wu,et al.  New Iterative Algorithms for Solving Coupled Markovian Jump Lyapunov Equations , 2015, IEEE Transactions on Automatic Control.

[11]  Ai-Guo Wu,et al.  Current-estimation-based iterative algorithms for solving periodic Lyapunov matrix equations , 2016 .

[12]  Huamin Zhang,et al.  Reduced-rank gradient-based algorithms for generalized coupled Sylvester matrix equations and its applications , 2015, Comput. Math. Appl..