A Finite Iterative Method for Solving the General Coupled Discrete-Time Periodic Matrix Equations

Analysis and design of linear periodic control systems are closely related to the discrete-time periodic matrix equations. In this paper, we propose an iterative algorithm based on the conjugate gradient method on the normal equations (CGNE) for finding the solution group of the general coupled periodic matrix equations {A(1), (i) X-i B-1,B- i + C-1,C- i Xi+1D1, i = E-1,E- i, A(2, i) X-i B-2,B- i + C-2,C- i Xi+1D2, i = E-2,E- i, for i = 1, 2, 3, ... By proving some properties of the algorithm, we show that the solution group of the periodic matrix equations can be obtained within a finite number of iterations in the absence of roundoff errors. Numerical examples are given to illustrate the efficiency and accuracy of the proposed algorithm.