Periodic conjugate direction algorithm for symmetric periodic solutions of general coupled periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The conjugate direction (CD) method is a famous iterative algorithm to find the solution to nonsymmetric linear systems $Ax=b$. In this work, a new method based on the CD method is proposed for computing the symmetric periodic solutions $(X_1,X_2,\dots ,X_{\lambda })$ and $(Y_1,Y_2,\dots ,Y_{\lambda })$ of general coupled periodic matrix equations

for $i=1,2,\dots ,\lambda$. The key idea of the scheme is to extend the CD method by means of Kronecker product and vectorization operator. In order to assess the convergence properties of the method, some theoretical results are given. Finally two numerical examples are included to illustrate the efficiency and effectiveness of the method.