Solution of Positive Periodic Discrete Lyapunov Equations with Applications to the Balancing of Periodic Systems

The discrete-time positive periodic Lyapunov equations have important applications in the balancing and potentially also in the model reduction of discrete-time periodic systems. Efficient numerically reliable algorithms based on periodic Schur decomposition are proposed for the solution of these equations. The proposed algorithms are extensions of the method of Hammarling for the case of positive semidefinite solution. Special methods were developed to solve efficiently small order periodic Lyapunov and Sylvester equations. 1 Introduction In the last few years there has been a constantly increasing interest for the development of numerical algorithms for the analysis and design of linear periodic discrete-time control systems [2, 8, 10, 12] of the form x k+1 = A k x k +B k u k y k = C k x k +D k u k (1) where the matrices A k 2 R n\Thetan , B k 2 R n\Thetam , C k 2 R p\Thetan and D k 2 R p\Thetam are periodic with period K 1. Of particular interest in many applications is..