Decay estimates of Green & apos;s matrices for discrete-time linear periodic systems

We study periodic Lyapunov matrix equations for a general discrete-time linear periodic system $ B_px_p-A_px_{p-1}=f_p $ Bpxp-Apxp-1=fp, where the matrix coefficients $ B_p $ Bp and $ A_p $ Ap can be singular. The block coefficients of the inverse operator of the system are referred to as the Green matrices. We derive new decay estimates of the Green matrices in terms of the spectral norms of special solutions to the periodic Lyapunov matrix equations. The study is based on the periodic Schur decomposition of matrices.

Automated summary

This paper investigates the periodic Lyapunov matrix equations for a general discrete-time linear periodic system, where the matrix coefficients can be singular. New decay estimates of the Green matrices are derived in terms of the spectral norms of special solutions to the periodic Lyapunov matrix equations, based on the periodic Schur decomposition of matrices. The results are of great significance for stability and control problems in discrete-time linear periodic systems.