On the periodic Sylvester equations and their applications in periodic Luenberger observers design

In this note, we dedicate to solving the forward-time discrete periodic Sylvester matrix equations (FDPSE) and the reverse-time discrete periodic Sylvester matrix equations (RDPSE), which are frequently encountered in the analysis and design of linear discrete-time periodic systems. Explicit and complete parametric solutions to these equations are derived without any constraints on the coefficient matrices. As one of the most important applications of the periodic Sylvester matrix equations, periodic Luenberger-type observers design problem for linear discrete-time periodic systems is solved in the paper. Necessary and sufficient conditions for the existence of Luenberger observers are established, and a parametric design algorithm for Luenberger observers is presented. A numerical example is worked out to illustrate the effectiveness of our results.