Projected generalized discrete-time periodic Lyapunov equations and balanced realization of periodic descriptor systems

From the necessary and sufficient conditions for complete reachability and observability of periodic descriptor systems with time-varying dimensions, the symmetric positive semidefinite reachability/observability Gramians are defined. These Gramians can be shown to satisfy some projected generalized discrete-time periodic Lyapunov equations. We propose a numerical method for solving these projected Lyapunov equations, and give an illustrative numerical example. As an application of our results, the balanced realization of periodic descriptor systems is discussed.