Abstract
This dissertation is consisted of two parts. The first part treats of applications of the structure-preserving doubling algorithm (SDA) to solve various algebraic Riccati equations, while the second part concerns with the problem of balanced realization for discrete-time periodic descriptor systems. In the first part, we investigate structure-preserving algorithms for computing the symmetric positive semi-definite solutions to the periodic discrete-time algebraic Riccati equations (P-DAREs), continuous-time algebraic Riccati equations (CAREs) and generalized discrete-time algebraic Riccati equations (G-DAREs), respectively. All are based on the SDA algorithm for solving the discrete-time algebraic Riccati equations (DAREs). In Section 2 of Chapter 1, we develop the SDA algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability. With several numerical results, the algorithm is shown to be efficient, out-performing other algorithms on a large set of benchmark problems. In the second part, necessary and sufficient conditions are derived for complete reachability and observability of periodic time-varying descriptor systems. Applying these conditions, the symmetric positive semi-definite reachability/observability Gramians are defined and 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 an illustrative numerical example is given. As an application of our results, the balanced realization of periodic descriptor systems is discussed.