9.1 Introduction

Various results pertaining to discrete-time systems are similar to those of continuous-time systems. In this chapter, results on controllability, observability, stability, stabilizability, and optimal control for discrete-time systems are discussed. These results are analogous to those of continuous-time systems presented in Chapters 5–8, with minor changes. For the computational study of a continuous-time system, it is often necessary to discretize the system equations and analyze the equivalent discrete-time system. Thus, the study of discrete-time systems becomes important in applications.

In Section 9.2, the state transition matrix and solutions to discrete-time linear autonomous and nonautonomous systems are defined. Sections 9.3 and 9.4 provide results on controllability and observability for discrete-time linear systems. A method for obtaining a discrete-time system from a continuous-time system while preserving the controllability property is presented in Section 9.5. In Section 9.6, the dual system and the duality theorem are briefly discussed. In Section 9.7, stability and stabilizability of discrete-time systems using eigenvalues are analyzed, and in Section 9.8, Lyapunov theory is explained in detail. Optimal control problems and the matrix Riccati equation analogous to those for continuous-time systems are presented in Sections 9.9–9.11.