Basics and Preliminaries of Time-Delay Systems

In this chapter, we review the basic characteristics and properties of dynamic systems with time delays, and revisit remarkable research achievements made by many researchers during the last decades. In particular, we explore remarkable mathematical equations, inequalities, and lemmas which are widely used in system analysis to deal with state variables including time delays that are essentially encountered when the stability and control problems of dynamic systems and networks with time delays have been considered. We will also look at the results of Lyapunov function selection for the stability analysis of dynamic systems because the Lyapunov stability analysis method is almost the only alternative for solving the stability and control problems. Finally, we end this chapter by introducing the main issues related to stabilization of the time delay system.

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