Stability Analysis Framework of Delay Systems

In this chapter, we first provide some motivations on the importance of time-delays in dynamical systems. More precisely, we show how time-delays can have intriguing and at times counter-intuitive effects on the behavior of dynamical systems. This discussion is followed by the definitions of linear stability and the spectrum of time-delay systems, together with two key notions, namely D- and tau-partitioning methods, with which stability is studied with respect to partitions in the space of time-delays and system parameters. The stability analysis and the relevant tools are summarized with illustrative examples to demonstrate the main ideas. The focus of the presentation of this chapter is on stability, while the definition of slinky effects is left to Chap. 6 .