Chapter 3
Lyapunov Stability and Stability Regions of Nonlinear Dynamical Systems
3.1 INTRODUCTION
Stability is a fundamental subject that unifies engineering and the sciences. This subject has been regarded by many as a fascinating and difficult problem of human culture. As such, there are at least 50 different terms for stability concepts used in the literature. Stability is a very broad subject, and the concept of stability can be formulated in a variety of ways depending on the intended use of stability analysis and design (Alberto and Chiang, 2007; DeCarlo et al., 2000; Hahn, 1967; La Salle and Lefschetz, 1961; May, 1973; Michel et al., 1982). In this chapter, we review some relevant stability concepts from the nonlinear dynamical systems theory. Some of these concepts and their implications will be discussed in detail.
Another important subject related to stability is the stability region of nonlinear dynamical systems. Indeed, the problem of determining stability regions (regions of attraction) of nonlinear dynamical systems is of fundamental importance for many disciplines in engineering and in the sciences (Athay et al., 1984; Chiang et al., 1988; Genesio et al., 1985; Loparo and Blankenship, 1978; Saha et al., 1997; Sastry, 1999; Vu and Liu, 1992). For instance, knowledge of the stability region is essential in the development of direct methods for power system transient stability analysis. Indeed, this problem is at the heart of direct methods. A comprehensive theory of ...
