Nonlinear Control Sy.. - Free

Chapter 3
Lyapunov Stability I: Autonomous
Systems
In this chapter we look at the important notion of stability in the sense of Lyapunov.
Indeed, there are many definitions of stability of systems. In all cases the idea is: given a
set of dynamical equations that represent a physical system, try to determine whether such
a system is well behaved in some conceivable sense. Exactly what constitutes a meaningful
notion of good behavior is certainly a very debatable topic. The problem lies in how to
convert the intuitive notion a good behavior into a precise mathematical definition that can
be applied to a given dynamical system. In this chapter, we explore the notion of stability
in the sense of Lyapunov, which applies to equilibrium points. Other notions of stability
will be explored in Chapters 6 and 7. Throughout this chapter we restrict our attention to
autonomous systems. The more general case of nonautonomous systems is treated in the
next chapter.
3.1 Definitions
Consider the autonomous systems
x=f(x) f:D -Rn
where D is an open and connected subset of Rn and f is a locally Lipschitz map from D
into Rn. In the sequel we will assume that x = xe is an equilibrium point of (3.1). In other
words, xe is such that
f(xe) = 0.
'Notice that (3.1) represents an unforced system
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