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52 Introduction to Control Theory

Figure 2.6.1: (a) Stability of x e at t 0 ; (b) Instability of x e at t 0 .

x e is stable in the sense of Lyapunov if it is stable for any given t 0.
See Figure 2.6.1a.

2. Instability: x e is unstable in the sense of Lyapunov (UL), if no matter
how close to x e the state starts, it will not be confined to the vicinity
of x e at some later time. In other words, x e is unstable if it is not
stable at t 0. See Figure 2.6.1b for an illustration.

3. Convergence: x e is convergent (C) at t 0, if states starting close to x e
will eventually converge to x e. In other words, x e is convergent at t 0
if for any positive there exists a positive 1(t 0) and a positive T( 1, x 0,
t 0) such that if

then

Copyright © 2004 by Marcel Dekker, Inc.