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2.6 Stability Theory 63

bounded. More precisely, x e is UUB if for any δ>0, >0, there exists a
finite time T( ,δ) such that whenever ||x 0 -x e ||<δ, the following is
satisfied

for all t T( ,δ).

4. Global Uniform Ultimate Boundedness: x e is said to be globally, uniformly,
ultimately bounded (GUUB) if for >0, there exists a finite time T( )
such that

for all t T( ) See Figure 2.6.12 for an illustration of the boundedness
stability concepts.

EXAMPLE 2.6–4: Boundedness
1. The second-order system given by

has a uniformly bounded equilibrium point at the origin as shown in
Figure 2.6.13.

2. The second-order system given by

has an UUB equilibrium point at x e=0, as shown in Figure 2.6.14.

Note that in general, we are interested in the stability of the motion x(t)
when the system is perturbed from its trajectory. In other words, how far
does x(t) get from its nominal trajectory if the initial state is perturbed?
This problem can always be reduced to the stability of the origin of a non-

Copyright © 2004 by Marcel Dekker, Inc.