Nonlinear Control Sy.. - Free

3.5. ASYMPTOTIC STABILITY IN THE LARGE 77
Thus V(x) is negative semi-definite. It is not negative definite since V(x) = 0 for x2 = 0,
regardless of the value of xl (thus V (x) = 0 along the xl axis). According to this analysis, we
conclude that the origin is stable by Theorem 3.1, but cannot conclude asymptotic stability
as suggested by our intuitive analysis, since we were not able to establish the conditions of
Theorem 3.2. Namely, V(x) is not negative definite in a neighborhood of x = 0. The result
is indeed disappointing since we know that a pendulum with friction has an asymptotically
stable equilibrium point at the origin.
This example emphasizes the fact that all of the theorems seen so far provide sufficient but
by no means necessary conditions for stability.
Example 3.6 Consider the following system:
±2
-XI + x2(x1 + x - a2).
To study the equilibrium point at the origin, we define V(x) = 1/2(x2 +x2). We have
V (x) = VV f (x)
[xl//, x2] [xl (X2 + x - /32) + x2, -xlx2 (2 + x2 - Q2)]T
xllxl+2-'32)+x2(xi+2-'32)
(xi + x2)(xl + 2 - Q2).
Thus, V(x) > 0 and V(x) < 0, provided that (xl +x2) < Q2, and it follows that the origin
is an asymptotically stable equilibrium point.
3.5 Asymptotic Stability in the Large
A quick look at the definitions of stability seen so far will reveal that all of these concepts
are local in character. Consider, for example, the definition of stability. The equilibrium xe
is said to be stable if
JIx(t) - xell < e, provided that 11x(0) - xell < b
or in words, this says that starting "near" xei the solution will remain "near" xe. More
important is the case of asymptotic stability. In this case the solution not only stays within
e but also converges to xe in the limit. When the equilibrium is asymptotically stable,
it is often important to know under what conditions an initial state will converge to the
equilibrium point. In the best possible case, any initial state will converge to the equilibrium
point. An equilibrium point that has this property is said to be globally asymptotically stable,
or asymptotically stable in the large.