Nonlinear Control Sy.. - Free

72 CHAPTER 3. LYAPUNOV STABILITY I: AUTONOMOUS SYSTEMS
(i) V(O) = 0,
(ii) V(x) > 0 in D - {0},
(iii) V(x) < 0 in D - {0},
thus x = 0 is stable.
In other words, the theorem implies that a sufficient condition for the stability of the
equilibrium point x = 0 is that there exists a continuously differentiable-positive definite
function V (x) such that V (x) is negative semi definite in a neighborhood of x = 0.
As mentioned earlier, positive definite functions can be seen as generalized energy
functions. The condition V (x) = c for constant c defines what is called a Lyapunov surface.
A Lyapunov surface defines a region of the state space that contains all Lyapunov surfaces
of lesser value, that is, given a Lyapunov function and defining
S21 = {x E Br : V(x) < c1}
522 = {x E Br : V(x) < c2}
where Br = {x E R" : JJxlJ < rl }, and c1 > c2 are chosen such that 522 C Br, i = 1, 2,
then we have that 522 C 521. The condition V < 0 implies that when a trajectory crosses a
Lyapunov surface V (x) = c, it can never come out again. Thus a trajectory satisfying this
condition is actually confined to the closed region SZ = {x : V(x) < c}. This implies that
the equilibrium point is stable, and makes Theorem 3.1 intuitively very simple.
Now suppose that V(x) is assumed to be negative definite. In this case, a trajectory
can only move from a Lyapunov surface V (x) = c into an inner Lyapunov surface with
smaller c. This clearly represents a stronger stability condition.
Theorem 3.2 (Asymptotic Stability Theorem) Under the conditions of Theorem 3.1, if
V (O) = 0,
(ii) V(x) > 0 in D - {0},
(iii) V(x) < 0 in D - {0},
thus x = 0 is asymptotically stable.