Nonlinear Control Sy.. - Free

88 CHAPTER 3. LYAPUNOV STABILITY I: AUTONOMOUS SYSTEMS
between theorems 3.6 and theorem 3.2 is that in Theorem 3.6 is allowed to be only
positive semi-definite, something that will remove part of the conservativism associated
with certain Lyapunov functions.
Theorem 3.6 The equilibrium point x = 0 of the autonomous system (3.1) is asymptotically
stable if there exists a function V(x) satisfying
(i) V(x) positive definite Vx E D, where we assume that 0 E D.
(ii) 1 (x) is negative semi definite in a bounded region R C D.
(iii) V(x) does not vanish identically along any trajectory in R, other than the null solution
x=0.
Example 3.18 Consider again the pendulum with friction of Example 3.5:
Again
xl
x2
X2
g .
- l slnx1 - -x2.
k
m
V (X) > 0 Vx E (-7r, -7r) x IR,
(x) _ -kl2x2 (3.13)
which is negative semi definite since V (x) = 0 for all x = [x1i 0]T . Thus, with V short of
being negative definite, the Lyapunov theory fails to predict the asymptotic stability of the
origin expected from the physical understanding of the problem. We now look to see whether
application of Theorem 3.6 leads to a better result. Conditions (i) and (ii) of Theorem 3.6
are satisfied in the region
R
with -7r < xl < 7r, and -a < X2 < a, for any a E IR+. We now check condition (iii) of
the same theorem, that is, we check whether V can vanish identically along the trajectories
trapped in R, other than the null solution. The key of this step is the analysis of the condition
V = 0 using the system equations (3.11)-(3.12). Indeed, assume that V(x) is identically
zero over a nonzero time interval. By (3.13) we have
V (x) = 0 0 = -k2 l2x b X2 = 0
X2
l