Nonlinear Control Sy.. - Free

3.8. THE INVARIANCE PRINCIPLE
Step 3: Find V:
V(x) = VV f(x)
= 9(x) f(x)
= [hix1, h2x2]f (x)
= -ahixi +h2 (b + xlx2)x2.
Step 4: Find V from VV by integration. Integrating along the axes, we have that
Step 5:
JO
x1
hiss dsl +
V(x) = I hix2 + 1 h2x2
V(x) -ahixi+h2(b+xlx2)x2
From (3.9), V (x) > 0 if and only if hi, h2 > 0. Assume then that hi = h2 = 1. In this case
V(x) = -axe + (b+x1x2)x2
assume now that a > 0, and b < 0. In this case
V(x) = -axi - (b - x1x2)x2
and we conclude that, under these conditions, the origin is (locally) asymptotically stable.
3.8 The Invariance Principle
i
1
x1
1 2
2h1x1
91(sl, 0) dsl +
1 2X2 + 2h2x2.
Verify that V > 0 and V < 0. we have that
fx2
92(x1, s2) ds2
0
X2
h2s2 ds2
10
Asymptotic stability is always more desirable that stability. However, it is often the case
that a Lyapunov function candidate fails to identify an asymptotically stable equilibrium
point by having V (x) negative semi definite. An example of this is that of the pendulum
with friction (Example 3.5). This shortcoming was due to the fact that when studying the
85