Nonlinear Control Sy.. - Free

84 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
Proof: See the Appendix.
We now put these ideas to work using the following example.
Example 3.10 Consider the following system:
.i1 = -axl
2 = bx2 + xlx2.
Clearly, the origin is an equilibrium point. To study the stability of this equilibrium point,
we proceed to find a Lyapunov function as follows.
Step 1: Assume that VV(x) = g(x) has the form
Step 2: Impose the symmetry conditions,
In our case we have
02V
x1 + h2x2]. (3.8)
g(x) = [hlxl + hix2, hl2 2
2
as,, - ag,
= a V or, equivalently
axtax, ax, ax, ax; - ax,
1991
ahl
xl ahl + h2 + x2
ax2 ax2 ax2
19x1
= h2 + x1
1 2
ax,
+ x2 axl .
To simplify the solution, we attempt to solve the problem assuming that the J, 's are constant.
If this is the case, then
ahi = ahi = ah2 = ah2 = 0
ax2 ax2 19x1 axl
and we have that:
In particular, choosing k = 0, we have
1991 1992 = hl 2 =1 h2 = k
ax2 19x1
g(x) = [hlxl + kx2i kxl + h2x2].
9(x) = [91, 92] = [hlxl, h2x2]