Nonlinear Control Sy.. - Free

190 CHAPTER 7. INPUT-TO-STATE STABILITY
or, equivalently
IxI > ( )1/3
a
It follows that the system is globally input-to-state stable with ^j(u) _ "
( a B
Example 7.3 Now consider the following system, which is a slightly modified version of
the one in Example 7.2:
x=-ax3+x2u a>0
Using the same ISS Lyapunov function candidate used in Example 7.2, we have that
= -ax4 +x3u
= -ax4 + aOx 4 - aBx4 + x3u 0 < 0 < 1
= -a(1 - 0)x4 - x3(a0x - u)
< -a(1 - 0)x4, provided
x3(a0x - u) > 0 or,
lxI > I .
Thus, the system is globally input-to-state stable with 7(u) = .
Example 7.4 Now consider the following system, which is yet another modified version of
the one in Examples 7.2 and 7.3:
i=-ax3+x(1+x2)u a>0
Using the same ISS Lyapunov function candidate in Example 7.2 we have that
-ax 4 + x(1 + x2)u
-ax4 + aOx 4 - aOx 4 + x(1 + x2)u
-a(1 - 0)x4 - x[aOx 3 - (1 + x2)u]
< -a(1 - 9)x, 4 provided
1/3
0
0