Nonlinear Control Sy.. - Free

5.2. INTEGRATOR BACKSTEPPING 143
Define
where
z = - O(x) (5.11)
z = - fi(x) = u- fi(x) (5.12)
q= a0x=
ao[f(x)+9(x).]
(5.13)
This change of variables can be seen as "backstepping" -O(x) through the integrator,
as shown in Figure 1(c). Defining
the resulting system is
v = z (5.14)
i = f(x) + 9(x)O(x) + 9(x)z
(5.15)
i = v (5.16)
which is shown in Figure 1(d). These two steps are important for the following reasons:
(i) By construction, the system (5.15)-(5.16) is equivalent to the system (5.7)-(5.8).
(ii) The system (5.15)-(5.16) is, once again, the cascade connection of two subsystems,
as shown in Figure 1(d). However, the subsystem (5.15) incorporates the
stabilizing state feedback law and is thus asymptotically stable when the
input is zero. This feature will now be exploited in the design of a stabilizing
control law for the overall system (5.15)-(5.16).
To stabilize the system (5.15)-(5.16) consider a Lyapunov function candidate of the
form
V = V (x, ) = Vi (x) + Z z2. (5.17)
We have that
We can choose
V _ -
av, -
E--[f(x)
+ 9(x)O(x) + 9(x)z] + zi
x
_AX f (x) + -
ax
a-x g(x)z + zv.
v - ((x) + kz) , k > 0 (5.18)