Nonlinear Control Sy.. - Free

276 CHAPTER 10. FEEDBACK LINEARIZATION
which does not contain u. Differentiating once again, we obtain
Thus, letting
we obtain
or
u f
x2 = -xl -axix2+(x2+1)u.
1 [v + axi x2]
X2 + 1
y+y=v
(X2 54 1)
which is a linear differential equation relating y and the new input v. Once this linear
system is obtained, linear control techniques can be employed to complete the design. 11
This idea can be easily generalized. Given a system of the form (10.29) where f,g : D C
l -4 1R' and h : D C R" -+ R are sufficiently smooth, the approach to obtain a linear
input-output relationship can be summarized as follows.
Differentiate the output equation to obtain
There are two cases of interest:
y
Oh.
8x x
ah Oh
axf (x) + 8xg(x)u
= Lfh(x) + L9h(x)u
- CASE (1): Qh # 0 E D. In this case we can define the control law
U = L91(x) [-Lfh + v]
that renders the linear differential equation
y=v.
- CASE (2): = 0 E D. In this case we continue to differentiate y until u
D-X
appears explicitly: