Nonlinear Control Sy.. - Free

272 CHAPTER 10. FEEDBACK LINEARIZATION
which is of the form i = j (x) + g(x)u. To find a feedback linearizing law, we seek a
transformation T = [T1 T2 T3]T such that
with
a
5g(x) l = 0 (10.22)
2 g ( x) = 0 ( 10 . 23 )
2 ( ) 0 (10 . 24 )
g x
1f(x) = T2
a 1(X) 2 = T's a 2 AX) = -q(x)w(x).
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Equation (10.22) implies that
0
0r1+ i)
c711
8x3 ) = 0
A
1+µxi
(10.25)
(10.26)
(10.27)
so T1 is not a function of x3. To proceed we arbitrarily choose T1 = x1. We will need to
verify that this choice satisfies the remaining linearizing conditions (10.22)-(10.27).
Equation (10.25) implies that
ax 1(X) = T2 = [1 0 011(X) = T2
and thus T2 = x2. We now turn to equation (10.26). We have that
and substituting values, we have that
af(x)=T3
2
k aµx3
T3 =g-mx2 2m1+
(
{tx1)2.