Nonlinear Control Sy.. - Free

274 CHAPTER 10. FEEDBACK LINEARIZATION
(ii) The distribution A = span{g, ad fg, , adf-2g} is involutive in Do.
Proof: See the Appendix.
Example 10.12 Consider again the system of Example 10.10
We have,
Thus, we have that
and
r ex2 - 1
L
axe
i
+ 1 u = f(x) + 9(x)u.
adfg = [f,9] = axf(x) - 'f g(x)
rank(C) = rank
Also the distribution A is given by
adfg =
x2
e0
{g,adfg}={LOJ'
[er2
J}
0 ex2
( 1 0 2, Vx E 1R2.
span{9} = span C 1
L 10
which is clearly involutive in R2. Thus conditions (i) and (ii) of Theorem 10.2 are satisfied
VxEIR2. E)
Example 10.13 Consider the linear time-invariant system
i=Ax+Bu
i.e., f (x) = Ax, and g(x) = B. Straightforward computations show that
adf9 = [f, 9] = ex f (x) - of g(x) = -AB
af9 = [f, [f, 9]] = A2B
g = (-1)tRB.