Nonlinear Control Sy.. - Free

10.7. CONDITIONS FOR INPUT-OUTPUT LINEARIZATION 287
10.7 Conditions for Input-Output Linearization
According to our discussion in Section 10.6, the input-output linearization procedure depends
on the existence of a transformation T(.) that converts the original system of the
form
fx = f (x) + g(x)u, f, g : D C R" -> R"
(10.41)
y=h(x), h:DCRn ->R
into the normal form (10.37)-(10.39). The following theorem states that such a transformation
exists for any SISO system of relative degree r < n.
Theorem 10.3 Consider the system (10.41) and assume that it has relative degree r <
n Vx E Do C D. Then, for every xo E Do, there exist a neighborhood S2 of xo and smooth
function It,, , /t,,, _r such that
(i) Lgpi(x) = 0, for 1 < i < n - r, dx E S2
(ii)
is a diffeomorphism on Q.
Proof: See the Appendix.
10.8 Exercises
10.1) Prove Lemma 10.2.
T(x) =
f i i ( ) 1
An-r(x)
01
Lr J
01 = h(x)
02 = Lfh(x)
Or = Lf lh(x)