Nonlinear Control Sy.. - Free

10.5. INPUT-OUTPUT LINEARIZATION 277
dt[axf(x)l = Lfh(x)+L9Lfh(x)u.
ydef y(2)
If L9Lfh(x) = 0, we continue to differentiate until, for some integer r < n
with L9Lf -llh(x) $ 0. Letting
we obtain the linear differential equation
y(r) = Lfh(x) + L9Lf-ll h(x)u
u= 1 [-Lrh+v]
L9Lf llh(x)
y(r) = v. (10.30)
The number of differentiations of y required to obtain (10.30) is important and is
called the relative degree of the system. We now define this concept more precisely.
Definition 10.10 A system of the form
x = f (x) + g(x)u f, g : D C Rn R"
y=h(x) h:DCR"R
is said to have a relative degree r in a region Do C D if
Remarks:
L9L fh(x) = 0 Vi, 0 < i < r - 1, Vx E Do
L9Lf lh(x) # 0 Vx E Do
(a) Notice that if r = n, that is, if the relative degree is equal to the number of states,
then denoting h(x) = Tl (x), we have that
y = 5[f 1 (x) + g(x)u].
The assumption r = n implies that Og(x) = 0. Thus, we can define
C9T1 de f
y = ax f(x) = T2
y(2) = a 22[f(x) + g(x)ul, 22g(x) = 0