Nonlinear Control Sy.. - Free

10.5. INPUT-OUTPUT LINEARIZATION 279
where
A=
0 1 0 ... 0
0 0 1 0
0 0 0 1
-qo -Q1 .. -9n-1
nxn
, B=
C= [ Po P1 ... Pm 0 ... 11xn' Pin 0.
The transfer function associated with this state space realization can be easily seen to be
H(S) = C(SI - A) 1B =
Pmsm +Pm-lsm-1 + ... +Po
sn+qn-lss-1+...+qo
0
0
0
1
nxl
m 0.
We now calculate the relative degree using definition 10.10. We have
CB = [ Po P1 ... Pm 0 .
y=Ci=CAx+CBu
0
0
IO
1
_ pm, if m=n-1
0, otherwise
Thus, if CB = p.,,,,, we conclude that r = n - m = 1. Assuming that this is not the case, we
have that y = CAx, and we continue to differentiate y:
CAB = [ Po P1 ... Pm 0 ... ]
y(2) = CAi = CA2X + CABu
0
0
J Pm, if m=n-2
l 0, otherwise
If CAB = pm, then we conclude that r = n - m = 2. Assuming that this is not the case,
we continue to differentiate. With every differentiation, the "1" entry in the column matrix
A'B moves up one row. Thus, given the form of the C matrix, we have that
CA'-1B= 1 0 for i=
l pm for i=n - m