Nonlinear Control Sy.. - Free

268 CHAPTER 10. FEEDBACK LINEARIZATION
From here we conclude that any coordinate transformation T(x) that satisfies the
system of differential equations (10.11)(10.12) for some 0, w, A and B transforms, via the
coordinate transformation z = T(x), the system
into one of the form
z = f(x) + g(x)u (10.13)
i = Az + Bw(z) [u - O(z)] (10.14)
Moreover, any coordinate transformation z = T(x) that transforms (10.13) into (10.14)
must satisfy the system of equations (10.11)-(10.12).
Remarks: The procedure just described allows for a considerable amount of freedom when
selecting the coordinate transformation. Consider the case of single-input systems, and
recall that our objective is to obtain a system of the form
i = Az + Bw(x) [u - ¢(x)]
The A and B matrices in this state space realization are, however, non-unique and therefore
so is the diffeomorphism T. Assuming that the matrices (A, B) form a controllable pair, we
can assume, without loss of generality, that (A, B) are in the following so-called controllable
form:
Letting
Ar _
0 1 0
0 0 1
0 1
0 0 0 0
T(x)
nxn
Ti(x)
T2(x)
Tn(x)
, B, _
nx1
with A = Ac, B = B,, and z = T(x), the right-hand side of equations (10.11)-(10.12)
becomes
A,T(x) - Bcw(x)4(x) _
0
0
1
nxl
(10.15)