Nonlinear Control Sy.. - Free

260 CHAPTER 10. FEEDBACK LINEARIZATION
This transformation was made possible by the nonsingularity of T. Also because of the existence
of T-1, we can always recover the original state space realization. Given a nonlinear
state space realization of an affine system of the form
x = f (x) + g(x)u
a diffeomorphism T(x) can be used to perform a coordinate transformation. Indeed, assuming
that T(x) is a diffeomorphism and defining z = T(x), we have that
z = a
x = [f (x) + g(x)u]
Given that T is a diffeomorphism, we have that 3T-1 from which we can recover the original
state space realization, that is, knowing z
x = T-1(z).
Example 10.3 Consider the following state space realization
0
x1
X2 - 2x1x2 + X1
and consider the coordinate transformation:
Therefore we have,
and
x1
x = T(x) = xl +X2
x 2+X3
1 0 0
OIT (x) = 2x1 1 0
0 2x2 1
1 0 0 0 1 0 0 1
z = 2x1 1 0 x1 + 2x1 1 0 - 2x1
0 2x2 1 x2 - 2x1x2 + xi 0 2x2 1 4x1x2
0 1
z = x1 + 0 u
X2 +X2 0
U
U