Nonlinear Control Sy.. - Free

10.5. INPUT-OUTPUT LINEARIZATION 275
Thus,
{g(x), ad fg(x), ... , adf-19(x)} _ {B, -AB, A2B, ... (-1)n-'An-'B}
and therefore, the vectors {g(x), adfg(x), , adf-lg(x)} are linearly independent if and
only if the matrix
C = [B, AB, A2B, - -, An-1B]nxn
has rank n. Therefore, for linear systems condition (i) of theorem 10.2 is equivalent to the
controllability of the pair (A, B). Notice also that conditions (ii) is trivially satisfied for
any linear time-invariant system, since the vector fields are constant and so 6 is always
involutive.
10.5 Input-Output Linearization
So far we have considered the input-state linearization problem, where the interest is in
linearizing the mapping from input to state. Often, such as in a tracking control problem,
our interest is in a certain output variable rather than the state. Consider the system
= f (x) + g(x)u
f, g : D C Rn -* lR'
y=h(x) h:DCRn-4 R
(10.29)
Linearizing the state equation, as in the input-state linearization problem, does not necessarily
imply that the resulting map from input u to output y is linear. The reason, of course,
is that when deriving the coordinate transformation used to linearize the state equation we
did not take into account the nonlinearity in the output equation.
In this section we consider the problem of finding a control law that renders a linear
differential equation relating the input u to the output y. To get a better grasp of this
principle, we consider the following simple example.
Example 10.14 Consider the system of the form
- axix2 + (x2 + 1)u
We are interested in the input-output relationship, so we start by considering the output
equation y = x1. Differentiating, we obtain
=x2