Nonlinear Control Sy.. - Free

10.2. INPUT-STATE LINEARIZATION 267
10.2.2 Systems of the Form 2 = f (x) + g(x)u
Now consider the more general case of affine systems of the form
x = f(x) + g(x)u. (10.5)
Because the system (10.5) does not have the simple form (10.4), there is no obvious way to
construct the input-state linearization law. Moreover, it is not clear in this case whether
such a linearizing control law actually exists. We will pursue the input-state linearization of
these systems as an extension of the previous case. Before proceeding to do so, we formally
introduce the notion of input-state linearization.
Definition 10.9 A nonlinear system of the form (10.5) is said to be input-state linearizable
if there exist a diffeomorphism T : D C r -> IR defining the coordinate transformation
and a control law of the form
z = T(x) (10.6)
u = t(x) + w-1(x)v
that transform (10.5) into a state space realization of the form
i = Az + By.
(10.7)
We now look at this idea in more detail. Assuming that, after the coordinate transformation
(10.6), the system (10.5) takes the form
where w(z) = w(T-1(z)) and (z) = O(T-1(z)). We have:
i = Az + Buv(z) [u - (z)]
= Az + Bw(x) [u - O(x)] (10.8)
z- axx-
Substituting (10.6) and (10.9) into (10.8), we have that
ax[f(x)+g(x)u]. (10.9)
[f (x) + g(x)u] = AT(x) + Bw(x)[u - O(x)] (10.10)
must hold Vx and u of interest. Equation (10.10) is satisfied if and only if
f (x) = AT(x) - Bw(x)O(x) (10.11)
ag(x) = Bw(x). (10.12)