Nonlinear Control Sy.. - Free

10.2. INPUT-STATE LINEARIZATION 265
and thus
223 -21 -4X3
fl f2 [fl, f2] 1 = -1 -2X2 2
0 X3 0
which has rank 2 for all x E R3. It follows that the distribution is involutive, and thus it is
completely integrable on D, by the Frobenius theorem.
10.2 Input-State Linearization
Throughout this section we consider a dynamical systems of the form
± = f(2) + g(x)u
and investigate the possibility of using a state feedback control law plus a coordinate
transformation to transform this system into one that is linear time-invariant. We will
see that not every system can be transformed by this technique. To grasp the idea, we start
our presentation with a very special class of systems for which an input-state linearization
law is straightforward to find. For simplicity, we will restrict our presentation to single-input
systems.
10.2.1 Systems of the Form t = Ax + Bw(2) [u - cb(2)]
First consider a nonlinear system of the form
2 = Ax + Bw(2) [u - q5(2)] (10.3)
where A E Ilt"', B E II2ii1, 0 : D C II2" -4 )LP1, w : D C 1(P" -4 R. We also assume that
w # 0 d2 E D, and that the pair (A, B) is controllable. Under these conditions, it is
straightforward to see that the control law
renders the system
u = O(2) + w-1 (2)v (10.4)
i=A2+Bv
which is linear time-invariant and controllable.
The beauty of this approach is that it splits the feedback effort into two components
that have very different purpose.