Nonlinear Control Sy.. - Free

10.4. CONDITIONS FOR INPUT-STATE LINEARIZATION 273
To verify that (10.23)-(10.24) we proceed as follows:
aT2
9(x) = [0
so that (10.23) is satisfied. Similarly
ax1( )
0
=
1 0] 0 u 0
39(x) = m,(1µ+ix1)
provided that x3 # 0. Therefore all conditions are satisfied in D = {x E R3 : X3 # 0}. The
coordinate transformation is
The function q and w are given by
xl
T = X2
k
g mx2
(aT3/ax)f(x)
w(x) _ 0 g(x) O(x)
ax (aT3/ax)g(x)
10.4 Conditions for Input-State Linearization
In this section we consider a system of the form
0
± = f(x) + g(x)u (10.28)
where f , g : D -+ R' and discuss under what conditions on f and g this system is input-state
linearizable.
Theorem 10.2 The system (10.28) is input-state linearizable on Do C D if and only if
the following conditions are satisfied:
(i) The vector fields {g(x), adfg(x), , ad f-lg(x)} are linearly independent in Do.
Equivalently, the matrix
has rank n for all x E Do.
aµx2
2m(l+µxi)
C = [g(x), adfg(x),... adf-lg(x)]nxn