Nonlinear Control Sy.. - Free

330 APPENDIX A. PROOFS
A.7 Chapter 10
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), , adf-lg(x)} are linearly independent in Do.
Equivalently, the matrix
has rank n for all x E Do.
C = [g(x), adfg(x), ... , adf-lg(x)}nxn
(ii) The distribution A = span{g, ad fg, , adf-2g} is involutive in Do.
Proof: Assume first that the system (10.28) is input-state linearizable. Then there exists
a coordinate transformation z = T(x) that transforms (10.28) into a system of the form
.z = Az + By, with A = Ac and B = B,, as defined in Section 10.2. From (10.17) and
(10.18) we know that T is such that
and
ax f (x) = T2(x)
axf(x) = T3(x)
ax 1 f W = Tn(x)
T. f(x) i4 0
ag(x) l = 0
ag(x) 2 = 0
8x
1 g(x) = 0
a ng(x) 54 0.
(A.62)
(A.63)