Nonlinear Control Sy.. - Free

332 APPENDIX A. PROOFS
Assume now that conditions (i) and (ii) of Theorem 10.2 are satisfied. By the Frobenius
theorem, there exist Tl (x) satisfying
L9T1(x) = LadfgTl = ... = Lad,._2gT1 = 0
and taking into account the Jacobi identity, this implies that
but then we have that
L9T1(x) = L9L fT1(x) _ ... = L9L' -2Ti (x) = 0
VT1(x)C=OT1(x)[9, adf9(x), ..., adf-lg(x)] = [0, ... 0, Ladf-19Ti(x)]
The columns of the matrix [g, adfg(x), , adf-lg(x)] are linearly independent on Do
and so rank(C) = n, by condition (i) of Theorem 10.2, and since VT1(x) # 0, it must be
true that
adf-19Ti(x) 0
which implies, by the Jacobi identity, that
Proof of theorem 10.3:
LgL fn-1T1(x) 4 0.
The proof of Theorem 10.3 requires some preliminary results. In the following lemmas
we consider a system of the form
f x = f (x) + g(x)u, f, g : D C R" - Rn
Sl
y=h(x), h:DClg"->R
and assume that it has a relative degree r < n.
Lemma A.1 If the system (A.69) has relative degree r < n in S2, then
VxEc, Vj0.
k - f 0
Lad)fgLfh(x) t (-1) L9Lf lh(x) 54 0
Proof: We use the induction algorithm on j.
0