Nonlinear Control Sy.. - Free

10.1. MATHEMATICAL TOOLS 263
Definition 10.7 (Involutive Distribution) A distribution A is said to be involutive if gi E i
and 92 E i = [91,92] E A.
It then follows that A = span{ fl, f2,. , fp} is involutive if and only if
rank ([fi(x), ... , fn(x)]) --rank
Example 10.6 Let D = R3 and A = span{ fi, f2} where
([fi(x),...
x2 1 1
, fn(x), [f i, f,]]) , Vx and all i,j
Then it can be verified that dim(O(x)) = 2 Vx E D. We also have that
Therefore i is involutive if and only if
This, however is not the case, since
rank
[f1, f21 = a2 f1 - afl f2 =
1 0 1 0 0
rank 0 xi = rank 0 xi 1
x2 1 x2 1 0
and hence A is not involutive.
0 1 0 0
xi = 2 and rank 0 xi 1 = 3
1 1) ( 1 1 0
Definition 10.8 (Complete Integrability) A linearly independent set of vector fields fi, , fp
on D C II2n is said to be completely integrable if for each xo E D there exists a neighborhood
N of xo and n-p real-valued smooth functions hi(x), h2(x), , hn_p(x) satisfying the
partial differentiable equation
8hj
8xf=(x)=0
and the gradients Ohi are linearly independent.
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