Nonlinear Control Sy.. - Free

262 CHAPTER 10. FEEDBACK LINEARIZATION
Definition 10.4 (Distribution) Given an open set D C 1R' and smooth functions fl, f2i , fp
D -> R", we will refer to as a smooth distribution A to the process of assigning the subspace
spanned by the values of x c D.
A = span{f1, f2,. - -fp}
We will denote by A(x) the "values" of A at the point x. The dimension of the distribution
A(x) at a point x E D is the dimension of the subspace A(x). It then follows that
dim (A(x)) = rank ([fi (x), f2(x), ... , fp(x)])
i.e., the dimension of A(x) is the rank of the matrix [fi(x), f2(x), , fp(x)].
Definition 10.5 A distribution A defined on D C Rn is said to be nonsingular if there
exists an integer d such that
dim(O(x)) = d Vx E D.
If this condition is not satisfied, then A is said to be of variable dimension.
Definition 10.6 A point xo of D is said to be a regular point of the distribution A if there
exist a neighborhood Do of xo with the property that A is nonsingular on Do. Each point
of D that is not a regular point is said to be a singularity point.
Example 10.4 Let D = {x E ii 2 : xl + X2 # 0} and consider the distribution A
span{ fl, f2}, where
fl=[O], f2=Lx1 +x2]
We have
dim(O(x)) = rank
1 1
0 xl+x2 ])
Then A has dimension 2 everywhere in R2, except along the line xl + x2 = 0. It follows
that A is nonsingular on D and that every point of D is a regular point.
Example 10.5 Consider the same distribution used in the previous example, but this time
with D = R2. From our analysis in the previous example, we have that A is not regular
since dim(A(x)) is not constant over D. Every point on the line xl + X2 = 0 is a singular
point.