Nonlinear Control Sy.. - Free

10.1. MATHEMATICAL TOOLS 261
and substituting xi = z1i x2 = z2 - zi, we obtain
10.1.5 Distributions
0 1
i = z1 + 0 u.
z2 0
Throughout the book we have made constant use of the concept of vector space. The
backbone of linear algebra is the notion of linear independence in a vector space. We recall,
from Chapter 2, that a finite set of vectors S = {xi, x2i , xp} in R' is said to be linearly
dependent if there exist a corresponding set of real numbers {) }, not all zero, such that
E Aixi = Aixi + A2x2 + - + APxP = 0.
On the other hand, if >i Aixi = 0 implies that Ai = 0 for each i, then the set {xi} is said
to be linearly independent.
Given a set of vectors S = {x1, X2, , xp} in R', a linear combination of those vector
defines a new vector x E R", that is, given real numbers A1, A2, Ap,
x=Aixl+A2x2+...+APxP Ellen
the set of all linear combinations of vectors in S generates a subspace M of Rn known as
the span of S and denoted by span{S} = span{xi, x2i )XP} I i.e.,
span{x1 i x2, , xp} _ {x E IRn' : x = A1x1 + A2x2 + . + APxp, Ai E R1.
The concept of distribution is somewhat related to this concept.
Now consider a differentiable function f : D C Rn -> Rn. As we well know, this
function can be interpreted as a vector field that assigns the n-dimensional vector f (x) to
each point x E D. Now consider "p" vector fields fi, f2,. , fp on D C llPn. At any fixed
point x E D the functions fi generate vectors fi(x), f2 (x), , fp(x) E and thus
O(x) = span{ fi(x), ... f,(x)}
is a subspace of W'. We can now state the following definition.