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Stabilization of Nonholonomic Systems 199

where g i (x 1 , x 2 ) denotes the ith column of the matrix

g 11 (x 1 , x 2 ) 0
g(x 1 , x 2 ) =
g 21 (x 1 , x 2 ) g 22 (x 1 , x 2 )
is nonsingular in U. Finally, assume, without lack of generality, that the set U
¯
¯
n
contains the origin of R .
Then the following holds:
1. Set u 1 = u 1 (x 1 , x 2 ) with

u 1 (0, x 2 ) = 0 (5.13)

for all x 2 . Then, for every u 2 , the n − p-dimensional manifold M =
¯
{x ∈ U: x 1 = 0} is invariant for the system
˙ x 1 = g 11 (x 1 , x 2 )u 1 (x 1 , x 2 )
(5.14)
˙ x 2 = g 21 (x 1 , x 2 )u 1 (x 1 , x 2 ) + g 22 (x 1 , x 2 )u 2

¯
2. If the matrix function g 11 (x 1 , x 2 ) has constant rank equal to p in U
and there exists a smooth scalar function φ(x 1 ) such that the matrix
¯
function φ(x 1 )g 21 (x 1 , x 2 ) is smooth in U, then the n−p-dimensional
distribution

0 p×(n−p)
= span
I n−p
is controlled invariant. 11

Remark 5.2 As discussed in Reference 17, under mild hypotheses and with
a proper choice of coordinates, it is always possible to write the kinematic
equations of a nonholonomic system with equations having the form (5.12), with

0 I m
g 11 (x 1 , x 2 ) = I p , g 21 (x 1 , x 2 ) = , g 22 (x 1 , x 2 ) =
∗(x 1 , x 2 ) ∗(x 1 , x 2 )
This form is known as normal form [17].

Lemma 5.l is instrumental to yield a necessary condition and a certain
number of sufficient conditions for asymptotic stabilizability of nonholonomic
systems described by equations of the form (5.12).

11 0 p×(n−p) denotes the zero matrix of dimensions p × (n − p) and I s denotes the identity matrix
of dimension s.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 199 — #13