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Stabilization of Nonholonomic Systems 207

there exists no finite time T such that x 1 (T) = 0. Thus, the singular plane
x 1 = 0 is never crossed, but is just approached asymptotically. Moreover, every
n
n
trajectory starting in + ={x ∈ R |x 1 > 0} ( − ={x ∈ R |x 1 < 0})
−
+
remains in ( ) for every finite t and approaches the border of ( )
−
+
as t goes to infinity.
5.3.4 An Algorithm to Design Almost Stabilizers
In this section we propose a procedure to design discontinuous control laws for
smooth nonholonomic systems described by equations of the form (5.12). The
procedure is composed of the following steps.

(I) Transform a given smooth nonholonomic system, by means of a
σ process, into a discontinuous system.
(II) Check if the discontinuous system admits a smooth control law
yielding asymptotic stability. In case of positive answer proceed
to step III, otherwise return to step I and use a different σ process.
(III) Build a smooth stabilizer for the transformed system.
(IV) Apply the anti-σ process to the obtained stabilizer to build a
discontinuous control law for the original system.

The crucial points of the algorithm are the selection of the σ process (step I)
and the design of the smooth asymptotically stabilizing control law for the
transformed system (step III). In particular, step III can be easily solved for
low dimensional systems; whereas there is no constructive or systematic way
to perform step I successfully; that is, to select a σ process which allows to
conclude positively the algorithm.
Finally, to obtain a discontinuous nonholonomic system described by
equations of the form (5.12), with g 21 (x 1 , x 2 ) fulfilling condition (5.15), the
following simple result may be useful.

Proposition 5.1 [20] Consider a nonholonomic system described by equa-
tions of the form (5.12). Assume that g 11 (x 1 , x 2 ) = I p and that the
n
matrices g 21 (x 1 , x 2 ) and g 22 (x 1 , x 2 ) have smooth entries in R . Consider a
coordinates transformation (σ process) described by equations of the form
2 (x 1 , x 2 )
ξ 1 = x 1 , ξ 2 =
σ(x 1 )

where 2 (x 1 , x 2 ) is a smooth mapping such that 2 (0, x 2 ) = 0 and σ(x 1 )
is a smooth function which is zero at x 1 = 0. Then the transformed system
is always described, in the new coordinates, by equations of the form (5.12)

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 207 — #21