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Stabilization of Nonholonomic Systems 205

x 2 ξ 2

x 1 ξ 1

2
2
FIGURE 5.1 The anti-σ process x 1 = ξ 1 , x 2 = ξ 1 ξ 2 does not map the ball ξ +ξ = R 2
1 2
into a neighborhood of the origin in the x 1 x 2 -plane.
The continuously differentiate control law which stabilizes a given
discontinuous nonholonomic system needs to be transformed back to the ori-
ginal coordinates via inversion of the σ process (in Reference 45 such procedure
is denoted anti-σ process). Note that the anti-σ process yields a discontinuous
control law

α n−1
anti-σ x α 1 x 2,n−1
21
u(ξ 1 , ξ 21 , ... , ξ 2,n−1 ) −→ u x 1 , , ... ,
β 1 β n−1
x x
1 1
Such a control law cannot be directly implemented, because it is not defined
at x 1 = 0. Nevertheless, it is implementable provided that some conditions are
fulfilled.
Theorem 5.6 [20] Consider a smooth nonholonomic system

˙ x = g(x)u (5.18)

m
n
with x ∈ R ,u ∈ R , and n > m. Assume that x 1 (0) = 0. Apply the σ
process (5.17) and suppose there exists a continuously differentiable control
law u = u(ξ) globally exponentially stabilizing the transformed system, that
is, |ξ 1 (t)|≤ c 1 exp −λ 1 t and |ξ 2i (t)|≤ c 2i exp −λ 2i t , for some positive λ 1 , λ 2i , c 1 ,
and c 2i and for all i = 1, ... , n − 1. Assume moreover that there exist positive
constants c 0 ≤ c 1 and λ 0 ≥ λ 1 such that 14 c 0 exp −λ 0 t ≤|ξ t (t)|. Assume
finally that

β i ≥ 0 (5.19)

14 This implies that the state ξ 1 does not converge to zero in finite time.

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