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204 Autonomous Mobile Robots

and it is discontinuous if one of the g i (z, zw) is such that g i (0, 0) = 0. If the
system (5.16) is not discontinuous we can further transform it with a second σ
process. 12

5.3.3 The Issue of Asymptotic Stability
Theorems 5.3, 5.4, and 5.5 yield sufficient conditions for stabilizability of
discontinuous nonholonomic systems, while the σ process allows to map a
continuously differentiable system into a discontinuous one. To have a practic-
ally useful result, we have to show that asymptotic stability of the transformed
(discontinuous) system implies almost asymptotic stability of the original
(continuously differentiable) system. Moreover, to implement a discontinuous
control we must define it on the points of singularity.
Consider a continuously differentiable nonholonomic system described
by equations of the form (5.1). Set x = col(x 1 , x 2 ) with x 1 ∈ R and
x 2 = col(x 21 , ... , x 2,n−1 ) ∈ R n−1 and define the σ process 13

ξ 1 x 1
ξ = = (5.17)
ξ 2 σ(x 1 , x 2 )
where ξ 2 = col(ξ 21 , ... , ξ 2,n−1 ), σ(x 1 , x 2 ) = col(σ 1 (x 1 , x 2 ), ... , σ n−1 (x 1 , x 2 )),
α i
and σ i (x 1 , x 2 ) = x /x β i with α i ≥ 1 and β i ≥ 0, for all i = 1, ... , n − 1.
2i 1
The application of the σ process (5.17) to the system (5.1) yields a new system
which is, in general, not defined for ξ 1 = 0. Suppose now that the transformed
system, with state ξ,is exponentially stabilized by a control law u = u(ξ),
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. Then |x 1 (t)|≤ c 1 exp −λ 1 t and
|x 2i (t)|≤ (c 1 c 2 ) 1/α i exp( −λ 1 β 1 +λ 2i t) for all i = 1, ... , n − 1. We conclude
α i
that exponential convergence to zero of the state ξ of the transformed system
implies exponential convergence to zero of the state x of the original system.

Remark 5.4 The previous conclusions also remain valid if the stabilizer is
dynamic. This fact is useful to design dynamic, output feedback, discontinuous
stabilizers for nonholonomic systems [46].
Remark 5.5 Asymptotic stability of the system with state ξ does not imply
asymptotic stability of the system with state x, as the inverse of the coordinates
transformation (5.17) does not map neighborhood of ξ = 0 into neighborhood
of x = 0, as illustrated in Figure 5.1. Therefore, exponential stability (in the
sense of Lyapunov) of the closed loop system with state ξ implies only almost
exponential stability of the closed loop system with state x.

12 Note that the composition of σ processes yields a σ process.
13 The coordinates transformation (5.17) defines a σ process only if i β i ≥ 1.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 204 — #18