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206 Autonomous Mobile Robots

for all i = 1, ... , n − 1. Then for every > 0 there exists a δ> 0 (depending
on ) satisfying δ c 0 ≤|x 1 (0)|=|ξ 1 (0)|≤ c 1 , such that the trajectories of
0
the system in closed loop with the C control law

α 1 α n−1
x x
u x 1 , , ... , if |x 1 | >δ
 21 2,n−1
u = x β 1 x β n−1 (5.20)
1
1
0 elsewhere

n
converge to the set ={x ∈ R | x ≤ } in some finite time T ∗ and remain
therein for all t ≥ T ∗ .
At this point the reader may argue whether it is possible or not to let δ go
to zero, that is, what we can conclude about the (discontinuous) control law

α n−1
x α 1 x
u x 1 , , ... , if x 1 = 0
 21 2,n−1
u = x β 1 x β n−1 (5.21)
1
1

0 p×1 if x 1 = 0
Observe that the control law (5.21) is discontinuous at x 1 = 0 as a function of
x, but it is continuous as a function of t, since x 1 (t) = 0 only asymptotically
(if x 1 (0) = 0, which is without lack of generality). Moreover, by hypothesis,
β i
α i
the variables ξ 2i = x /x tend to zero when t goes to infinity. Thus
2i 1
α n−1
α 1
x (t) x 2,n−1 (t)
21
lim u x 1 (t), , ... , = u(0, 0, ... ,0) = 0
β 1 β n−1
x (t) x (t)
t→∞
1 1
As a consequence, the control law (5.21) is well defined and bounded,
along the trajectories of the closed loop system, for all t ≥ 0 and, viewed
0
as a function of time, is even continuous (i.e., it is at least C )as t
goes to infinity. Finally, using Theorem 5.6, with δ = 0, and assuming
that the conditions (5.19) hold, we conclude that the control law (5.21)
almost exponentially stabilizes the system (5.18) on the open and dense set
n
={x ∈ R |x 1 = 0}.
Remark 5.6 The assumption x 1 (0) = 0 is without lack of generality,
as it is always possible to apply preventively an open loop control, for
example, a constant control, driving the system away from the hyperplane
x 1 = 0 [32,33,47].
Remark 5.7 By a general property of one dimensional dynamical systems,
we conclude that the state variable x 1 = ξ 1 evolving from a nonzero initial
condition approaches the equilibrium x 1 = 0 without ever crossing it, that is,

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 206 — #20