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218 Autonomous Mobile Robots

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Lemma 5.3 There exists a continuous function ρ g : R → R satisfying
ρ g (x)> 0, ∀x = 0 such that, for any initial condition, the considered perturbed
system where e and d satisfies the regularity assumptions in Section 5.2 and
Equation (5.7) with ρ = ρ g (x), admit a unique Carath´eodory solution, defined
for all t ≥ 0. Moreover there exists a function δ g of class K ∞ such that, for any
r > 0 and for any M > 0 there exists a time T g = T g (M, δ g (r)) such that, for
all Carathe´eodory solutions X with initial condition x 0 with |x 0 |≤ δ g (r), one
has z(X(t)) ≤ M for all t ≥ T g , and |X(t)|≤ r for all t ≤ T g .
Lemma 5.3 states that, for any M > 0, the trajectories of the perturbed sys-
tem enter the region z(x) ≤ M in finite time, while remaining bounded for all t.

5.7.3 Definition of the Hybrid Controller and Main
Result

We are now ready to define the hybrid controller robustly stabilizing system
(5.22). To this end, for any strictly positive number M, we define the subset
n
M of R as M ={x, x 1 = 0, z < M}, where z is defined by (5.33). Let
M 2 > M 1 > 0. The hybrid controller (k, k d ) is defined making a hysteresis
, that is,
between u l and u g on M 2 and M 1

u l (x) if s d = 1 and x 1 = 0

k(x, s d ) = 0 if s d = 1 and x 1 = 0 (5.34)

u g (x) if s d = 2



1 if x ∈ M 1 ∪{0}

k d (x, s d ) = s d if x ∈ M 2 \ M 1 (5.35)

2 if x ∈ M 2 ∪{0}

Theorem 5.7 [32] The hybrid controller (k, d k ), described in Section 5.7.1,
Section 5.7.2, and Section 5.7.3 robustly globally exponentially stabilizes
system (5.22).

5.7.4 Discussion
A hybrid control law globally robustly exponentially stabilizing a chained
system has been proposed. This controller retains the main features of the
discontinuous controller proposed in Section 7.4, while allowing to counteract
(small) exogenous disturbances and measurement noise. A similar, but local,
result was developed in Proposition 3 of Reference 33. Note finally that the idea
of switching between a local and a global controller to achieve stabilization in

© 2006 by Taylor & Francis Group, LLC

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