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196 Autonomous Mobile Robots

consider two functions e and d satisfying the following regularity assumptions:
n
n
∞
e and d are in L (R ×[0, +∞); R ), and are continuous in x for each t.We
loc
6
introduce these functions as a measurement noise e and an external noise d
and define the perturbed system with u given by Equation (5.3), that is,
˙ x = g(x)k(x + e(x, t), s d (t)) + d(x, t)
(5.5)
s d = k d (x + e(x, t), ¯s d )
In this context the definition of global exponential stability is as follows.

Definition 5.2 [32] Let e and d be two functions satisfying our standing
regularity assumptions. The origin of the system (5.5) is said to be a globally
n
exponentially stable equilibrium on R if the following two properties hold: 7

n
(i) For every (x 0 , s 0 ) ∈ R ×{1, 2}, there exists a solution of (5.5)
starting from (x 0 , s 0 ). Moreover all maximal solutions of (5.5) are
defined on [0, +∞).
(ii) There exists δ of class K ∞ and C > 0 such that, for all r > 0 and
n
for all (x 0 , s 0 ) ∈ R ×{1, 2} with |x 0 |≤ δ(r) and for all maximal
solutions (X, S d ) of (5.5) starting from (x 0 , s 0 ), one has

|X(t)|≤ re −Ct , ∀t ≥ 0 (5.6)

Finally, we characterize robustly stabilizing controllers. 8

Definition 5.3 [32] The controller (k, k d ) is a robustly globally exponentially
n
stabilizing controller if there exists a continuous function ρ : R → R such that
ρ(x)> 0, for all x = 0, and such that for any two functions e and d satisfying
our standing regularity assumptions and

sup |e(x, ·)|≤ ρ(x), ess sup |d(x, ·)|≤ ρ(x) (5.7)
R ≥0
R ≥0
n
for all x ∈ R , the origin of (5.5) is a globally exponentially stable equilibrium
n
on R .
6 Using similar arguments we could also consider an actuator noise.
7
A function γ : R ≥0 → R ≥0 is of class K if it is continuous, strictly increasing, and zero at zero.
It is of class K ∞ , if it is of class K and unbounded. A continuous function β : R ≥0 × R ≥0 → R ≥0
is of class KL if β(·, τ) is of class K for each τ ≥ 0 and β(s, ·) is decreasing to zero for each s > 0:
8 Note that our notion of robust stability is closely related to the classical notion of Input-to-State
stability [38].

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 196 — #10