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Stabilization of Nonholonomic Systems 217

¯

¯
is Hurwitz. 18 Let P = P > 0 be such that A P+PA < 0, and let z be a variable
in R ∪ {+∞} defined by

Y PY if x 1 = 0
z = z(x) = (5.33)
+∞ if x 1 = 0
with 19

x 2 x 3 x n n
Y = Y(x) = , , ... , , ∀x ∈ R , x 1 = 0
x 1 x 2 x n−1
1 1
Consider now the perturbed closed loop system composed of the chained
system (5.22) perturbed by an additive disturbance d and in closed loop with
u = u l (x + e), where e represents a measurement noise. For such a perturbed
system the following fact holds.

Lemma 5.2 There exists a continuous function ρ l : R → R satisfying ρ l (ξ) >
0, ∀ξ = 0, such that, for all e and d satisfying the regularity assumptions in
Section 5.2 and Equation (5.7) with ρ = ρ l (x 1 ), and for all x 0 satisfying
z(x 0 ) ≤ M, there exists a Carath´eodory solution X starting from x 0 and all
such Carath´eodory solutions are maximally defined on [0, +∞). Moreover
there exists a function δ l of class K ∞ and C > 0 such that, for all r and M, and
√
for all x 0 satisfying |x 0 |≤ δ l (r) and z(x 0 ) ≤ M, we have |X(t)|≤ r Me −Ct
and z(t) ≤ Me −Ct , for all t ≥ 0.

Lemma 5.2 states that, for any M > 0, the region z(x) ≤ M is robustly
forward invariant, that is, it is positively invariant in the presence of a class of
measurement noise and external additive disturbances. Moreover, any trajectory
in such a region converges exponentially to the origin.

5.7.2 The Global Controller
Let µ> 0 and consider the control law u g defined as u 1g = 1 and u 2g =
−µx 2 . Consider the perturbed closed loop system composed of the chained
system (5.22) perturbed by an additive disturbance d and in closed loop with
u = u g (x + e), where as before e represents a measurement noise. For such a
perturbed system the following fact holds.

18
The eigenvalues of the matrix ¯ A can be arbitrarily assigned by a proper selection of the
coefficients p i .
19 The variable Y differs from the variable used in the σ-process in Section 5.4 and in References 20
and 57. It is not difficult to show that using the σ-process therein it is possible only to prove a
weaker version of Theorem 5.7.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 217 — #31