Page 216 - Autonomous Mobile Robots

Stabilization of Nonholonomic Systems 201

(iii) The matrix function g 22 (x 1 , x 2 ) depends on x 2 only, that is,
g 22 (x 1 , x 2 ) =¯g 22 (x 2 ) for some function ¯g 22 (·).
(iv) There exists a smooth vector function u 1 (x 1 , x 2 ), zero for x 1 = 0
and for all x 2 , that is, u 1 (0, x 2 ) = 0, such that

−∞ < x Xg 11 (x 1 , x 2 )u 1 (x 1 , x 2 )< 0
1
for some positive definite matrix X and for all nonzero x 1 in U.
¯
¯
Moreover g 21 (x 1 , x 2 )u 1 (x 1 , x 2 ) is smooth in U and it is a function
of x 2 only, that is, g 21 (x 1 , x 2 )u 1 (x 1 , x 2 ) = f 2 (x 2 ), for some function
¯
f 2 (·) such that f 2 (0) = 0.
¯
¯
(v) There exists a smooth function u 2 (x 2 ) that renders the equilibrium
x 2 = 0 of the system
˙ x 2 = f 2 (x 2 ) +¯g 22 (x 2 )u 2 (x 2 )
¯
locally asymptotically stable.

Then, the smooth control law

u 1 (x 1 , x 2 )
u = u(x 1 , x 2 ) =
u 2 (x 2 )
locally asymptotically stabilizes the system (5.12).
As should be clear from Theorem 5.3, the possibility of rendering the
manifold x 1 = 0 invariant for the closed loop system, allows the asymptotic
stabilization problem to be solved in two successive steps. Hypothesis (iv)
determines the component u 1 of the control law; whereas the component u 2
must be chosen to fulfill hypothesis (v). Observe that the choice of u 1 is crucial,
as the existence of a smooth function u 2 (x 2 ) fulfilling hypothesis (v) depends
on such a choice. The hypotheses of Theorem 5.3 may be easily strengthened
to obtain a global result.

Theorem 5.4 Consider the system (5.12) defined in an open and dense set
n
¯
U, such that U = R . Suppose (i), (ii), (iii), and (iv) of Theorem 5.3 hold.
Moreover, suppose that the following holds:

(v) There exists a smooth function u 2 (x 2 ) which renders the equilibrium
x 2 = 0 of the system
¯
˙ x 2 = f 2 (x 2 ) +¯g 22 (x 2 )u 2 (x 2 )
globally asymptotically stable.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 201 — #15