Page 215 - Autonomous Mobile Robots

200 Autonomous Mobile Robots

Theorem 5.2 [20] Consider a system described by equations of the form
¯
(5.12). Let U be a neighborhood of the origin. Assume there exists a smooth
control law

u 1 (x 1 , x 2 )
u = u(x 1 , x 2 ) =
u 2 (x 1 , x 2 )
¯
defined on U, which locally asymptotically stabilizes the resulting closed loop
system. Moreover, assume that:
(i) The control u 1 (x 1 , x 2 ) satisfies the condition (5.13)
(ii) The vector field g 21 (x 1 , x 2 )u 1 (x 1 , x 2 ) is a smooth n−p-dimensional
vector field defined in U ¯
(iii) The matrix functions g 11 (x 1 , x 2 ) and g 22 (x 1 , x 2 ) are smooth in U.
¯
a
Then there exists a smooth matrix function g (x 1 , x 2 ), defined on U, such that
¯
a
b
¯
g (0, x 2 ) = 0 (n−p)×p , and a smooth scalar function g (x 1 , x 2 ), defined on U,
b
such that g (0, x 2 ) = 0, having the property that
a
g (x 1 , x 2 )
g 21 (x 1 , x 2 ) = (5.15)
b
g (x 1 , x 2 )
that is, the matrix function g 21 (x 1 , x 2 ) is not defined for x 1 = 0.
Remark 5.3 Strictly speaking, it is not correct to discuss the asymptotic
stability of the origin for a system described by equations of the form (5.12)
with g 21 (x 1 , x 2 ) fulfilling condition (5.15), as such a system is not defined at
the origin. Hence, the origin is not an equilibrium. However, it is possible to
overcome this problem using the following definition of asymptotic stability. We
say that a smooth control law locally (globally) asymptotically stabilizes system
n
(5.12) if the closed loop system is smooth in a neighborhood of the origin (in R )
and the origin is a locally (globally) asymptotically stable equilibrium of the
1
closed loop system. For example, the system ˙x = u is globally asymptotically
x
2
stabilized by the smooth control u =−x .
We now discuss sufficient conditions for asymptotic stabilizability of
systems described by equations of the form (5.12).

Theorem 5.3 [20] Consider the system (5.12) defined in an open and dense
set U, such that U contains the point x = 0. Consider the following hold.
¯
(i) The matrix functions g 11 (x 1 , x 2 ) and g 22 (x 1 , x 2 ) are smooth in U.
¯
(ii) The matrix function g 21 (x 1 , x 2 ) is smooth in U.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 200 — #14