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202 Autonomous Mobile Robots

Then the smooth control law

u 1 (x 1 , x 2 )
u = u(x 1 , x 2 ) =
u 2 (x 2 )
globally asymptotically stabilizes the system (5.12).

Example 5.1 The following simple example illustrates the obtained results.
Consider the system

2 2
˙ x 1 = (x + x )u 1
1 2
x 2
˙ x 2 =− u 1 + u 2
x 1
2
defined on U ={x ∈ R |x 1 = 0} and fulfilling hypotheses (i), (ii), and (iii)
of Theorem 5.4. Setting u 1 =−x 1 we fulfill also (iv) and (v). Hence, simple
calculations show that the smooth (linear) control law

−x 1
u =
−2x 2
yields a globally asymptotically stable closed loop system.

Before concluding thissection we discussanotherextension ofTheorem5.3.
Theorem 5.5 [20] Consider the system (5.12) defined in an open and dense
¯
set U, such that U contains the point x = 0. Suppose (i), (ii), and (iii) of
Theorem 5.3 hold. Suppose, moreover, that the following holds:
(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 )< −x Qx 1
1
1
for some positive definite matrices X and Q 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 ) which renders the equilib-
rium x 2 = 0 of the system
˙ x 2 = f 2 (x 2 ) +¯g 22 (x 2 )u 2 (x 2 )
¯
locally exponentially stable.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c005” — 2006/3/31 — 16:42 — page 202 — #16