Page 296 - Autonomous Mobile Robots

Adaptive Control of Mobile Robots 283

˜
˜ ˜ ˜
˙
We have V ≤ 0. Accordingly, ˜u, ˜ θ, Z 2 , e I , L, R, K a , K N are all bounded in the
sense of Lyapunov.
From Equation (7.55), using the Corollary of Baralart’s theory [31], ˜u → 0
→ 0 (1 ≤ j ≤ m − 1) as t →∞, and e I → 0as t →∞.
as t →∞, z j,n j
Next, let us prove the asymptotic stability of Z.
The first equation of the controlled system is
(7.56)
˙ z 1 =−k u1 z 1 + h(Z 2 , t) +˜u 1

From Assumption 7.3, we know that h(Z 2 , t) is uniformly bounded. In addition,
with ˜u 1 converging to zero, (7.56) is a stable linear system subjected to the
bounded additive perturbation h(Z 2 , t) + e u1 . Therefore, z 1 (t) is also bounded
uniformly.
Because z 1 and h(Z 2 , t) are bounded, it is clear that u d1 is bounded from
(7.30). Together with ˜u converging to zero, u 1 is bounded. Since u 1 and Z 2
are bounded, ˜u goes to zero, u dj and u j (2 ≤ j ≤ m) are bounded. Under
the condition that Z 2 , u 1 , and u j (2 ≤ j ≤ m) are bounded, ˙z j,n j and ˙z j,i
(1 ≤ j ≤ m − 1, 2 ≤ i ≤ n j − 1), from (7.25), are bounded.
In the following, let us show that u d1 Z 2 tends to zero. For 1 ≤ j ≤ m − 1,
2
d1
since u d1 is bounded and z j,n j tends to zero, u z j,n j tends to zero. Taking the
2
d1
time derivative of u z j,n j ,wehave
d
2 2
d1
d1
(u z j,n j ) = u (−k u j +1 z j,n j − ρ j,n j −2 u d1 z j,n j −1
dt
d 2
(u d1 )
+˜u 1 L h1 z j,n j +˜u 2 ) + z j,n j
dt
3 2
d1 d1
=−ρ j,n j −2 u z j,n j −1 + (2˙u d1 u d1 z j,n j − k u j +1 u z j,n j
2 2
d1 d1
+ u ˜u 1 L h1 z j,n j + u ˜u 2 ) (7.57)
Since
d 3 3 2
u z j,n j −1 = u ˙z j,n j −1 + 3u ˙u d1 z j,n j −1
d1
d1
d1
dt
is bounded, the first term in (7.57) is uniformly continuous. Together
and
with the fact that all other terms in (7.57) tend to zero (since u d1 z j,n j
2
˜ u tend to zero), from the extended version of Barbalat’s Lemma, (d/dt)(u z j,n j )
d1
3
tends to zero. Therefore, u z j,n j −1 also tends to zero. So u d1 z j,n j −1 also tends
d1
to zero.
© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 283 — #17