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284 Autonomous Mobile Robots

2
Differentiating u z j,n j −1 yields
d1
d
2 2
d1
d1
(u z j,n j −1 ) = 2u d1 ˙u d1 z j,n j −1 + u (−ρ j,n j −3 u d1 z j,n j −2 + u d1 z j,n j
dt
)
− ρ j,n j −3 z j,n j −2 ˜u 1 +˜u 1 z j,n j
3 2
=−ρ j,n j −3 u z j,n j −2 + (2˙u d1 u d1 z j,n j −1 + u (−ρ j,n j −3 z j,n j −2 ˜u 1
d1 d1
3
+˜u 1 z j,n j ) + u z j,n j ) (7.58)
d1
where the first term is uniformly continuous since its time derivative is bounded,
the other terms tend to zero. From the extended version of Barbalat’s Lemma,
3
2
(d/dt)(u z j,n j −1 ) tends to zero. Therefore, u z j,n j −2 and u d1 z j,n j −2 tend
d1 d1
to zero.
2
Taking the time derivative of u z j,i ,2 ≤ i ≤ n j −2 and repeating the above
d1
procedure iteratively, one obtains that u d1 z j,i ,2 ≤ i ≤ n j tends to zero. From
being a linear combination of z j,i ,2 ≤ i ≤ n j ,
(7.25) and considering L h1 z j,n j
we know u d2 , Z 2 tends to zero.
˙
Differentiating u d1 z j,i ,2 ≤ i ≤ n j − 1 yields
d
(u d1 z j,i ) =˙u d1 z j,i + u d1 ˙z j,i
dt
˙
u
= z j,i h + (−k u 1 d1 z j,i − k u 1 1 z j,i + u d1 ˙z j,i )
˜ u
where the first term is uniformly continuous, the other terms tend to zero. From
˙
the extended version of Barbalat’s Lemma, z j,i h tends to zero. By condition (ii)
in Assumption 7.3 on h, it can be concluded that z j,i tends to zero, which
leads to h tending to zero. By examining (7.30), noting ˜u 1 tending to zero and
condition (i) in Assumption 7.3, z 1 tends to zero. From (7.25) and condition (i),
u d1 tends to zero, therefore u = u d +˜u tends to zero. The theorem is
proved.

Remark 7.2 In Theorem 7.2, it has been proven that Z is globally asymp-
totically stabilizable, and all the signals in the closed-loop are bounded.
Accordingly, we can only claim the boundedness of the estimated parameters
and no conclusion can be made on its convergence. In general, to guarantee the
convergence of the parameter estimation errors, persistently exciting traject-
ories are needed [31,36], which is hard to meet in practice. Therefore, for the
globally asymptotical stability of Z, it is an advantage to remove the stringent
requirement of persistent excitation conditions for parameter convergence in
actual implementation.

© 2006 by Taylor & Francis Group, LLC

FRANKL: “dk6033_c007” — 2006/3/31 — 16:43 — page 284 — #18