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Then, (
54
) becomes
e
1
b
V
3
¼
C
2
l
k
1
e
2
k
2
e
3
ð
c
1
þ
a
6
Þ
c
ð
57
Þ
þ r
1
e
r
2
t
þ r
3
e
r
4
t
One can write (
57
) as follows
V
3
k
2
kk
E
þ1ð
t
Þ
ð
58
Þ
n
o
h
i
T
e
3
; C
l
Þ;
c
; k
2
; k
3
where
k
¼
min
ð
c
1
þ
a
6
;
E
¼
e
1
;
e
2
;
, and
1ð
t
Þ
¼
r
1
e
r
2
t
þ r
3
e
r
4
t
.
Note that
1ð
t
Þ
veri
es the following nice properties:
1ð
t
Þ2
L
1
and lim
t
!1
1ð
t
Þ
¼
0
L
2
Those properties will be exploited later in the stability analysis.
1ð
t
Þ2
5.1 Study of the Tracking Error Convergence
The study of the asymptotic convergence of tracking errors is divided into three
parts.
5.1.1 Proof of the Boundedness and Square Integrability
of the Tracking Errors
V
3
can be rewritten as
V
3
k kk
2
By inequality (
58
),
þr
1
þ r
3
. Choosing
V
3
k
[
r
1
þr
3
v
for any small
v
[
0,
there exists a constant
k
0
such that
2
2
k
0
E
for all t
T.
This implies that the tracking errors are uniformly ultimately bounded (UUB), i.e.
ð
e
1
;
e
2
;
e
3
; C
l
Þ2
L
1
(Khalil
2001
). According to the standard Lyapunov theorem,
we conclude that
h
1
; j
1
; h
2
and
kk
0 for all E
kk
[
v
. Thus, there is a T
[
0, such that E
kkv
\
j
2
are all UUB. The boundedness of
h
1
; j
1
; h
2
and
j
2
is respectively established from that
h
1
; j
1
; h
2
and
j
2
. Also, From (
58
) and
ð
e
1
;
e
2
;
e
3
; C
l
Þ2
L
2
.
since
1ð
t
Þ2
L
2
, one can easily show that
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