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C
l
Þ2
ð
_
e
1
;
_
e
2
;
_
5.1.2 Proof of
e
3
;
L
1
and the Boundedness of All Signals
in the Closed Loop
Because e
1
;
e
3
2
L
1
and x
1d
;
x
3d
2
L
1
, therefore x
1
;
x
3
2
L
1
. From (
14
), one can
write the dynamics of the tracking errors of the stator
fl
fluxes as follows:
e
5
¼
_
a
1
e
5
þ x
s
e
4
þ
e
3
e
4
¼
_
a
1
e
4
x
s
e
5
with e
4
¼
/
sq
and e
5
¼
/
sd
.
From those dynamics and since e
3
2
L
1
, we can easily prove the boundedness
of e
4
, e
5
and x
4
. From x
4
;
x
3d
;
e
1
;
x
1d
;
x
1
2
L
1
, it can be concluded that
t
2
2
L
1
based on (
23
). Because x
2
¼
ð
e
2
þ t
2
Þ=
a
5
x
5
, e
2
; t
2
2
L
1
, x
5
[
0, we can show that
u
sd
and x
5
follows that of x
2
and e
5
. Due to the
boundedness of x
1
;
x
2
;
x
3
;
x
4
;
x
5
; C
l
and since
h
1
; j
1
; h
2
; j
2
2
L
1
, we can conclude
that the controls (u
1
and u
2
)are also bounded. The boundedness of states, reference
signals,
x
2
2
L
1
. The boundedness of
tracking errors and adaptation parameters implies the boundedness of
C
l
(i.e. this implies that
C
l
Þ2
e
1
; _
_
e
2
; _
e
3
;
ð_
e
1
; _
e
2
; _
e
3
;
L
1
:
)
5.1.3 Proof of the Asymptotic Convergence of the Tracking Errors
e
3
,
e
3
; C
l
Þ2
C
l
Þ2
Because
s
lemma (Khalil
2001
), we can conclude that all tracking errors and the estimation
error
C
l
converge asymptotically to zero, despite the presence of the uncertainties
and perturbations.
ð
e
1
;
e
2
;
L
1
\
L
2
and (
e
1
; _
_
e
2
; _
L
1
, and using Barbalat
'
5.2 An Implementable Version of the Load Torque Estimator
Now, let us consider the load torque adaptation law (
28
) that can be written in the
following form
C
l
¼
bC
l
bC
l
c
a
7
e
1
ð
59
Þ
As the actual load torque
C
l
is unknown, the
first equation in (
18
) will be used to
C
l
is given by
compute its value. Consequently,
C
l
¼
_
ð
x
1
þ
a
5
x
5
x
2
a
5
x
4
x
3
þ
a
6
x
1
Þ
ð
60
Þ
a
7
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