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2.
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ad
an
d
t
h
e p
ar
a
m
e
t
er
s
o
f
t
h
e
r
e
m
ai
n
i
n
g
co
m
p
o
n
e
n
t
s
d
ep
en
d
s
o
n
t
h
ei
r
d
es
i
g
n
.
T
h
e n
o
m
i
n
al
r
an
g
e o
f
t
h
e p
ar
a
m
et
er
s
ar
e as
f
o
l
l
o
w
s
[
6
].
1
0
< K
a <
40;
0.
0
2s
<
T
a
<
1s
1
< K
e <1
0
;
0
.
4
s
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e <
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s
K
g
a
n
d T
g
de
pe
n
ds
on
t
h
e
l
oa
d 0.
7<
K
g
<
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<
T
g
<
2s
0.
01s
<
T
s
<
0
.
06s
a
n
d K
s
=
1
2.
2.
P
a
rt
i
cl
e S
w
a
rm
O
p
t
i
m
i
za
t
i
o
n
P
a
r
t
i
c
l
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s
w
a
r
m
opt
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m
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z
a
t
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on
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e
t
h
od
w
a
s
i
n
t
r
odu
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e
d b
y
K
e
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e
d
y
a
n
d E
be
r
h
a
r
t
i
n
1
995.
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t
i
s
e
vo
l
ut
i
o
na
r
y
opt
i
m
i
z
a
t
i
o
n
t
e
c
h
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i
q
u
e
a
n
d s
t
oc
h
a
s
t
i
c
m
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t
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v
e
l
ope
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obs
e
r
v
i
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g t
h
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s
oc
i
a
l
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m
e
n
t
o
f
s
w
a
r
m
s
s
u
c
h
a
s
f
i
s
h
s
c
h
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i
n
g
a
n
d bi
r
d f
l
oc
k
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ng
.
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h
i
s
m
e
t
h
od i
s
r
obu
s
t
i
n
s
ol
v
i
ng pr
obl
e
m
s
f
e
a
t
u
r
i
ng
n
o
n
li
n
e
a
r
it
y
,
n
o
n
d
i
f
f
e
r
e
n
tia
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ilit
y
,
m
u
ltip
le
o
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ti
m
a
a
n
d
h
ig
h
d
i
m
e
n
s
i
o
n
al
i
t
y
.
I
t
h
as
s
t
ab
l
e co
n
v
er
g
e
n
ce
c
h
a
r
a
c
te
r
is
tic
s
w
it
h
g
o
o
d
c
o
m
p
u
ta
t
io
n
a
l e
f
f
ic
ie
n
c
y
a
n
d
e
a
s
il
y
i
m
p
le
m
e
n
ta
b
le
.
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n
lik
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o
th
e
r
e
v
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et
h
o
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s
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o
l
u
t
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o
n
ar
y
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er
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o
r
s
m
a
n
i
p
u
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e t
h
e
p
ar
t
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cl
e,
each
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ar
t
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cl
e
i
n
P
S
O
f
l
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es
i
n
t
h
e s
ear
c
h
sp
a
c
e
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i
t
h
ve
l
o
c
i
t
y
w
h
i
c
h
i
s
d
y
na
m
i
c
a
l
l
y
a
d
j
us
t
e
d
a
c
c
o
r
d
i
ng t
o
i
t
s
o
w
n
f
l
yi
ng e
xp
e
r
i
e
nc
e
a
nd
f
l
yi
n
g
ex
p
er
i
en
ce o
f
i
t
s
co
m
p
an
i
o
n
s
’
.
A
t
t
h
e
be
g
i
nn
i
n
g
P
S
O
a
l
g
or
i
t
hm
i
n
t
r
odu
c
e
s
’
N
’
num
be
r
of
pa
r
t
i
c
l
e
s
r
a
n
dom
l
y
.
T
h
e
ob
j
e
c
t
i
v
e
f
u
n
ct
i
o
n
v
al
u
e i
s
o
b
t
ai
n
ed
f
o
r
each
p
ar
t
i
c
l
e
.
T
he
n b
a
s
e
d
o
n t
he
f
l
yi
ng
ve
l
o
c
i
t
y
o
f
t
he
p
a
r
t
i
c
l
e
a
nd
i
t
s
gr
o
up
t
he
n
e
w
p
o
p
u
la
tio
n
o
f
p
a
r
tic
le
s
a
r
e
g
e
n
e
r
a
te
d
f
o
r
n
e
x
t g
e
n
e
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a
tio
n
in
s
e
e
k
i
n
g
s
til
l b
e
tte
r
s
o
lu
ti
o
n
.
T
h
e
b
e
s
t v
a
lu
e
o
b
t
ai
n
ed
b
y
t
h
e p
ar
t
i
cl
e s
o
f
a
r
i
s
cal
l
ed
p
b
es
t
an
d
t
h
e b
e
s
t
v
al
u
e o
b
t
ai
n
ed
a
m
o
n
g
al
l
t
h
e p
ar
t
i
cl
es
i
s
cal
l
ed
gb
e
s
t
.
E
a
c
h p
a
r
t
i
c
l
e
i
n t
he
gr
o
up
up
d
a
t
e
s
t
he
i
r
ve
l
o
c
i
t
y b
a
s
e
d
o
n t
he
p
b
e
s
t
a
nd
gb
e
s
t
a
s
gi
ve
n
i
n
e
q
ua
t
i
o
n (
1
)
a
n
d (
2)
.
L
et
u
s
as
s
u
m
e j
t
h
p
ar
t
i
cl
e i
s
r
ep
r
es
en
t
ed
as
x
j
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x
j,
1
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j,
2
,
……x
j,
n
)
i
n
n d
i
m
e
ns
i
o
na
l
s
p
a
c
e
.
T
h
e
pr
e
v
i
ou
s
be
s
t
pos
i
t
i
on of
t
h
e
j
t
h
pa
r
t
i
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l
e
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s
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c
or
de
d a
s
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s
t
j
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p
b
es
t
j,
1
,
pbe
s
t
j,
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…
…
pbe
s
t
j,
n
)
.
T
h
e b
es
t
p
ar
t
i
cl
e am
o
n
g
t
h
e
g
r
o
u
p
i
s
r
ep
r
es
en
t
ed
b
y
g
b
es
t
g
.
T
h
e v
el
o
ci
t
y
o
f
t
h
e p
ar
t
i
cl
e j
i
s
r
ep
r
es
e
n
t
ed
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v
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v
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1,
v
j,
2
…… v
j,
n
)
.
T
h
e cal
cu
l
at
i
o
n
o
f
m
odi
f
i
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d
v
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l
oc
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t
y
a
n
d pos
i
t
i
on
of
e
a
c
h
pa
r
t
i
c
l
e
u
s
i
n
g v
e
l
oc
i
t
y
a
n
d di
s
t
a
n
c
e
t
hr
o
u
gh p
b
e
s
t
j
,
g t
o
gb
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s
t
g i
s
d
o
ne
a
s
s
ho
w
n i
n t
he
f
o
l
l
o
w
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ng
f
o
r
m
ul
a
s
:
V
j,
n
(
t
+1
)
= w
.
v
j,
n
(t
) +
c
1
*
ra
n
d
()*
(p
b
e
s
t
j,
n
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x
j,
n
(t
)) +
c
2
*r
a
n
d
(
)
*(
g
b
e
s
t
g
-
x
j,
n
(t
))
(1
)
x
j,
n
(
t
+1
)
= x
j,
n
(t
)+
v
j,
n
(
t+
1
)
(2
)
j
=
1
,
2
,
……………,
N
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
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2088
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at
i
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p
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2.
3.
F
ra
ct
i
o
n
a
l
O
rd
er s
y
s
t
e
m
s
2.
3.
1.
F
ra
ct
i
o
n
a
l
O
rd
er C
a
l
cu
l
u
s
S
o
m
e
o
f
th
e
p
r
a
c
tic
a
l s
y
s
te
m
s
c
o
u
ld
b
e
w
e
ll d
e
s
c
r
ib
e
d
u
s
in
g
f
r
a
c
tio
n
a
l o
r
d
e
r
d
if
f
e
r
e
n
tia
l e
q
u
a
tio
n
s
r
at
h
er
t
h
a
n
i
n
t
eg
er
o
r
d
er
d
i
f
f
e
r
en
t
i
al
eq
u
at
i
o
n
s
,
i
n
1
6
9
5
,
L
’
H
o
p
i
t
al
co
i
n
ed
t
h
e
w
o
r
d
f
r
act
i
o
n
al
o
r
d
er
cal
cu
l
u
s
[1
1
]
.
S
i
n
ce t
h
e
n
E
u
l
er
,
L
ap
l
a
ce,
F
o
u
r
i
er
,
A
b
l
e
,
R
i
e
m
a
n
n,
a
nd
L
ur
e
l
w
o
r
ke
d
o
n
t
hi
s
.
T
he
r
e
s
e
a
r
c
h o
n
t
he
f
r
act
i
o
n
al
o
r
d
er
cal
cu
l
u
s
i
s
a
ccel
er
at
ed
f
r
o
m
1
8
8
4
.
T
h
e b
as
i
o
p
er
at
o
r
i
n
t
h
e f
r
act
i
o
n
al
o
r
d
er
cal
cu
l
u
s
i
s
d
i
f
f
er
i
n
t
e
g
r
al
.
T
h
i
s
n
a
m
e
h
as
co
m
e b
ecau
s
e a s
i
n
g
l
e o
p
er
at
o
r
r
ep
r
es
en
t
s
t
h
e
f
r
act
i
o
n
al
o
r
d
er
d
er
i
v
at
i
v
e an
d
f
r
act
i
o
n
al
o
r
d
er
i
n
t
eg
r
at
o
r
.
T
h
e d
i
f
f
er
i
n
t
e
g
r
al
i
s
r
ep
r
es
en
t
ed
as
f
o
l
l
o
w
i
n
g
:
a
=
⎩
⎪
⎨
⎪
⎧
(
)
>
0
1
(
)
=
0
∫
(
)
−
(
)
<
0
(
3
)
W
h
e
r
e
‘
a
’
a
n
d
‘
t
’
a
r
e
t
h
e
l
i
m
i
t
s
of
t
h
e
ope
r
a
t
or
.
T
h
e
ope
r
a
t
or
‘
α’
i
s
t
h
e
or
de
r
of
t
h
e
ope
r
a
t
i
on
a
nd
be
l
on
g
s
t
o
R
(
a
ny
r
a
t
i
on
a
l
n
um
be
r
)
bu
t
‘
α’
c
ou
l
d
a
l
s
o
be
a
c
om
pl
e
x
n
um
be
r
[
12]
.
T
w
o de
f
i
n
i
t
i
on
s
u
s
e
d f
or
t
h
e
g
en
er
al
f
r
act
i
o
n
al
d
i
f
f
er
i
n
t
eg
r
al
ar
e t
h
e G
r
u
n
w
al
d
-
L
e
t
n
ik
o
v
(
G
L
)
d
e
f
i
n
itio
n
a
n
d
th
e
R
ie
m
a
n
n
-
L
o
u
v
ill
e
(
R
L
)
d
e
fi
n
i
t
i
o
n
[1
3
],
[1
4
]
.
T
he
G
L
i
s
gi
ve
n he
r
e
:
a
f
(t
)=
l
im
ℎ
→
0
ℎ
−
∑
(
−
1
)
−
ℎ
=
0
f(
t
-
j
h)
(
4
)
T
h
e f
r
act
i
o
n
al
d
i
f
f
er
i
n
t
eg
r
al
d
ef
i
n
ed
b
y
R
L
i
s
a
f
(t
)=
1
(
−
)
∫
(
)
(
−
)
(
−
+
1
)
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I
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8708
IJ
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
20
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8708
IJ
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R
EF
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C
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S
[
1]
J
.
G
.
Zie
g
le
r
a
n
d
N
.
B
.
N
ic
h
o
ls
,
“
O
p
tim
u
m
s
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ttin
g
s
f
o
r
a
u
to
m
a
tic
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o
n
tr
o
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s
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ra
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s.
A
S
M
E
,
vo
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64
(8
)
,
p
p
.
759
/
7
68
–
7
59/
76
8,
19
42
.
[
2]
A
.
V
is
io
li,
“
T
u
n
in
g
o
f
P
I
D
c
o
n
tr
o
lle
r
s
w
ith
f
u
z
z
y
lo
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ic
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”
2001
I
E
E
P
r
oc
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e
di
n
gs
I
n
C
o
nt
r
ol
T
he
or
y
a
nd
A
ppl
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c
at
i
ons
,
vo
l
/
i
ssu
e
:
1
48
(
1
)
, p
p
.
1
-
8,
2
001
.
[
3]
T
.
L
. S
e
n
g
,
et
a
l
.
,
“
T
uni
ng
of
a
n
e
ur
o
-
f
uz
z
y
c
ont
r
ol
l
e
r
by
ge
ne
t
i
c
a
l
g
or
i
t
hm
,
”
I
E
E
E
T
r
ans
ac
t
i
o
ns
o
n Sy
s
t
e
m
s
,
M
an,
a
n
d
C
yb
er
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et
i
cs
,
P
a
r
t
B
:
C
y
b
er
n
et
i
cs
,
vo
l
/is
s
u
e
:
29
(
2
)
,
pp.
2
26
-
2
36,
1
99
9.
[
4]
K
.
Y
av
ar
i
an
,
e
t a
l.
,
“
D
e
s
i
g
n of
I
nt
e
l
l
i
g
e
nt
P
I
D
c
ont
r
ol
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r
f
or
A
V
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y
s
t
e
m
us
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ve
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ur
o F
uz
z
y
I
n
f
er
en
ce S
y
s
t
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m
,
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nt
e
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nat
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al
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our
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al
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om
put
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J
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C
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)
,
v
ol
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ssu
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:
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(
5
)
, p
p.
703
-
71
8
,
20
14
.
[
5]
D.
G
ol
dbe
r
g
,
“
G
en
et
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c
A
l
g
o
r
i
t
h
m
s
i
n
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ear
ch
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t
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m
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zat
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o
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M
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ch
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n
e L
ear
n
i
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g
,
”
A
dd
i
s
on
-
w
is
le
y
,
198
9.
[
6]
Z
.
L
G
a
i
ng
,
“
A
P
a
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t
i
c
l
e
S
w
a
r
m
O
pt
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m
i
z
a
t
i
on A
ppr
oa
c
h f
or
O
pt
i
m
u
m
D
e
s
i
g
n of
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C
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e
r
i
n A
V
R
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y
s
t
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m
,
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E
E
E
T
r
a
ns
ac
t
i
ons
on
E
ne
r
gy
C
o
nv
e
r
s
i
on
,
vo
l
/
i
ssu
e
:
19
(
2
)
,
pp
.
3
84
-
3
91,
2
004
.
[
7]
O
.
Ib
ra
h
i
m
,
e
t a
l.
,
“
P
e
r
f
or
m
a
nc
e
E
v
a
l
ua
t
i
on of
t
hr
e
e
P
I
D
C
on
t
r
ol
l
e
r
T
uni
ng
A
l
g
or
i
t
hm
on a
pr
oc
e
s
s
pl
a
nt
,
”
I
nt
e
r
nat
i
o
nal
J
our
n
al
of
E
l
e
c
t
r
i
c
al
an
d C
om
pu
t
e
r
E
n
gi
ne
e
r
i
n
g
(
IJ
E
CE
)
,
v
ol
/
i
ssu
e
:
5
(
5
)
,
20
15
.
[
8]
D
.
D
ev
ar
aj
an
d
B
.
S
el
v
ab
al
a,
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(
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)
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64
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09
.
[
9]
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2]
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3]
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r
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-
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