Int
ern
at
i
onal
Journ
al of
R
obot
ic
s
and
Autom
ati
on (I
JRA)
Vo
l.
8
,
No.
3
,
S
eptem
ber
201
9,
pp.
174
~
183
IS
S
N:
20
89
-
4856,
DOI: 10
.11
591/
i
jra
.
v
8
i
3
.
pp17
4
-
183
174
Journ
al h
om
e
page
:
http:
//
ia
escore.c
om/j
ourn
als/i
ndex.
ph
p/IJRA
Optimi
zation
an
d comp
arativ
e analysis
of
PID
and
FOPID
co
nt
rolle
r for
BL
DC
m
otor
P.
Vimala
1
, C.
R.
Bal
amu
ru
gan
2
, A.
Subr
am
an
ia
n
3
, T.
Vishwan
ath
4
1
,3,4
Depa
rtment
o
f
EEE,
IFET
Col
le
ge
of Engineer
ing,
Vi
ll
upura
m
,
India
2
Depa
rtment of
EE
E
,
Karp
aga
m
Coll
ege o
f
Enginee
ring
,
Co
imbatore
,
Indi
a
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
A
pr
22
, 201
9
Re
vised
Jun
24
, 201
9
Accepte
d
J
ul
10
, 2
01
9
The
FO
PID
and
PID
con
trol
l
er
are
design
ed
to
con
trol
the
spee
d
of
the
BLDC
m
oto
r.
The
par
ameters
k
p
,
k
i
,
k
d
,
λ
and
µ
of
th
ese
cont
ro
ll
er
a
r
e
opti
m
iz
ed
b
ase
d
on
gen
et
i
c
al
g
orit
hm
.
The
op
t
imize
d
co
eff
icie
nts
ke
ep
in
tra
ck
with
ze
ro
err
or
signa
ls.
T
he
outpu
t
of
th
e
con
trol
l
er
is
g
ive
n
to
th
e
var
ia
b
le
dc
sour
ce
whi
ch
v
aries
the
input
vo
ltag
e
to
th
e
thr
ee
ph
ase
i
nv
erter
depe
nding
on
th
e
input
signal.
T
he
thre
e
ph
ase
i
nver
te
r
giv
es
th
e
vo
lt
ag
e
to
the
BLDC
m
otor
which
e
nhanc
es
th
e
stabi
lit
y
of
t
he
s
y
st
em.
The
eff
e
ct
iv
ene
s
s of
the c
on
trol
l
e
r
is de
m
onstra
ted b
y
sim
ula
t
ion.
Ke
yw
or
d
s
:
BLDC m
oto
r
FO
P
ID
PI
D
Stabil
it
y
Copyright
©
201
9
Instit
ut
e
o
f Ad
vanc
ed
Engi
n
ee
r
ing
and
S
cienc
e
.
Al
l
rights re
serv
ed
.
Corres
pond
in
g
Aut
h
or
:
P.
Vim
a
la
,
Dep
a
rtm
ent o
f EEE
,
IF
ET
Colle
ge
of Enginee
rin
g,
Vill
upur
am
, I
ndia
.
Em
a
il
:
vi
m
a
la
.eee@gm
ai
l
.co
m
1.
INTROD
U
CTION
Brushle
ss
dc
m
oto
r
has
wi
ndin
gs
in
t
he
st
at
or
a
nd
pe
rm
anen
t
m
agn
et
in
the
ro
t
or.
A
s
it
do
es
not
hav
e
brus
hes
and
c
omm
uta
t
or
the
ef
fici
en
cy
of
the
m
ot
or
inc
reases
with
decr
ea
se
in
t
he
ohm
ic
loses
.
Brushle
ss
dc
m
oto
r
are
a
va
il
able
in
si
ngl
e
phase,
tw
o
and
th
ree
ph
a
se
co
nf
i
gurati
on.
B
ut
thre
e
phase
config
ur
at
io
n
are
com
m
on
ly
us
e
d.
Gen
e
rall
y,
Hall
Ef
fect
sens
or
s
a
re
us
e
d
to
se
ns
e
t
he
po
sit
io
n
of
t
he
ro
t
or
.
But
in
this
proj
ect
t
he
r
otor
posit
ion
is
de
te
rm
ined
f
r
om
excit
at
ion
c
urren
t
a
nd
vol
ta
ge
acr
os
s
the
sta
to
r
windin
gs
.
The
BLDC
m
otor
re
quires
in
ve
rter
as
the
re
is
no
brus
h
an
d
c
omm
ut
at
ion
a
rr
a
nge
m
ent.
The
in
ver
te
r
use
s
transisto
r
f
or
lo
w
dri
ve
a
ppli
cat
ion
a
nd
T
hyristo
r
f
or
high
po
wer
dr
i
ves
.
MOSF
ET/D
I
OD
E
switc
hes
are
use
d
i
n
t
he
i
nv
e
rter.
The
c
omm
uta
ti
on
se
qu
ence
t
o
t
he
M
OS
FE
T/D
IOD
E
de
pe
nds
on
t
he
ro
t
or
po
sit
io
n.
BLD
C
m
oto
r
re
qu
ir
e
le
ss
m
ai
ntenan
ce,
ge
ner
at
e
le
ss
noise
,
ha
ve
low
ine
rtia
,
hav
e
hi
gh
ef
fici
ency,
has
l
ong o
per
at
ing
li
fe
.
The
c
on
t
ro
ll
er
is
us
e
d
to
im
pr
ov
e
th
e
dy
nam
ic
response
of
the
syst
e
m
.
The
P
I
D
an
d
F
OPID
con
t
ro
ll
er
pa
ra
m
et
ers
are
t
uned
by
gen
et
ic
al
gorithm
[1
]
,
m
or
e
opti
m
izati
on
te
c
hn
i
ques
are
us
e
d
to
tun
e
the
c
oeffici
ent
s
ofF
OPID
a
nd
PID
c
ontr
oller
[
2
]
.
I
n
this
pap
e
r
the
proc
ess
to
c
on
t
ro
l
is
s
peed
of
the
BLDC
m
oto
r.
The
B
LDC
m
oto
r
is
m
od
el
ed
[
3
,
4
]
.
T
he
F
OPI
D
c
on
t
ro
ll
er
t
r
ansf
e
r
functi
on
is
form
ed
by
us
in
g
FO
MC
O
N
to
ol
box
[
5
,
6
]
.
T
he
ge
netic
al
gorithm
opti
m
i
zat
ion
to
tu
ne
P
ID
a
nd
FOPID
is
done
by
su
i
ng
op
ti
m
iz
ation
to
olbox.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
R
ob
&
A
uto
m
IS
S
N:
20
89
-
4856
Op
ti
miz
atio
n a
nd com
pa
r
ativ
e ana
ly
sis
of P
ID
and F
OPID
controll
er fo
r
BLD
C
m
oto
r
(
P.
Vim
ala
)
175
2.
SPEE
D CO
N
TROL O
F BL
DC M
OTO
R
The
bl
ock
diagr
am
for
sp
ee
d
c
ontr
ol
of
B
LDC
m
oto
r
is
sh
ow
n
i
n
Fig
ure
1
.
T
he
r
oto
r
posit
io
n
a
n
d
sp
ee
d
of
t
he
B
LDC
m
oto
r
is
com
pu
te
d
f
rom
the
volt
age
and
c
urren
t
s
upplied
t
o
t
he
sta
tor
wi
nd
i
ng
s
of
BL
DC
m
oto
r.
Eac
h
phase win
ding
is
excit
e
d
in
a
s
equ
e
nce
to
r
un
the
BLDC
m
oto
r.
T
he
phase win
ding o
f
the
BLDC
m
oto
r
is excit
e
d
in
a se
quence
b
y se
quence
c
omm
uta
ti
on
of the M
OS
FE
T/DI
ODE in
the i
nv
e
rter
[
7
,
8
]
.
Figure
1
.
Bl
oc
k diag
ram
o
f
th
e pro
posed
sys
tem
That
s
witc
hing
seq
uen
ce
is
gi
ven
t
o
the
i
nverter
f
ro
m
posit
ion
to
gate
s
ign
al
blo
c
k
a
s
show
n
i
n
F
igure
1.
T
he
sp
eed
of
the
BLDC
m
oto
r
is
directl
y
propo
rtion
al
to
the
volt
age.
T
her
e
fore,
t
o
con
t
ro
l
the
sp
ee
d
of
t
he
BLDC
m
oto
r,
the
i
nput
vol
ta
ge
m
us
t
be
con
t
ro
ll
ed
.
T
he
sp
ee
d
com
pu
te
d
is
com
par
e
d
with
the
refe
re
nce
s
peed
by
the
c
om
par
at
or
an
d
t
he
er
ror
si
gn
al
is
ob
ta
ine
d.
T
he
er
ror
si
gn
al
is
giv
e
n
as
in
pu
t
to
the
co
ntr
oller.
The
outp
ut
of
the
c
ontr
oller
is
c
on
tr
ol
sign
al
.
This
c
on
t
ro
l
si
gn
al
is
gi
ven
as
i
n
pu
t
t
o
the v
a
riable
D
C so
ur
ce
.
T
here by the
in
pu
t
DC volt
age
is
con
t
ro
ll
ed
ba
se
d on the BL
DC
m
oto
r
sp
ee
d.
2
.
1.
PI
D
c
ont
rolle
r:
The
PID
co
ntr
oller
c
onsist
s
of
the
propo
rtion
al
,
inte
gr
al
and
der
i
vative
con
t
ro
ll
e
r.
H
e
pro
portion
a
l
con
t
ro
ll
er
giv
e
s
propo
rtion
al
re
sp
onse
of
the
in
pu
t
er
ror
va
lue.
T
he
in
te
gr
al
c
ontr
oller
is
pro
port
io
nal
t
o
m
agn
it
ud
e
a
nd
the
durati
on
of
t
he
e
rror.
It
i
m
pr
oves
t
he
set
tl
ing
po
i
nt
an
d
el
im
inates
ste
ady
sta
te
error.
The
i
nteg
ral
t
erm
giv
es
res
pons
e
w
hich
a
lso
depen
ds
on
t
he
s
um
of
pr
e
vious
er
ror
val
ues.
T
he
s
um
of
the
pr
e
vious
e
r
ror
value
is
m
ulti
plied
with
t
he
gain
ki.
T
he
de
rivati
ve
co
nt
ro
l
giv
es
the
s
lop
e
of
t
he
e
rror
f
or
sp
eci
fied
du
rati
on
a
nd
m
ulti
ples
it
with
the
gain
kd.
Th
e
su
m
of
this
pr
oport
ion
al
,
int
egr
al
a
nd
der
i
vative
con
t
ro
l
giv
es
t
he
P
ID c
on
tr
ol
le
r.
T
he
c
on
t
rol
sign
al
of P
ID
con
t
ro
ll
er i
n
ti
m
e d
om
ai
n
.
(
)
=
(
)
+
∫
(
)
0
+
(
)
Takin
g
La
plac
e trans
f
or
m
o
f a
bove
e
quat
io
n, the t
ran
s
fer f
unct
ion o
f PI
D
c
on
t
ro
ll
er is
g
i
ve
n by,
(
)
=
(
)
(
)
=
+
+
Wh
e
re,
k
p
is pr
oport
io
nal g
ai
n,
k
i
is i
nteg
ral
gain,
k
d
is
der
ivati
ve
g
ai
n,
e
(t)
is a
n
e
rror sig
nal
.
The
blo
c
k
diag
ram
of
PID
c
ontr
oller
is
sho
wn
i
n
F
i
gure
2.
The
pa
ram
et
e
r
k
p
,
k
i
,
k
d
ha
ve
to
be
tun
e
d
by
us
in
g
ge
netic
al
gorithm
.
T
he
PID
c
ontr
oller
ca
n
im
ple
m
ent
in
MA
TL
A
B/
Si
m
ulink
by
PI
D
blo
c
k
a
vai
la
ble
in conti
nuous
tim
e too
lbox
.
Figure
2
.
Bl
oc
k diag
ram
o
f
P
ID
co
ntr
oller
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2089
-
4856
I
nt
J
R
ob
&
A
uto
m
,
Vo
l.
8
, No
.
3
,
Septem
ber 2
019
:
174
–
183
176
2
.
2.
FOP
ID
c
on
t
rolle
r
Fr
act
io
nal
ord
er
PID
c
on
t
rol
le
r
is
com
mo
nly
in
th
e
f
or
m
P
I
λ
D
µ
.
In
this
the
integ
rato
r
an
d
diff
e
re
ntiat
or
i
s
in
the
or
der
of
λ
an
d
µ
res
pecti
vely
[
9
]
.
The
ti
m
e
do
m
ai
n
r
ep
rese
ntati
on
of
FOPI
D
con
t
ro
l
sign
al
is
,
(
)
=
(
)
+
−
(
)
+
µ
(
)
Wh
e
re
,
D
−
λ
-
F
rac
ti
on
al
orde
r
i
nt
egr
at
or
,
D
µ
-
F
ra
ct
ion
al
orde
r
di
ff
ere
ntiat
or
.
T
akin
g
La
place
trans
form
fo
r
above e
quat
io
n, the t
ran
s
fer f
unct
ion o
f FO
PID c
on
t
ro
ll
er is
giv
e
n by
,
(
)
=
(
)
(
)
=
+
+
µ
Wh
e
re,
λ
-
Order
of
inte
gr
at
or
,
µ
-
Order
of
di
ff
e
ren
ti
at
or
.
T
he
blo
c
k
diagr
am
of
FOPI
D
c
on
t
ro
ll
er
is
show
n
in
F
ig
ure
3.
Figure
3. Bl
oc
k diag
ram
o
f
F
OP
I
D
c
ontr
oller
3.
DESIG
N OF
BL
DC
MOT
OR
The
r
otor
an
d
sh
aft
of
the
BL
DC
m
oto
r
is
as
su
m
ed
to
be
ri
gid
[
10
]
.
The
m
od
e
is
as
su
m
ed
as
ha
v
in
g
visco
us
fr
ic
ti
on
m
od
el
.
T
herefo
re
the
fr
ic
ti
on
t
orq
ue
is
pr
oport
ion
al
to
the
s
haf
t
a
ngul
ar
vel
ocity
.
Ba
sed
on
New
t
on
sec
ond
la
w
a
nd
Kirc
hhoff
’s
vo
lt
a
ge
la
w,
th
ree
phase
sta
r
co
nne
ct
ed
BL
DC
m
otor
ca
n
be
de
scribe
d
by foll
owin
g
(1
-
4)
.
=
Ө
+
2
Ө
2
+
(1)
=
(
−
)
+
(
−
)
+
−
(2)
=
(
−
)
+
(
−
)
+
−
(
3)
=
(
−
)
+
(
−
)
+
−
(
4)
Wh
e
re
T
e
,
Ө
m
, b
, J,
T
L
,V
, R,
L,
I,
e are
elec
tric
al
tor
qu
e
, m
echan
ic
al
r
otati
on
al
s
pe
ed,
visco
us
fri
ct
ion
const
ant,
the
r
otor
inerti
a,
m
echan
ic
al
loa
d
tor
qu
e
,
phase
to
phase
vo
lt
a
ge,
ph
a
se
resis
ta
nce,
phase
c
urren
t,
ph
a
se in
duct
an
ce an
d ph
a
se
ba
ck
em
f
res
pec
ti
vely
.
The v
oltage a
nd c
urren
t e
quat
ion
is
g
i
ven b
y
(5
-
6)
,
+
+
=
0
(
5)
+
+
=
0
(6)
To
sim
plify t
he
m
od
el
ing
,
on
ly
two
vo
lt
ag
e
equ
at
io
n
a
re
ne
ed,
t
hey are
(7
-
8)
.
2
+
=
3
+
3
+
2
−
−
(7)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
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R
ob
&
A
uto
m
IS
S
N:
20
89
-
4856
Op
ti
miz
atio
n a
nd com
pa
r
ativ
e ana
ly
sis
of P
ID
and F
OPID
controll
er fo
r
BLD
C
m
oto
r
(
P.
Vim
ala
)
177
−
+
=
3
+
3
+
2
−
−
(8)
The
t
orqu
e
g
e
ne
rated
by the
B
LDC m
oto
r
is
giv
e
n by
(
9)
,
=
(
+
+
)
/
Ө
(9)
The
tra
pez
oid
a
l back em
f
can
be writt
en
as
(
10
-
12)
:
=
2
Ө
(
Ө
)
(
10)
=
2
Ө
(
Ө
−
2
3
)
(
11)
=
2
Ө
(
Ө
−
4
3
)
(
12)
Wh
e
re
,
Ө
is t
he back
em
f
con
st
ant and elec
tric
al
an
gle
res
pec
ti
vely
.
Ө
=
Ө
Tra(
Ө
e
)
is t
he
tra
pezo
i
dal w
a
ve
f
or
m
u
nctio
n
a
nd
on
e
p
e
rio
d of t
he fu
nctio
n
c
an be
descr
i
be
d
as
foll
ow
(13)
:
Tra
(
Ө
)
=
{
1
,
0
≤
Ө
<
2
3
1
−
6
(
Ө
−
2
3
)
,
2
3
≤
Ө
<
−
1
+
6
(
Ө
−
5
3
)
,
5
3
≤
Ө
<
2
(
13)
Substi
tuti
ng (1
0
-
12)
i
n (
9)
we
torq
ue
as
f
ollo
ws
(14)
:
=
2
[
(
Ө
)
+
(
Ө
−
2
3
)
+
(
Ө
−
4
3
)
]
(
14)
The
a
ngle
of t
he
rotor an
d spe
ed of
the
ro
t
or
is relat
ed
as
fol
lows
(
15
-
16)
:
=
Ө
(
15)
=
(
16)
Wh
e
re
,
-
Ele
ct
rical
sp
ee
d,
-
Me
cha
nic
al
sp
ee
d
.
F
r
om
the
e
qu
at
i
on
(
1)
an
d
(
15
),
w
e
get
the
sp
ee
d
equ
at
io
n
a
s (1
7)
:
=
−
−
(
17)
Fr
om
(
7)
,
The
current
is
giv
e
n
as
(
18)
:
=
2
+
−
2
+
+
−
3
3
(
18)
Fr
om
(
8)
,
as
=
−
+
−
2
+
+
−
3
3
(
19)
Fr
om
(
6)
,
The
current
is
giv
e
n
as
(
20)
:
=
−
−
(
20)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
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:
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I
nt
J
R
ob
&
A
uto
m
,
Vo
l.
8
, No
.
3
,
Septem
ber 2
019
:
174
–
183
178
4.
INV
E
RTER
TOPOL
OG
Y
The
in
ver
te
r
use
d
h
e
re
is
a
t
hr
ee
ph
ase
in
ve
rter.
It
co
ns
is
t
of
t
hr
ee
le
g
and
si
x
M
OS
F
ET/DI
O
DE.
It
is
im
ple
m
ented
by
us
in
g
unive
rsal
bri
dge
avail
able
in
too
l
box.
T
he
ga
te
sign
al
i
n
t
his
univ
ersal
bri
dge
decides
t
he
M
OS
FE
T
com
m
uta
ti
on
. Th
e
i
nverter
is
s
how
n
in
F
ig
ur
e 4
A
nd
for
Ci
rc
uit
diag
ram
of
dc
m
oto
r
is
sh
ow
n
in
F
i
gur
e
5
.
Figu
re
4. Th
re
e phase M
OS
F
ET/DI
O
DE
i
nv
erter
Figure
5. Ci
rcui
t diagr
am
o
f
DC
m
oto
r
5.
SY
STE
M MO
DELL
ING
Transfe
r
f
unct
ion o
f
BL
DC m
otor: Usi
ng K
i
r
chho
ff’s
vo
lt
ag
e law
(
2
1)
,
=
+
+
(
2
1)
Wh
e
re
,
R,
I,
L,
e
are
t
he
source
volt
age,
r
esi
sta
nce
in
ohm
,
cur
ren
t
to
t
he
m
oto
r,
in
du
ct
ance
of
the
c
oil,
back em
f
resp
e
ct
ively
.
The
t
orqu
e
of t
he dc
m
oto
r
from
second
la
w of m
ot
ion
is
g
i
ven b
y
(2
2
)
,
=
+
+
(
2
2
)
Wh
e
re
,
,J,
,
are
the
el
ect
rical
t
orq
ue,
the
f
rict
ion
co
ns
ta
nt,
t
he
r
oto
r
i
ner
ti
a
the
a
ngular
ve
locit
y
an
d
the s
upposed
m
echan
ic
al
loa
d resp
ect
ively
.
The
el
ect
rical
t
orq
ue
a
nd the
back
em
f
cou
l
d be
wr
it
te
n
as
,
e =
and
=
Wh
e
re
is
bac
k
em
f
con
sta
nt
and
is
torque
const
ant.
F
ro
m
equ
at
io
n
(
2
1)
and
(
2
2)
.
T
he
trans
fer
functi
on
for dc m
oto
r
is
g
ive
n
a
s
(
2
3)
:
(
)
=
=
2
+
(
+
)
+
+
(
2
3
)
Con
si
der
i
ng th
e f
ollow
i
ng ass
um
ption
,
−
The frict
io
n
c
onsta
nt is
sm
all (
=
0
)
−
≫
−
≫
And
m
ulti
plyi
ng
by
×
1
in
(
2
3)
.
The
tra
nsfer
fu
nct
ion o
f dc m
otor is
giv
e
n
as
(
2
4)
:
(
)
=
1
2
+
+
1
(
2
4)
Wh
e
re m
echani
cal
tim
e con
sta
nt
=
Ele
ct
rical
tim
e
const
ant
=
But f
or the BL
DC m
oto
r
=
∑
=
∑
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
R
ob
&
A
uto
m
IS
S
N:
20
89
-
4856
Op
ti
miz
atio
n a
nd com
pa
r
ativ
e ana
ly
sis
of P
ID
and F
OPID
controll
er fo
r
BLD
C
m
oto
r
(
P.
Vim
ala
)
179
Ther
e
f
or
e,
si
nc
e
the
re
is
a
sy
m
m
e
tric
al
arr
a
ng
em
ent
an
d
a
thre
e
ph
a
se
t
he
m
echan
ic
al
and
el
ect
rical
c
on
sta
nt
beco
m
e
(
2
5)
,
=
3
=
3
(
2
5)
Con
si
der
i
ng th
e phase e
ff
ect
(
2
6)
,
=
3
(
−
)
√
3
(
2
6)
Wh
e
re
,
=
(
−
)
√
3
.
From
the
MA
XON
EC
45
FL
AT
BLDC
m
oto
r
da
ta
sh
eet
,
the
va
lue
of
phase
r
esi
sta
nce,
ph
a
se in
duct
an
ce, roto
r
ine
rtia
and m
echan
ic
al
tim
e con
sta
nt
are
s
how
n
in
T
able
1.
Table
1
.
M
AXON EC
45 FL
AT
BL
DC m
oto
r
d
at
as
heet
Para
m
eter
Sy
m
b
o
l
Valu
e
Un
it
Ph
ase re
sis
tan
ce
1
.2
Oh
m
Ph
ase in
d
u
ctan
ce
L
0
.56
0
Mh
Ro
to
r
in
ertia
J
92
.
5
×
10
−
6
2
Mechan
ical ti
m
e
co
n
stan
t
1
7
.1
Ms
Subs
ti
tuti
ng th
e ab
ov
e
v
al
ues i
n
(
2
5)
,
=
155
.
56
×
10
−
3
Fr
om
(
2
6)
,
=
0
.
076
−
,
Substi
tuti
ng th
e v
al
ue
s
of
,
,
in
(
2
4) w
e
g
et
t
he
tra
nsfer
f
unct
ion
of BLDC
m
oto
r
as
(
2
7)
:
1
(
)
=
13
.
11
2
.
66
×
10
−
6
2
+
0
.
01
7
1
+
1
(
2
7)
Transfe
r
f
unct
ion o
f
i
nv
e
rter:
The
tra
nsfer
fu
nction o
f
t
he
in
ver
te
r
is
giv
e
n as
(
2
8)
:
2
(
)
=
1
+
(
2
8)
Delay
tim
e
=
0
.
004
Gain
=
2
=
0
.
65
×
200
20
=
6
.
36
Ther
e
f
or
e
(
2
9)
,
2
(
)
=
6
.
36
1
+
0
.
00
4
(
2
9)
Finall
y, the tra
ns
fe
r funct
io
n of t
he
syst
em
i
s g
i
ven b
y
(
3
0)
,
(
)
=
1
(
)
×
2
(
)
(
)
=
83
.
38
1
.
06
4
×
10
−
08
3
+
7
.
1
06
×
10
−
05
2
+
0
.
0
21
1
+
1
(
3
0)
The param
et
ers
are t
un
e
d f
or
this sy
ste
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2089
-
4856
I
nt
J
R
ob
&
A
uto
m
,
Vo
l.
8
, No
.
3
,
Septem
ber 2
019
:
174
–
183
180
6.
GENET
IC
AL
GOR
IT
HM
Gen
et
ic
al
gorithm
is
an
optim
iz
at
ion
te
ch
ni
que.
This
te
ch
ni
qu
e
is
us
e
d
t
o
t
un
e
the
par
am
et
e
r
of
PID
and F
OP
I
D
c
ontr
oller
[
11, 1
2
]
.
The flo
wc
ha
rt of the
g
e
neti
c algori
thm
is sh
ow
n
in
F
ig
ure
6
.
Figure
6. Flo
w
char
t
of g
e
neti
c algori
thm
It
is
an
it
erati
ve
process
with
the
popula
ti
on
in
eac
h
it
er
at
ion
cal
le
d
a
gen
e
rati
on.
Fi
r
st
the
in
it
ia
l
popula
ti
on
is
gen
e
rated
.
T
he
popula
ti
on
m
ay
r
ang
e
from
hu
ndre
d
t
o
th
ousan
d.
F
ro
m
the
popu
la
ti
on
the
fitness
so
l
ut
ion
to
each
i
ndivid
ual
is
c
hosen
by
ob
j
ect
iv
e
functi
on.
I
n
the
sel
ect
io
n
p
r
ocess
t
he
po
rtion
of
the
ex
ist
in
g
popula
ti
on
is
s
el
ect
ed.
I
n
th
e
cro
ss
over/m
utati
on
,
a
ne
w
ge
ner
at
i
on
is
gen
e
rated.
Finall
y,
it
ch
ecks
w
het
her it
m
ee
t t
he
crit
eria.
I
f
it
m
eet
s the cr
it
eria
it
g
et
s term
inate
o
the
rw
ise
t
he
it
erati
on
re
pe
at
s.
6
.
1.
Op
timi
z
at
ion
of
th
e P
I
D and
FOP
ID
using g
enetic
algorithm
The
ge
netic
a
lgorit
hm
can
be
im
ple
m
ented
by
us
in
g
op
ti
m
iz
ation
t
oo
l
box.
I
n
F
OP
I
D
tu
ning
the
pa
ram
et
er
,
,
,
λ,
an
d
µ
s
ho
uld
be
tu
ne
d
to
the
pr
opos
e
d
s
yst
e
m
[
13
]
.
T
he
fitness
functi
on
sho
uld
be
cal
le
d
as
@
(
x)
f
un
nam
e(x
)
.
Th
e
lo
wer
a
nd
up
per
li
m
it
of
the
c
os
t
f
un
ct
io
n
is
sho
wn
in
t
he
T
a
ble
2
.
The param
et
er o
f
g
e
netic
alg
ori
thm
is cho
se
n a
s in
T
able
3.
Table
2
.
L
im
it
of FOPI
D para
m
et
er
FOPID
para
m
ete
r
Lower li
m
it
Up
p
er
li
m
it
0
2
0
2
Λ
0
1
0
2
µ
0
1
Table
3
.
C
rite
ria of
gen
et
ic
alg
or
it
hm
Po
p
u
latio
n
200
Initial p
o
p
u
latio
n
10
Selection
Sto
ch
astic u
n
if
o
r
m
Mu
tatio
n
Co
n
strain
t dep
en
d
en
t
Cro
ss
o
v
er
Co
n
strain
t dep
en
d
en
t
In
PID
tu
ni
ng,
,
,
are
the
cost
functi
on.
The
lim
i
t
of
P
I
D
c
on
t
ro
ll
er
i
s
c
hose
n
as
i
n
T
a
bl
e
4
.
Si
m
il
arly
by
choosi
ng
c
rite
ria
in
op
ti
m
izati
on
to
olbo
x
the
pa
ram
et
er
are
tu
nes
as
show
n
in
T
a
ble
5.
The gra
phic
al
re
pr
ese
ntati
on i
s g
i
ven
as s
ho
wn in Fi
gure
7.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
R
ob
&
A
uto
m
IS
S
N:
20
89
-
4856
Op
ti
miz
atio
n a
nd com
pa
r
ativ
e ana
ly
sis
of P
ID
and F
OPID
controll
er fo
r
BLD
C
m
oto
r
(
P.
Vim
ala
)
181
Table
4
.
L
im
it
of PID c
ontr
oller
PID
p
ar
a
m
e
ters
Lower li
m
it
Up
p
er
li
m
it
0
2
0
2
0
2
Table
5
.
T
une
d value
of
par
a
m
et
er o
f
c
ontr
oller
Λ
µ
PID
1
.98
1
.78
2
1
.86
9
1
1
FOPID
1
.90
4
1
.78
1
8
1
.70
6
1
0
.18
3
5
0
.22
1
7
Figure
7
.
Com
par
is
on p
l
ot
The
gr
a
ph
ge
ne
rated
wh
il
e
t
un
i
ng
the
par
a
m
et
er
of
F
OPI
D
co
ntr
oller
is
s
how
n
in
F
ig
ur
e
8
a
nd
f
or
PI
D
con
t
ro
ll
er is
shown i
n
F
i
gure
9
.
Figure
8. F
OP
I
D
ste
p
re
spo
nse
b
y
gen
et
ic
al
gorithm
Figure
9. PID s
te
p
res
ponse
by
g
eneti
c al
gor
it
h
m
7.
SIMULATI
O
N AND
RES
U
LT
The
sim
ulati
o
n
m
od
el
of
th
e
sp
eed
c
ontr
ol
of
BL
DC
m
oto
r
is
sh
own
i
n
F
ig
ur
e
10.
The
PID
con
t
ro
ll
er
is
i
m
ple
m
ented
by
us
in
g
P
ID
bl
ock
avail
able
in
MA
TLAB
.
The
tran
sfe
r
f
un
ct
io
n
of
FOPID
i
s
gen
e
rated
b
y
usi
ng FO
MC
O
N
to
olbo
x.
The
outp
ut s
peed r
esp
on
se
for P
I
D
c
on
t
ro
ll
er is
sh
ow
n
in
F
i
gur
e 10
.
An
e
xter
nal
di
sturbance
is
a
pp
li
ed
t
o
the
BLDC
m
oto
r
at
0.
5
se
co
nd.
The
P
ID
c
ontr
oller
co
ntr
ols
the
in
put
volt
a
ge
a
nd
the
B
LDC
m
oto
r
re
aches
t
he
ste
a
dy
sta
te
at
8.5
seco
nd.
Th
us
the
s
et
tl
ing
ti
m
e
of
the
BL
DC
m
oto
r
us
in
g
PID
c
on
t
ro
ll
er
is
8
s
econd.
T
he
out
pu
t
s
peed
re
spon
s
e
of
the
sy
stem
du
e
to
F
OP
I
D
con
t
ro
ll
er is
shown i
n
F
i
gure
11.
An
e
xter
nal
disturbance
is
ap
plied
to
t
he
BL
DC
m
oto
r
at
0.
5
sec
ond.
The
FO
P
ID
co
ntr
ol
le
r
co
ntr
ols
the
in
put
volt
a
ge
a
nd
the
B
LDC
m
oto
r
re
aches
t
he
ste
a
dy
st
at
e
at
1.3
seco
nd.
Th
us
the
s
et
tl
ing
ti
m
e
of
the
BLDC
m
oto
r
us
in
g
F
OPID
c
ontrolle
r
i
s
0.8
sec
ond.
Ther
e
f
or
e
t
he
set
tl
ing
tim
e
of
the
s
pee
d
du
e
to
the
F
OP
I
D
c
ontr
oller
is
10.
5
ti
m
es
faster
than
t
he
P
ID
con
t
ro
ll
er.
Co
m
par
ison
of
P
ID
a
nd
F
OPI
D
s
peed
respo
ns
e is
sho
wn in
F
ig
ur
e
12.
0
0.5
1
1.5
2
2.5
PID
FOP
ID
Kp
Ki
Kd
λ
µ
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2089
-
4856
I
nt
J
R
ob
&
A
uto
m
,
Vo
l.
8
, No
.
3
,
Septem
ber 2
019
:
174
–
183
182
Fi
gure
10
. S
pe
ed resp
onse
du
e to PI
D
c
on
tr
ol
le
r
Figure
11. Spe
ed resp
onse
du
e to FOP
ID c
ontr
oller
Figure
12. C
om
par
ison
of PID a
nd F
OP
I
D spee
d
re
spo
ns
e
In
this
sect
i
on,
it
is
ex
pl
ai
ned
the
res
ults
of
resear
ch
a
nd
at
th
e
sam
e
tim
e
is
gi
ven
the
c
om
pr
ehe
nsi
ve
disc
us
sio
n.
Re
s
ults
can
be
presente
d
in
F
ig
ur
es
,
gr
aphs,
T
a
bles
and
oth
e
rs
that
m
ake
the r
ea
der u
nd
erstan
d
easi
ly
[
14
,
4
]
. T
he dis
cussion ca
n be
m
ade in
se
vera
l su
b
-
c
hap
te
rs.
8.
CONCL
US
I
O
N
The
F
OPID
con
t
ro
ll
er
e
nh
ances
the
dy
nam
ic
resp
on
se
of
t
he
B
LDC
m
oto
r
com
par
ed
to
the
c
onve
ntio
na
l
PID
co
ntr
oller.
T
he
set
tl
ing
ti
m
e
of
the
sp
ee
d
res
ponse
is
ver
y
fast
wh
e
n
we
us
e
FO
P
ID
con
t
ro
ll
er.
S
o
the
pe
rfor
m
ance
of
the
F
OPID
base
d
s
pee
d
c
on
tr
oller
is
bette
r
a
nd
e
ffi
ci
ent.
Stabil
it
y
and
rob
us
tness
of
the
BLDC
m
oto
r
ca
n
be
im
pro
ved.
S
o
thi
s
pro
posed
co
ntr
oller
can
be
exten
de
d
to
ot
he
r
industrial
c
on
tr
ol syst
em
.
REFERE
NCE
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Khubalk
a
r,
A.
S.
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are
,
"D
esign
a
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tuni
ng
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f
fra
ct
ion
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ord
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PI
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cont
roller
for
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cont
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p
ermane
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agnet
brushless
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otor,
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L
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chnol
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“
Design
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a
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Design
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rm
anc
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ev
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at
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Frac
ti
on
al
ORD
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rna
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ct
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i
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p
lj
ak
ov,
Edua
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tlenkov
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“
FO
MCO
N:
Frac
ti
o
nal
-
Order
Mode
li
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Contro
l
Tool
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TL
AB,
”
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2011,
18th
I
nte
rna
ti
ona
l
Con
fer
ence
"M
ixe
d
Design
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gr
at
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C
irc
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I
nt
J
R
ob
&
A
uto
m
IS
S
N:
20
89
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4856
Op
ti
miz
atio
n a
nd com
pa
r
ativ
e ana
ly
sis
of P
ID
and F
OPID
controll
er fo
r
BLD
C
m
oto
r
(
P.
Vim
ala
)
183
[8]
Sundara
vadi
vu
.
K,
B.
Arun,
K.
Sar
ava
nan
,
“
Des
ign
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Frac
t
ional
Order
PID
Cont
roll
er
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qu
id
Le
v
el
Contro
l
o
f
Spheric
a
l
T
ank,”
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te
rna
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o
nal
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ere
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puti
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[9]
Olunda
y
o
john
ogunto
y
inbo,
“
PID
cont
rol
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rushless
dc
m
ot
or
and
ROBO
T
tra
j
ec
tor
y
pl
anning
and
sim
uli
nk
with
MA
TL
AB/
SIM
ULINK
,
”
Technol
og
y
and
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omm
unic
ati
on
,
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009.
[10]
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ee
r
L
.
Sa
le
h
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“
Speed
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DC
Motor
base
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on
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ti
on
al
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PID
Contro
ll
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,”
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[11]
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ojc
ie
chGi
ern
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“
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r
to
Op
ti
m
al
Design
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µ
Frac
t
ional
-
Orde
r
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Controll
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PID
)
For
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Rotor
Si
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“
Tuni
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Appl
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ct
i
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PID
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”
2n
d
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rn
at
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onfe
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aVa
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“
Frac
ti
on
a
l
-
Order
PID
Con
trol
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Design
fo
r
Speed
Contro
l
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”
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hird
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ern
ationa
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e
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imal
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ctional
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ord
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rollers
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par
t
ic
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s
warm
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iz
ati
on
al
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C
m
otor
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y
ste
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,
"
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ce
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rm
ati
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Evaluation Warning : The document was created with Spire.PDF for Python.