Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
Vol.
4, No. 4, Decem
ber
2014, pp. 461~
473
I
S
SN
: 208
8-8
6
9
4
4
61
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJPEDS
Perform
a
nce Ind
i
ces
B
a
sed Opti
mal Tunining Criterion for
Speed Control of DC Drives Using GA
D
eept
i
S
i
ng
h*,
Brij
es
h S
i
ngh
*
*
,
N
i
t
i
n S
i
ng
h*
**
* Depart
em
ent o
f
El
ectr
i
cal
Engi
neering
,
Ashoka
Ins
titute of
Tech
nolog
y
and
Man
a
gem
e
nt, Var
a
nasi-India
** Departement
of Electr
i
cal
Eng
i
n
eer
ing, IIT (B
HU) Varanasi-In
dia
*** Departemen
t of
Electr
i
cal
En
gin
eer
ing, MNNIT, Allah
a
bad-In
dia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
May 27, 2014
Rev
i
sed
Ju
l 2
,
2
014
Accepte
d
J
u
l 25, 2014
This
pap
e
r pr
es
e
n
ts
a fr
am
ework to
car
r
y
out
a s
i
m
u
lation
to
tun
e
th
e spe
e
d
controller gains for known input of DC
drive s
y
stem. The obj
ectiv
e is to find
the optimal con
t
roller gains (proporti
onal
and integr
al) in a closed loo
p
s
y
s
t
em
. Various
perform
ance i
ndices
hav
e
be
en cons
ider
ed
as
optim
al
crit
erion
in this
work. Th
e opt
im
al gain
valu
e
s
have be
en ob
tain
ed
b
y
conventional
an
d Genetic Algor
ithm (
GA) based optimization
methods. The
stud
y
h
a
s been
conducted on a simuli
nk model of thr
ee ph
ase converter
controll
ed d
i
re
ct
curren
t
(DC) dr
ive wi
th curr
ent
and s
p
eed
con
t
r
o
l s
t
ra
teg
y
.
The r
e
sults show that
the GA
based t
unning
p
r
ovided b
e
tter s
o
lutions as
compared to
con
v
ention
a
l op
timization methods
based tunn
ing.
Keyword:
DC m
o
tor
dri
v
es
Gain
s tun
i
ng
Gen
e
tic Algo
ri
th
m
Op
tim
izat
io
n
Perform
a
nce indices
Spee
d c
ont
rol
l
er
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
B
r
i
j
es
h Si
ng
h,
Depa
rtem
ent of Elect
ri
cal
E
n
gi
nee
r
i
n
g,
IIT
(B
HU
) Var
a
nasi, In
dia-
2
2
5
0
0
1
.
Em
a
il: sin
g
h
b
1
9
81@g
m
ail.co
m
1.
INTRODUCTION
The i
n
t
r
od
uct
i
on
of va
ri
abl
e
-
s
pee
d
dri
v
es i
n
crease
s
t
h
e aut
o
m
a
t
i
on, pr
o
duct
i
v
i
t
y
and
effi
ci
ency
o
f
pr
ocess
an
d c
o
nt
r
o
l
i
n
dust
r
i
e
s. Nea
r
l
y
6
5
%
of t
h
e t
o
t
a
l
electric energy
ha
s bee
n
c
o
nsum
ed by
electric m
o
tors
wo
rl
d
-
wi
de. T
h
i
s
i
s
a k
n
o
w
n
fact
t
h
at
t
h
e en
ergy
c
ons
um
pt
i
on ca
n be
re
d
u
ced
by
dec
r
ea
si
ng t
h
e ene
r
g
y
i
nput
or
by inc
r
easing the
efficienc
y
of t
h
e m
echanical transm
ission
during
proce
sses. T
h
e sys
t
e
m
efficiency can
be
increases from
15
to 27
% us
ing varia
b
le
s
p
eed drives
in
place of c
o
nstant speed dr
ives
. At prese
n
t, most of
th
e electric driv
es
(75
-
80
%)
still ru
n
at constan
t
sp
ee
d
.
On
ly so
m
e
s
m
al
ler nu
m
b
ers of d
r
i
v
es
(20-25%) are
use
d
i
n
p
r
oces
s an
d c
o
nt
r
o
l
i
n
d
u
st
ri
es
w
h
o
s
e rat
e
of
c
h
a
nge
o
f
spee
d
and
t
o
rq
ue i
s
vari
e
d
t
o
m
a
t
c
h t
h
e
mechanical load. T
h
ese
dri
v
es are ba
si
cal
l
y
DC
dri
v
es
whi
c
h ha
ve
be
en us
ed i
n
el
e
c
t
r
i
c
t
r
act
i
on.
The
DC
m
o
tors can be
considere
d
as single input and single
output syste
m
s (SISO) syst
em
s.
In these m
o
tors, the
to
rq
u
e
/sp
eed
ch
aracteristics are co
m
p
atib
le with
m
o
st o
f
the
m
echanical loads. T
h
e
m
odern electric drives are
cap
ab
le to
con
t
ro
l th
e sp
eed
an
d
ob
tain
ed
va
riable spee
ds
for t
h
e loa
d
s.
Mainly, an electric drive
has
vari
ous
im
portant
pa
rts suc
h
as elec
tric m
o
tor,
power elect
ro
nic
co
n
v
erter (
P
EC), d
r
iv
e controllers and l
o
ad. A
num
ber o
f
m
o
der
n
i
n
d
u
st
ri
es
requi
re vari
a
b
l
e
speed d
r
i
v
es
for t
h
e
r
e effi
ci
ent
and ec
on
o
m
i
cal operat
i
o
ns. T
h
e
vari
a
b
l
e
spee
d DC
m
o
t
o
rs ha
ve been fre
q
u
e
n
t
l
y
pre
f
er
red
by
t
h
ese
i
n
d
u
st
ri
es. Al
so,
t
h
e br
us
hl
ess
DC
m
o
t
o
rs,
nd
uct
i
o
n m
o
t
o
rs an
d sy
nc
hr
o
n
o
u
s m
o
t
o
rs h
a
ve p
r
o
v
i
d
e
s
a vari
abl
e
s
p
ee
ds w
h
i
c
h i
s
wi
dl
y
used i
n
el
e
c
t
r
i
cal
tractio
n
.
Ho
wev
e
r, th
e
b
e
h
a
v
i
ou
r
of DC
m
o
to
rs with
resp
ect
to
d
yna
m
i
c
lo
ad
in
g co
nd
itio
ns
is g
ood
as
com
p
ared to thses m
o
tors and als
o
there s
p
eed c
ont
rol st
rategies are si
m
p
ler. Consiquenly, the DC
m
o
tors
have al
s
o
bee
n
p
r
o
v
i
d
es a
go
o
d
g
r
o
u
nd t
o
ap
pl
y
t
h
e adva
nce
d
co
nt
r
o
l
al
go
ri
t
h
m
s
i
n
i
t
s
speed c
ont
rol
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
4
,
No
.
4
,
D
ecem
b
er
2
014
:
46
1 – 473
46
2
ope
rat
i
o
ns. T
h
eref
ore
,
DC
d
r
i
v
es usi
n
g DC
m
o
t
o
rs are m
o
re preffere
d as
com
p
aterd to
AC drives i
n
proces
s
and control industries.
N
o
r
m
all
y
clo
s
ed
loop
op
er
ation
w
i
t
h
PI
con
t
ro
ller
s
in
t
h
e inn
e
r
cur
r
e
n
t
loop
an
d
t
h
e
o
u
t
er sp
eed
loop
i
s
em
pl
oy
ed f
o
r spee
d co
nt
r
o
l
.
In
fact
, t
h
e P
r
op
o
r
t
i
ona
l-In
teg
r
al-Deriv
ativ
e (PID) co
ntroll
ers are
widely use
d
i
n
t
h
e
p
r
oces
s
i
n
d
u
st
ri
es e
v
e
n
t
h
o
u
g
h
m
o
re
adva
nce
d
c
ont
r
o
l
t
ech
ni
q
u
es
have
bee
n
de
v
e
l
ope
d. M
o
st
l
y
, t
h
ese
adva
nce
d
c
ont
rol
st
ret
e
gi
es
have
bee
n
use
d
t
o
det
e
rm
in
e th
e
p
a
ram
e
ters of PID con
t
ro
ller in
sing
le in
pu
t
si
ngl
e o
u
t
put
(
S
IS
O) sy
st
em
s. In t
h
i
s
w
o
r
k
,
a com
p
ari
s
i
on has
bee
n
p
r
esent
e
d i
n
bet
w
een a
p
pl
i
cat
ions
o
f
con
v
e
n
t
i
onal
a
nd m
oder
n
o
p
t
i
m
i
zati
on t
ech
ni
q
u
es base
d
o
n
G
r
adi
e
nt
-D
e
cent
an
d Ge
nt
i
c
Al
go
ri
t
h
m
(GA
)
o
n
spee
d co
nt
r
o
l
s
t
rat
e
gi
es o
f
D
C
dri
v
es.
A c
o
nsi
d
e
r
abl
e
n
u
m
ber of
wo
rk
s
on
ap
pl
i
cat
i
o
n
of
G
A
i
n
pr
oc
ess an
d
cont
rol
i
n
d
u
st
r
i
es ha
ve b
een
r
e
po
rt
ed
by
var
i
ous
rec
h
erc
h
e
r
s i
n
di
f
f
ere
n
t
t
i
m
e
fram
e
s. Most
l
y
, t
h
e
GA
base
d
m
e
t
hod
ol
o
g
y
h
a
s bee
n
a
p
pl
i
e
d
fo
r i
d
ent
i
f
i
c
a
t
i
on
of
b
o
t
h
con
tin
uou
s and
discrete ti
m
e
syste
m
s [1]-[4].
First
t
i
m
e
, M
a
n & t
a
ng
was i
n
t
r
o
duce
d
a
ppl
i
cat
i
ons
of
GA i
n
engi
neeri
ng
f
i
el
ds [5]
.
A d
e
si
gn m
e
t
hod
whi
c
h
det
e
rm
i
n
es PI/
P
ID
pa
ram
e
t
e
r
s
of m
o
t
i
on c
ont
rol
sy
st
em
s based
o
n
ge
net
i
c
al
go
ri
t
h
m
s
(GAs
) ha
s
been
prese
n
t
e
d i
n
[
7
]
-[1
2]
. The
s
e p
a
pers
pr
op
ose
an
an
alytical p
r
o
c
edu
r
e to
ob
tain
th
e op
tim
a
l
PI/PID parameters.
The i
m
pl
em
ent
a
t
i
onal
i
ssue
rel
a
t
e
d t
o
pre
m
at
ure co
nve
r
g
ence
o
f
G
A
i
n
som
e
app
l
i
cat
i
ons ha
ve
bee
n
ex
am
in
ed
and
repo
rted
i
n
[13]-[16
]
. Th
ereafter, th
e app
licab
ility o
f
GAs
as an op
tim
iza
tio
n
t
o
o
l
for
pro
cess
cont
rol
l
e
rs a
n
d
t
h
e sol
u
t
i
o
n
of
prem
at
ure co
n
v
er
ges i
n
G
A
base
d o
p
t
i
m
i
z
at
i
on has
bee
n
e
xpl
ai
ne
d i
n
[6]
.
The
GA m
e
t
hod c
a
ns easi
l
y
i
n
terat
e
d wi
t
h
o
t
her o
p
t
i
m
i
zati
on t
ech
ni
q
u
e
s
. A G
A
and
neur
o
-
f
u
zzy
base
d
o
p
tim
izat
io
n
s
h
a
v
e
b
e
en
presen
ted
to
so
lve th
e sp
eed
con
t
ro
l p
r
ob
lem
s
o
f
DC m
o
to
rs [17
]
-[22
]. A
m
u
lti-
ob
ject
i
v
e a
n
d
per
f
o
r
m
a
nce i
n
di
ces
based
o
p
t
im
i
zat
i
on f
o
r
t
u
n
n
i
n
g
of
P
I
D
cont
rol
l
e
rs
ha
v
e
bee
n
si
m
u
l
a
ted
by
di
ffe
re
nt
resear
chers
[2
4]
-
[
3
2
]
.
The s
p
ee
d co
nt
r
o
l
of a l
i
n
ea
r b
r
us
hl
ess
DC
m
o
t
o
r u
s
i
n
g s
o
ft
com
put
i
n
g
bas
e
d
optim
ization for dete
rm
ining the optim
a
l
param
e
ters PI
D cont
roller ha
s been reporte
d in [33]. Recently, som
e
of t
h
e ne
w a
d
v
a
ncem
ent
s
hav
e
bee
n
car
ri
ed
out
i
n
t
h
e
fi
el
d
of
b
r
us
hl
ess
D
C
dri
v
e co
nt
r
o
l
usi
n
g a
d
apt
i
ve
and
r
obu
st con
t
r
o
l
th
eor
i
es [
4
0
]-[4
2
]
. Th
e wor
k
p
r
esen
ts a study o
f
stead
y–state b
e
h
a
v
i
or
of
D
C
m
o
to
r
s
sup
p
lied
fro
m
p
o
w
er co
nv
erters an
d
th
eir in
tegration
to
t
h
e
lo
ad
.
Th
e
p
a
p
e
r
was repo
trted
a co
m
p
arativ
e stud
y of
con
v
e
n
t
i
onal
P
I
co
nt
r
o
l
l
e
rs s
u
ch as P
I
s
p
eed
and c
u
rre
nt
co
nt
r
o
l
l
e
r o
v
er
G
A
ba
sed
PI c
o
n
t
rol
l
e
r
usi
n
g t
r
ansfe
r
fu
nct
i
o
n a
p
p
r
o
ach.
In fact, t
h
e s
p
e
e
d c
ont
rol m
e
thods
of a
DC
dri
v
es
a
r
e si
m
p
l
e
r a
n
d l
e
ss e
xpe
nsi
v
e i
n
co
m
p
ari
s
i
on
t
o
AC dri
v
es. The speed control
of DC
dri
v
es below and above rated spe
e
d
can
also be eas
ily achieved. The two
t
y
pes of co
nt
r
o
l
s
have bee
n
u
s
ed f
o
r co
nt
r
o
l
l
i
ng t
h
e spee
d of DC
d
r
i
v
es
, arm
a
t
u
re cont
r
o
l
and fi
el
d c
o
nt
r
o
l
.
Som
e
t
i
m
e
s, t
h
ese
m
e
t
hods
h
a
ve bee
n
com
b
i
n
ed t
o
y
i
el
ds a wi
der r
a
n
g
e
of s
p
eed c
o
nt
r
o
l
.
Us
ual
l
y
, t
h
e speed
co
n
t
r
o
l
op
er
atio
n of
D
C
dr
iv
es h
a
v
e
b
een classified int
o
three types; sing
le, two
an
d fo
ur qu
adr
a
n
t
op
eratio
ns.
In each
ope
ration a unique set voltage
and curre
nt have
bee
n
applied to th
e ar
m
a
ture and field windi
ng
of DC
m
o
t
o
rs. I
n
t
h
i
s
w
o
r
k
,
t
h
e m
a
in em
phasi
s
has
bee
n
gi
ve
n
o
n
t
w
o
q
u
a
d
ra
nt
ope
rat
i
o
n
of
D
C
dri
v
e
[3
4]
-[
3
6
]
.
I
n
two
q
u
a
d
r
an
t
op
eration
,
a conv
erter–
co
n
t
ro
ll
ed
separately ex
cited
DC m
o
to
r
h
a
s
b
een
u
s
ed
fo
r
ob
tain
ing
th
e
vari
a
b
l
e
spee
d
.
The c
u
r
r
ent
or s
p
ee
d si
g
n
al
s are
pr
oce
ssed t
h
ro
u
gh
a pr
op
o
r
t
i
onal
pl
us i
n
t
e
grat
or
(PI
)
cont
rol
l
e
r t
o
d
e
t
e
rm
i
n
e t
h
e c
ont
rol
c
o
m
m
a
nd
w
h
i
c
h
pr
o
v
i
des a de
si
red
spee
d o
p
ret
i
on
. I
n
t
h
i
s
ope
rat
i
on, t
h
e
co
n
t
ro
l co
mm
a
n
d k
e
p
t
with
in th
e safe limits
. Th
ese co
n
t
rol
comm
ands also re
quire
d
p
r
op
er
scalin
g
[3
7]
an
d
[38
]
. In
t
h
is, if th
e ro
tor ach
i
v
e
s a
reco
mm
e
n
d
e
d
v
a
lu
e t
h
en
th
e con
t
ro
l co
mman
d
will settle d
o
w
n
to a v
a
lu
e
wh
ich
is equ
a
l to
th
e su
m
o
f
lo
ad
t
o
rq
u
e
and o
t
h
e
r m
o
to
r losses. Th
is con
d
isio
n
is req
u
i
red
to
k
e
ep
th
e
m
o
to
r
in
stead
y state con
d
ition
.
Th
e
p
r
o
p
e
r selectio
n
s
o
f
g
a
ins and
tim
e co
n
s
tan
t
s
o
f
th
e sp
eed
an
d
cu
rrent
cont
rol
l
e
rs i
s
a
l
so ut
m
o
st
im
p
o
rt
a
n
t
cri
t
e
ri
on
for m
eet
i
ng t
h
e dy
nam
i
c speci
fi
cat
i
ons o
f
dri
v
es [
39]
-
[
4
5
]
. In
t
h
i
s
w
o
r
k
, t
h
e
desi
g
n
of c
o
nt
r
o
l
l
e
rs an
d t
h
ei
r
im
pl
em
ent
a
t
i
o
n f
o
r
DC
dri
v
e
s
ha
ve bee
n
pr
esent
e
d
.
T
h
e s
y
st
e
m
anal
y
s
i
s
an
d
d
e
si
gn
o
f
t
h
e
m
o
t
o
r d
r
i
v
e
are
kept
i
n
pers
p
e
ctiv
e with
regard
t
o
cu
rrent practice.
T
h
e prese
n
t
wo
rk
us
es t
h
e
per
f
o
rm
ance i
ndi
ces a
s
o
n
e
of t
h
e
opt
im
i
zati
on cri
t
e
ri
o
n
s f
o
r o
p
t
i
m
al
t
unni
n
g
of
P
I
D
co
n
t
ro
llers in
a DC d
r
iv
e sy
ste
m
. Th
e co
ntro
l alg
o
rith
m
s
an
d
an
alysis h
a
v
e
b
een
d
e
velo
p
e
d
to
facilitated
d
y
n
a
m
i
c si
m
u
l
a
tio
n
with
p
e
rso
n
al co
m
p
u
t
ers. Also
, th
e
ap
p
lication
s
of m
o
to
r d
r
iv
es
hav
e
illu
strated with
sel
ect
i
ons
fr
om
i
n
d
u
st
ri
al
e
n
vi
ro
nm
ent
.
2.
PROP
OSE
D
METHO
D
In
t
h
is wo
rk
,
sep
e
rataly ex
ici
t
ed
DC m
o
to
r
d
r
i
v
e h
a
s b
e
en co
nsid
ered
as a test syste
m
m
o
d
e
l. To
i
nvest
i
g
at
e t
h
e
effect
s
o
f
co
n
v
ent
i
o
nal
a
n
d
GA
base
d t
u
n
n
i
ng,
t
h
e M
A
T
L
AB
si
m
u
l
i
nk
m
odel
of a s
e
p
a
rat
e
l
y
exci
t
e
d DC
m
o
t
o
r wi
t
h
spee
d an
d cur
r
ent
cont
rol
l
e
rs h
a
ve bee
n
de
vel
ope
d o
n
t
h
e b
a
si
s of m
a
t
h
em
at
i
c
al
fo
rm
ul
at
i
ons. The m
a
t
h
em
at
ical
and si
m
u
l
i
nk m
odel
s
f
o
r
separat
e
l
y
exci
t
e
d dc d
r
i
v
e s
y
st
em
usi
ng t
r
ansfe
r
fu
nct
i
o
n a
p
p
r
o
ach
have
be
en
di
ssuc
u
sse
d i
n
sub
s
eq
ue
nt
sec
t
i
ons.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Perf
or
ma
nce
I
ndi
ces
Ba
sed
Opt
i
m
al
T
u
ni
ni
ng
C
r
i
t
e
ri
on
f
o
r S
p
ee
d C
o
nt
r
o
l
of
DC
Dri
ves
…
(
D
eept
i
Si
n
gh)
46
3
2.1. Mathematical Concepts
of
Spe
ed Con
t
rol of
DC Motor
usin
g Elec
tromec
hanical
Cover
s
ion
Fo
r sim
p
licit
y, th
e l
o
ad is m
o
d
e
led
as a m
o
men
t
o
f
in
ertia,
J
,
i
n
kg –
m
2
/sec
2
, with a
v
i
sco
u
s frictio
n
coefficient
B
1
i
n
N.m
/
(rad/sec
)
the
n
t
h
e accel
eration torque,
T
a
, i
n
N.m
dri
v
es t
h
e l
o
ad
an
d
i
s
gi
ven
by
:
J(d
ω
/d
t)
+ B
1
ω
m
=
T
e
–
T
1
=
T
a
(
1
)
Whe
r
e
T
1
is the lo
ad to
rqu
e
.
Equ
a
tio
n (1
) co
n
s
titu
tes t
h
e
dyn
amic
m
o
d
e
l o
f
t
h
e
DC m
o
t
o
r
with
l
o
ad.
No
w, the
m
o
to
r equ
a
tio
n can
b
e
rep
r
esen
ted
with
n
e
g
l
ectin
g
all th
e in
itial co
nd
itio
ns as:
I
a
(s
)
=
[V(s
)
–
K
b
ω
m
(s
)
]
/ [R
a
+ sL
a
]
(
2
)
ω
m
(s
)
=
[K
b
I
a
(s
)
–
T
1
(s)
]
/ [B
1
+ sJ
]
(
3
)
These e
q
u
a
t
i
o
ns ca
n be
re
pr
esent
r
e
d
i
n
bl
o
c
k–
di
ag
ram
fo
rm
s as show
n
i
n
Fi
g
u
re
1
[4
3]
. Th
us
, t
h
e t
r
ans
f
er
fu
nct
i
o
ns
ω
m
(s)
/ V(s
)
and
ω
m
(s
)
/ T
1
(s
)
can be
deri
ved
f
r
o
m
bl
ock di
a
g
ra
m
shown
i
n
Fi
gu
re
1.
These t
r
ans
f
er
fuctions a
r
e as:
G
ω
v
(s
)
=
ω
m
(s) / V(s)
=
K
b
/ [s
2
(JL
a
) +
s
(
B
1
L
a
+ J
R
a
) +
(
B
1
R
a
+ K
b
2
)]
(4
)
G
ω
1
(s)
=
ω
m
(s)
/ T
1
(s
)
= - (R
a
+ sL
a
) /
[s
2
(JL
a
) +
s
(
B
1
L
a
+ J
R
a
) +
(
B
1
R
a
+
K
b
2
)]
(5
)
Fi
gu
re
1.
B
l
oc
k
di
ag
ram
of t
h
e D.C
.
m
o
t
o
r
It is a k
n
o
w
n
fact th
at th
e separately–
e
x
c
ited DC m
o
tor is a linear syste
m
. There
f
ore, t
h
e
variation in
sp
eed
du
e to
si
m
u
ltan
e
o
u
s
v
o
ltag
e
i
n
pu
t
an
d
l
o
ad
t
o
rqu
e
d
i
st
u
r
b
a
n
c
e can
b
e
written
as a su
m
o
f
th
eir
respective
indi
vidual s
p
eeds
.
ω
m
(s
)
=
G
ω
V
(s)
V(s
)
+
G
ω
1
(s
) T
1
(s)
(
6
)
The i
n
d
u
ce
d
v
o
l
t
a
ge
due
t
o
fi
el
d fl
ux
an
d s
p
eed ca
n
be
deri
ved
as:
e = K
Φ
f
ω
m
(
7
)
Whe
r
e
e
=
bac
k
e.m
.
f.,
K
= m
o
t
o
r c
onst
a
nt
,
Φ
f
= field
fl
ux
and
ω
m
= m
o
tor s
p
eed.
Usu
a
lly, t
h
e
field
flux
is propo
rtion
a
l to th
e
fiel
d
curren
t i
f
th
e iron
is
no
t saturated
an
d is
represe
n
ted as
.
Φ
f
α
i
f
(
8
)
By su
b
s
titu
ting (7
) i
n
Equ
a
tion
(8), t
h
e sp
eed
is exp
r
essed
as:
(
9
)
Whe
r
e
v
a
nd
i
a
are the applie
d voltage a
nd
arm
a
ture curre
nt, res
p
ectiv
el
y. Fro
m
(9
), it is seen
th
at the ro
t
o
r
spee
d i
s
depe
n
ce on t
h
e a
ppl
i
e
d v
o
l
t
a
ge an
d
fi
el
d cur
r
e
n
t
.
S
i
nce, t
h
e v
o
l
t
a
ge d
r
o
p
i
n
resi
st
i
v
e arm
a
t
u
re
i
s
very
sm
al
l as co
m
p
ared t
o
t
h
e rat
e
d ap
pl
i
e
d v
o
l
t
a
ge an
d t
h
e arm
a
t
u
re cu
rre
nt
b
ecom
e
s a seconda
ry
effect
.
M
o
st
l
y
,
in a curre
nt cont
rol operat
ion, the arm
a
ture curr
ent shoul
d create dom
i
nating effects and to make a
dom
inating armature current
,
an extern
al
r
e
si
st
or has
bee
n
co
nnect
e
d
i
n
series with
armatu
re wind
ing
.
Th
e
spee
d
of
t
h
e m
o
t
o
r
has
bee
n
c
ont
rol
l
e
d
by
va
ry
i
n
g
t
h
e
val
u
e
o
f
e
x
t
e
r
n
al
res
i
st
or i
n
st
ep
wi
se.
As a
n
e
ffec
t
, t
h
e
po
we
r di
ssi
pat
i
on i
n
t
h
e ext
e
rnal
resi
st
o
r
l
eads t
o
l
o
we
r efficiency. T
h
ere
f
ore,
in
the p
r
esen
t work
th
e
con
v
e
r
t
e
r co
nt
rol
has
been
u
s
ed t
o
o
b
t
a
i
n
t
h
e desi
re s
p
ee
d usi
ng
opt
i
m
al
t
unni
n
g
o
f
PI cu
rre
nt
an
d
speed
cont
rol
l
e
rs
. In
t
h
e prese
n
t
w
o
rk
, t
w
o m
e
t
hods, arm
a
t
u
re v
o
l
t
a
ge co
nt
r
o
l
and
fi
el
d cur
r
e
n
t
cont
rol
ha
v
e
been
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
4
,
No
.
4
,
D
ecem
b
er
2
014
:
46
1 – 473
46
4
con
s
idere
d
fo
r
spee
d co
ntr
o
l
of a
DC m
o
tor
[3
6]
. It is
kn
own
t
h
at, th
e app
lied
arm
a
tu
re
v
o
ltag
e
is m
a
i
n
tain
ed
con
s
t
a
nt
du
ri
n
g
fi
el
d c
u
r
r
ent
cont
rol
m
e
t
hod
. T
h
en
t
h
e s
p
ee
d
of
m
o
t
o
r can
be
rep
r
ese
n
t
e
d
as:
ω
m
α
1 /
i
f
(
1
0
)
Th
is eq
u
a
tion
sh
ows t
h
at th
e ro
t
o
r
sp
eed
is in
v
e
rs
ely pr
op
or
tion
a
l to
the f
i
eld
cu
rr
en
t
.
Sin
c
e, by
v
a
rying
t
h
e
field
cu
rren
t, th
e ro
to
r sp
eed is ch
ang
e
d and
if th
e field curren
t is rev
e
rsed
then th
e
ro
tatio
n
a
l
di
rect
i
o
n ha
s
al
so bee
n
c
h
a
nge
d.
The
r
f
o
r
e
, t
h
e s
p
eed
can be
i
n
crea
sed
or
decre
s
s
e
d by
wea
k
en
i
ng
o
r
stran
ghtin
g t
h
e
field
flu
x
. Si
m
ilarly
,
th
e field
curren
t
is main
tain
ed
con
s
t
a
n
t
in
t
h
e arm
a
tu
re con
t
ro
l
meth
o
d
and
t
h
e
spee
d i
s
de
ri
ve
d
fr
om
(9
) as:
ω
m
α
(v – i
a
R
a
)
(
1
1
)
The s
p
ee
d
o
f
dri
v
e ca
n
be
v
a
ry
i
n
g
by
cha
ngi
ng
t
h
e
ap
pl
i
e
d
vol
t
a
ge
acr
oss t
h
e a
r
m
a
t
u
re
wi
n
d
i
n
gs.
Equ
a
tio
n (1
1)
sh
ows th
at th
e rev
e
rsal
o
f
app
lied
v
o
ltag
e
ch
ang
e
s th
e
d
i
rectio
n
o
f
ro
tatio
n of t
h
e m
o
to
r. The
arm
a
ture curre
n
t control m
e
thod
has
a
n
a
d
v
a
n
t
ag
e
to
co
ntr
o
l th
e a
r
ma
tu
r
e
cu
rr
e
n
t
r
a
p
i
d
l
y b
y
adj
u
s
t
in
g the
appl
i
e
d
v
o
l
t
a
g
e
. As a re
sul
t
,
a wi
de ra
n
g
e o
f
spee
d c
ont
r
o
l
i
s
possi
bl
e by
com
b
i
n
i
ng t
h
e arm
a
t
u
re and
fi
el
d
cont
rol
f
o
r sp
e
e
ds bel
o
w an
d
abo
v
e t
h
e rat
e
d spee
d re
sp
ectiv
ely. To
o
b
t
ain
th
e sp
eed
l
o
wer th
an
its rated
sp
eed
,
t
h
e app
lied
arm
a
tu
re vo
ltag
e
is
v
a
ried
wh
ile th
e field
cu
rren
t is k
e
p
t
at its rated v
a
l
u
e i
n
th
is
co
m
b
in
atio
n
.
O
n
t
h
e o
t
h
e
r
h
a
nd
, t
o
o
b
t
ai
n
sp
eed
s
above th
e r
a
ted speed
,
f
i
eld
cur
r
en
t is d
e
cr
eased
wh
ile
k
eep
i
n
g th
e
app
lied
arm
a
tu
re v
o
ltag
e
con
s
tan
t
.
Now, th
e t
o
rqu
e
o
f
th
e m
o
to
r can b
e
driv
ed
as:
T
e
= K
Φ
f
i
a
(
1
2
)
Equ
a
tio
n (1
2)
can
b
e
n
o
rm
ali
zed
if it is d
i
v
i
d
e
d b
y
rated torqu
e
, wh
ich is
ex
pressed
as:
T
er
= K
Φ
fr
i
ar
(
1
3
)
Wh
ere t
h
e ad
ditio
n
a
l sub
s
crip
t
r
den
o
t
e
s
t
h
e
rat
e
d o
r
nom
i
n
al
val
u
e
s
of t
h
e
co
rre
sp
o
n
d
i
ng va
ri
abl
e
s. Henc
e
th
e norm
a
l
i
zed
v
e
rsion
o
f
(1
2) is:
,
(
1
4
)
Wh
ere th
e additio
n
a
l su
b
s
cri
p
t
n
ex
pres
s t
h
e va
ri
abl
e
s i
n
norm
a
l
i
zed t
e
rm
s, co
m
m
only
kno
w
n
as pe
r uni
t
(p.u.) va
riables
.
2.
2.
T
r
ans
f
er Functi
on M
o
d
e
l
i
n
g
o
f
D
C
D
r
i
v
e
S
y
ste
m
In the
prese
n
t case study, a c
onta
n
t field flux has
b
e
e
n
co
ns
id
e
r
ed
fo
r
th
e D
C
mo
to
r
op
er
a
tio
n.
T
h
e
DC
m
o
t
o
r pa
ra
m
e
t
e
rs, rat
i
n
g
and t
h
e m
a
t
h
em
at
i
cal
m
odels of
di
f
f
ere
n
t
s
ubsy
s
t
e
m
s
of t
h
e t
e
st
m
a
odel
are as
fo
llows
DC mo
tor sp
ecifica
tio
n
s
:
DC m
o
to
r inpu
t vo
ltag
e
=
2
20V; Arm
a
tu
re curren
t
ratin
g =
8
.
3
A
; R
a
ted
speed
=
1
470
rp
m;
Arm
a
ture resis
t
ance R
a
= 4
Ω
;
M
o
m
e
nt
of i
n
ert
i
a
J = 0.0
6
07
kg
– m
2
; Arm
a
ture Induct
ance La = 0.072
h
;
Viscous
friction c
o
efficient B
t
=0.
0
8
6
9
N
– m
/
rad/
sec a
n
d T
o
r
q
ue c
onst
a
nt
K
b
=
1.
2
6
V/
ra
d/
sec.
Co
n
verter sp
ecifica
tio
n
s
:
Sup
p
lied
vo
ltag
e
= 23
0
V
,
3
–
ph
ase
A
.
C.;
Fr
eq
u
e
n
c
y = 60
H
z
; Max
i
m
u
m co
n
t
r
o
l input v
o
ltag
e
is ±
10
V. T
h
e spe
e
d ref
e
re
nce v
o
l
t
a
ge has a m
a
xi
m
u
m
of 18
V. t
h
e m
a
xim
u
m
curre
nt
per
m
i
t
t
e
d i
n
t
h
e m
o
t
o
r i
s
2
0
A. Fo
r th
e si
m
u
latio
n
,
th
e tran
sfer fun
c
tion
o
f
all sub
s
yste
m
s
o
f
g
i
v
e
n
plan
t m
o
d
e
l as fo
llo
ws:
Motor
-
Load connecte
d
syste
m
transfer function
(
1
5
)
(
1
6
)
1
22
0.
086
9
0.
044
9
1.
26
4
0
.
0
8
6
9
t
ba
t
B
K
kR
B
2
2
12
11
1
1
,
24
ta
t
a
b
a
t
aa
a
B
RB
R
K
R
B
T
T
JL
JL
J
L
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Perf
or
ma
nce
I
ndi
ces
Ba
sed
Opt
i
m
al
T
u
ni
ni
ng
C
r
i
t
e
ri
on
f
o
r S
p
ee
d C
o
nt
r
o
l
of
DC
Dri
ves
…
(
D
eept
i
Si
n
gh)
46
5
(
1
7
)
So
t
h
e m
o
to
r an
d lo
ad
sub
s
yste
m
tran
sfer fun
c
tio
ns are:
1
12
()
(
1
)
M
o
to
r tra
n
s
f
e
r
fu
n
c
ti
o
n
()
(
1
)
(
1
)
0.
044
9(
1
0
.
7
)
(
1
0.
0
208
)
(
1
0
.
1
0
7
7
)
am
a
Is
s
T
K
V
s
sT
sT
s
ss
(1
8)
()
/
14
.
5
L
o
a
d
t
r
a
n
s
f
e
r
f
unc
t
i
on
()
(
1
)
(
1
0
.
7
)
mb
t
am
sK
B
I
ss
T
s
(
1
9
)
Co
n
verter transfer fu
n
c
tion:
Th
e rated
DC m
o
to
r
vo
ltag
e
requ
ired
is 22
0
V
,
w
h
i
c
h c
o
r
r
e
s
po
n
d
s t
o
a c
o
nt
r
o
l
v
o
l
t
a
ge
o
f
7.
09
V.
(
2
0
)
(
2
1
)
Th
e tran
sfer fun
c
tio
n of t
h
e co
nv
erter is:
(
2
2
)
(
2
3
)
(2
4)
Cu
rren
t
con
t
roller tra
n
s
fer f
u
n
c
tio
n:
(2
5)
(
2
6
)
(
2
7
)
So,
(
2
8
)
(
2
9
)
1
2
0.
10
77
se
c
.
0.
02
0
8
s
e
c
.
0.
7
s
e
c
.
m
t
T
T
J
T
B
1.
35
1.
35
230
31.
05
/
10
r
cm
V
KV
V
V
(m
a
x
)
3
1
0
.
0
5
dc
VV
()
()
()
1
a
r
r
cr
Vs
K
Gs
Vs
s
T
60
/
2
(
t
i
m
e p
e
r
i
o
d
o
f
o
n
e
cy
cl
e)
360
r
T
11
,
s
ec
.
1
.388
.
0
.00
138
sec
.
12
s
ms
f
31.
05
()
1
0
.
0013
8
r
Gs
s
(1
)
()
cc
c
c
Ks
T
Gs
sT
2
0.
10
77
se
c
.
c
TT
2
2
r
T
K
T
0
.
001
38
r
T
0.
10
77
38.
8
2
0
.
00138
K
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
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-86
94
I
J
PED
S
Vo
l.
4
,
No
.
4
,
D
ecem
b
er
2
014
:
46
1 – 473
46
6
(
3
0
)
So,
(
3
1
)
(
3
2
)
(
3
3
)
C
u
rrent
co
nt
ro
l
l
o
o
p
ap
pr
oxi
m
at
i
o
n
:
(3
4)
(
3
5
)
(3
6)
(
3
7
)
(
3
8
)
(
3
9
)
(
4
0
)
So,
(
4
1
)
(
4
2
)
Spee
d
c
o
nt
rol
l
er t
r
a
n
sf
er f
u
n
c
t
i
on:
(1
)
()
s
s
s
s
K
sT
Gs
sT
(4
3)
2
24
1
;
2
ib
s
tm
KK
H
KK
KT
B
T
(4
4)
44
;4
iw
s
TT
T
T
T
(4
5)
1
2.
33
c
c
cr
m
KT
K
KH
K
T
1
38.
8
;
0.
208
se
c;
0.
0
449
;
0
.
3
55
/
cc
K
TK
H
V
A
3
1
.0
5
/
;
0
.7
s
e
c
.
rm
KV
V
T
2
1
.6
3
(
1
0
.1
0
7
)
T
h
erefo
r
e
(
)
c
s
Gs
s
*
()
()
1
ai
ai
Is
K
Is
s
T
1
.
(1
)
fi
i
cf
i
K
K
HK
1
38
.
8
cr
m
c
fi
c
KKK
T
H
K
T
2.
75
c
i
K
3
1
i
fi
T
T
K
31
wh
ere
0.
1
0
9
r
TT
T
0
.
00
27
s
e
c
.
i
T
()
1
i
c
i
K
Gs
sT
2.7
5
T
h
e
r
e
f
or
e
with
c
u
rre
nt l
o
op
a
p
p
r
ox
im
a
tio
n
(
)
1
0
.0
027
c
Gs
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Perf
or
ma
nce
I
ndi
ces
Ba
sed
Opt
i
m
al
T
u
ni
ni
ng
C
r
i
t
e
ri
on
f
o
r S
p
ee
d C
o
nt
r
o
l
of
DC
Dri
ves
…
(
D
eept
i
Si
n
gh)
46
7
4
0.00
27
0
.
00
2
0
.
0
04
7
iw
TT
T
(
4
6
)
2
3.
7
0
ib
tm
KK
H
K
BT
(4
7)
24
1
28
.
7
3
2
s
K
KT
(
4
8
)
4
40
.
0
1
8
8
s
e
c
.
s
TT
(4
9)
28
.
7
3
(
1
0
.
0
1
8
8
)
()
0.
01
88
s
s
Gs
s
(5
0)
Cu
rren
t
tran
sdu
cer
ga
in
:
The m
a
xim
u
m
safe c
o
nt
rol
v
o
l
t
a
ge i
s
18
V
, and
this has
t
o
corres
pond
t
o
th
e
ma
x
i
mu
m c
u
r
r
e
n
t
er
r
o
r
.
Here
in
pre
s
ent
case study, it
has
bee
n
acce
pted as
unity value.
The
r
efore
,
(
5
1
)
Th
e
Ta
cho
-
g
e
nera
to
r tran
sfer fu
n
c
tion
is g
i
ven
in prob
lem”
(
5
2
)
No
w,
de
vel
o
p
a Sim
u
l
i
nk p
l
ant
m
odel
i
n
M
A
TLA
B/SIMULINK
with
th
ese sub
s
yste
m
tran
sfer
fu
nct
i
o
ns,
w
h
i
c
h i
s
s
h
ow
n
i
n
Fi
g
u
re
2
an
d
f
i
nd t
h
e
di
f
f
ere
n
t
si
m
u
l
a
t
i
on r
e
sul
t
s
f
o
r
vari
o
u
s case
s
on
t
h
i
s
pl
a
n
t
m
odel for a
variable s
p
ee
d
ope
ration.
Als
o
, a
n
alyze the
effect
o
f
di
ff
erent
c
o
nt
rol
l
e
r
on
dy
nam
i
cs o
f
DC
m
o
t
o
r co
nt
rol
ope
rat
i
o
n.
(a)
(b
)
Fi
gu
re
2.
Ty
pi
cal
Pl
ant
m
ode
l
wi
t
h
(a)
spee
d a
n
d
(
b
)
cu
rre
nt
co
nt
r
o
l
base
d m
e
t
hod
ol
o
g
i
e
s
2.
3. Pro
b
l
e
m
O
bje
c
tive
and Optimal
Criterion
Th
e m
a
in
o
b
j
e
ctiv
e fun
c
tion
in
th
is
work is to
m
i
n
i
mize t
h
e stead
y state erro
r, rise time, ov
ershoo
t
an
d settlin
g
ti
me d
u
ring
sp
eed
co
n
t
ro
l
o
f
DC d
r
i
v
e.
Th
e ob
j
ective fun
c
tio
n can b
e
rep
r
esen
ted as:
Minimize
(5
3)
Whe
r
e,
1
In th
is, th
e
obj
ectiv
e
fun
c
tion
J
prov
ides an
o
p
e
rating
poin
t
wh
ich is
gen
e
rally a
relatio
n
o
f
fou
r
wei
g
ht
ed t
e
rm
s o
f
P
I
D
co
nt
r
o
l
l
e
r a
n
d
de
pe
ndi
ng
o
n
t
h
e
v
a
l
u
es
of
wei
g
h
t
s
α
1
,
α
2
,
α
3
and
α
4
. T
h
e
weights,
α
1
,
α
2
,
α
3
a
nd
α
4
are th
e wei
g
h
ting
factors
o
f
the stead
y state erro
r, rise ti
m
e
, ov
ersho
o
t
and
settlin
g
tim
e.
In
the
prese
n
t work,
the perform
a
nce indici
es of
ITAE, ISE, IAE, ITSE, a
n
d IT^2SE ha
ve been consi
d
ered as
optim
ization criteria. These
pe
rfor
m
a
nce indicies are as
follows:
1.
0
/
c
HV
A
()
1
Gs
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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94
I
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S
Vo
l.
4
,
No
.
4
,
D
ecem
b
er
2
014
:
46
1 – 473
46
8
a)
Int
e
gral
of
sq
u
a
re
of t
h
e e
r
r
o
r
∞
b)
In
teg
r
al
o
f
th
e
ab
so
lu
te m
a
g
n
itu
d
e
of th
e error
|
|
∞
c)
In
teg
r
al
o
f
ti
me m
u
lt
ip
lied
b
y
ab
so
lu
te erro
r
|
|
∞
d)
Int
e
gral
of
t
i
m
e
m
u
l
t
i
p
l
i
e
d by
sq
uare
d e
r
r
o
r
|
|
∞
e)
In
teg
r
al
o
f
ti
me squ
a
red
m
u
ltip
lied
b
y
squ
a
red
erro
r
^2
∞
The opt
i
m
i
zed
cont
rol
l
e
r par
a
m
e
t
e
rs
have b
een
o
b
t
a
i
n
e
d
b
y
m
i
nim
i
zi
ng
t
h
ese per
f
o
r
m
a
nce
i
n
di
ces
.
Th
e m
a
in
ob
j
e
ctiv
e of th
is wo
rk
is t
o
im
p
r
ov
e th
e p
e
rforman
ce
o
f
th
e test
syste
m
m
o
d
e
l.
3
.
R
E
SEARC
H M
ETHOD
In th
is
wo
rk
, Zig
e
ler-Nicho
l
s,
Grad
ien
t
-d
es
ce
nt
an
d
Genet
i
c
Al
g
o
r
i
t
h
m
based opt
i
m
i
zati
on
t
echni
q
u
es
ha
v
e
bee
n
used
t
o
t
une
t
h
e
PI c
ont
rol
l
e
r
p
a
rameters for th
e
test
m
o
d
e
l of
DC m
o
to
r
d
r
ive with
vari
ous
per
f
o
r
m
a
nce i
ndi
ces
based o
p
t
i
m
izat
i
on cri
t
e
ri
o
n
s. T
h
e res
u
l
t
s
obt
ai
ne
d t
h
r
o
u
g
h
si
m
u
l
a
tion
usi
n
g
propose
d
techniques have be
e
n
com
p
ared wi
th each ot
he
r.
The MATL
AB
PID optimizer tool has bee
n
use
d
t
o
ap
pl
y
t
h
ese opt
i
m
i
z
at
i
on t
echni
q
u
es
fo
r
t
uni
n
g
t
h
e P
I
cont
r
o
l
l
e
rs. T
h
e p
r
o
p
o
se
d sol
u
t
i
o
n m
e
t
hodol
og
y
base
d
pl
ant
m
odel
ha
s
been
sh
ow
n i
n
Fi
gu
re
3.
Fi
gu
re
3.
Pr
o
p
o
se
d s
o
l
u
t
i
o
n
m
e
t
hod
ol
o
g
y
b
a
sed
pl
ant
m
o
d
e
l
3.1. Z
i
eger-Nichols
(Z
-N) PI
D Tuning
Us
i
n
g Tri
a
l and
Error Based
Optimiz
a
tion
In th
e
presen
t
work, t
h
e Zieg
er-Nich
o
l
s (Z-N) tun
i
n
g
h
a
s b
e
en
u
s
ed
t
o
o
b
t
ain th
e in
itial tu
nn
i
ng
v
a
lu
es fo
r PID
con
t
ro
llers
and
th
en
d
e
sign
th
e
con
t
ro
llers for the stud
y
o
f
syste
m
. Once, th
e i
n
itial t
u
ned
val
u
es
o
f
PI
D
param
e
t
e
rs ha
ve
bee
n
obt
ai
n
e
d,
an
d t
h
en
i
t
has
bee
n
o
p
t
i
m
i
zed by
t
h
e
t
r
i
a
l
an
d e
r
r
o
r
m
e
t
hod
.
This m
e
thod is
base
d on calc
u
lation
of c
r
itical gain
K
er
a
nd c
r
i
t
i
cal
peri
od
P
er.
In
itiall
y, th
e in
tegral
ti
m
e
T
i
h
a
s
b
een set to in
fin
ity and
t
h
e d
e
ri
v
a
tiv
e time Td
is to
zero. Th
is h
a
s b
e
en
u
s
ed to
g
e
t
in
itial PID sett
in
g
for
t
h
e t
e
st
sy
st
em
. In Z-
N m
e
t
h
o
d
, o
n
l
y
t
h
e pro
p
o
rt
i
o
nal
cont
rol
act
i
on
wo
ul
d
be use
d
and t
h
e K
p
has b
e
en
increase
d
to a
critical value K
er
w
h
i
c
h has
been e
x
hi
bi
t
e
d t
h
e case
of
s
u
st
ai
ne
d osci
l
l
at
i
ons
of sy
st
e
m
out
p
u
t
.
In
th
is m
e
th
o
d
, if th
e syste
m
o
u
t
p
u
t
do
es not ex
h
i
b
itin
g
th
e su
stain
e
d
oscillatio
n
s
th
en
it
is n
o
t
u
s
efu
l
for th
e
appl
i
cat
i
o
n. T
h
ese are t
h
e
fol
l
owi
n
g
st
eps
t
o
obt
ai
n
t
h
e t
u
ne
d
val
u
e
o
f
P
I
D
param
e
t
e
r fo
r a
gi
ve
n
pl
ant
.
St
eps
1
:
Sub
s
titu
te
T
i
=
∞
and T
d
= 0
f
o
r
r
e
du
cing
th
e co
m
p
lete tr
an
sf
er
fu
n
c
tion
o
f
a clo
s
e loop
trans
f
er
syste
m
.
St
ep 2
:
Ch
eck th
at th
e
syste
m
is
m
a
rg
in
ally stab
le b
y
Rou
t
h
’
s Criterio
n
:
If
syte
m
o
u
t
pu
t
offeres su
stained
o
s
cillatio
n
,
th
e
n
system
is
marg
in
ally stab
le,
Else
no
t m
a
rg
in
ally stab
le.
St
ep 3
:
Determ
in
ed
the v
a
lu
e of
K
p
by Ro
u
t
h
’
s Stabilit
y criterio
n
an
d set
K
p
= Cri
tical g
a
in
K
er
.
St
ep 4
:
C
a
l
c
ul
at
e t
h
e
fre
que
ncy
(
ω
) o
f
sustain
e
d
o
s
cillatio
n
b
y
su
bstitu
tin
g
j
ω
in place
of s
in
characte
r
istic equation.
St
ep 5
:
Calcu
l
ate
th
e
period
o
f
su
stai
n
e
d
oscillatio
n
as
P
er
=
2
π
/
ω
.
St
ep 6
:
Esti
m
a
te th
e param
e
ters o
f
K
p
, T
i
an
d T
d
by
t
h
e sec
o
n
d
Z
-
N f
r
e
que
ncy
m
e
t
h
o
d
.
St
ep 7
:
Ob
tain th
e com
p
le
te tran
sf
e
r
f
unct
i
o
n
of
P
I
D c
ont
rol
l
e
r.
3.
2.
A
Cl
as
si
cal
Op
ti
mi
z
a
ti
o
n
T
echni
qu
e
–
Gra
d
i
e
nt
-De
s
cent
Opti
mi
z
a
ti
on
Al
gori
t
h
m
B
a
sed
A
p
p
r
oac
h
In
fact
, t
h
i
s
i
s
a fi
rst
-
or
de
r o
p
t
i
m
i
zat
i
on
m
e
t
h
o
d
. T
h
e m
a
i
n
i
d
ea
of t
h
i
s
opt
i
m
i
zati
on
m
e
t
hod i
s
t
o
reach the m
i
nima by the s
h
ortest path.
In
order t
o
ac
hi
eve t
h
e s
h
ortest pat
h
, t
h
e steepe
s
t
gra
d
ient
ha
ve
m
oving
d
o
wn
and
th
en th
is will lead
to
reach
th
e min
i
m
a
. Fu
nd
am
en
tally, wh
en
t
h
e grad
ien
t
chan
g
e
s fro
m
p
o
in
t to
poi
nt
t
h
e
n
si
g
n
i
fi
cant
l
y
cho
o
s
e
a ne
w
di
rect
i
on
an
d m
a
ke
changes acc
ordi
ngly to e
n
s
u
re
the steepest
pa
th. In
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Perf
or
ma
nce
I
ndi
ces
Ba
sed
Opt
i
m
al
T
u
ni
ni
ng
C
r
i
t
e
ri
on
f
o
r S
p
ee
d C
o
nt
r
o
l
of
DC
Dri
ves
…
(
D
eept
i
Si
n
gh)
46
9
t
h
i
s
m
e
t
hod, t
h
e m
i
nim
i
zation
of t
h
e er
r
o
r fu
nct
i
o
n has
been achi
e
ve
d by
anal
y
z
i
n
g t
h
e f
unct
i
o
n
of err
o
r
fun
c
tion
.
To fi
n
d
a l
o
cal m
i
n
i
m
u
m
o
f
a fu
nctio
n
,
select the step
s at th
e
cu
rren
t
po
in
t i
n
p
r
op
ortion
a
l
to
th
e
negat
i
v
e
o
f
t
h
e gra
d
i
e
nt
(o
r
of t
h
e a
p
p
r
o
x
i
m
at
e gradi
e
nt
)
fu
nct
i
o
n. T
h
i
s
m
e
t
hod i
s
al
s
o
k
n
o
w
n
as
st
eepest
desce
n
t
,
or t
h
e
m
e
t
hod of st
e
e
pest
desce
n
t
.
Fu
ndam
e
nt
al
l
y
, t
h
e gra
d
i
e
nt
desce
n
t
i
s
based o
n
t
h
e obse
r
vat
i
o
n
th
at if th
e
m
u
lt
iv
ariab
l
e fun
c
tio
n
F(x)
i
s
defi
ned a
nd
di
f
f
er
ent
i
a
bl
e i
n
a n
e
i
g
h
b
o
r
ho
o
d
o
f
a poi
nt
a
, the
n
F(x)
decrease
s
faste
s
t if one goes
from
a
in
th
e
d
i
rection
of the n
e
g
a
ti
v
e
grad
ien
t
of
F
at
a
,
()
Fa
. It
follo
ws
th
at, if
()
ba
F
a
for
0
a small enough
num
b
er, then
F(
a)
≥
F(b)
.
Usin
g
th
is
ob
serv
atio
n, th
e
so
lu
tion
starts with
a gu
ess X
0
fo
r a
local m
i
nim
u
m
of
F,
and consi
d
ere
d
that the
seque
n
ce
X
0
an
d X
1
, such
th
at
1
()
,
0
nn
n
n
XX
F
X
n
. Here
01
2
(
)
(
)
(
)
.
....
....
..
FX
FX
FX
and
d
u
e to th
is th
e sequ
en
ce (X
n
)
conve
r
ges t
o
t
h
e de
sired loca
l
m
i
nim
u
m
.
It
is
also noted t
h
at the
value
of the ste
p
size
γ
is cha
nge
s in every
iteratio
n
with
certain
assu
m
p
ti
o
n
s
on
th
e
fu
nctio
n
F
(f
or
e
x
am
pl
e,
F
convex and
F
Li
psc
h
t
z
) an
d
part
i
c
ul
ar
choice of
γ
(e
.g., c
h
osen
via a line search that satisfies th
e
Wol
f
e conditions). In th
is wa
y the conve
rge
n
ce to
a local
m
i
nimum
can be gua
r
an
teed
. Now, if
th
e
fu
n
c
tion
F
is co
nv
ex
th
en
all th
e lo
cal
m
i
n
i
m
a
h
a
ve also
been
gl
obal
m
i
nim
a
t
h
en i
n
t
h
i
s
case t
h
e
g
r
adi
e
nt
desce
n
t
has
bee
n
c
o
n
v
er
ge
d t
o
t
h
e
gl
o
b
al
sol
u
t
i
o
n
.
Thi
s
p
r
o
cess
h
a
s b
e
en
illu
strated
by Fig
u
r
e
4
.
Here it is assu
m
e
d
th
at th
e fun
c
tio
n
F
has
bee
n
de
fi
ne
d o
n
a
pl
ane
an
d its g
r
aph
has a bow
l sh
ape.
In the
Figure 4, t
h
e c
u
rves
show t
h
e
c
ont
our lines a
n
d t
h
ese a
r
e lies on that
regi
on i
n
w
h
i
c
h t
h
e val
u
e
o
f
F
i
s
const
a
n
t
. The arr
o
w
ori
g
i
n
at
i
n
g at
a poi
nt
sh
o
w
s
t
h
e di
rect
i
on
of t
h
e
n
e
g
a
tiv
e grad
i
e
n
t
at th
at po
in
t. It is
n
o
t
ed
th
at th
e (n
eg
ativ
e)
g
r
ad
ien
t
at
a p
o
i
n
t
is ortho
gon
al to
th
e co
n
t
our
lin
e g
o
i
ng
th
rou
g
h
th
at po
in
t. It h
a
s b
een
seen
th
at g
r
ad
ient d
e
scen
t
lead
s to
th
e b
o
tto
m
o
f
th
e bowl wh
ich
is
th
e po
in
t
wh
ere th
e
v
a
lu
e of t
h
e
fun
c
tion
F
is m
i
n
i
mal.
Fi
gu
re
4.
Il
l
u
st
rat
i
o
n
o
f
gra
d
i
e
nt
de
scent
o
p
t
i
m
i
zati
on t
ech
ni
q
u
e
3.
3.
Gene
ti
c
Al
g
o
ri
thm
Op
ti
mi
z
a
ti
on B
a
s
e
d A
ppr
oac
h
Genet
i
c
al
g
o
ri
t
h
m
s
(GAs)
ha
ve bee
n
ba
sed
on sea
r
ch m
echani
s
um
based o
n
bi
ol
o
g
i
cal
orga
ni
sm
s
whi
c
h h
a
ve
be
en a
d
apt
e
d an
d fl
ou
ri
she
d
c
h
an
gi
n
g
a
n
d h
i
ghl
y
com
p
et
i
t
i
v
e en
vi
r
onm
ent
.
T
h
i
s
can
al
so
b
e
ap
p
lied to op
t
i
m
i
ze th
e p
a
ra
m
e
ters o
f
com
p
lex
n
o
n
-
linear
p
r
o
cess co
n
t
ro
llers. Th
e adp
o
tib
ility of
n
o
n
-
lin
earity in
th
e co
m
p
u
t
atio
nal p
r
o
cess m
a
k
e
s it on
e of
the m
o
re effic
i
ent techni
que
s com
p
ared to othe
r
t
r
adi
t
i
onal
opt
i
m
i
zat
i
on t
echn
i
ques
.
G
A
s
pl
ay
s an im
port
a
nt
r
o
l
e
i
n
p
r
ocess c
ont
rol
appl
i
cat
i
o
ns f
o
r t
h
e
opt
i
m
i
zati
on o
f
param
e
t
e
rs. Thi
s
m
e
t
hod can q
u
i
c
kl
y
sol
v
e t
h
e va
ri
o
u
s
com
p
l
e
x opt
i
m
i
zat
i
on pr
o
b
l
e
m
s
such
as prob
lem
s
o
f
reliab
lity an
d
accu
racy. Thes
e are so
m
e
o
f
t
h
e m
o
j
o
r qu
alities of
GAs are
a)
Gen
e
tic al
g
o
rith
m
s
search
a po
pu
latio
n of
poi
n
t
s in
p
a
rallel, no
t fro
m
a sing
le po
in
t.
b)
Gen
e
tic
algorith
m
s
d
o
n
o
t
requ
ire d
e
ri
v
a
tiv
e in
fo
rm
atio
n
or o
t
h
e
r
au
x
iliary
kn
owledg
e; on
ly
th
e
ob
ject
i
v
e
f
unct
i
on a
n
d c
o
r
r
es
po
n
d
i
n
g
fi
t
n
ess
l
e
vel
s
i
n
fl
ue
nc
e t
h
e
di
rect
i
o
n
of
t
h
e sea
r
c
h
.
c)
Gen
e
tic al
g
o
rith
m
s
u
s
e pro
b
a
b
ilistic tran
sitio
n ru
les,
no
t determin
istic ru
les.
d)
Gen
e
tic al
g
o
rith
m
s
wo
rk
on
an
en
cod
i
ng of
a p
a
ram
e
ter set n
o
t
th
e
p
a
rameter set itself (ex
cep
t
whe
r
e
real-valued individuals
are
use
d
).
e)
Genet
i
c
al
go
ri
t
h
m
s
m
a
y pro
v
i
de a num
ber of p
o
t
e
nt
i
a
l
sol
u
t
i
ons t
o
a gi
ven
pr
obl
em
and t
h
e
ch
o
i
ce
of th
e fi
n
a
l is left
up
to th
e
u
s
er.
In t
h
e
pre
s
ent
wo
rk
, s
o
m
e
of
t
h
e i
m
port
a
nt
i
ssues
h
a
v
e
be
e
n
co
ns
id
e
r
ed f
o
r
op
ti
miz
i
n
g
th
e p
l
an
t
beha
vi
o
u
r
wi
t
h
pr
o
p
er i
m
pl
em
ent
a
t
i
on o
f
G
A
s
u
ch a
s
deci
si
on
of
p
o
pul
at
i
on si
ze.
M
o
st
l
y
, t
h
e p
o
pul
at
i
on
si
ze
has bee
n
co
nsi
d
ere
d
i
n
bet
w
e
e
n 2
0
t
o
3
0
ch
rom
o
som
e
s. It
i
s
a wel
l
know
n fact
t
h
at
t
h
e bi
g p
o
pul
at
i
o
n
si
ze
co
nsu
m
es
m
o
re co
m
p
u
t
atio
nal ti
me fo
r fi
nd
ing
th
e
o
p
timu
m
so
lu
tio
n
and
th
is m
a
y cau
se o
f
d
e
teriorat
io
n
in
p
e
rf
or
m
a
n
ce of
G
A
. So
m
e
ti
me
s, the
proble
m
of
prem
at
ure
co
n
v
er
ge
n
ce has
bee
n
ar
i
s
ed
due
t
o
i
m
pr
o
p
er
sel
ect
i
on
of
cr
oss
ove
r
rat
e
s.
The
prem
at
ure
co
nve
r
g
ence
pr
o
b
l
e
m
s
have
bee
n
m
i
nim
i
zed
by
co
nsi
d
er
i
ng t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
4
,
No
.
4
,
D
ecem
b
er
2
014
:
46
1 – 473
47
0
recom
m
ended hi
g
h
er rat
e
o
f
cross
o
ver o
f
a
b
o
u
t
8
5
perc
ent to
95
p
e
rcen
t. Th
e low m
u
tatio
n
rate
o
f
abo
u
t
0
.
5
p
e
rcen
t to 1
p
e
rcen
t is
g
e
n
e
rally reco
mmen
d
ed
to ob
tain
opti
m
ized
resu
lts fro
m
GA.
In
fact, th
e m
u
tati
o
n
is
an art
i
f
i
c
i
a
l
and fo
rce
d
m
e
t
hod
of cha
n
gi
n
g
t
h
e num
eri
cal
val
u
e of t
h
e
chrom
o
som
e
. M
u
t
a
t
i
on sh
o
u
l
d
be
avoi
ded as fa
r as possible because it is
totally rand
om
in nature. Sm
a
ll
m
u
ta
tion rates pre
v
ent ge
netic
alg
o
rith
m
s
from fallin
g
in
to lo
cal
m
a
x
i
m
a
o
r
m
i
n
i
m
a
.
Decid
i
ng
o
f
selectio
n
m
e
th
od
fo
r selecting
goo
d
ch
ro
m
o
so
m
e
s is ano
t
h
e
r im
p
o
rtan
t issu
e wh
ile app
l
yin
g
g
e
n
e
tic al
g
o
rith
m
s
fo
r
pro
cess con
t
ro
l app
licatio
ns.
R
a
nk
sel
ect
i
o
n
m
e
t
h
o
d
a
n
d
r
o
ul
et
t
e
w
h
eel
se
l
ect
i
on m
e
t
hod
s ha
ve s
h
ow
n
go
o
d
resul
t
s
o
v
er
ot
her
m
e
t
h
ods
of
selectio
n
.
4.
RESULTS
A
N
D
DI
SC
US
S
I
ON
In
th
e
p
r
esen
t work, conv
en
ti
o
n
a
l and
op
ti
mal
tu
n
i
ng
m
e
th
od
s of PI con
t
ro
ller with
bo
th
con
t
ro
l
strategies (s
pe
ed control and
current
c
ontrol
)
for DC m
o
tor drive system
have
bee
n
considere
d
as test cases.
In t
h
e c
o
nvet
i
onal
t
u
n
n
i
n
g
m
e
t
hod t
h
e t
r
i
a
l
and er
r
o
r
ba
sed Z-
N m
e
t
hod
ol
o
g
y
has
b
een use
d
t
o
t
u
ne t
h
e P
I
cont
rol
l
e
rs
. O
n
t
h
e ot
h
e
r
han
d
, t
h
e
G
r
adi
e
n
t
-Desce
nt
(
G
D
)
an
d
GA
base
d o
p
t
i
m
i
z
at
i
on
m
e
t
hods
ha
ve
bee
n
con
s
i
d
ere
d
fo
r
PI t
u
n
n
i
n
g. T
h
e pe
rf
o
r
m
a
nce of s
u
gge
st
ed
t
echni
q
u
es
ha
s bee
n
t
e
st
ed
on a
DC
m
o
t
o
r d
r
i
v
e
syste
m
. The overall transfe
r
function of syste
m
m
odel
h
a
s b
een
r
e
du
ced
in
to
second
ord
e
r
tr
an
sf
er
f
u
n
c
tion
sy
st
em
m
odel
usi
n
g t
r
u
n
cat
i
o
n base
d m
e
t
hodol
ogy
.
In t
h
i
s
m
e
t
hod, t
h
e
h
i
ghe
r or
de
rs h
a
ve bee
n
ne
gl
ect
ed
fro
m
th
e o
v
e
rall tran
sfer
fun
c
tio
n.
As
a
resu
lt, th
e ov
erall tran
sfer
fu
n
c
tio
n of con
v
erter an
d m
o
t
o
r-load
con
n
ect
ed
sy
st
em
or i
n
ot
her
wo
r
d
t
h
e
pl
an
t
m
odel
f
o
r c
u
rre
nt
co
nt
r
o
l
a
n
d
spee
d c
o
nt
r
o
l
st
rat
e
gy
has
bee
n
gi
ve
n as:
TF
Cur
r
ent
c
o
nt
r
o
l
.
.
.
.
TF
Speed cont
r
ol
.
.
.
.
C
a
se:
1
The PI Tu
ni
n
g
by usi
n
g
c
o
nve
n
t
i
o
n
a
l
t
r
i
a
l
a
n
d
err
o
r b
a
se
d Z-N
met
h
od
ol
o
g
y
The
perform
a
nce of t
h
e system w
ith
Z-N based
tunn
ing
in
th
e con
t
ro
l l
o
op
h
a
s
b
een
t
a
b
u
l
ated i
n
Tabl
e 1.
Tabl
e
1.
Per
f
o
r
m
a
nce o
f
sy
st
e
m
wi
t
h
t
r
i
a
l
and e
r
r
o
r
m
e
t
hod
base
d
o
n
Z
-
N
m
e
t
hod
Perf
o
r
m
a
n
ce p
a
ra
m
e
te
rs
Cu
rre
nt contr
o
l
Speed contr
o
l
Set value (
p
.
u
.)
1
1 p.
u.
Settling ti
m
e
(sec
)
12.228
12.10
Over
shoot
-
-
-
-
K
P
0.
0446
3
0.
0012
5
K
I
0.
5193
9
0.
0251
C
a
se:
2
Opt
i
m
al
t
u
ni
n
g
of
P
I
c
ont
r
o
l
l
er par
a
m
et
er
usi
n
g Ev
ol
ut
i
o
nar
y o
p
t
i
m
i
z
at
i
on
ba
sed
met
h
o
d
(
GD an
d GA
b
a
se
d)
In th
is sectio
n, th
e resu
lts
ob
tain
ed
u
s
ing
GD
an
d
G
A
base
d
opt
i
m
i
z
at
i
on al
go
ri
t
h
m
has bee
n
disscus
sed. T
h
ese res
u
lts ha
ve
been a
n
aly
zed for the
s
m
al
l
e
st
oversh
oot
,
fast
est
ri
s
e
t
i
m
e
and t
h
e fast
est
settlin
g
ti
m
e
re
sp
on
se
o
f
the desig
n
e
d
PI contro
ller fo
r te
st syste
m
. Th
e b
e
st tu
n
e
d
v
a
lu
es h
a
v
e
b
een
sel
ected
fo
r t
h
e sy
st
em
ope
rat
i
o
ns.
T
h
e res
p
on
ses
obt
ai
ne
d
by
G
D
de
si
g
n
ed
PI
and
G
A
desi
gne
d P
I
have
been
com
p
ared. T
h
e Tabl
e 2 an
d
3 sho
w
s t
h
e
per
f
o
r
m
a
nce of sy
st
em
wi
t
h
GD
based
o
p
t
i
m
i
zat
i
on
m
e
t
h
od
fo
r
current
and
s
p
eed loop
res
p
ectivel
y
.
Si
m
i
l
a
rl
y
,
Tabl
e
4 a
n
d
5 s
h
o
w
s t
h
e
per
f
o
r
m
a
nce o
f
G
A
base
d
opt
i
m
i
zati
on f
o
r cu
rre
nt
co
nt
rol
an
d spee
d cont
rol
.
T
h
e re
sul
t
s
sho
w
n i
n
Tabl
e 1 t
o
5 are revel
s
t
h
at
t
h
e G
A
base
d opt
i
m
i
z
at
i
on p
r
o
v
i
d
es
bet
t
e
r sol
u
t
i
o
ns usi
ng I
T
A
E
perf
o
r
m
a
nce i
ndex as an
opt
i
m
al
cri
t
e
ri
o
n
as
com
p
ared
t
o
G
D
base
d opt
i
m
izat
i
on
m
e
t
hod
.
Tabl
e
2.
Per
f
o
r
m
a
nce o
f
sy
st
e
m
usi
ng
GD
ba
sed
o
p
t
i
m
i
zati
on i
n
cu
rre
nt
c
o
nt
r
o
l
Opti
m
i
z
a
tion cri
t
erion
Kp
Ki
Mp (p.u
.)
Ts (Sec.
)
e
2
(t)
∫
e
2
(
t)
ITAE
0.
0390
0.
4962
1.
14
7.
286
0.
72
1.
68
ISE
0.
0781
0.
5227
1.
25
9.
36
0.
74
1.
10
IAE
0.
0504
0.
5088
1.
18
7.
33
0.
72
1.
50
ITSE
0.
0612
0.
5103
1.
2
7.
88
0.
72
0.
70
IT^2SE
0.
0448
0.
4983
1.
15
7.
07
0.
71
0.
70
Evaluation Warning : The document was created with Spire.PDF for Python.