TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 12, Decembe
r
2014, pp. 81
4
0
~ 815
1
DOI: 10.115
9
1
/telkomni
ka.
v
12i12.64
70
8140
Re
cei
v
ed
Jul
y
15, 201
4; Revi
sed O
c
tob
e
r 14, 201
4; Acce
pted No
vem
ber 1
2
, 2014
Design of Fractional Order PID Controller for DC Motor
using Genetic Algorithm
Ashu Ahuja*
1
, Bha
w
n
a
T
a
ndon
2
Electrical E
ngi
neer
ing D
e
p
a
rtment, Maharis
hi Mark
a
ndes
h
w
a
r
Eng
i
ne
eri
n
g Coll
eg
e, Mull
ana, Indi
a.
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: ashu.val
e
ch
a
.
ahuj
a@gm
ail.
com
1
, bha
w
n
a _an
eja
@
red
i
ffmail.com
2
A
b
st
r
a
ct
Desig
n
of fracti
ona
l ord
e
r PID (F
OPID) controller for DC
mot
o
r is prop
ose
d
in this p
aper. A
F
O
PID
(PI
λ
D
μ
) is a PI
D contro
ll
er w
hose
deriv
ative
and
inte
gral
or
ders ar
e fracti
o
nal
nu
mbers r
a
ther tha
n
i
n
teg
e
rs.
Desig
n
stag
e of such contro
ller co
nsists o
f
det
ermini
ng
six para
m
eter
s – proporti
on
al consta
nt (K
p
),
integr
al co
nsta
nt (K
i
), derivati
v
e consta
nt (K
d
), filter time constant (
τ
d
), in
tegral or
der (
λ
) and d
e
rivativ
e
order (
μ
). T
he prop
osed a
ppr
oach p
o
ses th
e probl
e
m
as
desi
gni
ng a D
C
motor sp
eed
controller o
n
th
e
conce
p
t of fix
e
d structure
ro
b
u
st
contro
ller
a
nd
mixed
se
nsi
t
ivity H
∞
metho
d
. T
he
unc
ertai
n
ty caus
ed
by
th
e
para
m
eter cha
nges of
motor
resistance,
motor in
d
u
ctanc
e and
loa
d
ar
e formulate
d
as multip
licativ
e
uncerta
inty w
e
ight, w
h
ich are
used in the o
b
jectiv
e f
unctio
n
in the des
ig
n. G
enetic Alg
o
rith
m (GA) an
d
Simulat
ed A
n
n
eali
ng (SA)
are
empl
oyed
to c
a
rry out
th
e af
ore
m
e
n
tion
ed
desi
gn
proce
d
ure. Co
mparis
on
s
are
ma
de
w
i
th a PID w
i
th
de
rivative first
or
der filter
co
ntroller
a
nd
it is
show
n that th
e
prop
ose
d
F
O
PID
controller c
an
highly
im
prov
e the system robustness
wi
th r
e
spect to
m
odel
uncer
tainties
. The
comparis
on
of PID and
F
O
PID control
l
ers
is al
s
o
be
en
d
one
on th
e b
a
s
i
s of T
i
me D
o
ma
in P
e
rfor
ma
nce i
ndex
i.e. ISE
(Integral of Sq
uare Error).
Ke
y
w
ords
:
fr
action
al
ord
e
r
contro
ller,
mi
xed s
ensitiv
ity,
sim
u
la
te
d ann
e
a
l
i
n
g
(SA),
a
l
go
ri
thm
,
ge
ne
ti
c
alg
o
rith
m, PID control
l
er.
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Propo
rtional
–
i
ntegral
–de
rivative (PID)
cont
rolle
r is
the mo
st wi
dely u
s
ed
controlle
r
stru
cture in industri
a
l appli
c
ation
s
[1]. Its stru
ct
u
r
al si
mplicity and
ability to solve many pra
c
tical
control p
r
obl
e
m
s h
a
ve
cont
ributed
to thi
s
wid
e
a
c
cept
ance. In PID
controlle
r the
derivative
an
d
the integral o
r
de
r are in int
eger. F
r
a
c
tio
nal order
PID (FOPID) is a
spe
c
ial
kind
of PID control
l
er
who
s
e
de
riva
tive and
inte
gral
orde
r a
r
e fra
c
tion
al
rather than
in
teger.
The
key ch
allen
g
e
of
desi
gning F
O
PID controlle
r is to dete
r
mine the two
key parame
t
ers
λ
(inte
g
ral orde
r)
and
μ
(de
r
ivative order) ap
art fro
m
t
he usual t
uning p
a
ram
e
ters
of PID
usin
g differe
n
t
tuning meth
ods.
Both
λ
and
μ
are i
n
fractio
n
whi
c
h
in
cre
a
s
e
s
the
ro
bu
stness of th
e
system an
d gi
ves a
n
o
p
timal
control [2-6]. This pa
per p
r
opo
se
s a no
vel tuning method for tuni
ng
λ
and
μ
o
f
FOPID usin
g
geneti
c
algo
ri
thms [7-9].
The sp
eed of
DC moto
r can be adju
s
t
ed to a gr
eat
extent as to provide cont
rollability
easy an
d hi
gh perfo
rma
n
ce [10, 11].
Dun
c
an M
c
Farla
ne [12] in 1992 intro
duced a de
sign
procedure which i
n
corporate loop shaping m
e
t
hods to obtain
performanc
e and robust stability
trade off and
a particula
r H
∞
optimization problem t
o
guarant
ee closed loop
stability. M.
D.
Minkova [
13]
in 199
8 ap
plied a
daptiv
e neu
ral
m
e
thod a
nd A.
A. El-Samah
y [14] in 20
00
descri
bed
ro
bust ad
aptive
discrete vari
able struct
u
r
e cont
rol sch
e
me for spee
d cont
rol of DC
motor. In
DC motor
sp
ee
d control, m
any engi
nee
rs attempt to
desi
gn a
ro
b
u
st
controller to
ensure b
o
th the stability and the pe
rforma
nce of
the system u
nder the p
e
rt
urbe
d co
nditi
ons.
One of the m
o
st pop
ula
r
tech
niqu
es i
s
H
∞
optimal control [10, 15
, 21]
in which
the uncertai
n
ty
and
perfo
rma
n
ce
can b
e
i
n
co
rpo
r
ate
d
i
n
to the
co
ntroller
de
sign.
A multi obje
c
tive formulati
on
[16] is introd
u
c
ed by Ta
pab
rata Ray in 2002. The
con
t
rollers of the
spe
ed that are con
c
eive
d for
goal to
co
ntrol the
spe
ed
of DC moto
r
are
nume
r
o
u
s
: PID
Contro
ller, Fu
zzy Lo
gic
Cont
rolle
r; or
the com
b
inati
on bet
ween t
hem [17]: Au
gmented
Lag
rangi
an Pa
rticle Swarm
O
p
timization [2
0],
Linea
r Matri
x
Inequality [21], Fuzzy-Swa
r
m [
22],
Fuzzy-Neu
r
al Net
w
orks, Fuzzy-G
e
n
e
tic
Algorithm [2
3], Fuzzy-
Ants Colo
ny, Fuzzy-S
lidi
ng mode control [24], Particle Swa
r
m
Optimiz
a
tion [18, 25], Neural Network
[26].
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
De
sign of Fra
c
tional O
r
de
r
PID Controlle
r for DC
Moto
r usin
g Gen
e
tic Algorithm
(Ashu Ahuj
a)
8141
In this p
ape
r, Geneti
c
Alg
o
rithm i
s
e
m
ployed to
de
sign
an
FOP
I
D controll
er
for DC
motor spee
d control. The
prop
osed con
t
roller
i
s
simu
lated with six
tuning pa
ram
e
ters
(K
p
, K
i
, K
d
,
τ
d
,
λ
,
μ
) and it
s pe
rform
a
n
c
e is comp
are
d
with tho
s
e
of an optimall
y
design
ed PI
D controll
er
with
four tuning p
a
ram
e
ters (K
p
, K
i
, K
d
,
τ
d
).
The re
sults
concl
ude that
the FOPID control is abl
e
to
signifi
cantly improve robu
stness of the syste
m with res
p
ec
t to s
y
s
t
em unc
ertainties
.
The pap
er is organi
ze
d as follows. Sectio
n
s
2, 3 and 4 overvi
ew the co
ncepts of
fraction
al cal
c
ulu
s
, Geneti
c
Algorithm
and Simu
late
d Annealing
(SA) Algorith
m
resp
ectivel
y
.
De
sign of
th
e
propo
se
d FOPID controller
fo
r
DC motor
u
s
in
g GA
and
SA, taking both Mixed
Sensitivity and Integral of
Square er
ro
r (ISE) as
co
st function
s is
d
e
scrib
ed in S
e
ction 5. Se
ction
6
is devoted to
com
puter simulatio
n
of the
pr
opo
sed
cont
rolle
r an
d its
comp
ari
s
on
with
a PID
controlle
r. Section 7 con
c
lu
des the p
ape
r.
2. Fractio
nal
Calculus
Fra
c
tional
ca
lculu
s
is
a g
eneralization
of t
he ordin
a
ry cal
c
ul
us.
The chief id
ea is to
develop a fu
nctioni
ng op
e
r
ator
D, asso
ciated to an
orde
r v not limited to integer nu
mbe
r
s,
that
gene
rali
ze
s the o
r
dina
ry concepts
of
d
e
rivative (fo
r
a po
sitive v)
and inte
gral
(for a n
egativ
e v)
[27]. There
are
different
definition
s
f
o
r fra
c
ti
on
al
derivatives.
The mo
st u
s
ual definitio
n
is
introdu
ce
d
by Riem
ann
and
Liouvi
lle [28]
tha
t
gene
rali
ze
s the
follo
wing
definiti
ons
corre
s
p
ondin
g
to integer o
r
de
rs:
N
n
dt
t
f
n
t
x
x
f
D
x
c
n
n
x
,
)
(
)!
1
(
)
(
)
(
1
0
(1)
The gen
erali
z
ed d
e
finitio
n
of D becomes
)
(
x
f
D
v
x
c
. The Lapla
c
e tra
n
s
form of D
pursue
s
the reno
wne
d
rule
)
(
)]
(
[
s
F
S
x
f
D
L
v
v
x
c
for zero initia
l condition
s. This me
ans t
hat, if
zero initial condition
s are assume
d, the sy
stem
s with dyna
mic beh
avio
ur de
scrib
e
d
by
differential
eq
uation
s
in
clud
ing fra
c
tion
al
derivatives
give rise to trans
f
er func
tions with frac
tional
orde
rs of s. More detail
s
a
r
e provide
d
in [29] and [30].
The mo
st co
mmon
way of
usin
g, in bot
h simul
a
tion
s and ha
rd
wa
re implem
enta
t
ions, o
f
transfe
r fun
c
tions in
cludi
n
g
fraction
al o
r
de
rs of
s is to approxim
ate them with usual
(inte
ger
orde
r) tra
n
sf
er fu
nction
s.
To
perfe
ctly app
roxima
te
a fractio
nal
tran
sfer fun
c
tion, a
n
int
eger
transfe
r fu
ncti
on
woul
d h
a
ve to i
n
volve a
n
infinite
num
ber of p
o
le
s a
nd
ze
ro
e
s
.
Nonethel
ess, it
is
possibl
e to o
b
tain logi
cal
approximatio
ns
with a
finite
num
ber
of zeroe
s
a
nd p
o
les. One of the
well
kno
w
n
a
pproxim
ation
s
i
s
cau
s
ed
b
y
Oustal
oup who
uses
the
r
e cu
rsive distribution of
po
les
and zero
es [3
]. The method is ba
sed o
n
the
approxim
ation of a function of the form:
R
s
s
H
,
)
(
(2)
By a rational function:
N
N
k
k
k
s
s
C
s
H
/
1
/
1
)
(
(3)
Usin
g the followin
g
set of synthesi
s
form
ulas,
1
;
;
1
5
.
0
0
5
.
0
0
k
k
u
n
(4)
log
log
;
log
log
;
0
;
0
0
1
N
k
k
k
k
N
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8140 – 81
51
8142
With
u
being
the unit gain
freque
ncy a
nd the ce
ntral freque
ncy
of a band of
freque
ncie
s
geomet
rically distribute
d
around it. That is:
b
h
u
(5)
Whe
r
e,
h
and
b
are the hig
h
a
nd low tra
n
siti
onal freq
uen
cies.
3. Gene
tic
Alg
o
rithm
The ge
netic
algorith
m
(G
A) [31] is an
optim
izatio
n tech
niqu
e that performs a
parall
e
l,
stocha
stic a
n
d
directe
d
se
arch to evolv
e
the fi
ttes
t
(bes
t) solution. Different from c
o
nventional
optimizatio
n method
s,
GA employs
the prin
ciple
s
of evolution,
nat
ural
sele
ction
and
geneti
c
s, as
inspi
r
ed by n
a
tural biol
ogi
cal sy
stem
s, in a
comp
uter
algorith
m
to simulate evolu
t
ion.
Thre
e main o
perato
r
s com
p
risi
ng GA
s a
r
e: rep
r
od
ucti
on, cro
s
sove
r, and mutation.
R
e
pr
o
d
u
c
tion:
- Evolution is, in effect, a method o
f
searchin
g among a
n
e
norm
o
u
s
numbe
r of p
o
ssibilitie
s for solution
s. F
o
r the an
alysis and
cont
ro
l of DC moto
r, PID control
l
er
para
m
eters selectio
n is u
s
ed fo
r reprodu
ction. A
string i
s
permitted reprod
uction ba
se
d
on
fitness fo
r p
r
odu
ctivity, where
pro
d
u
c
tivity of an
indiv
i
dual i
s
defin
ed a
s
the val
ue of a
stri
ng
’s
non-neg
ative obje
c
tive function.
Cro
s
so
ve
r:
- The cro
s
sover ope
rato
r e
x
chan
ge
s ge
netic informa
t
ion betwee
n
string
s.
There are a
numbe
r of co
mmonly use
d
cro
s
sover o
perato
r
s: such as bl
en
d crossover (B
LX),
simulated bi
nary crossover (SBX), unim
odal
norma
l distribution crossover
(UNDX) and
sim
p
lex
cro
s
sove
r (S
PX) and parent centri
c re
combi
nati
on
operator (P
CX) [32]. In th
e pre
s
ent pa
per
PCX operator has be
en u
s
ed be
cau
s
e t
h
is pa
rticul
ar
operator a
s
si
gns m
o
re p
r
o
bability kee
p
i
n
g
an offsp
r
ing close
r
to the parent
s than a
w
ay from pa
rents.
Mutation: -
Real code
d m
u
tation (RCM
) ope
rato
r [3
3] has
bee
n
use
d
to protect the
irre
cove
rabl
e
or prem
ature loss of important
notio
ns. Since
co
ntinuou
s vari
able
s
are
co
ded
dire
ctly, RCM
is flexible
in
nature. P
C
X
and
RCM o
p
e
rato
r h
a
ve b
een
used in
conj
un
ction a
nd
attain sea
r
ch
powe
r
simila
r to the individual metho
d
o
logie
s
, yet the overall al
gorithm p
e
rfo
r
ms
better than bi
nary-co
ded G
A
s.
4.
Simulated Annealing Alg
o
rithm
Simulated An
nealin
g (SA) is motivated b
y
an analogy to annealin
g in solid
s. The idea of
SA comes fro
m
a pape
r pu
blish
ed by Metropoli
s
etc
al in 1953 (M
etropoli
s
, 195
3). The alg
o
ri
thm
in this pap
er simulate
d the cooli
ng of
material
in a
heat bath. This i
s
a pro
c
e
ss
kno
w
n
as
anne
aling.
If a solid i
s
h
eated
pa
st m
e
lting p
o
int a
nd the
n
cool
it, the structu
r
al
pro
pertie
s
of the
solid
depe
nd
on the rate o
f
coolin
g. If the liquid i
s
cooled
slo
w
ly enou
gh, larg
e cry
s
tals
will
be
formed.
Ho
wever, if th
e liquid i
s
cool
ed qui
ckly (quen
ch
e
d
) the
cry
s
tals
will co
n
t
ai
n
imperfe
ction
s
. Metropoli
s
’
s
algo
rithm
simulate
d the material a
s
a syste
m
of particle
s
.
The
algorith
m
sim
u
lates th
e co
oling p
r
o
c
e
s
s by grad
ually
lowe
ring th
e
temperature
of the syste
m
until it converges to a
stea
dy, fr
ozen
sta
t
e. In 1982; Kirkp
a
tri
ck et
a
l
(Kirkpatri
ck,
1983
) too
k
the
idea of the
Metrop
olis al
gorithm a
nd
applie
d it
to optimisatio
n
probl
em
s. Th
e idea is to
use
simulate
d an
nealin
g to se
arch for fea
s
i
b
le sol
u
tion
s and converge
to an optimal solution.
5.
DC Mo
tor
De
sign using F
O
PID
5.1.
FOPID Controller
The differe
ntial equatio
n of a fractional o
r
de
r PI
λ
D
μ
co
ntrolle
r is de
scrib
ed by:
)
(
)
(
)
(
)
(
t
e
D
k
t
e
D
k
t
e
k
t
u
t
D
t
i
p
(6)
The
co
ntinuo
us t
r
an
sfer fu
nction
of F
O
PID is
obtai
n
ed th
roug
h
L
apla
c
e t
r
an
sf
orm
and
is gi
ve
n
by:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
De
sign of Fra
c
tional O
r
de
r
PID Controlle
r for DC
Moto
r usin
g Gen
e
tic Algorithm
(Ashu Ahuj
a)
8143
s
k
s
k
k
s
G
D
i
p
c
)
(
(7)
De
sign
of an
FOPID
co
ntroller i
n
volv
es de
sign
of three p
a
ra
meters
k
p
, k
i
, k
D
, and two
orde
rs
λ
,
μ
whi
c
h are no
t nece
s
sarily
integer. The
fractional o
r
der controller generalizes
the
conve
n
tional
integer o
r
d
e
r PID co
ntroller. Th
i
s
e
x
pansi
on ca
n provide m
o
re flexibility in
achi
eving co
ntrol obj
ectiv
e
s. A dra
w
b
a
ck
with de
rivative action i
s
that an ide
a
l derivative
has
very high
gai
n for hi
gh fre
quen
cy si
gna
ls. Thi
s
me
an
s that hi
gh freque
ncy m
e
a
s
ureme
n
t noi
se
will generate
large vari
ations
of
the control signal.
Th
e effect
of m
easurement
noise be reduced
by repla
c
ing t
he term
s
k
D
by
1
s
s
k
D
D
.
Therefore, th
e transf
e
r fun
c
tion of FOPID co
ntrolle
r is:
1
)
(
s
s
k
s
k
k
s
G
D
D
i
p
c
(8)
So, there are
six tuning pa
rameters to tune no
w.
]
,
,
,
,
,
[
D
D
i
p
k
k
k
p
(9)
5.2.
PID Con
t
roller
The P
r
op
ortio
nal-Integ
ral
-
Derivative (PI
D
)
c
ontroller [3
4-35]
is the m
o
st
comm
on f
o
rm
of
feedba
ck a
n
d
al
so
a req
u
isite
eleme
n
t of ea
rly g
o
verno
r
s. PID
control
with its th
ree
term
function
ality coveri
ng
bot
h tran
sie
n
t a
nd
steady-st
a
tes
re
spo
n
se, offers the
simple
st a
n
d
the
most
efficient
sol
u
tion to
many real
wo
rld
co
nt
rol
problem
s. Be
cause of
th
e
a
bove a
d
vanta
ges
PID controller with four tuni
ng paramete
r
s are
sele
cte
d
:
1
)
(
s
s
k
s
k
k
s
K
D
D
i
p
(10)
Tuning p
a
ra
meters of the controlle
r are:
]
,
,
,
[
D
D
i
p
k
k
k
p
(11)
5.3. DC Mo
to
r Model
DC machines are
charac
terized by
their
versatility. By me
ans of various combinations
of
shu
n
t, seri
es and
sepa
rat
e
ly excited field win
d
ing
s
they can b
e
desi
gned to
displ
a
y a wi
de
variety of volt-ampe
r
e o
r
spee
d-to
rqu
e
cha
r
a
c
teri
stics for
both
dynamic
an
d stea
dy sta
t
e
operation. Be
cau
s
e
of the
ease by with
whi
c
h t
hey can
be co
ntroll
ed,
sy
ste
m
s of
DC machi
nes
have been freque
ntly use
d
in many applicatio
ns
req
u
iring a
wide
rang
e of motor sp
eed
s an
d a
pre
c
ise outp
u
t
motor control. The
sche
matic dia
g
ra
m of a typical
armatu
re
co
ntrolled
DC
motor
is sh
own in Figure 1.
Figure 1. Sch
e
matic dia
g
ra
m of armature controlled
DC Moto
r
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046
TELKOM
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KA
Vol. 12, No. 12, Decem
ber 20
14 : 8140 – 81
51
8144
A well kno
w
n model of a
r
mature co
ntrolled DC mot
o
r for a spe
ed co
ntrol sy
stem is
sho
w
n in Fig
u
re 2 an
d its tran
sfer fun
c
ti
on is re
prese
n
ted by Equa
tion (1) a
nd (2) .
Figure 2. Block di
agram of
armatu
re co
ntrolled
DC m
o
tor
From the ab
ove diagram, the transfer
functi
on from
the input voltage, V(s), to
the
output velocit
y
,
ω
(s) and to
the output an
gle,
θ
(s) can
be written
with:
]
)
)(
[(
)
(
)
(
2
K
B
Js
R
Ls
s
K
s
V
s
(12)
Whe
r
e
J
(k
g.m
2
/s
2
) is th
e
moment of i
n
ertia of the
ro
tor,
B
is the
d
a
mping
ratio
of the me
cha
n
ical
sy
st
em,
R
(o
hm)
i
s
ele
c
tri
c
al re
sista
n
ce,
L
(H) i
s
el
ectri
c
al i
ndu
ctance,
and
K
(N
m/A) i
s
th
e
electromotive
force
con
s
ta
nt.
5.4. Cost Fu
nctions in Pr
oposed T
e
c
hnique
The
comp
ari
s
on
of PID
and F
O
PID
controlle
rs i
s
done
on th
e ba
sis of F
r
equ
en
cy
Domai
n
and
Time Dom
a
in
Performa
nce Indice
s i.
e. Mixed Sensi
t
ivity and Integral of Squ
a
re
Erro
r re
sp
ecti
vely. The cost function in the de
si
gn i
s
the infinity norm ba
sed o
n
the con
c
e
p
t of
robu
st mixe
d-sen
s
itivity co
ntrol i
s
given
by equation (14). In the mi
xed-sen
s
itivity method, firstly,
the weig
hting
function of t
he plant’
s
pe
rturb
a
tion an
d/or pe
rform
a
nce m
u
st b
e
spe
c
ified. In this
pape
r,
W
2 is spe
c
ified for the uncerta
inty weight of the plant and
W
1 is
sp
ecified for th
e
disturban
ce a
ttenuation of t
he syste
m
. The co
st functi
on [36-3
7
], ca
n be written a
s
:
1
2
1
T
W
S
W
J
(13)
Whe
r
e
T
is th
e plant’s
com
p
lementa
r
y sensitivity func
tion and S is the plant se
n
s
itivity function.
Assu
me that
the plant i
s
d
enoted
as
P.
The
cont
roll
er i
s
de
noted
as
K
and th
e syste
m
is the
unity negative feedba
ck control.
The
sensitivity and
compl
e
ment
ary se
nsitivity function
can
be
expre
s
sed a
s
:
1
)
1
(
PK
S
1
)
1
(
PK
PK
T
(14)
The co
st fun
c
tion in Equation (13
)
is b
a
sed on freq
ue
ncy domai
n specifi
c
ation
s
.
In Controller
desi
gn meth
o
d
s, the mo
st comm
on Tim
e
Dom
a
in Performan
c
e
crit
eria a
r
e
integrate
d
ab
solute e
rro
r (I
AE), the integrated of
time weight squ
a
re erro
r (IT
SE), integrat
ed o
f
squ
a
re
d erro
r (ISE) and Mean Squ
a
re Erro
r (MSE
).
These four integral p
e
rf
orma
nce criteria
have their o
w
n advantag
es
and disadva
n
tage
s.
ISE is propo
sed as
co
st function for d
e
signing
a
nd co
mpari
s
o
n
of PID and FOP
I
D cont
rolle
rs.
t
dt
t
e
ISE
0
2
)
(
(15)
Whe
r
e e is th
e steady stat
e error in
the
step re
sp
on
se of the syste
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
De
sign of Fra
c
tional O
r
de
r
PID Controlle
r for DC
Moto
r usin
g Gen
e
tic Algorithm
(Ashu Ahuj
a)
8145
6. Design
Ex
a
m
ple
The p
a
ramet
e
rs of F
O
PID a
nd PID controlle
rs h
a
v
e bee
n d
e
signed
u
s
ing
GA with
obje
c
tive fun
c
tion
s given
by Equation
(13
)
an
d (1
5
)
and
sim
u
la
tion ha
s bee
n don
e u
s
in
g
MATLAB. The para
m
eters of the DC m
o
tor are given
in Table 1.
Table 1. Para
meters of DC motor
Motor Para
meter
Val
ue
J 0.02(kg.m
2
/s
2
)
B 0.2
K 0.1(Nm/A)
R 2(ohm)
L 0.5(H)
Thus, the tra
n
sfer fu
nction
of the DC m
o
tor ca
n be written as:
41
.
0
14
.
0
001
.
0
1
.
0
)
(
)
(
2
s
s
s
V
s
(16)
6.1. Contr
o
ller design us
ing Mixed Sensitiv
it
y
The synthe
si
s
p
r
o
c
edu
re of
H
∞
controll
er
can
be d
o
ne only by
selectin
g prop
er
weight
function
s. Th
e sele
ction p
u
rely dep
end
s on the pl
ant
model. There are no ha
rd
and fast rule
s for
sele
cting th
e
perfo
rman
ce
weig
ht functi
on an
d th
e
ro
bustn
ess
wei
ghting fun
c
tio
n
s. An ite
r
ati
on
work
with assume
d initial
values i
s
con
ducte
d
to find out the wei
ght function
s.
The freq
uen
cy
depe
ndent
weighting fun
c
t
i
ons in
ca
se (i
) are:
001
.
0
10
5
.
0
1
s
s
W
and
7
.
106
28
.
26
06
.
19
649
.
5
2619
.
0
2
2
2
s
s
s
s
W
In c
a
se (ii),
W
2
is taken a
s
above
and
W
1
is tuned using optimization. In case (iii),
W
2
is
taken a
s
abo
ve and deno
minator of
W
1
is tuned usin
g optimizatio
n. In case (iv
)
,
W
2
is taken as
above a
nd n
u
merator of
W
1
is tun
ed
usin
g optimi
z
ation.The
si
ze of pop
ulati
on of GA is
often
cho
s
e
n
between [20,10
0]. For the p
r
op
ose
d
sim
u
la
ti
on, the si
ze
of populatio
n
is take
n a
s
20.
The nu
mbe
r
of gene
ration
is often cho
s
en between [
100,50
0]. For the pro
p
o
s
e
d
ca
se, nu
m
ber
of gene
ration
s is
equal to
100. The
mut
a
tion rate i
s
cho
s
e
n
to be
0.05. The
weight co-effici
ent
w1, w2 a
nd w3 are 0.98
8, 0.001 an
d 3.0
resp
ectively.
The GA
algo
rithm aim
s
to
find optim
al
value of
]
,
,
,
,
,
[
D
D
i
p
k
k
k
p
to
minimize
the obje
c
tive function give
n by (14
)
. Th
e initial value,
lowe
r and
up
per b
oun
d of solutio
n
varia
b
le
are set at [92.38 198.93 7.
24 0.0006 0
0], [
10 100 1 0.0001 0 0] and [1000 1
0
00 100 0.1 1
1
]
respe
c
tively. The GA conv
erge
s
with the opt
imal sol
u
tion, [670.23
01 482.4
493
99.944
3 0.06
73
0.9994 0.9
7
1
7
], which o
n
substitution to
Equation (9)
provide follo
wing co
ntrolle
r
K(s)
.
1
0673
.
0
9443
.
99
4493
.
482
2301
.
670
)
(
9717
.
0
9994
.
0
s
s
s
s
K
The infinity norm obtai
ned
by the evaluated co
ntrolle
r is 0.628
8 which i
s
less than 1.
Con
s
e
quently
, since this n
o
rm i
s
less th
an 1, then
th
e system i
s
robu
st acco
rdi
ng to the con
c
ept
of Mixed Se
n
s
itivity robu
st co
nt
rol. T
he
same
GA
sp
ecification
s
a
r
e u
s
e
d
to
tu
ne PID controller
para
m
eters
]
,
,
,
[
D
D
i
p
k
k
k
p
.
The initial value, lowe
r an
d uppe
r bou
nd of solutio
n
variabl
e
are set at [9
5.38 200.93
10.24 0.000
9
], [1
10 1
0.0001] and [10
0
1000 10
0 0.1] resp
ectiv
e
ly.
The GA
co
nverge
s
with th
e optimal
sol
u
tion, [
29.40
99, 825.4
733
, 9.515
1, 0
.
0158], whi
c
h
on
sub
s
titution to Equation (1
1) provide foll
owin
g co
ntroll
er
K(s
)
.
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8140 – 81
51
8146
1
0158
.
0
5151
.
9
4733
.
825
4099
.
29
)
(
s
s
s
s
K
The infinity norm obtai
ned
by the evaluated co
ntrolle
r is 0.722
1 which i
s
less than 1.
Con
s
e
quently
, since this n
o
rm i
s
less th
an 1, then
th
e system i
s
robu
st acco
rdi
ng to the con
c
ept
of mixed sen
s
itivity robust
control.
Figure 3. Clo
s
ed lo
op ste
p
resp
on
se wit
h
PID and FO
PID controllers takin
g
Mixed Sensitivity
as
an obje
c
tive functio
n
The co
mpa
r
ison dra
w
n b
e
twee
n the two
cont
roll
ers is explained in
the Table 2:
Table 2. Co
m
pari
s
on of PID and F
O
PID controll
er
Parameters
PID controlled pr
ocess
FOPID cont
rolled
process
Cases
Case1 Case2 Case3 Ca
se4 Case1 Case2
Case3 Case4
Rise
time
.053 .063 .056 .063 .017 .022
.027 .013
Settling
time
0.32 0.52
0.9
1.4 0.094
0.149
0.4
0.3
Stead
y
state
e
rro
r
0 0 0 0 0 0
0 0
Peak
overshoot
15.4%
13.9%
12.3%
11.2%
16.3%
17.2%
19.4%
19.7%
From th
e tabl
e above, th
e
rise
time
redu
ction a
nd
th
e
settling time
redu
ction i
s
calcul
ated a
n
d
is
written in the
followin
g
tables:
Table 3. Ri
se
time redu
ctio
n
Case 1
67.9%
Case 2
65.2%
Case 3
51.78%
Case 4
79.3%
Table 4. Settling time red
u
ction
Case 1
70.62%
Case2
71.34%
Case 3
55.5%
Case 4
78.57%
Table 3
and
4 sh
ows that
the rise time
and
settli
ng time are bein
g
redu
ce
d in case
of a FOP
I
D
controlle
r tha
n
in the ca
se
of a PID controller.
0
0.1
0.
2
0.
3
0.4
0.
5
0.
6
0.7
0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
cl
o
s
e
d
l
o
o
p
st
e
p
r
e
sp
o
n
se
Ti
m
e
(
s
e
c
)
A
m
pl
i
t
ude
PI
D
FOP
I
D
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
De
sign of Fra
c
tional O
r
de
r
PID Controlle
r for DC
Moto
r usin
g Gen
e
tic Algorithm
(Ashu Ahuj
a)
8147
Figure 4. Sensitivity function of DC moto
r with PI
D an
d FOPID co
ntrolle
r takin
g
Mixed Sensiti
v
ity
as an o
b
je
ctive function
Figure 5. Co
mplimenta
r
y Sensitivity fu
nction of
DC
motor with PI
D and F
O
PID controll
er ta
king
Mixed Sensiti
v
ity as an objective functio
n
Figure 4
and
5 sho
w
s Sen
s
itivity and
Complime
n
tary
Sensitivity function
s fo
r th
e DC m
o
tor
with
both PID and
FOPID co
ntro
llers u
s
in
g G
A
.
6.2. Contr
o
ller Desig
n
us
ing Integral
of Square Er
ror
The p
a
ra
met
e
rs of GA i
n
controlle
r de
sign u
s
ing
ISE as
an
obje
c
ti
ve functio
n
a
r
e taken
same
as in controlle
r d
e
si
gn u
s
ing Mix
ed Sen
s
itiv
ity as a
n
obj
ecti
ve function. T
he GA alg
o
rit
h
m
aims to find
optimal value
of
]
,
,
,
,
,
[
D
D
i
p
k
k
k
p
to minimize the o
b
je
ctive function gi
ven
by (16
)
. T
he
initial value, l
o
we
r a
nd
up
per
bou
nd
of sol
u
tion va
ri
able
are
set
at [92.38
198
.93
7.24 0.000
6
0 0], [10 100
1 0.0001
0
0] and [10
0
0
1000 1
00 0.
1 1 1] re
spe
c
tively. The GA
conve
r
ge
s wit
h
the optimal
solutio
n
,
[162
.8495
253.9
4
30 9
0
.764
7 0
.
0351
0.996
5
0.975
8], whi
c
h
on su
bstitutio
n
to Equation
(9) p
r
ovide followin
g
co
ntroller
K(s
)
.
1
0351
.
0
7647
.
90
9430
.
253
8495
.
162
)
(
9758
.
0
9965
.
0
s
s
s
s
K
The infinity norm obtai
ned
by the evaluated co
ntrolle
r is 0.503
4 which i
s
less than 1.
Con
s
e
quently
, since this n
o
rm i
s
less th
an 1, then
th
e system i
s
robu
st acco
rdi
ng to the con
c
ept
of Integral o
f
Square Error. The sam
e
GA spe
c
ification
s
are use
d
to tune
PID controll
er
para
m
eters
]
,
,
,
[
D
D
i
p
k
k
k
p
.
The initial value, lowe
r an
d uppe
r bou
nd of solutio
n
variabl
e
are set at [9
5.38 200.93
10.24 0.000
9
], [1
10 1
0.0001] and [10
0
1000 10
0 0.1] resp
ectiv
e
ly.
The
GA con
v
erge
s
with t
he o
p
timal
solution, [23.5
354
578.0
7
5
9
10.4
080
0.
0176],
whi
c
h
on
sub
s
titution to Equation (1
1) provide foll
owin
g co
ntroll
er
K(s
)
.
-2
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
2
2.
5
3
-100
-8
0
-6
0
-4
0
-2
0
0
20
s
ens
i
t
i
v
i
t
y
f
u
nc
t
i
on
PI
D
FO
P
I
D
-2
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
2
2.
5
3
-4
0
-3
5
-3
0
-2
5
-2
0
-1
5
-1
0
-5
0
5
c
o
m
p
li
m
e
n
t
r
y
s
e
n
s
i
t
iv
it
y
f
u
n
c
t
i
o
n
PI
D
FOP
I
D
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 12, Decem
ber 20
14 : 8140 – 81
51
8148
1
0176
.
0
4080
.
10
0759
.
578
5354
.
23
)
(
s
s
s
s
K
The infinity norm obtai
ned
by the evaluated co
ntrolle
r is 0.743
8 which i
s
less than 1.
Con
s
e
quently
, since this n
o
rm i
s
less th
an 1, then
th
e system i
s
robu
st acco
rdi
ng to the con
c
ept
of Integral of Square Erro
r.
Figure 6. Clo
s
ed lo
op ste
p
resp
on
se wit
h
PID
and FO
PID controller taking Integral of Square
Er
r
o
r
a
s
an
ob
je
c
t
ive
fu
nc
tio
n
Figure 7. Sensitivity functions of DC mot
o
r with
PID a
nd FOPID co
ntrolle
rs ta
kin
g
Integral of
Square Erro
r
as an o
b
je
ctive function
Figure 8. Co
mplimenta
r
y Sensitivity fu
nction of
DC
motor with PI
D and F
O
PID controll
ers ta
king
Integral of Sq
uare Erro
r as
an obje
c
tive functio
n
0
0.
5
1
1.
5
2
2.
5
3
3.
5
0
0.
2
0.
4
0.
6
0.
8
1
1.
2
1.
4
S
t
e
p
r
e
s
pon
s
e
Ti
m
e
(
s
e
c
)
M
a
gnit
u
de
FO
P
I
D
PI
D
-2
-1
.
5
-1
-0.
5
0
0.
5
1
1.
5
2
2.
5
3
-9
0
-8
0
-7
0
-6
0
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
F
r
e
que
nc
y
(rad/
s
e
c
)
M
agn
i
t
ude i
n
db
S
e
n
s
it
i
v
it
y
f
u
n
c
t
i
o
n
FO
P
I
D
PI
D
-2
-1
.
5
-1
-0
.
5
0
0.
5
1
1.
5
2
2.
5
3
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
F
r
equenc
y
(
r
ad/
s
e
c
)
M
agni
t
ude
i
n
db
Com
p
l
e
m
ent
r
y
S
e
n
s
i
t
i
v
i
t
y
f
unc
t
i
on
FOP
I
D
PI
D
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
De
sign of Fra
c
tional O
r
de
r
PID Controlle
r for DC
Moto
r usin
g Gen
e
tic Algorithm
(Ashu Ahuj
a)
8149
As con
c
lud
e
d
from the
Fig
u
re
6 that th
e Ri
se time
for F
O
PID
co
ntrolle
r is re
d
u
ce
d by
1.39%
and
Settling time
for FOPI
D i
s
re
duced
by
76% a
s
com
pare
d
to
PID co
ntrolle
r,
when
Integral of Square error i
s
take
n as t
he obje
c
tive function. Fu
rther comp
ari
s
on is don
e for
FOPID
cont
roller,
whe
n
th
e pa
ram
e
ters are
tune
d u
s
ing G
eneti
c
A
l
gorithm
(GA
)
and
Simulat
e
d
Annealin
g (S
A) Algorithm.
The obje
c
tive function to
be minimi
zed
is given by (14). The i
n
itial
value,
lo
wer and upp
er bo
und of
solutio
n
varia
b
le
are
set
at [92.3
8
198.9
3
7.2
4
0.0006
0
0], [10
100 1 0.00
01
0 0] and [100
0 1000 1
00 0.
1 1 1] respect
i
vely.
Figure 9. Sensitivity function of DC moto
r with FOPID
controlle
r usi
ng GA and S
A
Figure 10. Sensitivity function of DC mot
o
r with FOPI
D co
ntrolle
r u
s
ing GA an
d SA
Figure 11. Cl
ose
d
loop
ste
p
respon
se fo
r FOPID cont
rolle
r usi
ng G
A
and SA
The co
mpa
r
i
s
on d
r
a
w
n for FOPID
controlle
r u
s
i
ng GA and
SA as optimization
techni
que
s is
explained in t
he Table 5:
-2
-1.
5
-1
-0.
5
0
0.
5
1
1.
5
2
2.
5
3
-
100
-8
0
-6
0
-4
0
-2
0
0
20
F
r
equenc
y
(r
ad/
s
e
c
)
M
agni
t
ude i
n
db
S
ens
i
t
i
v
i
t
y
f
unc
t
i
on
GA
SA
-2
-1.
5
-1
-0
.
5
0
0.
5
1
1.
5
2
2.
5
3
-3
0
-2
5
-2
0
-1
5
-1
0
-5
0
5
F
r
eq
ue
nc
y
(
r
a
d/
s
e
c
)
M
agn
i
t
ude i
n
db
C
o
m
p
l
e
m
e
nt
r
y
S
e
ns
i
t
i
v
i
t
y
f
u
nc
t
i
on
GA
SA
0
0.
1
0.2
0.3
0.4
0.5
0.
6
0.7
0
0.2
0.4
0.6
0.8
1
1.2
1.4
S
t
e
p
r
e
s
p
ons
e
Ti
m
e
(
s
e
c
)
M
a
gni
t
ude
GA
SA
Evaluation Warning : The document was created with Spire.PDF for Python.