2DOF mul i-objec i e op imal uning o dis u bance ejec ac ional
o de PIDA con olle s acco ding o imp o ed consensus o ien ed
andom sea ch me hod
Neca i Ozbey
a
, Celaleddin Ye oglu
a,
⇑
, Ba is Baykan Alagoz
a
, No be He encsa
b
, Aslihan Ka ci
b
,
Roman So ne
b
a
Inonu Uni e si y, Facul y o Enginee ing, Depa men o Compu e Enginee ing, Mala ya, Tu key
b
B no Uni e si y o Technology, Facul y o Elec ical Enginee ing and Communica ion, Depa men o Telecommunica ions, B no, Czech Republic
g aphical abs ac
The consensus cu e MðEÞs a es a dynamic bounda y ha go e ns op imiza ion p ocess depending on he alue o E.AsEdec eases, i implies ha se
poin con ol pe o mance is ge ing be e , he alue o consensus cu e MðEÞinc eases o mee highe dis u bance ejec ion expec a ion. The loga i hmic
consensus coe icien ais used o scaling o dynamic bounda y o RDR objec i e. As he pa ame e ainc eases and dynamic bounda y MðEÞinc eases o
highe dis u bance ejec ion pe o mance. This leads a mechanism ha inc ease o se poin pe o mance imposes he inc ease o dis u bance ejec ion
pe o mance. The loga i hmic consensus coe icien can be exp essed as a¼
RDR

dB
log
10
E

min
whe e E

min
is a desi ed op imal alue o min Egand RDR

dB
is a
desi ed op imal alue o min
x
2½
x
min
;
x
max

RDR
dB
ðxÞg. De e mina ion o he loga i hmic consensus coe icien ade ines a consensus cu e o op imal sea ch
o mul i objec i e op imiza ion me hod. The ollowing igu e illus a es a consensus cu a u e o he loga i hmic consensus coe icien a¼2.
a icle in o
A icle his o y:
Recei ed 7 Feb ua y 2020
Re ised 24 Ma ch 2020
Accep ed 24 Ma ch 2020
A ailable online 4 Ap il 2020
Keywo ds:
F ac ional o de con ol
2DOF con olle design
Dis u bance ejec ion
abs ac
This s udy p esen s a F ac ional O de P opo ional In eg al De i a i e Accele a ion (FOPIDA) con olle
design me hodology o imp o e se poin and dis u bance ejec con ol pe o mance. The p oposed con-
olle uning me hod pe o ms a mul i-objec i e op imal ine- uning s a egy ha implemen s a
Consensus O ien ed Random Sea ch (CORS) algo i hm o e alua e ansien simula ion esul s o a se
poin il e ype Two Deg ee o F eedom (2DOF) FOPIDA con ol sys em. Con ibu ions o his s udy ha e
h ee olds: Fi s ly, i add esses uning p oblem o FOPIDA con olle s o i s o de ime delay sys ems.
Secondly, he s udy aims ine- uning o 2DOF FOPIDA con ol s uc u e o imp o ed se poin and dis u -
bance ejec ion con ol acco ding o ansien simula ions o implemen a ion models. This enhances
p ac ical pe o mance o heo e ical uning me hod acco ding o implemen a ion equi emen s.
h ps://doi.o g/10.1016/j.ja e.2020.03.008
2090-1232/Ó2020 The Au ho s. Published by Else ie B.V. on behal o Cai o Uni e si y.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
Pee e iew unde esponsibili y o Cai o Uni e si y.
⇑
Co esponding au ho .
E-mail add ess: [email p o ec ed] (C. Ye oglu).
Jou nal o Ad anced Resea ch 25 (2020) 159–170
Con en s lis s a ailable a ScienceDi ec
Jou nal o Ad anced Resea ch
jou nal homepage: www.else ie .com/loca e/ja e
Re e ence o dis u bance a io
Random sea ch algo i hm
Thi dly, he pape p esen s a hyb id con olle uning me hodology ha inc eases e ec i eness o he
CORS algo i hm by using s abilizing con olle coe icien s as an ini ial con igu a ion. Acco dingly, he
CORS algo i hm pe o ms he ine- uning o 2DOF FOPIDA con olle s o achie e an imp o ed se poin
and dis u bance ejec ion con ol pe o mances. This ine- uning is ca ied ou by conside ing ansien
simula ion esul s o 2DOF FOPIDA con olle implemen a ion model. Mo eo e , Re e ence o
Dis u bance Ra io (RDR) o mula ion o he FOPIDA con olle is de i ed and used o measu emen o
dis u bance ejec ion con ol pe o mance. Illus a i e design examples a e p esen ed o demons a e
e ec i eness o he p oposed me hod.
Ó2020 The Au ho s. Published by Else ie B.V. on behal o Cai o Uni e si y. This is an open access a icle
unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
In oduc ion
Se e al esea ch wo ks ha e been highligh ed me i s o ac-
ional o de dynamical sys em modeling o mo e ealis ic ep e-
sen a ion o eal wo ld sys ems when compa ed o in ege o de
dynamical modeling [1–4]. Hence, ac ional o de dynamics and
ac ional o de con ol ha e been u ned in o a majo opic o con-
ol sys em esea ch s udies in las wo decades [5]. In o de o u i-
lize ad an ages o ac ional o de dynamics in closed loop con ol
sys ems, F ac ional O de PID (FOPID) con olle s, which allow un-
ing o non-in ege o de in eg al and de i a i e elemen s, ha e
been conside ed as a subs i u e o con en ional PID con olle s in
he ield o classical con ol. A cen al mo i a ion in he esea ch
wo ks o FOPID con olle s was o ha ness in ini e uning op ions
o ac ional o de s dynamics o ob ain mo e con ol pe o mance
me i s in con ol laws.
U iliza ion o ac ional o de dynamics in con ol ield ha e
been pa icula ly ocused on enhancemen o obus con ol pe -
o mance, which is so called ‘‘ ac al obus ness” in he ield
[6,7]. Many s udies e ealed bene i s o ac ional o de con olle s
ela i e o hei in ege o de coun e pa s and hese indings ha e
ini ia ed discussions on indus ial use o FOPID con olle s, namely
indus ializa ion o FOPID con olle s [8]. In gene al, obus ness
associa ed wi h ac ional o de con olle s ha e been add essed
in wo olds: (i) imp o emen s o he con ol pe o mance obus -
ness agains pa ame ic pe u ba ions o con ol sys ems [9,10], (ii)
enhancemen o he dis u bance ejec ion con ol pe o mance
agains en i onmen al dis u bances [11–16]. These wo majo con-
olle design objec i es ha e been widely conside ed in con ol
sys em esea ches o imp o e eal wo ld con ol pe o mance.
Bounds o inhe en dis u bance ejec ion capaci y o nega i e
eedback loops we e discussed o unknown addi i e inpu dis u -
bance models, and RDR measu emen was p oposed o exp ess dis-
u bance ejec ion capaci y o closed loop FOPID con ol sys ems
[13,14,16]. This was a use ul s ep o igu e ou bounds o dis u -
bance ejec ion capaci y o closed loop sys ems [16]. Fo mula ion
o RDR index was de i ed by assuming a closed loop con ol sys-
em as a communica ion channel and RDR spec um o he con ol
sys em was exp essed as he a io o powe densi y o e e ence
signal ela i e o he powe densi y o dis u bance signal a he
plan ou pu . I esembles Signal o Noise Ra io (SNR) ha was
de ined o e alua e signal ansmission capaci y o a noisy commu-
nica ion channel. Alagoz e al. showed ha RDR pe o mance o
closed loop con ol sys ems depends on spec al powe densi y
o con olle s, and p ac ical RDR pe o mance is bounded by s abil-
i y o con ol sys ems [16].
Al hough inc ease in spec al powe densi y o he con olle
unc ion con ibu es o RDR index and imp o es dis u bance ejec-
ion pe o mance o nega i e eedback con ol loops, i de e io a es
s ep esponse pe o mance because he inc easing ou pu powe o
con olle s causes highe o e shoo s and ipples ha appea while
se ling o a se poin . Fu he inc ease o RDR alues inally leads
o ins abili y o closed loop con ol sys ems. The e o e, s abili y
bounda y o con olle coe icien s becomes a na u al bounda y
o RDR pe o mance, namely an inhe en limi a ion o dis u -
bance ejec ion capaci y o closed loop sys ems [16]. Consequen ly,
he e exis s a design adeo be ween se poin pe o mance and
dis u bance ejec ion con ol pe o mance. This adeo b ings
ou an essen ial p oblem o dis u bance ejec ion con olle un-
ing app oaches. A easible solu ion o his p oblem was o use a
se poin il e ype 2DOF con ol sys ems. These sys ems pe o m
a e e ence inpu shaping s a egy by using a p e- il e unc ion a
e e ence inpu [16–18]. This p e- il e unc ion is also known as
he se poin il e .
In con ol p ac ice, he RDR spec um analysis was used o
e alua ion o dis u bance ejec ion pe o mance o a closed loop
FOPID con ol o magne ic le i a ion sys em, and an expe imen al
alida ion o dis u bance ejec ion pe o mance imp o emen s
was illus a ed in [15]. On he o he hand, minimum RDR con-
s ain has been used as a dis u bance ejec ion objec i e in
mul i-objec i e uning p oblems o PID and FOPID con olle s
[19–21]. Howe e , he design adeo be ween dis u bance ejec-
ion con ol and se poin con ol educes e ec i eness o con-
olle uning me hods in p ac ice. To add ess his design
adeo , a se poin il e ype 2DOF FOPID con olle s uc u e
was implemen ed o enhance s ep esponse pe o mance in case
o dis u bance ejec ion con ol [21]. This s udy also demons a ed
a mul i-objec i e pa e o op imal uning o FOPID con olle s by
in oducing CORS algo i hm. The CORS algo i hm implemen s a
consensus cu e o deal wi h he design adeo ha appea s
be ween se poin and RDR pe o mances [21]. Findings o his
s udy become a mo i a ion o he cu en s udy ha ex ends his
app oach o op imal ine- uning o 2DOF FOPIDA con ol sys ems
acco ding o ansien con ol simula ion esul s.
An accele a o e m (second de i a i e e m) was i s ly
adap ed o PID con olle s. This con olle can espond he second
o de dynamical changes in con ol e o and hus PIDA bene i s
om accele a o e m o espond highe o de dynamical changes
in con ol e o o closed loop con ol sys ems. This p ope y can be
expec ed o imp o e dis u bance ejec ion con ol pe o mance so
ha dis u bance can be conside ed as an in e mi en , highe o de
ex e nal dynamics ha empo a ily a ec plan unc ion dynamic
esponse. In li e a u e, uning p oblem and applica ion o PIDA
con olle s has been s udied a a limi ed ex en [22–24]. PIDA con-
olle s a e no highly complica ed con olle s uc u es howe e
uning o his con olle can be pe o med by using me aheu is ic
sea ch algo i hms such as pa icle swa m op imiza ion, a i icial
bee colony e c [24]. Due o hei highe compu a ional complexi y,
hese sea ch algo i hms may no be easible o implemen a ion on
low cos con ol ca ds o onsi e au o uning con ol applica ions.
Since possible ad an ages o FOPIDA con olle o deal wi h high
o de dynamics, Puangdown eong ha e sugges ed uning o
FOPIDA con olle s [25]. Pa icle swa m op imiza ion algo i hm
was implemen ed o uning FOPIDA con olle s and con ol pe -
o mance imp o emen s we e illus a ed in [26,27]. To he bes
o ou knowledge, uning p oblem o FOPIDA con olle s in o de
160 N. Ozbey e al. / Jou nal o Ad anced Resea ch 25 (2020) 159–170
o ob ain imp o ed dis u bance ejec ion con ol pe o mance o
la ge ime delay sys ems has no been a sol ed p oblem. In he cu -
en s udy, we add ess a s aigh o wa d solu ion o uning p ob-
lem o 2DOF FOPIDA and aim a easible solu ion o he dis u bance
ejec ion con ol p oblem o la ge ime delay con ol sys ems. Fo
his pu pose, in addi ion o a se poin pe o mance objec i e, an
RDR pe o mance objec i e is u ilized in op imal uning o FOPIDA
con olle s. Acco dingly, he RDR spec um o mula ion is de i ed
o closed loop FOPIDA con olle s in he ollowing sec ion. In u -
he sec ions, he CORS uning me hod is imp o ed by ini ializing
con olle coe icien s acco ding o esul s o an analy ical uning
me hod. The well known Zeigle Nichols uning me hod is u ilized
o ini ial con igu a ion o he CORS uning me hod. Thus, analy i-
cal Zeigle Nichols uning me hod p o ides a s able solu ion o pe -
o m a Random Sea ch (RS) o ine- uning con olle coe icien s.
This ine- uning scheme s a s wi h esul s o Zeigle Nichols
me hod, and con inues sea ching o con olle coe icien s ha
p o ide a be e se poin and RDR pe o mances acco ding o an-
sien simula ion esul s o con ol sys ems.
The RS algo i hm is a undamen al, low compu a ional com-
plexi y and s aigh o wa d s ochas ic sea ch me hod o ind local
minimum poin s acco ding o andom walk ype s a egy [28–32].
To employ his algo i hm in a mul i-objec i e con olle uning
p oblems, RS algo i hm was modi ied by adop ing a consensus
cu e o pa e o op imal sea ch o solu ions in case o con lic ing
mul i-objec i es [21]. One con ol objec i e equi es minimiza ion
o se poin e o o imp o ed s ep esponse and s abili y. The
o he objec i e maximizes RDR index o inc ease dis u bance
ejec ion capaci y o esul ing con ol sys ems. As a consequence,
CORS algo i hm can sea ch in a guidance o a consensus cu e ha
en o ces sea ch di ec ion owa ds highe RDR alues while keeping
he se poin con ol e o s a low le els [21]. A majo complica ion
o me ahe eus ic algo i hms is he inding a s able ini ial con igu-
a ion o con olle coe icien s o p og essi ely imp o e hem
acco ding o simula ion esul s. This p oblem is also sol ed in
he cu en s udy by de ising a hyb id algo i hm ha combines
an analy ical uning me hod o ob ain a s able ini ial solu ion,
and a andom sea ch algo i hm o imp o e his solu ion acco ding
o implemen a ion equi emen s.
RDR analysis o FOPIDA and heo e ical backg ound
RDR spec um was p oposed o quan i a i e assessmen o
inpu dis u bance ejec ion capaci y o closed loop con ol sys ems.
I esembles SNR index, which is a undamen al measu e o e al-
ua ion o signal ansmission pe o mance in communica ion
channels. The RDR analysis was ca ied ou o closed loop con ol
sys ems by conside ing addi i e inpu dis u bance model
[13,14,16] and exp essed in he o m o
RDRð
x
Þ¼ Cðj
x
Þ
jj
2
;ð1Þ
whe e CðjxÞs ands o equency esponse o con olle ans e
unc ions CðsÞ. The CðjxÞcan be ob ained by using s¼jxin he con-
olle ans e unc ions CðsÞ. RDR is exp essed in decibel (dB)
[14,16],
RDR
dB
ð
x
Þ¼10log Cðj
x
Þjj
2
:ð2Þ
Fo mo e heo e ical de ails on he o mula ion o RDR index,
one can conside e e ences [14] and [16]. RDR spec um, de ined
by Eq. (2), p o ides a use ul measu e o assess dis u bance ejec-
ion a es o con ol sys ems o each equency componen s. I is
no ewo hy o s a e ha Eq. (2) allows spec al assessmen o addi-
i e inpu dis u bance ejec ion capaci y o he closed loop con ol
sys ems depending on only con olle pa ame e s. In gene al, p ac-
ical con ol sys ems wo k in low equency egion and highe RDR
alues a low equency egion is p ominen o ob ain sa is ac o y
dis u bance ejec ion con ol agains en i onmen al dis u bances.
En i onmen al dis u bances such as al e a ions in ope a ing condi-
ions change slowly ela i e o con olle ou pu . Hence, a highe
RDR alue a low equency egion is p ominen o ejec ion o
slowly de eloping en i onmen al dis u bances. In eg al elemen
o con olle unc ion pa icula ly enhances he low equency pa
o RDR spec um as shown in Fig. 1(a). (RDR spec um o in eg al
e m
k
i
s
is 10logðk
2
i
=
x
2
Þ) Inc easing RDR spec um a highe e-
quency egion makes con ol sys em mo e obus agains high e-
quency dis u bances, o ins ance sys em noises (e.g. quan iza ion
noise, senso noises e c.) o whi e noises. Whi e noise signals a e
andom and i s spec al powe densi y sp eads o he whole spec-
um. De i a i e elemen o con olle unc ion pa icula ly
enhances he high equency pa o RDR spec um as shown in
Fig. 1(a). (RDR spec um o de i a i e e m k
d
sis 10logðk
2
d
x
2
Þ).
High RDR a highe equencies is p e e able o ejec ion o sys em
noise o whi e noises. Fo a ai compa ison, con olle coe icien s
a e aken equal o 1 in he igu e.
T ans e unc ion o FOPID con olle is commonly w i en in
gene al o m o
C
FOPID
ðsÞ¼k
p
þk
i
s
k
þk
d
s
l
;ð3Þ
Fig. 1. (a) RDR spec ums o k
i
¼1 and k
d
¼1. (b) RDR spec um o FOPIDA and
FOPID con olle o he same coe icien s.
N. Ozbey e al. / Jou nal o Ad anced Resea ch 25 (2020) 159–170 161
whe e pa ame e s k
p
,k
d
and k
i
a e gain coe icien s and he pa am-
e e s kand
l
a e ac ional o de s o FOPID con olle s. Design o a
FOPID con olle in ol es uning o hese i e design pa ame e s in
o de o ob ain a desi ed con ol esponse. The RDR o closed loop
FOPID con ol sys ems was de i ed as [14],
RDR
opid
ð
x
Þ¼ k
p
þk
i
x
k
cosð
p
2
kÞþk
d
x
l
cosð
p
2
l
Þ

2
þk
d
x
l
sinð
p
2
l
Þk
i
x
k
sinð
p
2
kÞ

2
:
ð4Þ
T ans e unc ion o FOPIDA con olle is w i en in gene al
o m by adding accele a o e m k
a
s
2
o FOPID con olle unc ion
as
C
FOPIDA
ðsÞ¼k
p
þk
i
s
k
þk
d
s
l
þk
a
s
2
;ð5Þ
whe e he addi ional pa ame e k
a
is he accele a o coe icien . The
accele a o e m k
a
s
2
conside s changes in eloci y o con ol e o
o closed loop con ol. Thus, he second o de dynamics in con ol
e o can con ibu es o he con ol law o FOPIDA con olle s. A
bene i o he accele a o e m appea s a high equency dis u -
bance ejec ion pe o mance because his e m inc eases RDR pe -
o mance a high equencies mo e han he de i a i e elemen o
con olle . (RDR spec um o accele a o elemen k
a
s
2
is
10logðk
2
a
x
4
Þ). FOPIDA con olle design equi es uning o hose
six design pa ame e s, whe e i e o hose pa ame e s a e coe i-
cien s o FOPID and an addi ional pa ame e is he accele a o coe -
icien . The RDR o closed loop FOPIDA con ol sys em can be
de i ed by using s¼jxin equa ion (1).
RDR
opida
ð
x
Þ¼ k
p
þk
i
x
k
cosð
p
2
kÞþk
d
x
l
cosð
p
2
l
Þk
a
x
2

2
þk
d
x
l
sinð
p
2
l
Þk
i
x
k
sinð
p
2
kÞ

2
ð6Þ
Fo dis u bance ejec con olle design, he ollowing RDR con-
s ain s can be used o speci y a lowe bounda y o dis u bance
ejec ion capaci y o he esul ing con ol sys em a an ope a ing
equency ange o
x
2½
x
min
;
x
max
.
min
x
2½
x
min
;
x
max

RDR
dB
ð
x
Þg PM;ð7Þ
whe e M2Ris a design speci ica ion. This cons ain in e s ha he
lowes RDR pe o mance should be equal o g ea e han he lowe
bounda y M[16].
To in es iga e e ec s o accele a o e m o dis u bance ejec-
ion capaci y o he closed loop con ol sys em, we compa e RDR
spec ums o con en ional FOPID con olle and FOPIDA con olle
o equal alues o coe icien s k
p
¼1, k
d
¼1, k
i
¼1, k¼1 and
l
¼1 and k
a
¼1. Fig. 1(b) e eals ha RDR pe o mance o FOPIDA
con olle is equal o g ea e han RDR pe o mance o FOPID
con olle excep RDR alues a ound he angula equency
x
¼1
ad/sec. Inse o Fig. 1(b) is a close iew o his pa o spec um.
This cha ac e is ic implies ha a ha monic dis u bance a 1
ad/sec de e io a es dis u bance ejec ion pe o mance o FOPIDA
con olle . Such pe o mance de e io a ions come ou a need o
special conside a ion o low equency dis u bance ejec ion
pe o mance when designing FOPIDA con olle s. The in eg al
compensa o s (
k
c
s
) a e widely used o emo al o s eady s a e
e o s [33]. As i is shown in Fig. 1(a), he in ege o de in eg al
compensa o con ibu es RDR pe o mance a low equency
egion. (RDR spec um o in eg al elemen
k
c
s
is 10logð1=
x
2
Þ). A
u u e s udy can add ess enhancemen o low RDR pe o mance
a he low equency egion by using an in eg al compensa o pa -
allel o FOPIDA con olle s.
A p ac ical and gene al solu ion o he low RDR p oblems is o
pe o m ine- uning o FOPIDA con olle implemen a ions acco d-
ing o he minimum RDR cons ain (Eq. (7)). To add ess he low
RDR p oblems, he cu en s udy implemen s his ine- uning
op ion by using he CORS algo i hm. To e i y alidi y o ine-
uning op ions o RDR enhancemen p ocess, one should heo e -
ically demons a e he exis ence o FOPIDA con olle coe icien
con igu a ions ha can su pass RDR o FOPID con olle s. Fo his
eason, a su icien condi ion is igu ed ou o alida e imp o e-
men o RDR pe o mance o FOPIDA con olle s ela i e o RDR
pe o mance o FOPID con olle s. This su icien condi ion can be
exp essed as RDR
opida
ð
x
ÞRDR
opid
ð
x
Þ>0. By using equa ions (6)
and (4), his condi ion can be ob ained as
2k
p
þ2k
i
x
k
cosð
p
2kÞþ2k
d
x
l
cosð
p
2
l
Þ<k
a
x
2
:ð8Þ
This su icien condi ion e i ies he exis ence o an in ini e se
o FOPIDA con olle coe icien s ha can su pass RDR pe o mance
o FOPID con olle a any desi ed equency componen . (See
appendix sec ion o he de i a ion o he su icien condi ion) This
heo e ical conside a ion alida es he ine- uning op ion o
FOPIDA con olle s.
FOPIDA con olle design by consensus cu e o ien ed RS
algo i hm
Fig. 2 shows a block diag am o se poin il e ype 2DOF closed
loop con ol s uc u e ha can be a p e e able solu ion o
enhancemen o se poin pe o mance in case o dis u bance ejec-
ion con ol [16]. In his con ol s uc u e, a se poin il e FðsÞis
employed o smoo h e e ence inpu signal ð Þ ia il e ing ou
high equency componen s om he e e ence inpu ð Þ. In case
o a powe ul con olle , which is also an indica ion o high RDR
pe o mance, high equency componen s o ð Þleads o as al e -
a ions (e.g. high o e shoo s, mul iple ipples) a he sys em ou pu
du ing se ling pe iod. High o e shoo s, ipples (also known as
inging e ec in elec onics) and longe se ling pe iods a e no
desi able o sensi i e se poin con ol applica ions such as le el
con ol applica ions. The se poin il e FðsÞcan smoo h he e e -
ence inpu signal ð Þand his allows mo e consis en and asymp-
o ical se ling e ec . This ype smoo h se ling educes
unnecessa y ipples ha can cause longe se ling pe iods and
mo e ene gy consump ion in con ol ac ions. To allow none-
o e shoo smoo h se ling cha ac e is ic in con ol sys em
esponse, a i s o de p e- il e unc ion [16] is implemen ed as
FðsÞ¼ a
sþa;ð9Þ
whe e cons an a¼1=
s
and he pa ame e
s
is he ime cons an
o he il e . S ep esponse o his il e unc ion yields a i s o de
dynamic esponse ha se les o i s inpu alue wi hou p oducing
any o e shoo . Such a il e desc ibes p e e able s ep esponse cha -
ac e is ics, which can be pa icula ly desi able o p ecise le el o
alignmen con ol applica ions o ins ance empe a u e con ol,
liquid le el con ol o con ol o smoo hly alignmen asks o an
equipped heads o ehicles. P e iously, u iliza ion o his ype se
poin p e- il e s as a e e ence model was shown o shaping he
e e ence inpu in adap i e con ol [34]. Essen ially, he unc ion
FðsÞis employed o desc ibe a desi ed ajec o y o s ep esponse,
o which he closed loop con ol sys ems can ack. As shown in
he block diag am in Fig. 2, closed loop con ol sys em acks he il-
e ou pu
, and his p e- il e ac s as a e e ence model. On he
Fig. 2. Block diag am o 2DOF FOPIDA con ol sys em [16].
162 N. Ozbey e al. / Jou nal o Ad anced Resea ch 25 (2020) 159–170
o he hand, he 2DOF closed loop con ol s uc u e is used o deal
wi h design adeo appea ing be ween se poin con ol and dis-
u bance ejec ion con ol pe o mances [16]. This e ec can be
explained as ollowing:
High dis u bance ejec ion equi es s ong o agg essi e con-
olle s, which is possible by using con ol laws wi h high powe
densi y. Such a high powe densi y con ol law can easily de e io-
a e se poin con ol pe o mance because o o ming high o e -
shoo s and ipples while se ling o he se poin [16]. To educe
hose o e shoo s and ipples in se ling, a i s o de p e- il e
FðsÞis used o elimina e e y high equency componen s in s ep
wa e o m and o smoo h e e ence inpu signal be o e applying
o he closed loop con ol sys em [16,21]. Thus a oids exci a ion
o high equency componen s a he con olle ou pu and conse-
quen ly, diminishes high o e shoo s and ipples a he ou pu o
con ol sys em while se ling o se poin s [16]. We assumed an
addi i e inpu dis u bance model o ep esen impac s o en i on-
men al dis u bance on he con ol sys em.
The mean squa ed con ol e o (MSCE) om ansien simula-
ion [36] is used o measu e se poin con ol pe o mance ha is
gi en by
E¼1
TZ
T
0
eð Þ
2
d :ð10Þ
The p ima y con ol objec i e o he con olle uning is com-
monly he minimiza ion o MSCE, which is w i en by min Eg
[36]. The pa ame e Tis he obse a ion ime o MSCE calcula-
ions. We pe o med ansien simula ion o he con ol sys em
and ob ained ins an con ol e o s eð Þin o de o calcula e E.
The obse a ion ime T is con igu ed o he o al simula ion ime
in hese simula ions. The minimiza ion o Eleads o dec ease he
magni ude o con ol e o s signal ha is w i en by
eð Þ¼
ð Þyð Þ. This en o ces eð Þ o app oxima e o ze o, which
implies se ling o he plan ou pu y o he desi ed e e ence inpu
. This p ima y objec i e assu es se poin acking and s abili y o
he closed loop con ol sys em.
The seconda y con ol objec i e is o inc ease dis u bance ejec-
ion pe o mance wi hou deg ading se poin con ol pe o -
mance. To pe o m he dis u bance ejec ion con ol objec i e o
closed loop con ol sys ems, he minimum RDR cons ains, gi en
by Eq. (7), is u ilized as a seconda y objec i e o mul i-objec i e
op imal uning p oblem. A consensus cu e, which desc ibes a
dynamic bounda y o accep able RDR pe o mance depending
on E, is de ined as
MðEÞ¼
a
logE:ð11Þ
Then, he minimum RDR alue in he RDR spec um is limi ed
by he consensus cu e MðEÞ. This condi ion is exp essed as
min
x
2½
x
min
;
x
max

RDR
dB
ð
x
Þg PMðEÞ:ð12Þ
The consensus cu e MðEÞs a es a dynamic RDR bounda y ha
go e ns op imiza ion p ocess depending on he alue o E.AsE
dec eases, i implies ha he se poin con ol pe o mance is ge -
ing be e , he alue o consensus cu e MðEÞinc eases o mee
highe dis u bance ejec ion expec a ions. This p ope y leads a
mechanism such ha an imp o emen in se poin pe o mance
imposes he inc ease in dis u bance ejec ion pe o mance. The
loga i hmic consensus coe icien
a
is used o scaling o dynamic
bounda y o RDR objec i e. When he pa ame e
a
is se o highe
alues, he dynamic bounda y MðEÞinc eases o p o ide highe
dis u bance ejec ion pe o mance. A sui able loga i hmic consen-
sus coe icien can be ound by
a
¼ RDR

dB
logE

min
;ð13Þ
whe e E

min
is a desi ed op imal alue o min Eg, and RDR

dB
is a
desi ed op imal alue o min
x
2½
x
min
;
x
max

RDR
dB
ðxÞg. De e mina ion o
he loga i hmic consensus coe icien acon igu es a consensus
cu e o op imal sea ch o mul i-objec i e op imiza ion me hod.
Fig. 3 illus a es a consensus cu e o he loga i hmic consensus
coe icien , a¼2. The upda e condi ion in s ep 5 allows op imiza-
ion o con olle coe icien s in he allowed design egion, which
is abo e he consensus cu e in Fig. 3. This egion ep esen s a se
o accep able solu ions o deal wi h adeo be ween opposing
design objec i es. The low pe o mance egion, which is below
he consensus cu e, is o bidden because designs in his egion
a e no accep able in e m o mul i-objec i e design pe o mance.
In his s udy, he dis u bance ejec con ol p oblem o he la ge
ime delay sys ems is conside ed. These sys ems can be ep e-
sen ed by a i s o de ime delay ans e unc ion
GðsÞ¼ K
dc
s
sþ1e
Ls
;ð14Þ
whe e he pa ame e K
dc
is s a ic gain o plan unc ion,
s
is he ime
cons an o domina ing i s o de dynamics o sys ems, and Lis he
ime delay, which is also known as dead ime o appa en ime
delay o he sys em. Due o la ge ime delay, op imal uning o in e-
g al componen o FOPID con olle s yields e y low alues o coe -
icien o in eg a o elemen (k
i
) ela i e o o he gain coe icien s.
Such a low in eg a o gain causes weak in eg al ope a ion and i
may lea e s eady s a e e o s in se poin con ol applica ions
[33]. The e o e, p ac ical con olle design ask o a la ge ime
delay plan needs a special conce n o se poin con ol [37].
In he p e ious s udy, CORS algo i hm was p oposed by modi y-
ing a classical RS algo i hm in o de o pe o m op imiza ion in
guidance o a consensus cu e [21]. In he cu en s udy, a signi i-
can modi ica ion o imp o e design pe o mance o CORS algo-
i hm is ha ini ial alues o design coe icien a e con igu ed
acco ding o esul s o an op imal uning me hod. This p o ides a
good ini ial design poin o u he op imize con ol sys ems o
imp o ed dis u bance ejec ion con ol pe o mance. The e o e,
o implemen analy ical uning, we con igu e ini ial coe icien s
o FOPIDA con olle designs acco ding o Zeigle Nichols me hod
in he cu en s udy. Zeigle Nichols me hod is a well-known and
widely accep ed analy ical uning me hod. Coe icien s o Zeigle
Nichols me hod is ine- uned by he CORS algo i hm. Since, he e
is no sugges ion o Zeigle Nichols me hod o he accele a o
Fig. 3. Consensus cu a u e, allowed and o bidden design egions o a¼2.
N. Ozbey e al. / Jou nal o Ad anced Resea ch 25 (2020) 159–170 163

coe icien and ac ional o de s, he ini ial alue o accele a o
coe icien (k
ao
) is se o ze o, and he ini ial alues o ac ional
o de s (k
o
and
l
o
) a e se o one. Consequen ly, PID design o Zei-
gle Nichols me hod is p og essi ely e ol ed o a FOPIDA con-
olle design. Fo 2DOF design o FOPIDA con olle , he p e-
il e pa ame e ais de e mined ega ding he ime delay and ime
cons an o plan unc ions. Thus, FOPIDA designs can ack he
i s o de dynamics o he p e- il e , p ope ly. A easible ime con-
s an o he p e- il e unc ion was empi ically ound as he ime
delay plus a ac ion o he ime cons an o plan unc ion
(
s
¼Lþ
s
c
). Typical alue o he
c
>0 is a ound 1–5. Then, a ele-
an p e- il e coe icien ais w i en by
a¼
c
c
Lþ
s
:ð15Þ
S eps o he imp o ed CORS algo i hm o ine- une 2DOF design
o FOPIDA con olle a e as ollows:
S ep 1 (Ini ial Con igu a ion): Se ini ial alues k
p
¼k
po
,k
d
¼k
do
,
k
i
¼k
io
,k
a
¼k
ao
,k¼k
o
,
l
¼
l
o
acco ding o an op imal con olle
design me hod. (Use Zeigle Nichols me hod o ini ial alues o
k
po
,k
do
,k
io
and se k
a
¼0, k
o
¼1 and
l
o
¼1) Se p e- il e pa ame-
e aacco ding o Eq. (15) and se E
min
o a la ge alue ( ypically
1000). Con igu e andom sea ch leng hs c
p
,c
d
,c
i
,c
k
,c
l
and c
a
.
S ep 2 (Random Sea ch): Gene a e new candida e o con olle
coe icien s by using andom walks in pa ame e sea ch space as
ollows
k
pn
¼k
p
þð and 0:5Þc
p
;ð16Þ
k
dn
¼k
d
þð and 0:5Þc
d
;ð17Þ
k
in
¼k
i
þð and 0:5Þc
i
;ð18Þ
k
n
¼kþð and 0:5Þc
k
;ð19Þ
l
n
¼
l
þð and 0:5Þc
l
;ð20Þ
k
an
¼k
a
þð and 0:5Þc
a
:ð21Þ
S ep 3 (Pe o mance E alua ion): Pe o m ansien simula ion
o hese candida e coe icien s and calcula e he e o unc ion E
o a s ep esponse wi h a Tsimula ion ime. Then, calcula e
min RDR
dB
g o he ope a ing equency ange o
x
2½
x
min
;
x
max
.
S ep 4 (Consensus and Coe icien Upda e): I he upda e condi ion
(E<E
min
and min RDR
dB
gPMðE
min
Þ) is sa is ied, hen upda e he
cu en con olle coe icien s by using candida e coe icien s;
k
p
¼k
pn
,k
d
¼k
dn
,k
i
¼k
in
,k¼k
n
,
l
¼
l
n
,k
a
¼k
an
. Then, upda e
he minimum e o as E
min
¼E.
S ep 5 (Upda e o Dynamic Lowe Bounda y): Calcula e he
dynamic RDR bounda y MðE
min
Þ¼
a
logE
min
o he cu en mini-
mum e o E
min
.
S ep 6 (S opping C i e ia):I E
min
is adequa ely small o a maxi-
mum i e a ion coun is exceeded, end he op imiza ion. O he wise
go o s ep 2.
The cons an s c
p
,c
d
,c
i
and c
a
a e RS leng hs o each gain coe -
icien s and, c
k
and c
l
a e andom sea ch leng hs o ac ional
o de s. These RS leng hs speci y a maximum bouncing ange o
each coe icien . Du ing op imiza ions, he minimum alue o Eis
s o ed in E
min
pa ame e . The e o e, E
min
should be se e y high
alues a ini ializa ion o op imiza ion.
Illus a i e design examples
This sec ion p esen s h ee design examples o demons a e
applica ions o p oposed design me hod. Fig. 4 illus a es a low
cha ha depic s inco po a ion o imp o ed CORS algo i hm and
ansien con ol simula ions o 2DOF FOPIDA con ol sys ems.
The CORS algo i hm sends candida e con olle coe icien s o Ma -
lab Simulink (MS) simula ion en i onmen in o de o ca y ou
ansien con ol simula ions. MSCE o each candida e solu ion is
calcula ed acco ding o simula ion esul s. F ac ional o de de i a-
i e and in eg al elemen s we e implemen ed in hese simula ions
acco ding o Ous aloup’s me hod by using FOTF Ma lab oolbox
[38].
Example 1 (La ge Time Delay Sys ems): Le ’s design a se poin
il e ype 2DOF FOPIDA con ol sys em o a la ge ime delay plan
model
GðsÞ¼ 3:13
433:33sþ1e
50s
ð22Þ
o a loga i hmic consensus coe icien a¼1. This plan unc ion
ep esen s a linea model o he expe imen al pla o m Basic P o-
cess Rig 38-100 Feedback Uni , which was used by Monje e al. o
demons a e pe o mance o ac ional o de con olle s in indus-
ial applica ions [11]. Acco ding o model pa ame e s o his plan
unc ion, he expe imen al sys em p esen s 50 sec ime delay in
esponding o a change in he e e ence inpu . A e his appa en
ime delay, he sys em se les acco ding o a domina ing i s o de
sys em pole wi h 433.33 sec ime cons an and a DC gain o 3.13.
Such a la ge ime delay plan complica es he closed loop con olle
design due o he equi emen o e y small in eg a o coe icien s,
which make i e y sensi i e o ealiza ion issues. Non-ideal ealiza-
ion o ac ional o de elemen s may ail esul s o analy ical uning
me hods in eal con ol applica ions because analy ical op imal un-
ing models ely on an ideal and heo e ical model o ac ional o de
elemen s. The e o e, a ine- uning wi h espec o p ac ical ealiza-
ion model o op imal con olle s imp o es eal wo ld pe o mance
o con ol sys em implemen a ions in he case o analy ical op imal
uning.
By using pa ame e s o Rig 38-100 eedback uni , which a e
K
dc
¼3:13,
s
¼433:33 and L¼50, ini ial alues o con olle
design coe icien s we e ob ained k
po
¼3:3227, k
do
¼78:25,
k
io
¼0:0313, k
o
¼1,
l
o
¼1 acco ding o Ziegle -Nichols uning
me hod and a¼0:0041 acco ding o Eq. (15). These alues we e
con igu ed as ini ial alue o coe icien s in he CORS algo i hm.
The p oposed CORS algo i hm was pe o med o 50 i e a ions.
Fo a as esponse o con ol sys em,
c
pa ame e o p e- il e
was se o 5. The MS simula ions o p oposed 2DOF FOPIDA con ol
sys em we e un 5000 sec. Se poin o basic p ocess ig 38-100
eedback uni was 0.47 [11]. Hence, a s ep inpu wi h he ampli-
ude o 0.47 was applied o e e ence inpu in he simula ions. A
he simula ion ime 2500 sec, a s ep dis u bance wi h ampli ude
o 0.3 was applied o he inpu o plan model. Based on MS simu-
la ion esul s, MSCEs o each candida e design was calcula ed and
sen back o he CORS algo i hm a each i e a ion o op imiza ion
p ocess. When he op imiza ion was comple ed, a ine- uned
FOPIDA con olle unc ion was ob ained as
C
FOPIDA
ðsÞ¼3:3817 þ0:0283
s
0:96764
þ80:0205s
1:0162
0:0108s
2
:ð23Þ
Pa ame e s o con olle unc ions, which we e used o pe o -
mance compa isons, a e lis ed in Table 1.Fig. 5 shows pe o mance
o con olle s. Table 2 summa izes se poin and dis u bance ejec-
ion con ol pe o mances o hese con olle s. The 2DOF FOPIDA
con olle se les in 663 sec, which is he sho es se ling ime
wi hou any o e shoo and ipples. A s ep dis u bance was applied
a e se ling, he 2DOF FOPIDA con ol sys em was se led back o
he se poin 0.47 in 300 sec wi h 21% o e shoo and 3 sligh ip-
ples. The 2DOF FOPIDA con ol was he as es in ese ling and
he sho es in o e shoo s in dis u bance simula ions. These pe -
o mance analyses indica e ha 2DOF FOPIDA con olle can p e-
sen much be e se poin con ol and dis u bance ejec ion
164 N. Ozbey e al. / Jou nal o Ad anced Resea ch 25 (2020) 159–170
Fig. 4. A low cha ha depic s inco po a ion o he CORS algo i hm and he ansien con ol simula ions.
Table 1
Coe icien s o con olle s designed o GðsÞ.
Tuning Me hod k
p
k
d
K
i
k
a
k
l
FOPID (Monje e al. [11]) 0.61 4.38 110
2
0 0.8968 0.4773
Op imal PID (Ma lab) 0.55 57.69 1.4910
3
011
2DOF FOPIDA 3.38 80.02 2.38 10
2
1.0810
-2
0.9676 1.0162
N. Ozbey e al. / Jou nal o Ad anced Resea ch 25 (2020) 159–170 165
con ol pe o mances han hose o o he con olle s in his exam-
ple. Fig. 5(a) compa es RDR pe o mances o con olle s o alida e
dis u bance ejec ion imp o emen s ia RDR spec um. Fig. 5(b)
shows s ep and dis u bance esponses o he p oposed 2DOF
FOPIDA con ol sys em o 5000 sec. One can obse e in Fig. 5(b)
ha he p oposed con ol sys em se les wi hou any o e shoo
in a sa is ac o y pe iod. Fo dis u bance ejec ion simula ion, he
con ol sys ems we e dis u bed a 2500 sec by a s ep dis u bance
and, dis u bance ejec ion pe o mance o he p oposed FOPIDA
con ol is mo e sa is ac o y han hose o o he con ol sys ems.
These simula ion esul s clea ly demons a e ha he p oposed
2DOF FOPIDA con ol sys em can imp o e bo h se poin con ol
pe o mance and dis u bance ejec ion con ol pe o mance. The
esul s in igu e also con i m he da a in Table 2. The igu e also
indica es he dis u bance ejec ion pe o mance imp o emen o
he FOPIDA con olle compa ed o he op imal FOPID con olle
designed by Monje e al. in [11].Fig. 5(c) shows e olu ion o con-
ol e o s and esul s e eals pe o mance imp o emen s o he
2DOF FOPIDA con ol sys em in e m o obus con ol pe o -
mance. This obse a ion indica es ha bo h se poin con ol and
dis u bance ejec ion pe o mance can be u he enhanced by
he imp o ed CORS algo i hm.
To es con olle in mo e ealis ic simula ions, an addi i e ype
whi e noise wi h powe o 510
–6
was inse ed o eedback loop o
mimic senso measu emen noise. Fig. 6 shows esponses o each
con ol sys em unde a s ep dis u bance and senso noise condi-
ions. Va iances o sys em ou pu s a e compu ed o compa ison
o o e all se poin con ol pe o mances as;
2
= 0.0053 o FOPID
(Monje e al. [11]),
2
= 0.0157 o Op imal PID (Ma lab) and
2
= 0.0044 o 2DOF FOPIDA. The a iance o 2DOF FOPIDA con-
ol sys em ou pu is measu ed lowe han a iance o o he con-
olle ’s ou pu s, and i is an indica ion o obus con ol
pe o mance imp o emen .
Example 2 (TRMS Nonlinea Model): This example demons a es
pe o mance o 2DOF FOPIDA con ol o a nonlinea model o
TRMS expe imen al se up. This nonlinea model o he main o o
was p o ided by p oduce o TRMS expe imen al se up [35,36].In
his con ol p oblem, he e ical angle o he main o o is con-
olled by egula ing e minal ol age o he DC elec ic mo o . This
con ol ac ion adjus s o a ional eloci y o p opelle o ho e he
main o o a he desi ed angle. Due o nonlinea ae odynamics
o p opelle blades, his example in oduces a nonlinea se poin
con ol p oblem. In his example, we es ed pe o mance o h ee
con olle s. These a e a classical PID con olle , a con en ional
FOPID con olle and he p oposed 2DOF FOPIDA con olle . The
op imal PID con olle o he main o o con ol o TRMS se up
is p o ided by Feedback Inc as [35,36]
C
PID
ðsÞ¼5þ8
sþ10s:ð24Þ
The FOPID con olle was uned acco ding by he CORS algo-
i hm as
C
FOPID
ðsÞ¼5:04 þ7:96
s
0:86
þ10:022s
1:13
:ð25Þ
The 2DOF FOPIDA con olle was designed by he ine- uning o
imp o ed CORS algo i hm as
C
FOPIDA
ðsÞ¼9:87 þ7:12
s
0:84
þ11:78s
1:10
0:95s
2
:ð26Þ
The CORS algo i hm was ini ialized by using he pa ame e s o
op imal PID con olle . When he op imiza ion is comple ed,
min RDR
dB
gwas ob ained 21.68 dB o a minimum MSCE
E
min
¼3:81 10
-3
.Fig. 7(a) shows s ep and dis u bance esponses
o hese con olle s. The 2DOF FOPIDA con olle can enhances
se poin and dis u bance ejec ion con ol pe o mances
compa ed o esponses o o he con olle s. Fig. 7(b) e eals
imp o emen s o dis u bance ejec ion con ol ia 2DOF FOPIDA
con olle . Fig. 7(c) shows changes o con ol e o s and con i ms
imp o emen in dis u bance ejec ion con ol.
Example 3 (Au oma ic Vol age Regula o (AVR) Model): This
example illus a es con ol o an AVR model by using he p oposed
2DOF FOPIDA con ol scheme. The AVR sys ems a e impo an
componen s o powe sys ems ha con ibu e o powe quali y
Fig. 5. (a) RDR spec ums o p oposed FOPIDA and FOPID (Monje e al. [11])
con olle s. (b) Compa ison o s ep and dis u bance esponses. (c) E olu ion o
con ol e o s o each con olle .
166 N. Ozbey e al. / Jou nal o Ad anced Resea ch 25 (2020) 159–170
o an elec ici y g id by s abilizing e minal ol age o gene a o s
[39]. Howe e , a numbe o ac o s such as load a iabili y o
demand luc ua ion in powe sys ems can dis u b AVR e minal
ol age. The p ese a ion o ol age s abili y is impo an o a eli-
able powe gene a ion. The objec i e o AVR sys em con ol is
keeping he e minal ol age o a gene a o a a desi ed se poin
le el [39]. Ramezanian e al. used Pa icle Swa m Op imiza ion
(PSO) and chao ic an swa m (CAS) op imiza ion me hods o design
an op imal FOPID con olle o he linea AVR model in [39].
C
FOPID PSO
ðsÞ¼1:26 þ0:55
s
1:18
þ0:23s
1:25
ð27Þ
C
FOPID CAS
ðsÞ¼1:05 þ0:44
s
1:06
þ0:25s
1:11
ð28Þ
The 2DOF FOPIDA con olle is e uned by using imp o ed CORS
algo i hm.
C
FOPIDA
ðsÞ¼1:50 þ0:65
s
1:179
þ0:27s
1:25
0:000287s
2
ð29Þ
The s ep and dis u bance esponses o hese con olle s a e
illus a ed in Fig. 8. Figu e e eals se poin and dis u bance ejec-
ion con ol pe o mance imp o emen s by using 2DOF FOPIDA
con ol. Main eason o hese imp o emen s is he ine- uning o
op imal FOPID con olle o ob ain be e dis u bance ejec ion
acco ding o ansien simula ion o he AVR model.
These illus a i e examples e eal ha p ac ical con ol pe o -
mance o op imal uning me hods can be u he enhanced by pe -
o ming ine- uning acco ding o he ansien simula ion o
con ol sys ems. A majo complica ion in his ype o me aheu is ic
op imiza ion p oblems is in e up ion o ansien simula ions due
o uns able design poin s. The uns able design poin s mainly esul
in o e low o simula ion pa ame e s, and i leads o in e up ion
o me aheu is ic op imiza ion asks be o e a success ul comple ion.
Such in e up ions in ansien con ol simula ion can be a se ious
conce n o implemen a ion o me aheu is ic op imiza ion me h-
ods in he op imal uning o con ol sys ems. To add ess his com-
plica ion in he cu en s udy, a hyb id uning app oach is
implemen ed, which combines s able solu ions o analy ical op i-
mal uning me hod wi h lexibili y o he s ochas ic sea ch: Analy -
ical uning me hods p o ide a s able design poin , and he
p oposed CORS algo i hm pe o ms ine- uning o he design poin
by conside ing ansien simula ions o implemen a ion models o
con ol sys ems. This s a egy allows ine- uning o con ol sys-
ems a ound he op imal design poin s and con ibu es o p ac ical
pe o mance o op imal con olle design me hods.
Conclusions
This s udy in oduced a compu e -aided con olle design
me hodology o imp o emen o dis u bance ejec con ol pe -
o mance o con ol sys ems. The CORS uning algo i hm becomes
mo e e ec i e by coope a ion o op imal uning me hods. The
imp o ed CORS algo i hm s a s wi h con olle coe icien s o
op imal uning me hods and u he op imizes con olle coe i-
cien s o inc ease dis u bance ejec ion pe o mance acco ding o
he consensus cu e. The consensus cu e is p oposed o go e n
he op imiza ion p ocess owa ds con olle solu ions ha esul s
in highe dis u bance ejec ion pe o mance and lowe se poin
e o . To measu e dis u bance ejec ion pe o mance o con ol
loops, RDR spec um o FOPIDA con olle s was ob ained. Then,
con ibu ions o FOPIDA o dis u bance ejec ion con ol pe o -
mance we e in es iga ed.
Simula ion esul s indica e ha he p oposed CORS algo i hm
can deal wi h wo sho -coming o analy ical op imal uning
me hods:
(i) Due o inc easing complexi y and di icul ies in inding ana-
ly ical solu ions o complica ed equa ion sys ems, analy ical
uning me hods do no conside sophis ica ed design speci-
ica ion and cons ain s. The CORS algo i hm can ine- une
Table 2
Pe o mances o con olle s designed o op imal con olling o GðsÞ.
Tuning Me hod S ep Response Dis u bance Response
O e shoo
a io
Numbe o ipples a ound
se poin s
Se ing ime o wi hin 2%
(0.46–0.48)
O e shoo
a io
Numbe o ipples a ound
se poin s
Se ing ime o wi hin 2%
(0.46–0.48)
FOPID (Monje
e al. [11])
23% 2 978 sec 34% 2 862 sec
Op imal PID
(Ma lab)
8% 1 1031 sec 66% 1 1955 sec
2DOF FOPIDA 0% 0 663 sec 21% 3 300 sec
Fig. 6. Responses o he con olle s in he case o s ep dis u bance and whi e noise
(senso noise model).
N. Ozbey e al. / Jou nal o Ad anced Resea ch 25 (2020) 159–170 167