2DOF multi-objective optimal tuning of disturbance reject fractional order PIDA controllers according to improved consensus oriented random search method

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