Optimal approximation of analog PID controllers of complex fractional-order

F ac ional Calculus and Applied Analysis h ps://doi.o g/10.1007/s13540-023-00168-x ORIGINAL PAPER Op imal app oxima ion o analog PID con olle s o complex ac ional-o de Shibendu Maha a1·No be He encsa 2·Guido Maione3 Recei ed: 3 Augus 2022 / Re ised: 3 May 2023 / Accep ed: 4 May 2023 © The Au ho (s) 2023 Abs ac Complex ac ional-o de (CFO) ans e unc ions, being mo e gene alized e sions o hei eal-o de coun e pa s, lend g ea e lexibili y o sys em modeling. Due o he absence o comme cial complex-o de ac ance elemen s, he implemen a ion o CFO models is challenging. To alle ia e his issue, a cons ained op imiza ion app oach ha mee s he a ge ed equency esponses is p oposed o he a ional app oxima ion o CFO sys ems. The echnique gene a es s able, minimum-phase, and eal- alued coe icien s based app oximan s, which a e no always easible o he cu e- i ing app oach epo ed in he li e a u e. S abili y and pe o mance s udies o he CFO p opo ional-in eg al-de i a i e (CFOPID) con olle s o he Podlubny’s, he in e nal model con ol, and he El-Khazali’s o ms a e conside ed o demons a e he easibili y o he p oposed echnique. Simula ion esul s highligh ha , o a p ac- ically easonable o de , all he designs achie e good ag eemen wi h he heo e ical cha ac e is ics. Pe o mance compa isons wi h he CFOPID con olle app oximan s de e mined by he Ous aloup’s CFO di e en ia o based subs i u ion me hod jus i y he p oposed app oach. Keywo ds Complex ac ional-o de sys em (p ima y) ·Complex ac ional-o de PID con olle ·App oxima ion ·Cons ained op imiza ion ·Di e en ial e olu ion BNo be He encsa [email p o ec ed] Shibendu Maha a [email p o ec ed] Guido Maione [email p o ec ed] 1Depa men o Elec ical Enginee ing, D . B. C. Roy Enginee ing College, Du gapu , Wes Bengal 713206, India 2Depa men o Telecommunica ions, Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, Technicka 12, 616 00 B no, Czechia 3Depa men o Elec ical and In o ma ion Enginee ing, Poly echnic Uni e si y o Ba i, Via E. O abona, 4, 70125 Ba i, I aly 123 S. Maha a e al. Ma hema ics Subjec Classi ica ion 26A33 (p ima y) ·93B50 ·93C99 ·93E11 1 In oduc ion Applica ions o ac ional calculus, he b anch o ma hema ics ha is ega ded as a gene aliza ion o classical calculus, a e e e -inc easing ac oss all domains [37, 38]. F ac ional-o de (FO) di e en ia ion/in eg a ion (di e in eg a ion) gene alizes he adi ional de i a ion om an in ege o any eal o complex numbe [58]. Va ious de ini ions o FO de i a i es a e a ailable in he ac ional calculus li e a u e. Capu o’s de ini ion o ac ional de i a i e allows a simila ep esen a ion o ini ial condi ions o ha o he in ege -o de de i a i e because ini ial condi ions a e de e mined by in ege de i a i es. Hence, i p o ides an in e p e a ion o ini ial condi ions o be used in eal-wo ld p oblems mo e easily han he Riemann-Liou ille de ini ion [36]. Howe e , many esea che s dispu e he abili y o Capu o’s de ini ion o ake p ope ly in o accoun ini ial condi ions i used o de ine o simula e ac ional-o de sys ems. In syn hesis, he e is a discussion on he espec i e bene i s and d awbacks o he wo men ioned (o o he ) de ini ions. In de ail, he Capu o and he Riemann-Liou ille de i a i es o a unc ion ( )a e espec i ely de ined as C aDα ( )=1 (n−α)  a (n)(τ) ( −τ)α+1−ndτ, (1.1) RL aDα ( )=1 (n−α) dn d n a (τ) ( −τ)α+1−ndτ, (1.2) whe e nis an in ege , n−1≤α<n,and (·)deno es he gamma unc ion [17]. F om he inpu –ou pu con ol poin o iew, ini ial condi ions a e usually se o ze o such ha ans e unc ions a e conside ed, hen he p oblem o ini ial condi ions is no signi ican . The dynamical sys ems can be ep esen ed by FO di e en ial equa ions; hence, he linea ime-in a ian models can be ep esen ed by using ac ional-o de ans e unc ions (FOTFs). Due o he p esence o addi ional uning knobs, he FO models can mo e e ec i ely cap u e he dynamics o eal-wo ld sys ems. Applicabili y o FO sys ems is epo ed in con ol sys ems, signal p ocessing, ci cui heo y, biomedical enginee ing, e c. [26,56]. The majo i y o he li e a u e has deal wi h he design and implemen a ion o FOTFs comp ising eal- alued exponen o s. The simples ans e unc ion in his ega d is H(s)=sα, which can be ob ained by aking he Laplace ans o m o (1.1) and (1.2) wi h ze o ini ial condi ions. Assuming α∈ +,we may w i e F(s)=spsα−p=spsγ, whe e pis an in ege such ha p=α. The e o e, γ=α−pimplies γ∈(0,1). Fi ing, nume ical, op imiza ion, and con inued- ac ion expansion echniques help o app oxima e he equency esponses o sγ using a ini e-o de a ional ans e unc ion o a speci ied bandwid h [1,19,21,43, 44,48,54]. F om he pe spec i e o signal p ocessing, FR(s)=sγ ep esen s a eal- o de FO di e en ia o (RFOD) o γ∈(0,1), whe eas γ∈(−1,0)co esponds o 123 Op imal app oxima ion o analog... i s in e se unc ion [33]. The ans e unc ion o he classical p opo ional-in eg al- de i a i e (PID) con olle has been gene alized as CR(s)=Kp+Kis−λ+Kdsδ, whe e 0 ≤λ, δ ≤1;Kp,Ki, and Kda e he p opo ional, in eg al, and de i a i e gains [55]. The adi ional il e s, including he special ypes, a e also ex ended o he FO domain [6]. The FOTF can be p ac ically implemen ed by eplacing one o mo e adi ional capaci o /induc o wi h he cons an phase elemen (CPE), such as he FO capaci o /induc o [15]. Se e al wo ks expe imen ally demons a ed he pe o mances o FO ci cui s and sys ems (con olle s, il e s, e c.) employing he CPEs [16,28]. The CPEs a e no ye a ailable as comme cial de ices in he ma ke ; hough, p oposals o implemen he FO sys ems o indus ial applica ions a e d awing a ac ion [50,66]. The impedance cha ac e is ics o he CPE can be emula ed using passi e o ac i e emula o ci cui s [30,31]. Howe e , he ha dwa e complexi y o he emula o s is subs an ial compa ed o a single elemen ac o . An al e na i e app oach ha a oids using CPEs in ealizing he FO sys ems in ol es de e mining he in ege -o de app oxima ion o he FOTF. Such models may be ob ained h ough di ec subs i u ion o he a ional app oximan o sγope a o in he FOTF [67]. Fo he non-commensu a e ype o mo e complica ed FOTFs, he sub- s i u ion me hod incu s la ge o e head. Hence, nume ical o op imal app oaches a e he p e e ed app oxima ion ools [11,42]. 1.1 Complex ac ional-o de con olle The gene ali y in oduced by ac ional calculus allows complex FOTF-based sys em modeling, i.e., he exponen o smay be a complex numbe . Fo ins ance, he RFOD is a subse o he complex FO di e en ia o (CFOD) FC(s)=sα+jβ, whe e α∈[0,1] and β∈[68]. The complex FOTFs p o ide mo e deg ees-o - eedom in modeling as compa ed o hei eal-o de coun e pa s. The complex-o de de i a i e o a sine unc ion a s eady-s a e is exp essed as Dα+jβsin(ω )=ωα[cos{βln(ω)}+jsin{βln(ω)}] ×sin ω +απ 2cosh βπ 2+jcos ω +απ 2sinh βπ 2,(1.3) whe e j=√−1 and ωis he angula equency a iable exp essed in adians pe second ( ad/s) [40]. The hi d-gene a ion CRONE con olle is a seminal wo k ha deals wi h he design and applica ion o complex-o de models [35]. O he applica ion a eas o complex- o de de i a i es can be ound in he locomo ion con ol o obo s [51,61], chao ic oscilla o s [52,53], sys em iden i ica ion [5,29], il e modeling [2], iscoelas ici y [8], op imiza ion algo i hms [7,39] and neu al ne wo ks [32]. The e ec i eness o complex ac ional-o de PID (CFOPID) con olle was demons a ed on ime-delay p ocess [3]. The imp o ed ime-domain esponses yielded by he gene ic algo i hm based complex FO con olle s as compa ed o he classical and eal-o de FO con- olle s we e jus i ied [64]. Tuning ules o he CFOPID and CFOPI con olle s we e p oposed in [23,59]. Modeling o CFODs and CFOPIDs in he disc e e- ime domain 123 S. Maha a e al. was also ca ied ou [9,40,57]. S abili y analysis o complex-o de ac ional di - e ence equa ion was epo ed in [10], while [47] in es iga ed he geome ical and physical in e p e a ion o he complex-o de ac ional in eg al. 1.2 Mo i a ions o his wo k The e ec i eness o complex FOTF modeling has no been expe imen ally alida ed since implemen ing such ans e unc ions equi es he complex-o de ac ance ele- men s; ealizing such elemen s has no ye ecei ed much a en ion in he li e a u e [22, 60]. The e o e, he a ional app oxima ion o he complex FOTFs emains a easible choice. The mo i a ion o ca ying ou his esea ch is based on he limi a ions o he exis ing me hods, as discussed below. 1.2.1 Limi a ions o i ing ou ines Cu e- i ing echniques we e employed o he app oxima ion o analog CFOD [14], CFOPID [13], and complex-o de il e [12]. The i ing echnique is based on he Sana hanan-Koe ne algo i hm wi h Le y’s cos unc ion [14]. The i h-o de app oxima ion o he CFOPID con olle epo ed in (28) o [14]isgi enby FC(s)=1+1 s0.8+0.1j+s0.8+0.1j ≈−2353s5+2.14 ×106s4+7.53 ×106s3+3.77 ×106s2+2922s−55.91 s5−1.43 ×104s4+4.58 ×106s3+1.63 ×105s2−100s−0.03 .(1.4) The app oxima ed model in (1.4) comp ises eal- alued coe icien s o bo h he nume a o and denomina o polynomials. Howe e , he poles o he a io- nal app oximan a e loca ed a {13972,327.8301,8.19 ×10−4,−0.0362,−2.21 × 10−4}, whe eas he ze os lie a {912.9844,−2.9027,−0.6036,−0.0043,0.0035}.The epo ed CFOPID app oximan is an uns able and non-minimum-phase sys em, since h ee poles (13972, 327.8301, 8.19 ×10−4) and wo ze os (912.9844, 0.0035) exis on he igh -hal s-plane. Vec o Fi ing (VF) [24,25] is ano he popula echnique o i he equency- domain esponses wi h a ional unc ions. The VF me hod is applied o a bandwid h o [0.001, 1000] ad/s wi h 1000 sample poin s o gene a e he ou h and i h-o de app oxima ions o he heo e ical CFOPID con olle gi en in (1.4). The co esponding app oximan s a e gi en by FC(s)≈307s4−12272s3+1781582s2+256374s+71.9481 s4+1383s3+418560s2+1247s+0.0851 ,(1.5) FC(s)≈1389.97s5+68427s4−819142s3+3320489s2+52141s+62.24 s5+5875s4+1273888s3+109639s2+315.95s+0.2286 .(1.6) The ze os o he VF-based ou h-o de app oximan lie a {−0.000281,−0.1434,20.0588±73.5296 j}; he poles eside a {−0.000069, −0.002909,−447.338,−935.659}. Fo he i h-o de app oximan , he loca ions 123 Op imal app oxima ion o analog... Fig. 1 Magni ude and phase plo s o he VF-based ou h and i h-o de app oxima ions o he heo e ical CFOPID con olle gi en in (1.4) o ze os and poles a e ob ained as {−59.7594,5.2730±3.5107 j,−0.0143,−0.0013} and {−5649.5172,−225.3966,−0.0831,−0.0016,−0.0013}, espec i ely. The VF echnique can gene a e a s able app oximan since all he poles eside on he le - hal s-plane. Howe e , he p esence o wo igh -hal plane ze os (occu ing in complex-conjuga e o m) o bo h he app oximan s highligh non-minimum phase beha io . Such sys ems may lead o s abili y issues in a eedback con ol mechanism. The magni ude and phase plo s o he VF-based app oximan s a e p esen ed in Fig. 1. I can be seen ha bo h he app oximan s ail o a ain good con o mi y wi h he heo e ical esponse. 1.2.2 Limi a ions o a buil -in MATLAB unc ion Using he MATLAB unc ion in eqs wi h minimiza ion o he no m o he g adien (wi h pa ame e i e se as 30 and conside ing 1000 sample poin s), he ou h and i h-o de app oxima ions o he heo e ical CFOPID con olle unc ion gi en by (1.4) a e espec i ely de e mined as FC(s)≈8.26 ×1017s4+2.49 ×1021s3+8.25 ×1022s2+4.15 ×1024s+3.86 ×1023 s4+1.26 ×1016s3+1.02 ×1022s2+7.02 ×1023s+1.78 ×1021 , (1.7) 123 S. Maha a e al. Fig. 2 Magni ude and phase plo s o he in eqs-based ou h and i h-o de app oxima ions o he heo e ical CFOPID con olle gi en in (1.4) FC(s)≈ 1.84 ×1018s5+4.59 ×1021s4+5.91 ×1023s3 +3.06 ×1025s2+7.47 ×1026s+7.91 ×1025 s5+4.27 ×1017s4+1.87 ×1022s3+2.47 ×1024s2 +1.28 ×1026s+3.23 ×1023 .(1.8) No e ha he app oximan s gene a ed by in eqs wi hou he i e pa ame e a e bo h uns able and non-minimum-phase. The poles o he ou h and i h-o de a io- nal unc ions a e loca ed a {−1.26 ×1016,−8.09 ×105,−68.8268, −0.0025} and {−4.27×1017,−4.36×104,−66.1630±49.8776 j,−0.0025}, espec i ely; he ze os eside a {−16.4200±37.5811 j,−0.0931, −2981.59} and {−2361.5033, −67.4670, −32.7442±38.2736 j,−0.1063}, espec i ely. Al hough s able and minimum-phase app oxima ions a e ob ained, his me hod gi es an ill-condi ioned unc ion (coe i- cien s wi h ex emely la ge o small alues), such ha i s implemen a ion will lead o imp ac ical alues o passi e componen s. The magni ude and phase esponses o he in eqs-based ou h and i h-o de app oximan s a e p esen ed in Fig. 2. These plo s highligh a signi ican de ia ion om he heo e ical esponses o a wide ange o he conside ed bandwid h. 123 Op imal app oxima ion o analog... 1.2.3 Applicabili y o he Ous aloup’s ecu si e algo i hm An al e na i e echnique o app oxima e he heo e ical CFOD cha ac e is ics was epo ed by Ous aloup e al. [48]. Acco ding o he Ous aloup’s me hod, he a io- nal app oximan o a CFOD comp ises o complex poles and ze os. The CFOD app oximan o s , whe e =a+jb, based on his echnique is gi en by D M(s)=μ− /2 M  k=−M 1+s ω k 1+s ωk ,(1.9) whe e Mis a posi i e in ege , μ=ωh ωb , and he bandwid h o app oxima ion is in he in e al [ωb,ω h] ad/s. The alues o ωkand ω ka e espec i ely gi en by ωk=ρkejθ,(1.10) ω k=ρ ke−jθ,(1.11) whe e ρk=ωbωh ωbk+M+1/2+a/2 2M+1,ρ k=ωbωh ωbk+M+1/2−a/2 2M+1, and θ=b 2(2M+1)log(μ). The Ous aloup’s CFOD app oximan may be e-w i en as a a io o complex unc ions as gi en by DM(s)=P(s)+jQ(s) P(s)+jQ(s),(1.12) whe e P(s), P(s), Q(s), and Q(s)a e eal coe icien s based polynomials. Al e na i ely, (1.12) may be w i en as DM(s)=PM(s) QM(s)+jP M(s) QM(s),(1.13) whe e PM(s)=P(s)P(s)+Q(s)Q(s),P M(s)=P(s)Q(s)−Q(s)P(s), and QM(s)=P2(s)+Q2(s). A F ench pa en [49] employing ope a ional ampli ie s o he ci cui implemen a- ion o (1.13) was epo ed h ee decades ago. Howe e , i may be mo e s aigh o wa d o p ac ically implemen a eal- alued unc ion as compa ed o (1.13), which is a com- plex unc ion. Mo eo e , as shown in Sec .3, he Ous aloup’s app oxima ion e en wi h M= 1 may lead o a CFOPID con olle o high o de , wi h coe icien s ha ing la ge alues. The e o e, i is wo h in es iga ing he ollowing ques ion: Wi hin he bandwid h app oxima ion limi s whe e six (o less) decades may be su icien (in gene al) o p ocess con ol applica ions, is i possible o achie e easonable accu acy using a eal- alued a ional ans e unc ion wi h smalle o de s (e en and odd) o app oxima e he cha ac e is ics o he CFOPID con olle ? 123 S. Maha a e al. 1.3 Con ibu ions o his wo k In his pape , he app oxima ion o he CFOPID con olle is p esen ed whe ein he app oximan is s ic ly a eal unc ion. The eal- alued coe icien s o he a ional app oximan a e de e mined op imally, while en o cing he poles and ze os o lie on he le -hal s-plane. Thus, he p oposed wo k helps in alle ia ing he issues associa ed wi h he exis ing app oaches. The p ima y con ibu ions o his wo k a e he ollowing: 1. A cons ained op imiza ion app oach based on a s a e-o - he-a e olu iona y me hod is p esen ed ha sa is ies he condi ions o s abili y, minimum-phase, and achie es eal- alued coe icien s eal a ional app oximan s o bo h e en and odd o de s o he CFOPID con olle . 2. The easibili y and e ec i eness o he p oposed app oach is alida ed on h ee di e en a ian s o he CFOPID con olle , namely he Podlubny’s o m [55], he in e nal model con ol (IMC) o m [65], and he El-Khazali’s o m [20]. E ec s o he app oxima ion o de on he app oxima ion pe o mance a e e alua ed using a ious equency esponse e o me ics. 3. The accu acy o he p oposed echnique is compa ed wi h he Ous aloup’s me hod. Resul s demons a e ha he p oposed app oach p o ides a compe i i e pe o mance when compa ed agains he much highe o de o he compe ing designs. While he Ous aloup’s CFOD app oximan s a e subs i u ed in he heo- e ical CFOPID ans e unc ion o ob ain he co esponding app oxima ion, he p oposed me hod di ec ly de e mines he CFOPID con olle coe icien s. In he es o he pape , Sec . 2p esen s he p oposed op imiza ion p oblem o mula ion and he solu ion me hodology. In Sec .3, design pe o mance s udies a e ca ied ou , and he simula ion esul s a e discussed. The pape concludes in Sec .4. 2 P oposed echnique 2.1 P oblem o mula ion The gene alized ans e unc ion o a eal FOTF is de ined as FR(s)=Bmsλm+Bm−1sλm−1+...+B0sλ0 Ansμn+An−1sμn−1+...+A0sμ0,(2.1) whe e Bi(i=0,1,...,m),Ak(k=0,1,...,n),λi(i=0,1,...,m), and μk(k=0,1,...,n)a e eal numbe s; λm>λ m−1>··· >λ 0≥0; and μn>μ n−1>···>μ 0≥0. Replacing he eal- alued exponen s o sin (2.1) wi h complex numbe s, such as λi= i+jy i(i=0,1,...,m)and μk=wk+jzk(k= 0,1,...,n), leads o he complex FOTF, as de ined below, which p o ides u he gene aliza ion and mo e deg ees-o - eedom han (2.1): FC(s)=Bms m+jym+Bm−1s m−1+jym−1+...+B0s 0+jy0 Answn+jzn+An−1swn−1+jzn−1+...+A0sw0+jz0.(2.2) 123 Op imal app oxima ion o analog... Any classical con inuous- ime sys em can be ep esen ed as a a io o polynomials in s, as de ined by HN P(s)=PN(s) QN(s)=aNsN+aN−1sN−1+...+a0 sN+bN−1sN−1+bN−2sN−2+...+b0 ,(2.3) whe e ak(k=0,1,...,N)and bk(k=0,1,...,N−1)a e he eal- alued coe i- cien s o he nume a o and denomina o polynomials, espec i ely, o HN P(s);and N is he o de o he sys em. HN P(s)is bounded-inpu bounded-ou pu (BIBO) s able i all he poles eside in he le -hal s-plane; i all he ze os lie on he le -hal s-plane, hen he sys em a ains minimum-phase beha io . In he case o complex conjuga e poles o ze os, hei eal pa mus be nega i e. A necessa y condi ion o BIBO s a- bili y and minimum-phase esponse is ha all he coe icien s o QN(s)and PN(s), espec i ely, mus possess he same sign wi h no missing e ms. The equency-domain ans e unc ions o (2.2) and (2.3) can be ob ained by subs i u ing swi h jω, as espec i ely de ined below FC(jω) =Bm(jω) m+jym+Bm−1(jω) m−1+jym−1+...+B0(jω) 0+jy0 An(jω)wn+jzn+An−1(jω)wn−1+jzn−1+...+A0(jω)w0+jz0 ,(2.4) HN P(jω) =aN(jω)N+aN−1(jω)N−1+...+a0 (jω)N+bN−1(jω)N−1+bN−2(jω)N−2+...+b0 .(2.5) Table 1shows he ans e unc ion and equency domain exp essions o he di - e en CFOPID con olle s conside ed o app oxima ion in his wo k. The design p oblem can be sol ed in he op imal sense, whe e he pu pose o he sea ch ou- ine is o minimize he equency esponse e o be ween he heo e ical ( a ge ed) model FC(s)and he a ional app oximan HN P(s), subjec o sa is ying he s abili y and minimum-phase c i e ia. The objec i e unc ion o he p oposed cons ained op imiza ion (minimiza ion) p oblem can be o mula ed as =1 L L  i=1|FC(jωi)|−HN P(jωi,X)+ ∠FC(jωi)−∠HN P(jωi,X), (2.6) Subjec o :σz<0 and σp<0(p, z =1,...,N), whe e Lis he o al numbe o log-spaced da a poin s in he bandwid h o app oxi- ma ion [ωmin,ω max] ad/s; X ep esen s he decision a iables ec o , i.e., X=[aN aN−1…a0bN−1bN−2…b0]; σzand σpdeno e he eal pa o he oo s o PN(s) and QN(s), espec i ely. The o al numbe o decision a iables o his op imiza ion p oblem is 2N+1, which ep esen s he dimension (D). 2.2 Solu ion me hodology A me aheu is ic algo i hm acili a es he in eg a ion o a ious cons ain handling s a egies o gene a e a easible solu ion [34]. Al hough (2.6) may be minimized using 123 S. Maha a e al. Fig. 3 Flowcha o he p oposed design me hod as {105,10 5}, {104,10 4}, {104,10 4}, and {105,10 6}, espec i ely, based on he ial-and-e o me hod men ioned p e iously. No e ha inc easing Nshould imp o e he app oxima ion by educing he e o s in he Bode gain and phase plo s. A highe o de Nis needed o ep esen he long memo y o he sys em e y accu a ely, bu he compu a ional bu den inc eases. On he con a y, a lowe Np oduces mo e ipples in gain and phase plo s, bu is ad an ageous, because he changes in coe icien s due o ole ances o componen s (in an analog implemen a ion) o due o he limi a ions o mic op ocesso wo ds and he quan iza ion e ec s ( o a digi al implemen a ion) can be con ained [45]. Hence a low sensi i i y o pa ame e a ia ions is ob ained. Mo eo e , an accu a e implemen a ion o e a wide equency ange is di icul o a con en ional ha dwa e ealiza ion due o compu a ional complexi y [27] and ha dwa e cos s can be limi ed i he accu acy is equi ed in a es ic ed equency ange. Finally, om he p ac ical poin o iew, app oxima ions a e no equi ed o e a e y la ge equency ange. Namely, in sys ems and con ol enginee ing applica ions, app ox- ima ions should wo k wi hin a bounded equency ange, whe e a ew decades a e usually sa is ac o y. 123 Op imal app oxima ion o analog... Fig. 4 The magni ude and phase- equency compa ison plo s o he p oposed op imal app oximan s o he CFOPID con olle s wi h he heo e ical models o (a) case I, (b) case II, (c) case III, (d) case IV The s abili y and minimum-phase o he designs can be con i med om Table 4, which shows ha all he poles and ze os eside on he le -hal s-plane. This is an impo an p ope y since he occu ence o non-minimum-phase ze os in a con olle may lead o s abili y issues in a closed-loop con ol sys em, including app oximan s o such con olle . The equency esponses o he designed con olle s a e compa ed wi h he heo- e ical models o he ou cases, as shown in Figs. 4(a)–(d). I may be obse ed ha : (i) o he case I, he six h-o de design achie es lowe magni ude e o han he i h- o de model in he equency ange [0.001, 0.028] ad/s. The i h-o de app oximan ou pe o ms he six h-o de one in e ms o phase esponse accu acy in he in e al [0.001, 0.145] ad/s. F equency esponses o bo h he app oximan s a e simila a high equency egions. Inc easing he o de o he p oposed app oximan u he sligh ly educes he e o in phase app oxima ion bu does no signi ican ly a ec he e o in magni ude app oxima ion; (ii) o case II wi h bo h he app oxima ion o de s, a simila accu acy in magni ude is ob ained in he equency ange [0.181, 6.884] ad/s, whe eas he same accu acy o phase cha ac e is ic is ob ained be ween 0.774 and 3.4 ad/s. O e all, an imp o ed accu acy in magni ude and phase is espec i ely a ained using 123 S. Maha a e al. Table 4 Loca ions o ze os and poles o he p oposed op imal app oximan s o he CFOPID con olle s Case NZe os Poles I5−2.1488 −0.9432, −0.1116, −0.0074, –59290, −0.1733, −0.0202, −0.0037, −0.0015 −0.0007 6−2.5457, −0.0865±1.0970j, −0.8593, –39646, −0.0941±1. 0932j, −0.1997, −0.1136, −0.0103 −0.0172, −0.0009 II 4 −43.22, −0.7714, −0.0756, −0.0048 −9991.7, −0.2740, −0.0220, −0.0019 5−192.73, −0.6720, −0.0877, −0.0138, −9496.7, −0.2209, −0.0248, −0.0045, −0.0026 −0.0006 III 5 −1236.5, −2.8835, −0.5352, −0.0279, −1238.4, −3.5570, −0.0707, −0.0088, −0.0029 −0.0010 6−2217.0, −0.8230±0.6719j, −0.3221, −2328.1, −0.8829±0.2645j, −0.0749, −0.0372, −0.0036 −0.0114, −0.0011 IV 4 −7.0858, −0.4112, −0.0327, −0.0019 –19642, −0.0671, −0.0035, −0.00015 5−6.4104, −0.4287, −0.0728, −0.0117, –183440, −0.1225, −0.0180, −0.0018, −0.0003 −0.00015 123 Op imal app oxima ion o analog... Table 5 Compa ison o e o me ics o he p oposed op imal app oximan s o he CFOPID con olle s (bes pe o mance is highligh ed in bold ace) Case NMax Absolu e E o Mean Absolu e E o Magni ude (dB) Phase (◦) Magni ude (dB) Phase (◦) I 5 49.841 22.469 29.939 6.029 649.153 28.412 25.023 12.405 II 4 34.570 43.250 14.499 4.668 5−0.795 51.978 −21.040 20.903 III 5 46.443 5.696 23.371 0.913 645.982 3.265 22.728 1.580 IV 4 42.075 36.109 17.247 15.584 5 45.052 21.782 21.639 11.798 he designs wi h N= 5 and N= 4; (iii) o case III, he equency cha ac e is ic plo s o bo h he app oximan s exhibi simila i y. While he phase esponse exhibi s p ox- imi y wi h he heo y h oughou he bandwid h, he magni ude plo s a s de ia ing a he lowe equency end below 0.04 ad/s; (i ) o case IV, he magni ude- equency p o iles o he p oposed ou h and i h-o de ap oximan s a e simila , whe eas he i h-o de design ma kedly ou pe o ms he ou h-o de model in he ange [0.001, 0.026] ad/s ega ding he phase esponse accu acy. Table 5p esen s he pe o mance me ics o all he designed CFOPIDs, which show ha : (i) o case I, maximum APE o N=5 (49.841 dB) and N=6 (49.153 dB) is simila . The i h-o de design ou pe o ms he six h-o de con olle abou he maximum and mean APE, whe eas he six h-o de design achie es smalle mean AME; (ii) o case II, he i h-o de design is in e io o i s lowe -o de coun e pa ega ding he maximum and mean APE me ics, bu signi ican ly ou pe o ms he ou h-o de app oximan abou bo h he magni ude esponse indices; (iii) o case III, he pe o mances o bo h he designed con olle s a e simila ; (i ) subs an ial imp o e- men in maximum APE is achie ed by he i h-o de app oximan as compa ed o he ou h-o de one o case IV. Howe e , i is ema ked ha he high alues o maximum absolu e e o s a e ypically ob ained a he ex emum low and high equencies o he conside ed in e als, whe e he app oxima ions a e ob iously wo s . As a consequence, mean absolu e e o s a e a ec ed by such alues. On he con a y, he app oxima ions gi e sui able esul s inside he equency in e al [ωmin,ωmax]. 3.2 Compa ison wi h he li e a u e The CFOPID con olle can be app oxima ed using Ous aloup’s CFOD app oximan by a simple subs i u ion me hod. Fo ins ance, he no malized CFOD app oximan o s0.8+0.1jbased on he Ous aloup’s me hod wi h M=1, ωb=0.001 ad/s, and 123 S. Maha a e al. ωh=1000 ad/s, is gi en by s0.8+0.1j≈D0.8+0.1j 1=1 1000(0.8+0.1j)⎛ ⎜ ⎜ ⎝ 1+s ω −1 1+s ω−1 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 1+s ω 0 1+s ω0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 1+s ω 1 1+s ω1 ⎞ ⎟ ⎟ ⎠ , (3.3) whe e ω−1=0.0614+0.0144 j,ω −1=0.0015−0.0004 j,ω0=6.1430+1.4400 j, ω 0=0.1543 −0.0362 j,ω1=614.3+144 j, and ω 1=15.4306 −3.6172 j. Thus, he heo e ical CFOPID con olle gi en by (1.4) can be app oxima ed using he Ous aloup’s echnique as FC(s)=1+s0.8+0.1j+1 s0.8+0.1j≈1+D0.8+0.1j 1(s)+1 D0.8+0.1j 1(s) .(3.4) Then, (3.4) can be ep esen ed in he o m o (1.13), whe e he exp essions o PM(s),P M(s), and QM(s)a e ob ained as PM(s)=194.5053s12 +1.4910 ×105s11 +8.2897 ×106s10 +1.7707 ×108s9 +1.7046 ×109s8+6.7896 ×109s7+8.8736 ×109s6+6.6288 ×109s5 +1.6625 ×109s4+1.7320 ×108s3+8.1395 ×106s2+1.4701 ×105s +187.2878,(3.5) P M(s)=159.9975s12 +8.0683 ×104s11 +3.3008 ×106s10 +4.7865 ×107s9 +2.4461 ×108s8+8.9531 ×107s7−3.9970 ×108s6−1.6790 ×108s5 +1.5350 ×108s4+3.6652 ×107s3+2.7079 ×106s2+6.8829 ×104s +144.7765,(3.6) QM(s)=s12 +1.2548 ×103s11 +4.3522 ×105s10 +1.7476 ×107s9 +2.6860 ×108s8+1.7739 ×109s7+4.5571 ×109s6+1.7733 ×109s5 +2.6834 ×108s4+1.7441 ×107s3+4.3339 ×105s2+1.2199 ×103s +0.9594.(3.7) Eqns. (3.5), (3.6), and (3.7) e eal ha e en wi h M=1, he subs i u ion o Ous aloup’s CFOD model yields a CFOPID con olle o wel h o de (N= 12) o bo h he eal and imagina y pa s o he app oximan . The coe icien s also a ain la ge alues as obse ed om he abo e h ee exp essions. No e ha he Ous aloup’s me hod yields only an e en o de o a ional app oximan o he CFOPID con olle . Fo compa ison pu pose, he CFOPID con olle s o all he ou cases a e ob ained by subs i u ing he Ous aloup’s CFOD app oximan (wi h M=1in(1.9)) in he heo e ical con olle unc ions. 123 Op imal app oxima ion o analog... Fig. 5 The magni ude and phase- equency compa ison plo s o he p oposed CFOPID con olle s wi h he published li e a u e o (a) case I, (b) case II, (c) case III, (d) case IV The magni ude and phase esponses compa ison plo s o he con olle s ob ained by using he Ous aloup’s CFOD-based subs i u ion and he p oposed op imiza ion wi h la ge Na e illus a ed in Figs. 5(a)-(d) o cases I-IV, espec i ely. I is obse ed ha : 1. o case I, he p oposed and Ous aloup’s con olle achie e supe io accu acy in he magni ude and phase esponses, espec i ely, a he lowe equency alues, whe eas hei pe o mance is simila in he mid-band egion. A he highe equen- cies, he p oposed con olle a ains be e phase accu acy, whe eas he Ous aloup’s app oximan yields supe io accu acy in magni ude; 2. o case II, he p oposed con olle ’s phase beha io is in e io o Ous aloup’s. Howe e , he p oposed app oximan a ains be e p oximi y o he heo e ical magni ude plo o e he en i e bandwid h; 3. o case III, he magni ude esponse accu acy o he Ous aloup’s CFOPID app ox- iman is supe io o ha o he designed one o equencies in [0.001, 0.019] ad/s. The phase esponse o he p oposed app oxima ion su e s lowe de ia ion in he in e al [0.001, 0.64] ad/s, while he Ous aloup’s model achie es be e con o - mi y wi h he heo e ical magni ude- equency beha io . A he highe equencies, he beha io is simila ; 123 S. Maha a e al. Table 6 Compa ison o e o me ics o he p oposed CFOPID con olle s wi h he epo ed li e a u e (bes pe o mance is highligh ed in bold ace) Case Model Max Absolu e E o Mean Absolu e E o Magni ude (dB) Phase (◦) Magni ude (dB) Phase (◦) I Ous aloup 34.665 44.338 12.262 8.517 P oposed 49.153 28.412 25.023 12.405 II Ous aloup 20.814 39.216 −0.419 8.143 P oposed −0.795 51.978 −21.040 20.903 III Ous aloup 34.665 36.745 5.992 3.817 P oposed 45.982 3.265 22.728 1.580 IV Ous aloup 34.641 44.357 12.309 7.990 P oposed 45.052 21.782 21.639 11.798 4. o case IV, in he low equency egion, he p oposed model yields an in e io phase esponse. Howe e , he p oposed app oximan achie es a smalle alue o he maximum de ia ion in phase esponse beyond 335 ad/s. In Table 6, compa isons abou he AME and APE me ics be ween he p oposed and Ous aloup’s CFOPID app oximan s a e p esen ed. Resul s e eal ha : (i) o case I, he Ous aloup’s model yields be e pe o mance o maximum and mean AME and mean APE, whe eas he p oposed design achie es signi ican ly lowe maximum APE; (ii) o case II, he magni ude e o s a e signi ican ly smalle o he p oposed design, al hough he Ous aloup’s model ou pe o ms he p oposed one abou he phase esponse e o s; (iii) o case III, signi ican educ ion in maximum APE (3.265◦) is ob ained by he p o- posed app oximan as compa ed o he Ous aloup’s app oximan (36.745◦), al hough he Ous aloup’s design yields a be e magni ude esponse accu acy; (i ) o case IV, he p oposed model p o ides ma kedly imp o ed accu acy o e Ous aloup’s o he maximum APE (21.782◦ e sus 44.357◦) bu is ou pe o med o he o he h ee e o indices. O e all, i may be in e ed ha he p oposed app oach does no ou pe o m he Ous aloup’s me hod o bo h he magni ude and phase esponses simul aneously. How- e e , he p oposed echnique esul s in eal coe icien s based eal a ional ans e unc ions, which is in con as o he eal coe icien s based complex ans e unc ion yielded using he Ous aloup’s me hod. The e o e, ha dwa e implemen a ion o he p oposed con olle s can be ealized using s anda d ci cui design echniques eadily a ailable in he li e a u e. Fu he mo e, unlike he Ous aloup’s me hod, he e is no es ic ion on he app oxima ion o de (N: e en o odd) o he p oposed con olle s. Among he a ious echniques a ailable o con inuous- ime a ional app oxima- ion o FOTFs, i is no possible o es ablish which one is he bes [46]. While some me hods p o ide be e accu acy ega ding he equency esponse, o he s may achie e imp o ed ime esponse pe o mance. The app oxima ion pe o mances can also depend on he non-in ege o de . The eal and imagina y pa o he complex- alued impulse esponse o he p oposed six h-o de app oximan and he Ous aloup’s 123 Op imal app oxima ion o analog... 0 0.005 0.01 0.015 0.02 0.025 0.03 −2.5 −2 −1.5 −1 −0.5 0 0.5 x 106 Time (s) Impulse Response (Real Pa ) Theo e ical Ous aloup’s app oximan P oposed app oximan Fig. 6 Impulse esponse ( eal pa ) compa ison plo s be ween he p oposed (N= 6) and Ous aloup’s (N= 12) app oximan s o he CFOPID con olle o case I app oximan o wel h o de o he CFOPID con olle pe aining o case I a e com- pa ed in Figs. 6and 7, espec i ely. I is e iden ha he Ous aloup’s app oximan a ains supe io impulse esponses ma ching wi h he heo e ical ones as compa ed o he p oposed one. Howe e , he de ia ion o he impulse esponse o he p o- posed app oximan diminishes wi h espec o he heo e ical a e a sho ime (abou 0.005s). 4 Conclusions An op imal echnique ha gua an ees s able poles, minimum-phase ze os, and eal- alued coe icien s o a ional app oximan s o he complex ac ional-o de PID con olle s is p esen ed in his pape . The d awbacks o he epo ed cu e- i ing echniques a e elimina ed h ough (i) inco po a ion o cons ain s, (ii) app op ia e selec ion o he lowe bound o decision a iables pe aining o he coe icien s o he nume a o and denomina o polynomials o he p oposed model, and (iii) u i- lizing a eal-pa ame e cons ained op imiza ion algo i hm. The e ec i eness o he sugges ed echnique is e i ied on he Podlubny’s, IMC, and El-Khazali’s o ms o CFOPID con olle s. Pe o mance compa isons o di e en o de s (odd and e en) o app oxima ion a e demons a ed o he design examples. Compa isons wi h he Ous aloup’s CFOPID app oximan shows ha he p oposed app oach may achie e be - e magni ude o phase esponse i ing, hough simul aneous imp o emen s in bo h a e 123 S. Maha a e al. Fig. 7 Impulse esponse (imagina y pa ) compa ison plo s be ween he p oposed (N= 6) and Ous aloup’s (N= 12) app oximan s o he CFOPID con olle o case I no possible. The p oposed app oach may be conside ed as an e ec i e al e na i e o he Ous aloup’s me hod ha esul s in complex a ional app oximan s wi h eal coe - icien s; especially, when he bandwid h conside a ions a e limi ed o 3∼4 decades. Fo he conside ed non-linea op imiza ion p oblem, wo limi a ions o he p oposed app oach a e (i) an inabili y o gua an ee he gene a ion o a global op imal solu ion, which is in gene al ue when employing a me aheu is ic me hod, and (ii) la ge dispe - sion o coe icien s (especially o he denomina o polynomial) when Nis inc eased. While his wo k solely concen a ed on he app oxima ion o CFOPID con olle s, i will be in e es ing o conside unc ions such as he uns able, non-minimum phase, and conjuga ed-o de CFO sys ems [4,68] in he u u e. Sol ing such p oblems may necessi a e inco po a ion o addi ional design cons ain s ha may in ol e employing he mul i- o many-objec i e cons ained e olu iona y me hods. Funding Open access publishing suppo ed by he Na ional Technical Lib a y in P ague. Con lic o in e es The au ho s decla e ha hey ha e no con lic o in e es . Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence, and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde . To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/. 123 Op imal app oxima ion o analog... Re e ences 1. AbdelA y, A.M., Elwakil, A.S., Radwan, A.G., Psychalinos, C., Maundy, B.J.: App oxima ion o he ac ional-o de Laplacian sαas a weigh ed sum o i s -o de high-pass il e s. IEEE T ans. Ci cui s Sys . II: Exp . B ie s 65, 1114–1118 (2018). h ps://doi.o g/10.1109/TCSII.2018.2808949 2. AbdelA y, A.M., Sol an, A., Ahmed, W.A., Radwan, A.G.: On he analysis and design o ac ional- o de Chebyshe complex il e . Ci cui s Sys . Signal P ocess. 37, 915–938 (2018). h ps://doi.o g/10. 1007/s00034-017-0570-1 3. Abdulwahhab, O.W.: Design o a complex ac ional o de PID con olle o a i s o de plus ime delay sys em. ISA T ans. 99, 154–158 (2020). h ps://doi.o g/10.1016/j.isa a.2019.10.010 4. Adams, J.L., Ha ley, T.T., Adams, L.I.: A solu ion o he undamen al linea complex-o de di e en ial equa ion. Ad . Eng. So w. 41, 70–74 (2010). h ps://doi.o g/10.1016/j.ad engso .2008.12.014 5. Adams, J.L., Ha ley, T.T., Lo enzo, C.F.: Iden i ica ion o complex o de -dis ibu ions. J. Vib. Con ol 14(9–10), 1375–1388 (2008). h ps://doi.o g/10.1177/1077546307087443 6. Ali, A.S., Radwan, A.G., Soliman, A.M.: F ac ional o de Bu e wo h il e : ac i e and passi e ealiza- ions. IEEE J. Eme g. Sel. Top. Ci cui s Sys . 3(3), 346–354 (2013). h ps://doi.o g/10.1109/JETCAS. 2013.2266753 7. A anacko ic, T.M., Jane , M., Pilipo ic, S., Zo ica, D.: Eule -Lag ange equa ions o Lag angians con aining complex o de ac ional de i a i es. J. Op im. Theo y Appl. 174, 256–275 (2017). h ps:// doi.o g/10.1007/s10957-016-0873-6 8. A anacko ic, T.M., Konjik, S., Pilipo ic, S., Zo ica, D.: Complex o de ac ional de i a i es in is- coeleas ici y. Mech. Time-Depend. Ma e . 20, 175–195 (2016). h ps://doi.o g/10.1007/s11043-016- 9290-3 9. Ba bosa, R.S., Machado, J.T., Sil a, M.F.: Disc e iza ion o complex-o de algo i hms o con ol applica ions. J. Vib. Con ol 14(9–10), 1349–1361 (2008). h ps://doi.o g/10.1177/1077546307087445 10. Bhaleka , S., Gade, P.M., Joshi, D.: S abili y and dynamics o complex o de ac ional di e ence equa ions. Chaos Soli ons F ac . 158, 112063 (2022). h ps://doi.o g/10.1016/j.chaos.2022.112063 11. Bingi, K., Ib ahim, R., Ka si i, M.N., Hassan, S.M., Ha ind an, V.R.: F ac ional-O de Sys ems and PID Con olle s. Sp inge , Cham (2020) 12. Bingi, K., De an, P.A.M., P us y, B.R.: Design and analysis o ac ional il e s wi h complex o de s. In: P oceedings o he 3 d IEEE In e a ional Con e ence on Ene gy Powe En .: Towa ds Clean Ene gy Tech. Shillong, Meghalaya, India, 5–7 Ma (2021) h ps://doi.o g/10.1109/ICEPE50861.2021. 9404431 13. Bingi, K., Kulka ni, R.R., Man i, R.: Design and analysis o complex ac ional-o de PID con olle s. In: P oceedings o he IEEE Mad as Sec ion Con e ence (MASCON), Chennai, India, 27–28 Aug (2021) h ps://doi.o g/10.1109/MASCON51689.2021.9563468 14. Bingi, K., Singh, A.P., P us y, B.R.: Cu e i ing-based app oxima ion o ac ional di e en ia o wi h complex o de . In: P oceedings o he 3 d In e na ional Con e ence Ene gy Powe En i onmen : Towa ds Clean Ene gy Tech. Shillong, Meghalaya, India, 5–7 Ma (2021) h ps://doi.o g/10.1109/ ICEPE50861.2021.9404417 15. Biswas, K., Bohannan, G., Capone o, R., Lopes, A.M., Machado, J.A.T.: F ac ional-O de De ices. Sp inge , Cham (2017) 16. Busca ino, A., Capone o, R., G aziani, S., Mu gano, E.: Realiza ion o ac ional o de ci cui s by a cons an phase elemen . Eu . J. Con ol 54, 64–72 (2020). h ps://doi.o g/10.1016/j.ejcon.2019.11.009 17. Das, S.: Func ional F ac ional Calculus. Sp inge -Ve lag, Be lin Heidelbe g (2011) 18. Deb, K.: An e icien cons ain handling me hod o gene ic algo i hms. Compu . Me hod Appl. Mech. Eng. 186, 311–338 (2000). h ps://doi.o g/10.1016/S0045-7825(99)00389-8 19. Deniz, F.N., Alagoz, B.B., Tan, N., Koseoglu, M.: Re isi ing ou app oxima ion me hods o ac ional o de ans e unc ion implemen a ions: s abili y p ese a ion, ime and equency esponse ma ching analyses. Annu. Re . Con ol. 49, 239–257 (2020). h ps://doi.o g/10.1016/j.a con ol.2020.03.003 20. El-Khazali, R.: F ac ional-o de PIλDμcon olle design. Compu . Ma h. Appl. 66(5), 639–646 (2013). h ps://doi.o g/10.1016/j.camwa.2013.02.015 21. El-Khazali, R.: On he biquad a ic app oxima ion o ac ional-o de Laplacian ope a o s. Analog In eg . Ci cui s Signal P ocess. 82, 503–517 (2015). h ps://doi.o g/10.1007/s10470-014-0432-8 22. Elwakil, A., Psychalinos, C., Maundy, B., Allagui, A.: On he possible ealiza ion o a complex-o de capaci i e impedance and i s applica ions. In . J. Ci cui Theo . Appl. 51(1), 500–507 (2023). h ps:// doi.o g/10.1002/c a.3396 123