Auto-Tuning of PID Controllers for Second Order Unstable Process Having Dead Time

Transcription

1 Joural of Chmical Egirig of Jaa, Vol. 3, No. 4, , 1999 Rsarch Par Auo-uig of PID Corollrs for Scod Ordr Usabl Procss Havig Dad im HSIAO-PING HUANG AND CHAN-CHENG CHEN Darm of Chmical Egirig, Naioal aiwa Uivrsiy, aii 1617, aiwa, R.O.C. Kywords: Dadim, Rlay Fdback, Usabl Pol, Sabl Pol, PID A auo-uig rocdur for PID corollrs for a mor gral class of usabl rocss ha has scod-ordr dyamics is rsd. hs scod-ordr dyamics ar rrsd by a modl havig boh sabl ad usabl ols oghr wih a aar dad im. A biasd rlay fdback s is usd o gra a cosa limi cycl for idifyig his modl. Cririo o disiguish a scod-ordr modl from h firs-ordr o is dvisd. Uo fiishig h idificaio, siml uig ruls ar rovidd o u h aramrs of PID corollrs. hs siml uig ruls ar drivd from h auhors rvious work rgardig corollr dsig for o loo usabl rocsss. Iroducio I rc yars, dvloms i auo-uig of idusrial PID corollrs hav focusd o corol of o loo sabl rocsss. For auo-uig, h auouig s of Åsröm ad Hägglad (1984) is usually usd o obai simaios of ulima gai ad ulima frqucy. Afr obaiig such ulima rsuls, ihr Z-N or Z-N rlad uig mhods ar alid o calcula h aramr sigs for idusrial corollrs (Hägglad ad Åsröm, 1991; Åsröm ad Hägglad, 1995). Rcly, aramric modls hav also b usd so ha modl-basd uig mhods ca b alid o dsig corollrs (Huag al., 1996). I coras o h dvlom of auo-uig for sabl rocsss, hr is o as so much liraur (Wag al., 1995; Kavdia ad Chidamaram, 1996) addrssig auo-uig for h corol of o loo usabl rocsss. h dficicy of liraur i his asc is arly du o h fac ha rocsss wih usabl dyamics ar o courd so of as hos wih sabl os. Bu, o loo usabl dyamics idd ar courd i h oraios of may bioracors or xohrmic racors (Ual al., 1974; Ual al., 1976; Hoo ad Kaor, 1985; Agrawal ad Lim, 1986). hus, o hac racicig corol for hs rocsss, auo-uig for a PID corollr would b hlful. Bsids, o accou for h dficicy i dvlom i auo-uig for usabl rocsss, hr ar a fw chical bolcks o b addrssd: Rcivd o Jauary 18, Corrsodc cocrig his aricl should b addrssd o H.-P. Huag ( addrss: huaghc@ccms.u.du.w). 1. h ulima gai ad frqucy obaid from rlay fdback ss ar o dircly rlad o corollr uig such as Z-N ruls for sabl rocsss. As a rsul, a aramric modl bcoms idissibl for uig corollrs. I h rord works of Wag al. (1995) ad of Kavdia ad Chidambaram (1996), h modls usd ar of firs ordr havig dadim. Noic ha, i a o loo sabl rocss, dyamic lags ca b lumd io a firs-ordr lm havig a aar im cosa ad a aar dadim. Bu, for a usabl rocss, a firs-ordr modl is o suffici for dscribig is dyamics. Bcaus, ohr ha h usabl im cosa, hr ar sill sabl dyamic lags i h rocss. hs sabl dyamic lags ca usually b lumd io aohr firs-ordr modl wih dad im. hus, o ak io accou hs wo ars of dyamics, a scod-ordr modl would b br. Bu, modlig his scod-ordr modl for usabl rocss sill sms a o issu.. uig mhods for a gral usabl rocss havig dad im usually rquir dious quaio-solvig ss which ar o fasibl for o-li imlmaio. Bsids, almos all such uig mhods rord i h liraur for a usabl rocss wih dadim ar usually cofid o firs-ordr modls. For scodordr modls, hr ar oly a fw rsuls rord (Shafii ad Sho, 1994; Huag ad Ch, 1996). 3. Ulik h auo-uig s for o loo sabl rocsss, h rlay fdback sysm may fail o rach a limi cycl for a giv o loo usabl rocss. hus, h sabiliy of a rlay fdback sysm is rsriciv. Iformaio rgardig h dyamics of h rocss is hus rquird i riori. Du o h issus addrssd abov, w roos a auo-uig sysm for o loo usabl rocsss. W shall cofi our sco o hos ha hav oly si- 486 Coyrigh 1999 h Sociy of Chmical Egirs, Jaa

2 gl usabl ols firs. For rocsss wih mor usabl ols, h siuaio would b mor comlicad ad is a roblm for fuur sudy. I his roosd auouig sysm, idificaio is coducd usig a rlay fdback s. h ouus of h rlay ar asymmric. Modls of firs ordr or scod ordr ar dvlod wih o-li algorihms. I dvloig a scod-ordr modl, a lad/lag lm which has a vry small lag im ad a adjusabl ladim is aachd o h rlay o chag h rocss dyamics io a firsordr o. Paramr simaio for h firs-ordr modl is h rformd usig h basic rlaios ha hav b drivd from a rlay fdback xrim. Corollr uig is basd o som siml uig formula ha ar drivd from h work of Huag ad Ch (1996). Such ruls ar dvisd for PID corollrs usig ihr a firs-ordr or scod-ordr modl. his aricl is orgaizd as follows. Scio 1 discusss h ky issus i a rlay fdback sysm for o loo usabl rocsss. Scio discusss daild rocdurs i h idificaio has. h corollr uig rul is sablishd i Scio 3. Scio 4 givs xamls o illusra ad s h roosd mhod. A oliar usabl bioracor ad a oliar usabl coiuous sirrd ak racor ar usd o illusra h roosd mhod i Scio 5. Coclusios of his work ar summarizd i Scio Auo-uig Sysm wih Asymmric Rlay Fdback o hac h auo-uig of a PID corollr for a o-loo usabl rocss, h sysm as show i Fig. 1 is cosidrd. h auo-uig sysm is calld o fucio by closig h swich ha cocs h rror sigal o h rlay fdback corollr. h rlay fdback corollr cosiss of a asymmric rlay followd by a lad/lag lm wih a adjusabl lad im. h biasd rlay is dfid as follows: ()= u γ h, h, if if ()> ()< whr () is h rror bw h s-oi, R, ad h ouu, y, as show i Fig. 1. h facor γ is o rovid asymmric rlay ouus. h sysm is xcid ad wais uil h limi cycl occurs. Basd o h rsos daa from hs cosa cycls, aramrs of a modl ar simad. By usig h rsulig modl, aramrs of a PID corollr ar h calculad. As has b miod rviously, h rlay fdback sysm may fail o sabiliz a giv o-loo usabl rocss. hus, i would b ssial o kow udr wha codiios, such a auo-uig sysm wih rlay fdback would b fasibl. Fig. 1 W shall cosidr h followig dyamic modls for h rrsaios of o loo usabl rocsss for auo-uig: s K G()= s s 1 G ()= s Auo-uig sysm for o-loo usabl rocss s K s 1 as 1 ( ) () 1 ( ) Equaio (1) sads for hos rocsss ha hav b discussd i may corol liraur ha dal wih o-loo usabl rocsss. Bu, h usabl ol is o h oly dyamics ha occurs i ral rocsss. o accou for h ohr dyamic lag ha usually occurs i ral rocsss, a sabl firs-ordr-lus-dadim lm is aachd o h o i Eq. (1) o giv h scod-ordr dyamics of Eq. (). 1.1 Sabiliy codiio for rlay fdback I his roosd auo-uig rocdur, a siml rlay fdback s is o b coducd firs. I may h rquir iroducig a lad/lag lm io h loo i subsqu ss durig idificaio. hus, as a sarig s for his roosd auo-u, h sabiliy codiios of a rlay fdback loo ha cosiss of rocss modls such as Eqs. (1) ad () ad a biasd rlay is ssial. Wh a rocss of Eq. (1) or of Eq. () ca b sabilizd wih such a biasd rlay fdback corollr, a fasibl rgio ha would guara loo sabiliy afr aachig a lad/lag lm o h rlay ca b foud. I h followig, h sabiliy of h rlay fdback for a firs-ordr rocss or a scod-ordr rocss will b sudid. h rsul will srv as a guidac for h fasibiliy of rformig a rlay fdback s o a giv rocss Sabiliy codiio for firs-ordr rocsss Cosidr a firs ordr rocss of Eq. (1) ad is rsos o a rlay fdback as show i Fig.. A firs ordr rocss o b sabilizd wih a asymmric rlay fdback has o comly wih h followig codiio (Huag ad Ch, 1997): 1 γ < mil ( 1 γ), l () 3 γ 487

3 Fig. Rlay fdback of a firs-ordr usabl rocss Fig. 4 Fasibl rgio for rlay fdback wih a scodordr usabl rocss Fig. 3 Rlay fdback rsos curvs of a scod-ordr usabl rocss I Eq. (3), dsigas h raio of dadim o h usabl im cosa. his codiio is drivd as a cssary codiio for h occurc of limi cycl i rsos o a shif of rlay ouu. Ay firs-ordr rocss which violas his codiio would o b fasibl for rformig rlay fdback s for auo-uig Sabiliy codiios for scod ordr rocsss Cosidr h scod-ordr rocss of Eq. (). A gral rsos of his rocss o a rlay fdback as show i Fig. 3. o avoid dious mahmaical drivaios, a aroach of h followig is adod. Firs, Eq. () is rwri as h o wih dimsiolss im uis of h followig: h hl of his figur, if a rocss whos aramrs loca i h fasibl rgio, h, cosa cyclig ca b guarad wh coducig such a rlay fdback s Sabiliy codiios for iiial y A h vry bgiig of h rlay fdback s, w hav u = for <. Usually, h s is sard wih y, ad ẏ =. Udr his siuaio, h iiial valus y is also ssial o h sabiliy. h rsos of y a > i his iiial sag is: For y < ay a hk y = a a a γ a y hk a a γ γ hk 5 G ()= s s K ( s 1) as 1 ( 4) or For y > whr, a = ad a = If w cosidr h ouu y big ormalizd wih h rocss cosa K, i is h obvious ha his rlay fdback sysm is characrizd by, a, ad γ oly. By assumig ha h rocss is iiially a a sady sa, h sabiliy rgio for a a diffr ad γ ca b foud by dirc simulaio. h MALAB Simulik is usd as a simulaio ool for his uros. For a giv γ ad, h maximum valu of a which rsuls i sabl rlay fdback s is obaid. By scaig h fasibl a alog usig γ as a aramr, h rsuls ar lod ad giv i Fig. 4. Wih ay ahk y = a a a a y hk a a hk 6 h drivaivs of y bcoms: For y < y hk ẏ = a a a γ a y hk a a γ JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

4 or For y > y hk ẏ = a a a a y hk a a 8 If limi cycl occurs, i is cssary ha h drivaiv of y chags is sig a som im isa, *, (* > ) wh ẏ (*) =. From Eqs. (7) ad (8), h followig cssary codiios ca b cocludd: ad, y < mi hk a γ, γhk 9 y hk a hk < mi, 1 So ha () y hk a hk a hk hk < mi γ,, γ, 11 If a =, h abov iqualiy bcoms: y mi γ, 1 hk 1 < Accordig o h rsul i Eq. (1), i is hus clar ha, i ordr o imlm auo-uig for a usabl rocss, w d o kow h valus of K ad / of h rocss. If such valus ar o availabl i advac, som addiioal ss may b dd. For xaml, w may morarily assum ha all h sabl dyamic lags ca b lumd io h dadim so ha h rocss ca b rrsd as a firs-ordr modl. h, hrough h arly ar of h s rsos, h dlay ca b rad as h im wh ouu chags ca b dcd, ad h aar usabl im cosa ca b simad from his arly ar of y as follows: ˆ y () y ( 1) = l y ( 1) y ( ) which maks us of h dyamic quaio: y ()= y ( 1)K 1 u 1 ad u big a s iu. Hr, ca b ay im largr ha h aar dadim. Noic ha 1 ad do h rvious o ad wo samlig isas a im. hus, rgardig a giv usabl rocss, h fasibiliy for a rlay fdback s ca b simad roughly by usig Eq. (3) I h followig, w shall dic h idificaio ad corollr uig of his auo-uig sysm.. O-li Idificaio for Low Ordr Modl h firs s of his auo-uig rocdur is o idify a suiabl low ordr modl of Eq. (1) or of Eq. () basd o h daa from a rlay fdback s. hus, h rlay fdback loo is xcid from is origial sady sa o rovid cssary daa. Afr a rasi riod, rsis cosa cyclig would aar. From hs cosa cycls, h rocss cosa, K P, ad ohr aramrs i Eq. (1) ca b simad. h usabl rocss cosa, K, ca b simad as follows: K = G( ) ( 13 ) yd () = lim ud () P N y() d = lim P N N u() d P yd () = P ( 14) ud () whr ca b ay im isa afr rsis cyclig aars ad P is h riod of cyclig. h aroximaio of Eq. (14) holds oly if h rlay is asymmric (i.. γ 1). Wih asymmric rlay, h igraios of y ad u ovr h rasi zo ar gligibl comarig wih hos ovr h cosa cyclig zo. As a rsul, h simaio ca bas o h igraio ovr o sigl cyclig riod i his cosa cyclig zo. horically, γ ca b ay osiiv ral umbr. Howvr, as γ icrass or dcrass from uiy, h sabilizabl rgio subjcd o h raio of / would b dwidld. hus, o comromis bw a widr sabilizabl rgio ad h comuaio of K P, a valu aroud uiy is dsirabl. VOL. 3 NO

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6 Hr, / is comud from h rvious aiv firsordr modl, ad ν is a scalig facor o corol h udaig sd. If w ak a biggr valu of ν, h rlay fdback sysm will rach a rsis cyclig zo mor quickly, which is rfrabl for a o-li xrim. Howvr, if h ruly aarly sabl im cosa is of small valu, a biggr valu of ν will ak mor iraio ss bfor i ca covrg. A valu bw.4 ad.6 is hrfor rcommdd. h w rocd o driv a o-li algorihm for udaig h valu of b. L us dfi a o-gaiv fucio V as follows: Fig. 5 Fasibl rgio of lad im cosa, b V = 1 3 ssial o kow h fasibl rgio for h valu of b. Rgardig a rocss whos dyamic modl ca b rrsd by Eq. (), h fasibl rgios for diffr valus of a/ so as o rsul i a sabl limi cycl for idificaio ar as show i Fig. 5. Wih hs fasibl rgios, iraio ss h ca b rocdd o idify h sabl im cosa. Uo fiishig h iraios, h sabl im cosa will b aroximaly caclld by h lad lm, so ha h comsad rocss bcoms a firs-ordr o. As a rsul, h aramrs i his comsad rocss ca b drmid followig h rocdurs dicd rviously. Huag al. (1996) illusrad a rial ad rror rocdur o drmi h oimal lad im ad h rsulig scod-ordr modl, icludig h dadim,, ad h sabl lag cosa, a. o limia h ds for rsoal irvios i rovidig a w rial valu o udrak h rialad-rror rocdurs, i is dsirabl ha his rial-adrror rocdur ca b udad by h sysm islf. o achiv his, a o-li udaig algorihm for h valu of b is dvisd..1 O-li udaig algorihm h o-li algorihm is iiiad wih a iiial rial valu for h lad im, b. his iiial rial of b should comly wih h fasibl rgio for rlay fdback s. As ca b s from Fig. 4, for a giv /, h sabl im cosa a should b smallr ha (1 /). A rasoabl guss for / is hus adod from h rsuls of a aiv firs-ordr modl i h rvious s, ad is giv as follows: b = ν 1 ν 1 < Obviously, h raio of /δ rsuls from h comsad rocss is a fucio of b. L V(b ) = V(b 1 ) V(b ), h, w hav: Vb ( )= 1 δ = 1 b 4 δ b o mak V(b ) o-osiiv for ay, b is chos as: b = b b 1 1 η 5 whr, φ η = ( 6) φ φ = b ( 7) I ohr words, o hav a vr dcrasig valu of (1 /δ), h rial valu of b is udad accordig o: b w whr, = bold ληˆ 1 8 VOL. 3 NO

7 ˆη b b b1 b ( 9) Accordig o Huag ad Ch (1996), h rsulig uig abl for boh firs-ordr ad scod-ordr rocsss is giv as follows: s k For G()= s : s 1 K( f ) k = 5. ( KM Km), R =, K 1 ad D = b h abov algorihm will guara h covrgc of b o ach udad rial. Rsulig from may simulaio rsuls, i is obsrvd ha /δ dcrass as b icrass, hus h abov algorihm ca b furhr simlifid as followig: bw = bold λb 1 3 whr λ b is a adjusig facor of s siz. A valu bw.3 ad.7 is rcommdd. h slf udad rial-ad-rror rocdur gos as follows. Afr iroducig h iiial rial lad im, b, h rocss ouu is xcid uil limi cycl occurs. A aiv firs-ordr modl is simad usig Eqs. (17) ad (18). Rsuls ar h validad if h w raio of o δ is clos ough o uiy or o. If i is, h aar sabl domia im cosa, a, quals b, ad h usabl im cosa ad dlay i h scod-ordr modl ar adod dircly from h rsuld firs-ordr modl. If h raio of o δ is o wihi h olrac limi aroud uiy, a w valu of b is udad accordig o h algorihm i Eq. (3) ad h s is rad. h rocdur is carrid ou uil saisfacory rsul is obaid. 3. O-li uig for PID Corollrs h corollr uig mhod usd hr is origiad from h work of Huag ad Ch (1996). I hir work, a o-dgr-of-frdom (1-df) corollr i rms of PID algorihm is a simlifid rsul from hir wo-dgr-of-frdom (-df) corol sysm. his -df sysm maks us of hr lms o achiv sabilizaio, disurbac rjcio, ad srvo-rackig saraly. For sabilizaio, h origial o loo usabl rocss is closd wih a P or PD lm (wih a lad im of b) o obai h wids fasibl rgio of gai for corol. As a scod s, his lmary loo is h formulad io a IMC srucur o fid corollrs for ihr disurbac rjcio or srvo-rackig. Fially, h rsulig hr-lm sysm is h rformd io a -df srucur of which h corollr i h fdback loo is of a cascadd PID form. h 1- df PID corollr, ha is h o usd i his auo-uig sysm, is a scial rsul from h -df sysm. s k For G()= s : ( s 1) ( as 1) K( f ) k = 5. ( KM Km), R =, K 1 ad D = a h corollr dsig a h bgiig of his hr-lm aroach is o fid a P or PD lm ha ca rovid h largs K M = K c,max K. o achiv his, w hav o rfr o h followig quaios which dscrib h criical oi: ω = a 1 ω a 1 bω ( 31) Ad, K M is: KM = 1 ω 1 b ω ( 3) If w l: x = ω ad ˆb = b/. Eqs. (31) ad (3) ca b rwri as: ad x = 1 x 1 a a bx ˆ ( 33 ) x KM = 1 1 b x ˆ ( 34) hus, h maximum valu of K M for a giv raio of / ca b drivd by sig h drivaiv of K M wih rsc o / qualig o zro o giv: K M * = K M *[/] his abov rlaio has b foud for diffr valus of / ad is lod as a solid li i Fig. 6(b). A quaio is h foud (< / < 1) for his rlaio: K M = JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

8 Similarly, h valu of b o b ak is giv as h followig quaio. b 4 1 = As show i Figs. 6(a) ad 6(b), good agrm is obaid. h abov quaios ca b alid o a scod ordr sysm by firs caclig h sabl ol o bcom a firs-ordr rocss, of which K M is giv as Eq. (35) ad b =. h rsulig corollr sigs ar h obaid by subsiuig h abov wo quaios o giv: Cas A: Firs-ordr rocss ( < / < 1) KK c = r d f KK c = KK 1 c 4 1 = ( 38) ( 39) Cas B: Scod-ordr rocss ( < / <.8) KK c = r d f KK c = KK 1 c ( 41) a = ( 4) I h abov uig quaios, hr is o aramr lf. h scod s of dsig is h o fid a ror f ha ca giv good srvo-rackig rformac. Du o h rrors i aroximaio, f i Eqs. (38) ad (41) has a lowr boud, ad his lowr bod ca b asily drmid from h Nyquis s. For covic i a auo-uig sysm, such a boud is valuad ad h followig rlaios ar foud for firs-ordr ad scod-ordr rocsss, rscivly. Cas A: Firs-ordr rocss ( < / < 1) f mi 4 = ( 43) Cas B: Scod-ordr rocss ( < / <.8) f Fig. 6 mi = ( 44) hs rlaios for uig f ar also giv i Figs. 6(c) ad (d). I is obvious ha h valu of f should b chos o b largr ha f,mi. A valu of -ims h miimum valu is rcommdd. 4. Simulaio Examl Examl 1 Cosidr h followig scod-ordr rocss: G( s)= Aroximaio of K M, ˆ γ ad ( f /) m. 5s s 1 5. s 1 ( ) ( 45) If h modl is aivly assumd o b of firs ordr, h followig aramrs ar obaid from a rlay fdback s wih h = 1 ad γ = 1.5: K = 1.3; =.844; = 1.63 VOL. 3 NO

9 abl 1 O-li udaig b for Examl 1 b K /δ Fig. 7 O-li xrim for Examl 1 abl Rsulig modls for Examl 1 K a Exac Firs-ordr modl Scod-ordr modl Fig. 8 Corol rsoss of Examl 1: li(1)-(wih dordr modl), li()-(wih 1s-ordr modl) abl 3 O-li udaig b for Examl b K /δ o valida his aiv assumio, h calculad ad xrimal valus of aar dadim ar foud o hav a raio of.9. I is h vid ha his assumio has o b did. hus, idificaio rocdurs ar carrid ou o fid a scod-ordr modl. h ouu y hroughou h xrim is as show i Fig. 7, ad h rocss of o-li udaig h rial valus of b ar summarizd i abl 1. h rsuls of firs-ordr ad scod-ordr modls ar summarizd i abl. Corollr sigs rsulig from ihr modls ar lisd i h followig: For 1s-ordr modl: K c =.71; R = 4.43; D =.317 For d-ordr modl: K c = 3.13; R = 5.77; D =.56 Prformacs of corol usig ihr corollr o rack a s-oi chag ar lod i Fig. 8. his shows ha idificaio for scod ordr rocss usig his roosd mhod ar vry ffciv. Fig. 9 For his xaml, a accura scod ordr modl is rfrrd o a firs-ordr o for corollr dsig. Examl Cosidr h followig highr-ordr rocss: G( s)= Corol rsoss of Examl : li(1)-roosd (wih d-ordr modl), li()-roosd (wih 1sordr modl). 1s s 1. 5s 1. 1s 1 ( )( ) ( 46) Accordig o h idificaio rocdur dicd abov, a aiv firs-ordr modl wih h followig aramrs rsuls: K =.998; = 1.76; =.7 W hav o valida his aiv modl by xamiig h raio of o δ. I is h foud ha h valu of his raio is.741 which is ry far away from uiy. As a rsul, h idificaio rocdur has o b coiud o fid a scod-ordr modl. By usig h o-li adaig algorihm, h rsuls of aramr simaio for his scod-ordr modl ar giv i abl 3. h corollr sigs ar h calculad ad lisd as follow. 494 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

10 For firs-ordr modl: K c =.3; R = 4.; D =.4 For scod-ordr modl: K c = 4.51; R = 1.67; D =.67 S rsoss o h wo diffr sysms ar giv i Fig. 9. I his figur, i shows h sysm basd o a scod-ordr modl is sigificaly br ha h o which uss a firs-ordr modl. 5. Alicaio o Usabl Noliar Procsss I his scio, wo xamls of usabl oliar rocsss ar usd o illusra h roosd mhod. h firs oliar rocss is a bioracor sysm giv by Agrawal ad Lim (1986). h scod o is a coiuous sirrd ak racor sudid by Ual al. (1974) ad Ual al. (1976). 5.1 Alicaio o a usabl bioracor A oliar coiuous bioracor which xhibis ouu muliliciy ar giv (Agrawal ad Lim, 1986): dx d ds d Whr = ( µ DX ) ( 47) = Sf S D µ X / γ 48 ( 49) µ = µ S/ K S K S m m I h modl aramrs ar: γ =.4% g/g S f = 4.% g/g µ m =.53 h 1 K m =.1% g/g K I =.4545% g/g I cas of D =.3, h sysm of Eqs. (47) o (49) has hr mulil sady-sas: [X, S] 1 = [., 4.] (wash-ou codiio) [X, S] = [.9951, 1.51] (usabl) [X, S] 3 = [1.531,.1746] (sabl) I is assumd ha h diluio ra (D) is cosidrd as h maiulad variabl o corol h cll mass cocraio (X) a h usabl sady-sa, i.. X = A dlay of o hour is addd o h masurm of x o simula h ouu dlay du o h aalyical isrum ad rocdurs. I is obvious ha h rocss gai dds o h magiud of h iu sigal bcaus of h oliar characrisics. o fid h usabl gai a X =.9951, a asymmrical rlay wih h =.1 (abou.33% of h sady-sa valu) ad γ = 1.5 is usd. Wh rsis cyclig occurs, as show i Fig. 1(a), a aiv firs ordr Fig. 1 Auo-uig of usabl oliar bioracor modl of h followig is obaid: K = 6.43; = 6.81; = 1.1 h raio of o δ is foud o b 1.1, hus his aiv firs-ordr modl is accabl. Followig Eqs. (37) o (39), h followig corollr sigs ar suggsd: K c =.93; R =.3871; D =.663 A s chag of X from.9951 o 1.94 is iroducd o h sysm ad is ouu is lod i Fig. 1(b). h rsuls of his xaml shows h ffcivss of idificaio for a firs-ordr modl. h corol rformac hus obaid is a las as good as h o rord by Kavdia ad Chidambaram (1996) as show i h figur. 5. Alicaio o a usabl coiuous sirrd ak racor A coiuous sirrd ak racor (CSR) wih xohrmic racio, which is xlord by Ual al. (1974, 1976) is cosidrd as h ohr xaml for illusraio. A irrvrsibl firs-ordr racio occurs i a racor. h mraur ad cocraio of h raca a h xi of h racor ar h sa variabls, ad h coola mraur is ak o b h maiulad variabl. Afr iroducig dimsiolss grous io h govrig quaios, h sysm is dscribd as: dx d 1 dx d = x D 1 x 1 a 1 x x 1 x co d 1 d 5 / ν x = x BDa ( 1 x1) x 1 x / ν β( x x ) βu( ) d ( 51) VOL. 3 NO

11 abl 4 Dimsiolss variabls ad grous of CSR sysm B D a β ν d x co x 1 x u d 1 u dimsiolss ha of racio Damkohlr umbr dimsiolss ha rasfr coffici dimsiolss acivaio rgy dimsiolss im dimsiolss im dlay omial dimsiolss coola mraur dimsiolss comosiio dimsiolss mraur dimsiolss coola mraur dimsiolss fd mraur disurbac dimsiolss fd comosiio disurbac Fig. 11 Auo-uig of usabl oliar CSR abl 5 O-li udaig b for CSR sysm b K /δ whr x 1 ad x ar dimsiolss cocraio ad mraur of h raca, rscivly. h dimsiolss disurbacs d 1 ad d ar usd o rrs h flucuaios of h mraur ad cocraio i h fd sram. A lis of dimsiolss variabls or grous is giv i abl 4. I is assumd ha h mraur ad cocraio of fd sram ar cosa (i.. d 1 = d = ). A dimsiolss im dlay d of. i h maiulaio variabl u is also icludd. As a summary, h dimsiolss aramrs of his sysm ar lisd as follows. D a =.7 ν = B = 8. β =.3 x co = I h cas of u =, h sady-sa soluio of Eqs. (5) ad (51) givs h followig hr mulil sadysa soluios: [x 1, x ] 1 = [.144,.866] (sabl) [x 1, x ] = [.447,.7517] (usabl) [x 1, x ] 3 = [.7646, 4.755] (sabl) o idify h rasfr fucio modl a x 1 =.447 ad x =.7517, a asymmrical rlay wih γ = 1.5 ad h =.5 is iroducd. A aiv firsordr modl wih h followig aramrs rsuls, i..: K =.31; = 4.; = 1.45 Uo xamiig his firs-ordr modl, i is foud ha h raio of o δ is.811. As has b xlaid bfor, his aiv modl will o b usd. As show i abl 5, h iraiv rocdur will lad o h followig scod-ordr modl: K =.31; = 1.78; a = 1.389; =.9 Figur 11(a) shows h ouu of h sysm durig h idificaio has. G( s)=. s s 1. 3s 1 ( ) ( 5) Comard wih a liarizd modl, h rsulig scod-ordr modl is a rasoabl simaio a a usabl oi. Followig Eqs. (4) o (4), h followig corollr aramrs ar suggsd: K c = 7.86; R = 1.7; D = 1.39 o show ha a scod-ordr modl is surior, corollr aramrs basd o h rvious aiv firsordr modl ar: K c = 13.67; R = 4.48; D =.36 A soi chag of x 1 form.447 o.497 is iroducd o h sysm. As Fig. 11(b) shows, h rsos rsuld from a scod-ordr modl is much br ha ha from a firs-ordr modl. Coclusio I his work, a mhod o auou a PID corollr for o loo usabl rocsss havig a sigl usabl ol ad dadim is roosd. A asymmric rlay fdback s is usd o gra a cosa limi cycl. From his cosa limi cycl, a aiv firs-ordr modl is idifid. his modl is h vali- 496 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

12 dad by xamiig h raio of calculad ad masurd dadim. If h aiv modl is o saisfacory, a idificaio rocdur is carrid ou o fid a scod-ordr modl. his scod-ordr modl is obaid by raig h rlay fdback s wih a oli udaig algorihm. Corollr aramrs of a PID corollr ar h calculad from a s of siml uig ruls. wo usabl oliar chmical rocsss ar usd o illusra his roosd auo-uig mhod. Nomclaur a = sabl im cosa b = lad cosa D = Diluio ra G (s) = rocss rasfr fucio h = rlay magiud K M = maximum loo gai K m = miimum loo gai K P = rocss cosa K = usabl gai P = riod of limi cycl i rlay fdback sysm S = subsra cocraio = usabl im cosa u = rocss iu V = rformac fucio X = cll mass cocraio x = ulima loo gai y = rocss ouu β = ulima gai δ = simad dadim = Diffrc = dadim γ = biasd mulil Λ = s siz ω = frqucy <Subscri> d = drivaiv f = IMC filr = rocss r = rs Liraur Cid Agrawal, P. ad H. C. Lim; Aalysis of Various Corol Schms for Coiuous Bioracors, Adv. Biochm. Bioch., 3, 61 9 (1986) Åsröm, K. J. ad. Hägglad; Auomaic uig of Siml Rgulaors wih Scificaios o Phas ad Amliud Margis, Auomaica,, (1984) Åsröm, K. J. ad. Hägglad; PID Corollrs, d Ediio, ISA (1995) Hägglad,. ad K. J. Åsröm; Idusrial Adaiv Corollrs Bas o Frqucy Rsos chiqus, Auomaica, 7, (1991) Hoo, K. A. ad J. C. Kaor; A Exohrmic Coiuous Sirrd ak Racor is Fdback Equival o a Liar Sysm, Chm. Eg. Commu., 37, 1 1 (1985) Hoo, K. A. ad J. C. Kaor; Liar Fdback Equivalc ad Corol of a Usabl Biological Racor, Chm. Eg. Commu., 46, (1986) Huag, H. P. ad C. C. Ch; Corol Sysm Syhsis for O Loo Usabl Procss Havig im Dlay, IEE Procdigs Corol ad Alicaios, 144, (1996) Huag, H. P. ad C. C. Ch; A Nw Aroach o Idify Low Ordr Modl for Procss Havig Sigl Usabl Pol, Sysm Aalysis-Modlig-Simulaio, 9, (1997) Huag, H. P., C. L. Ch, C. W. Lai ad G. B. Wag; Auouig for Modl-Basd PID Corollrs, AIChE J., 4, (1996) Kavdia, M. ad M. Chidambaram; O-li Corollr uig for Usabl Sysms, Comurs Chm. Egg,, (1996) Shafii, Z. ad. Sho; uig of PID-y Corollr for Sabl ad Usabl Sysms wih im Dlay, Auomaica, 3, (1994) Ual, A., W. H. Ray ad A. B. Poor; O h Dyamic Bhavior of Coious Sirrd ak Racors, Chm. Eg. Sci., 9, (1974) Ual, A., W. H. Ray ad A. B. Poor; h Classificaio of h Dyamic Bhavior of Coious Sirrd ak Racors, Chm. Eg. Sci., 31, 5 14 (1976) Wag, Q. G.,. H. L ad K. K. a; Rlay-ud FSA Corol for Usabl Procsss Wih Dadim, Procdigs of h ACC, , Sal, USA (1995) Yuwaa, M. ad D. E. Sborg; A Nw Mhod for O-Li Corollr uig, AIChE J., 8, (198) Zhao, Y. ad S. Jayasuriya; Sabilizabiliy of Usabl Liar Plas Udr Boudd Corol, J. of Dyamic Sysms, Masurm, ad Corol, 117, (1995) VOL. 3 NO