An Enhanced Performance PID Filter Controller for First Order Time Delay Processes

Transcription

1 Jounal of Chial Engining of Jaan, Vol. 4, No. 6,., 7 ah Pa An Enhand Pfoan PID Filt Contoll fo Fit Od Ti Dlay Po Mohaad SHAMSUZZOHA and Moonyong LEE Shool of Chial Engining and Thnology, Yungna Univity, Da-dong 4-, Kyongan 7-749, Koa Kywod: PID Contoll Tuning, Fit Od Plu Dad Ti Po, Dituban jtion, Fit Od Lad/Lag Filt, Two-Dg-of-Fdo Contoll An analytial tuning thod fo a PID ontoll aadd with a lad/lag filt i ood fo FOPDT o bad on th IMC dign inil. Th ontoll i dignd fo th jtion of dituban and a two-dg-of-fdo ontol tutu i ud to lakn th ovhoot in th t-oint on. Th iulation tudy how that th ood dign thod ovid btt dituban jtion than th onvntional PID dign thod whn th ontoll a tund to hav th a dg of obutn. A guidlin of a ingl tuning aat of lod-loo ti ontant (λ) i ovidd fo val diffnt obutn lvl. Intodution Pootional intgal divativ (PID) ontoll hav bn th ot oula and widly ud ontoll in th o induti bau of thi iliity, obutn and wid ang of aliability with na-otial foan. Howv, it ha bn notid that any PID ontoll a oftn ooly tund and a tain aount of ffot ha bn ad to ytatially olv thi obl. Th fftivn of th intnal odl ontol (IMC) dign inil ha ad it attativ in th o induti, wh any attt hav bn ad to xloit th IMC inil to dign PID ontoll fo both tabl and untabl o (Moai and Zafiiou, 989). Th IMC-PID tuning ul hav th advantag of uing only a ingl tuning aat to ahiv a la tad-off btwn th lod-loo foan and obutn. Th PID tuning thod ood by iva t al. (986), Moai and Zafiiou (989), Hon t al. (996), and L t al. (998) a tyial xal of th IMC-PID tuning thod. Th dit ynthi (DS) thod ood by Sith t al. (975) and th dit ynthi fo th dituban (DSd) thod ood by Chn and Sbog () an alo b atgoizd into th a la a th IMC- PID thod, in that thy obtain th PID ontoll aat by outing th idal fdbak ontoll whih giv a dfind did lod-loo on. Although th idal ontoll i oftn o oliatd than th PID ontoll fo ti dlayd ivd on Novb 7, 6; atd on Fbuay 5, 7. Coondn onning thi atil hould b addd to M. L (E-ail add: ynl@yu.a.k). o, th ontoll fo an b dud to that of ith a PID ontoll o a PID ontoll aadd with a low od filt by foing aoiat aoxiation of th dad ti in th o odl. Th ontol foan an b ignifiantly nhand by aading th PID ontoll with a lad/ lag filt, a givn by Eq. (). a K D + τ τ + b I () wh K, τ I and τ D a th ootional gain, intgal ti ontant, and divativ ti ontant of th PID ontoll, tivly, and a and b a th filt aat. Th tutu of th PID ontoll aadd with a filt wa alo uggtd by iva t al. (986), Moai and Zafiiou (989), Hon t al. (996), L t al. (998) and Dwy (3). Th PID filt ontoll in Eq. () an aily b ilntd in odn ontol hadwa. It i ntial to haiz that th PID ontoll dignd aoding to th IMC inil ovid xllnt t-oint taking, but ha a luggih dituban on, ially fo o with a all ti-dlay/ti-ontant atio (Moai and Zafiiou, 989; Chin and Fuhauf, 99; Hon t al., 996; L t al., 998; Chn and Sbog, ; Skogtad, 3). Sin dituban jtion i uh o iotant than t-oint taking fo any o ontol aliation, a ontoll dign that haiz th fo ath than th latt i an iotant dign goal that ha ntly bn th fou of nwd ah. Coyight 7 Th Soity of Chial Engin, Jaan

2 (a) wh i th otion of th odl invtd by th ontoll, A i th otion of th odl not invtd by th ontoll and A (). Th noninvtibl at uually inlud th dad ti and/o ight half lan zo and i hon to b all-a. To obtain a good on fo o with ol na zo, th IMC ontoll q hould b dignd to atify th following ondition.. If th o P ha ol na zo at z, z,..., z, thn q hould hav zo at z, z,..., z.. If th o D ha ol na zo, z d, z d,..., z d, thn ( P q) hould hav zo at z d, z d,..., z d. Sin th IMC ontoll q i dignd a q f, th fit ondition i atifid autoatially. Th ond ondition an b fulfilld by digning th IMC filt f a Fig. (b) Blok diaga of IMC and laial fdbak ontol yt: (a) Th IMC tutu; (b) Fdbak ontol tutu i β i + i f ( λ + ) ( 4) wh λ i an adjutabl aat whih ontol th tadoff btwn th foan and obutn; i ltd to b lag nough to ak th IMC ontoll (i-)o; β i a dtind by Eq. (5) to anl th ol na zo in D. In th nt tudy, a il and ffiint thod i ood fo th dign of a PID filt ontoll with nhand foan. A lod-loo ti ontant (λ) guidlin i ondd fo a wid ang of ti-dlay/ti-ontant atio. A iulation tudy wa fod to illutat th uioity of th ood thod fo both noinal and tubd o.. IMC Contoll Dign Podu Figu (a) and (b) how th blok diaga of th IMC ontol and quivalnt laial fdbak ontol tutu, tivly, wh P i th o, P th o odl, q th IMC ontoll, f th toint filt, and th quivalnt fdbak ontoll. Fo th noinal a (i.., P P ), th t-oint and dituban on in th IMC ontol tutu an b ilifid a: y q+ q d P P D Aoding to th IMC aatization (Moai and Zafiiou, 989), th o odl P i fatod into two at: () 3 P A q P zd, L, zd i A βi + i ( λ + ) Thn, th IMC ontoll o to b q i i β + ( λ + ) i Thu, th lod-loo on i y A i βi + i + ( λ + ) A zd, L, zd ( 5) ( 6) i βi + i d D 7 ( λ + ) Fo th abov dign odu, on an ahiv a tabl lod-loo on by uing th IMC ontoll.. PID filt Contoll Dign fo FOPDT Po Th idal fdbak ontoll that i quivalnt to th IMC ontoll an b xd in t of th intnal odl P and th IMC ontoll q: JOUNAL OF CHEMICAL ENINEEIN OF JAPAN

3 q () 8 q P Subtituting Eq. (3) and (6) into Eq. (8) giv th idal fdbak ontoll: i βi + i ( λ + ) i A βi + i ( λ + ) ( 9) Lt u onid th fit od lu dad ti (FOPDT) o, whih i ot widly utilizd in th hial o induti, a a ntativ odl. P θ K D τ + wh K i th gain, τ th ti ontant, and θ th ti dlay. Th IMC filt tutu i f β + λ + It i notid that th IMC filt fo in Eq. () wa alo utilizd by L t al. (998) and Hon t al. (996). Th ulting IMC ontoll bo q ( τ+ ) ( β+ ) K( λ+ ) Thfo, th idal fdbak ontoll i obtaind a ( + ) ( + ) τ+ β θ K λ+ β [ ] ( 3) Sin th idal fdbak ontoll in Eq. (3) do not hav th PID filt ontoll fo, th aining iu i how to dign th PID filt ontoll that aoxiat th idal fdbak ontoll ot loly. Aoxiating th dad ti θ with a / Pad xanion θ θ θ + θ θ ( 4) ult in C a ( + ) + θ + θ τ ( β + ) θ θ θ θ K ( λ+ ) ( β + ) + ( 5) It i iotant to not that th / Pad aoxiation i i nough to onvt th idal fdbak ontoll into a finit dinional fdbak ontoll with baly any lo of auay. Exanding and aanging Eq. (5) giv θ θ ( τ+ ) β+ βθ + λθ + λ K( λ β + θ) + λ β + θ βθ λθ λ θ λθ λ β + θ λ β + θ 3 (6) A n in Eq. (6), th ulting ontoll ha th fo of th PID ontoll aadd with a high od filt. Th analytial PID foula an b obtaind a K θ θ θ, τ, τ ( 7) K λ β + θ 6 I D Th valu of th xta dg of fdo β i ltd o that it anl out th on-loo ol at /τ that au a luggih on to load dituban. Fo Eq. (5), thi qui [ (β + ) θ /(λ + ) ] /τ. Thu, th valu of β i obtaind a λ θτ β τ τ ( 8) Futho, it i obviou fo Eq. (5) that th aining at of th dnoinato in Eq. (6) ontain th fato (τ + ). Thfo, th filt aat b in Eq. () an b obtaind by taking th fit divativ of Eq. (9) blow + b + βθ βθ λθ λ θ λθ + λθ + λ λ β + θ λ β + θ λ β + θ τ + (9) VOL. 4 NO

4 and ubtituting a βθ + λθ + λ b τ λ β + θ Th filt aat in Eq. () an b aily obtaind fo Eq. (6) a a β Sin th high od t ha littl iat on th ovall ontol foan in th ontol lvant fquny ang, th aining at of th fation in Eq. (6) an b ufully aoxiatd to a il fit od lad/lag filt a ( + a)/( + b). Ou iulation ult (although not hown in thi a) alo onfi th validity of thi odl dution. Th lad t (β + ) in th lod-loo tanf funtion of Eq. (7) au xiv ovhoot in th t-oint on, whih an b adiatd by adding th t-oint filt f a: f γβ + β + wh γ. Th xt a with γ ha no lad t in th t-oint filt whih would au a low vo on. On th oth hand, γ an that th i no t-oint filt. γ an b adjutd onlin to obtain th did d of th t-oint on. Th ood tudy i alo aliabl to th o with ngligibl dad ti whil it i ainly foud on th fit od ti dlay o. 3. obut Stability Sin η q f fo th IMC ontoll, th ulting Eq. (4) bo: fl ( ω) < ω ( 5) Thu, th abov tho an b inttd a l < / η / f, whih guaant obut tability whn th ultiliativ odl o i boundd by () l. η () () < ( 6) wh () dfin th o ultiliativ untainty bound. i.., () ( )/. Thi untainty bound an b utilizd to nt th odl dution o, o inut atuato untainty, and o outut no untainty, t., whih a vy fqunt in th atual o lant. Fo th FOPDT o, th olntay nitivity funtion η () an b obtaind a η () ( β+ ) ( λ + ) θ ( 7) Subtituting Eq. (7) and β into Eq. (6) yild th obut tability ontaint quid fo tuning th adjutabl aat λ λ θτ τ τ + λ + < () Subtituting iω into Eq. (8) ult in ( 8) Th wll-known obut tability tho an b utilizd to analyz th obut tability of th ood ontoll. obut Stability Tho (Moai and Zafiou, 989): Lt u au that all lant P in th faily λ θτ τ ω τ + λω + < ιω ( 9) i i ω ω < : l ω 3 ( iω) hav th a nub of HP ol and that a atiula ontoll tabiliz th noinal lant. Thn, th yt i obutly tabl with th ontoll if and only if th olntay nitivity funtion η fo th noinal lant atifi th following bound: ηl u ηl ( ω ) < ( 4) ω It i oibl fo untainty to ou in any of th th o aat i.., θ, τ, and K. Conquntly, w hav to onid th untainty in th diffnt aat aatly. Lt u onid th FOPDT o having th untainty in all th aat a ( θ+ θ) K K ( + ) τ+ τ ( + ) ( 3) It i ot oon ati that th FOPDT odl aoxiatd fo th high od o in th al 4 JOUNAL OF CHEMICAL ENINEEIN OF JAPAN

5 o lant. Du to thi fo th ti ontant untainty it i aud that th all ti ontant τ i ngltd/iing in dvloing th noinal odl a onidd in Eq. (3) (Sbog t al., 4). Thn th o ultiliativ untainty bound bo K + K () τ + θ ( 3) Subtituting th abov ult into Eq. (9), w obtain th obut tability ontaint a follow: λ θτ τ ω θω τ + K + < K i, λω + τω i + ω > ( 3) Th abov obut tability ontaint i vy uful to adjut λ wh th i untainty in th o aat. Th obut tability ontaint in Eq. (3) an alo b ud to dtin th axiu allowabl valu of untainty in ± K, ± θ and ± τ o vaiou obination of th fo whih obut tability an b guaantd. Fo xal, a lot of η (ω) l (ω) v. ω an b ontutd fo a all valu of any aati untainty and/o obination of diffnt untainti. 4. Siulation Study Thi tion dal with th iulation tudy ondutd fo th ntativ FOPDT o: th lag ti doinant o, th qual dad ti and lag ti o, and th dad ti doinant o. To valuat th obutn of a ontol yt, th axiu nitivity, M, whih i dfind by M ax /[ + (iω)], i ud. Sin th M i th inv of th hott ditan fo th Nyquit uv of th loo tanf funtion to th itial oint (, ), a all M valu indiat that th tability agin of th ontol yt i lag. Th M i a wllknown obutn au and i ud by any ah (Skogtad and Potlthwait, 996; Åtö t al., 998; Chn and Sbog, ; Skogtad, 3). Tyial valu of M a in th ang of.. (Åtö t al., 998; Sbog t al., 4). To nu a fai oaion, it i widly atd fo th odlbad ontoll (, DS, and IMC) to tun by adjuting λ o that th M valu bo th a valu. Thfo, thoughout all ou iulation xal, all of th ontoll oad w dignd to hav th a obutn lvl in t of th axiu nitivity, M. To valuat th lod-loo foan, two foan indi w onidd in th a of both a t t-oint hang and a t load dituban, viz., th intgal of th ti-wightd abolut o (ITAE) dfind by ITAE t (t) dt, and th ovhoot whih at a a au of how uh th on xd th ultiat valu following a t hang in th toint and/o dituban. In thi a, th iulation tudy ha bn ondutd uing th PID ontoll in th fo of Eq. (). Howv, fo al ilntation, th aalll fo of th PID ontoll, () K { + /(τ I ) + τ D / [(.τ D ) + ]}( + a)/( + b), whih i widly ud in th al o, an b alid to aoxiatly th a foan. To valuat th uag of aniulatd inut valu, w out TV of th inut u(t), whih i th u of all of it ovnt of u and down. If w ditiz th inut ignal a a qun [u, u, u 3,..., u i,...], thn TV i u i+ u i hould b a all a oibl. TV i a good au of th oothn of a ignal (Skogtad and Potlthwait, 996; Chn and Sbog, ; Skogtad, 3). 4. Exal : Lag ti doinant o (θ/τ.) Conid th following FOPDT o (Chn and Sbog, ; Sbog t al., 4): D + ( 33) Th ood PID filt ontoll i oad with oth ontoll bad on xiting thod, uh a th thod, and tho ood by iva t al. (986), Hon t al. (996), L t al. (998) and L t al. (998) with a onvntional filt. Th ontoll aat, inluding th foan and obutn atix, a litd in Tabl. In od to nu a fai oaion, all of th ontoll oad a tund to hav M.94 by adjuting λ. Figu oa th t-oint and load on obtaind uing th ood thod, th thod, and th thod ood by L t al. (998) and Hon t al. (996). Th DOF ontoll uing th t-oint filt wa ud in th thod and th thod ood by L t al. (998) and Hon t al. (996) to obtain an nhand t-oint on. It i iotant to not that th t-oint filt ud fo th t-oint on ha a la bnfit whn th o i lag ti doinant. In thi a, it i obvd that.4 γ giv ooth and obut ontol foan. In th ood ontoll, γ in th t-oint filt i ltd a γ.45. Th lod-loo on fo both th t-oint taking and dituban jtion ignifi that th ood thod ovid a uio on fo th a obutn. VOL. 4 NO

6 Tabl PID ontoll aat and foan atix fo xal (θ/τ.) Tuning thod λ K τ I τ D St-oint Dituban ITAE Ovhoot TV ITAE Ovhoot TV Pood thod a L t al. (998) b Hon t al. (996) d iva t al. (986) L t al. (998) f a a K D + τ, wh a 3., b.39; f τ + b I b f f a d K D + τ, wh a 4.3, b.,.34; f τ + b + I K D τ, wh b.45 τ + b I f Th L t al. (998) thod bad on th onvntional IMC filt fo of f /(λ + ) *DOF ontoll i ud only fo th thod of iva t al. (986) and L t al. (998) f (a) Po vaiabl Po vaiabl Pood thod L t al. (998). Hon t al. (996) Ti [in] 9 5 (b) Pood thod L t al. (998) Hon t al. (996) Ti [in] Fig. Siulation ult fo xal Th obut foan i valuatd by inting a tubation untainty of % in all th aat in th wot dition iultanouly and finding th atual o a D. /(8 + ). Th iulation ult fo th odl iath fo vaiou thod a givn in Tabl. Th foan and obutn indi obviouly dontat that th ood thod ha o obut foan than th oth. 4. Exal : Equal lag ti and dad ti o (θ/τ ) Conid th o odl dibd by Chn and Sbog () a follow D + ( 34) 6 JOUNAL OF CHEMICAL ENINEEIN OF JAPAN

7 Tabl 3 PID ontoll aat and foan atix fo xal (θ/τ ) Tuning thod λ K τ I τ D St-oint Dituban ITAE Ovhoot TV ITAE Ovhoot TV Pood thod a L t al. (998) b Hon t al. (996) d iva t al. (986) L t al. (998) f a a.97, b.; f b f f d a.975, b.79,.79; f b.67 *DOF ontoll i ud only fo th thod of iva t al. (986) and L t al. (998) f Tabl obutn analyi fo xal Tuning thod St-oint Dituban ITAE Ovhoot ITAE Ovhoot Pood thod a L t al. (998) b Hon t al. (996) d iva t al. (986) L t al. (998) f Th ood PID filt ontoll i oad with th ontoll and th ontoll dignd by L t al. (998), Hon t al. (996), iva t al. (986) and L t al. (998) with a onvntional filt. Th ontoll aat valu a litd in Tabl 3 along with th foan atix, wh M.84 i ltd fo all ontoll dign. Unit t hang a intodud both in th t-oint and in th dituban fo th iulation. Th iulation ult in Figu 3 indiat that both th dituban and th toint on a fat in th ood ontoll. Th DOF ontoll tutu i ud fo ah dign thod xt iva t al. (986), and L t al. (998) with a onvntional filt. γ i ltd fo th ood ontoll. It i la fo Figu 3 and Tabl 3 that th ood ontoll xhibit btt foan fo both th t-oint and dituban on. 4.3 Exal 3: Dad ti doinant o (θ/τ 5) Conid th o with a long dad ti tudid by Luybn () and Chn and Sbog () 5 D + ( 35) Th ood and afontiond dign thod a oad. Th ontoll tting with th foan ati a givn in Tabl 4. All of th ontoll a dignd to hav M.74. Sin in th a of a dad ti doinant o, th DOF ontoll i uffiint to ahiv atifatoy ontol foan, no t-oint filt i ud fo any dign thod. Th t-oint and load on a hown in Figu 4. Fo thi figu, it i aant that th ood ontoll and th on dignd by L t al. (998) with th onvntional filt ovid iila on, whil th and Hon t al. (996) thod xhibit luggih on and tak a long ti to ttl th on. Th ood ontoll ha xllnt foan whn th lag ti doinat, but it foan bo iila to that of th thod bad on th onvntional filt whn th dad ti doinat. Whn θ/τ >>, th filt ti ontant hould b hon a λ θ >> τ fo th ak of lod-loo tability. Thfo, th o ol at /τ i not a doinant ol in th lod-loo yt. Intad, th ol at /λ dtin th ovall dynai. Thu, intoduing th lad t (β + ) into th IMC filt to onat th o ol at /τ ha littl iat on th dituban on. Futho, th lad t uually ina th olxity of th IMC ontoll, whih in tun dgad th foan of th ulting PID ontoll VOL. 4 NO

8 Tabl 4 PID ontoll aat and foan atix fo xal 3 (θ/τ 5) Tuning thod λ K τ I τ D St-oint Dituban ITAE Ovhoot TV ITAE Ovhoot TV Pood thod a L t al. (998) b Hon t al. (996) d iva t al. (986) L t al. (998) f a a.998, b.689 d a.64, b 3.55,.56 b.96 *DOF ontoll i ud fo all of th thod Po vaiabl Po vaiabl Pood thod L t al. (998). Hon t al. (996) Ti [in] (a) (b) Pood thod L t al. (998) Hon t al. (996) Ti [in] Fig. 3 Siulation ult fo xal by auing a lag diany btwn th idal fdbak ontoll and thu th PID ontoll. It i alo iotant to not that a th od of th filt ina, th ow of th dnoinato t (λ + ) alo ina, whih an au an unnaily low outut on. A a ult, in th a of a dad ti doinant o, th PID ontoll bad on th IMC filt that inlud no lad t off btt foan. 4.4 Exal 4: Polyization o An iotant vioity loo in a olyization o wa idntifid by Chin t al. () a follow: 3 D + ( 36) Th abov-ntiond o ha a lag onloo ti ontant of in and a dad ti of in, whih i alo quit notwothy. Chin t al. () dignd th PI ontoll with th odifid Sith Pdito (SP) by aoxiating th abov o in th fo of an intgating odl with a long dad ti. Figu 5 oa th noinal on by th ood PID filt ontoll and that by th odifid SP. In th ood ontoll, λ 8. i ltd and th ulting tuning aat a obtaind a K.6446, τ I 5., τ D.6667, a and b.978. Th iulation wa ondutd by inting th t t-oint hang at t followd by a load t hang of. at t 9. Th ood ontoll u a il fdbak ontol tutu without any dad ti onato. Nvthl, th ood PID filt ontoll ovid a uio foan, a hown in Figu 5. 8 JOUNAL OF CHEMICAL ENINEEIN OF JAPAN

9 Po vaiabl (a) Pood thod Hon t al. (996) L t al. (998) Po vaiabl 5 5 Ti [in] 5 3 (b).8 Pood thod.6 Hon t al. (996) L t al. (998) Ti [in] Fig. 4 Siulation ult fo xal Pood thod Chin t al. () Po vaiabl Po vaiabl.5. Pood thod. Chin t al. () Ti [in] Fig. 5 Siulation ult of th olyization o Ti [in] Fig. 6 Siulation ult of th olyization o with odl iath Th dituban jtion affodd by th ood ontoll ha a all ttling ti, wha th odifid SP ontoll dibd by Chin t al. () how a luggih and quid long ttling ti. A gad th t-oint on, th odifid SP ontoll ha an initially fat on, bau of th liination of th dad ti, but aftwad it bo low. On th oth hand, th d of th on fo th ood ontoll i unifo and th ttling ti i iila to that by th odifid SP. It i iotant to not that th SP ontol onfiguation ha a la advantag of liinating th ti dlay fo th haatiti quation, whih i vy fftiv to t-oint taking foan. Howv, thi advantag i lot if th o odl i inauat. In od to valuat th obutn againt odl untainty, a iulation tudy wa ondutd fo th wot a of odl iath by auing that th o ha a % iath in th th o aat in th wot dition, a follow 36. D 8 + ( 37) Th lod-loo on a ntd in Figu 6. Noti that th ood thod and th odifid SP thod dibd by Chin t al. () hav iila dituban jtion on fo th odl iath a. Howv, th t-oint on affodd by th odifid SP ontoll how v oillation, whil th ood ontoll giv a o obut on. VOL. 4 NO

10 . M.4 M.5 M.6 M.8 M.9 oad with th o ohitiatd ontoll, uh a th odifid Sith Pdito, in th a of th vioity loo in a olyization o. Th ult how that th ood ontoll giv atifatoy foan without th xtnal dad ti onato. A guidlin of lod-loo ti ontant λ wa alo ood fo a wid ang of θ/τ atio. Fig. 7 In th ood tuning ul, th lod-loo ti ontant λ ontol th tadoff btwn th obutn and foan of th ontol yt. A λ da, th lod-loo on bo fat and an bo untabl. On th oth hand, a λ ina, th lod-loo on bo tabl but luggih. A good tadoff i obtaind by hooing λ to giv an M valu in th ang of.. (Åtö t al., 998; Sbog t al., 4). Th λ guidlin fo val obutn lvl i lottd in Figu 7. Conluion... λ guidlin fo th ood tuning thod A il analytial dign thod fo a PID ontoll aadd with a lad/lag filt wa ood bad on th IMC inil in od to iov it dituban jtion foan. Th ood thod alo inlud a t-oint filt to nhan th t-oint on lik th DOF ontoll uggtd by L t al. (998), Hon t al. (996) and Chn and Sbog (). FOPDT o with th ntativ diffnt θ/τ atio w ud fo th iulation tudy. Th ood PID filt ontoll onitntly ovid uio foan ov th whol ang of th θ/τ atio, whil th oth ontoll bad on th IMC-PID dign thod tak thi advantag only in a liitd ang of th θ/τ atio. In atiula, th ood ontoll how xllnt foan whn th lag ti doinat. Th ood ontoll wa alo Aknowldgnt Thi ah wa uotd by th 6 Engy ou and Thnology Dvlont Poga of Koa. Litatu Citd Åtö, K. J., H. Panagooulo and T. Hägglund; Dign of PI Contoll Bad on Non-Convx Otiization, Autoatia, 34, (998) Chn, D. and D. E. Sbog; PI/PID Contoll Dign Bad on Dit Synthi and Dituban jtion, Ind. Eng. Ch.., 4, () Chin, I. L. and P. S. Fuhauf; Conid IMC Tuning to Iov Contoll Pfoan, Ch. Eng. Pog., 86, 33 4 (99) Chin, I.-L., S. C. Png and J. H. Liu; Sil Contol Mthod fo Intgating Po with Long Dadti, J. Po Contol,, () Dwy, A. O.; Handbook of PI and PID Contoll Tuning ul, Iial Collg P, London, U.K. (3) Hon, I.., J.. Aulandu, J.. Chitoh, J.. VanAntw and. D. Baatz; Iovd Filt Dign in Intnal Modl Contol, Ind. Eng. Ch.., 35, (996) L, Y., S. Pak, M. L and C. Boilow; PID Contoll Tuning fo Did Clod-Loo on fo SI/SO Syt, AIChE J., 44, 6 5 (998) Luybn, W. L.; Efft of Divativ Algoith and Tuning Sltion on th PID Contol of Dad Ti Po, Ind. Eng. Ch.., 4, () Moai, M. and E. Zafiiou; obut Po Contol, Pnti-Hall, Englwood Cliff, U.S.A. (989) iva, D. E., M. Moai and S. Skogtad; Intnal Modl Contol. 4. PID Contoll Dign, Ind. Eng. Ch. Po D. Dv., 5, 5 65 (986) Sbog, D. E., T. F. Edga and D. A. Mlliha; Po Dynai and Contol, nd d., John Wily & Son, Nw Yok, U.S.A. (4) Skogtad, S.; Sil Analyti ul fo Modl dution and PID Contoll Tuning, J. Po Contol, 3, 9 39 (3) Skogtad, S. and I. Potlthwait; Multivaiabl Fdbak Contol; Analyi and Dign, John Wily & Son, Nw Yok, U.S.A. (996) Sith, C. L., A. B. Coiio and J. Matin; Contoll Tuning fo Sil Po Modl, Intu. Thnol.,, (975) JOUNAL OF CHEMICAL ENINEEIN OF JAPAN