On-Line PI Controller Tuning Using Closed-Loop Setpoint Responses for Stable and Integrating Processes*

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1 On-Lin PI Controllr Tuning Using Closd-Loop Stpoint Rsponss for Stabl and Intgrating Prosss* Mohammad Shamsuzzoha a, Sigurd Skogstad a, Ivar J. Halvorsn b a Norwgian Univrsity of Sin and Thnology (NTNU), Trondhim, Norway, (shamsuzz@hmng.ntnu.no), (skog@ntnu.no) b SINTEF ICT, Applid Cybrntis, N-7465 Trondhim, Norway Abstrat: A simpl and nw produr has bn dvlopd for th PI ontrollr tuning of an unidntifid pross using losd-loop rsponss. Th mthod rquirs only on stp tst in th losd-loop systm to obtain th proportional gain and intgral tim. Th stp tst is a stpoint hang prformd with a proportional only ontrollr whil disabling any intgral and drivativ ation. From th stpoint rspons on obsrvs th ovrshoot and th orrsponding tim to rah th pak. In addition on obsrvs th proportional gain (k ) and th stady-stat offst. Basd on a rang of first-ordr with dlay tst prosss, a simpl analytial orrlation has bn dvlopd for th ontrollr gain (k /k ) as a funtion of th ovrshoot. Th intgral tim stting is mainly a funtion of th tim to rah th pak. Th sttings wr drivd to math th tuning rul (with τ =θ) whih givs good robustnss with a gain margin of about 3 and snsitivity pak (M s -valu) of about.6. Th proposd tuning mthod, originally drivd for first-ordr with dlay prosss, has bn tstd on a broad rang of othr stabl and intgrating prosss. Th rsults using th losd-loop data ar omparabl with th tuning rul using th opn-loop modl. Kywords: PI ontrollr, stp tst, losd-loop rspons,, Shams s stpoint mthod. INTRODUCTION Th proportional intgral (PI) ontrollr is widly usd in th pross industris du to its simpliity, robustnss and wid rangs of appliability in rgulatory layr. Svral paprs hav rportd that a larg numbr of PI ontrollrs ar poorly tund and on rason is that quit tdious plant tsts ar ndd for gtting pross paramtrs to finally obtain th appropriatd ontrollr stting. Th lassial mthod of Ziglr-Nihols (94) has th grat advantags of rquiring vry littl information about th pross and tsting undr losd-loop onditions. Howvr, it is wll known that th Ziglr-Nihols (94) sttings ar aggrssiv for lag dominant (intgrating) pross and slow for dlay dominant pross. Th othr and mor signifiant disadvantag of th Z-N mthod is that th systm is brought at th limit to instability and that a numbr of trials may b ndd to obtain th ultimat gain. An altrnativ is to indu sustaind osillation by using an on-off ontrollr, i.. rlay tuning (Åström and Hägglund, 984), but this is a bit diffiult to us in prati baus on nds to swith to an on/offontrollr. Th original IMC-PID tuning mthod of Rivra t al. (986) and othr rlatd dirt synthsis (Sborg t al., 4) mthods provid vry good prforman for stpoint hangs but giv poor rsponss for input (load) disturbans in lag tim dominant prosss. To improv th input disturban rjtion, Skogstad (3) proposd th tuning ruls whr th intgral tim is rdud for lag-dominant (intgrating) prosss. Th rul has on tuning paramtr, th losd-loop tim onstant τ, and for fast and robust ontrol is rommndd to hoos τ = θ, whr θ is th fftiv tim dlay. Th tuning rul rquirs that on first obtains a first-ordr plus dlay modl of th pross, whih involvs approximations. Oftn, an opn-loop xprimnt is usd for gtting th modl paramtrs whih may b tim onsuming and may upst th pross and vn lad to pross runaway. Thrfor, thr is nd of an altrnativ losd-loop approah for plant tsting and ontrollr tuning whih rdus th numbr of trails, avoids th instability onrn during tuning xprimnt and works for a wid rang of prosss. Th proposd nw mthod satisfis ths onrns:. Th proposd mthod rquirs only a singl xprimntal losd-loop tst instad of a trial-and-rror produr undr losd-loop ondition.. Th pross is not ford to th stability limit, unlik Ziglr-Nihols (94) yling mthod. 3. Th mthod is appliabl for both intgrating and stabl pross and givs satisfatory disturban rjtion prforman. 4. Th mthod is simplr in us than xisting approahs and allows th pross to b undr losd-loop ontrol.. PI TUNING RULES A first-ordr pross with tim dlay is a ommon rprsntation of dynamis for pross ontrol and is givn as: * Prsntd at IFAC onfrn on dynamis and ontrol of pross systms (DYCOPS), Blgium, July

2 s k gp s whr k is th pross gain, τ th dominant (lag) tim onstant and θ is th fftiv tim dlay. It is a fat that th majority of prosss in th hmial industris an b satisfatorily ontrolld using a PI ontrollr: t k I u t k t t dt whih has two adjustabl paramtrs, th proportional gain k and th intgral tim τ I. Th ratio k I K I is known as th intgral gain. Th tuning rul (Skogstad, 3) for th pross () givs k I k min, 4(τ +θ) (4) Th tuning rul is analytially basd and has found wid us in th industry. Th losd-loop tim onstant (τ ) is sltd to giv th dsird trad-off btwn prforman and robustnss. This study is basd on th fast and robust stting (5) whih givs a good robustnss with a gain margin of about 3 and snsitivity pak (M s -valu) of about.6. On dimnsionlss form, th tuning ruls bom k kk.5 (6) I min,8 (7) Not that w hav sald tim with rspt to th dlay θ whih is approximatly th sam as th losd-loop tim onstant (with τ =θ). It is also of intrst to onsidr th intgral gain (K I ) on dimnsionlss form, kk (8) K I min.5,, 6 I Th dimnsionlss gain of in Figur. k and () () (3) K I ar plottd as a funtion 3. CLOSED-LOOP EXPERIMENT As mntiond, th objtiv is to us losd-loop data as basis for th ontrollr tuning. For pratial purpos, th simplst losd-loop xprimnt is a stpoint stp rspons. Suh a tst is asy to mak and on maintains full ontrol of th pross and th hang in th output variabl. W propos th following produr; y p : Pak output hang t p : Tim from stpoint hang to rah pak output y : Stady-stat output hang aftr stpoint stp tst k : Controllr gain usd in xprimnt From this data omput th following paramtrs yp y Ovrshoot=, y b=, b kk (9) y y b s Not that a P-ontrollr is usd and (-b) is th rsulting rlativ stady-stat offst. Th xprssion for th ovrall loop gain (kk ) is drivd from th xprssion for th losdloop transfr funtion, b = kk /(+kk ). kk k Fig.. k and τ I for tuning rul. I K I From Figur w not that th intgral trm ( K ) is most important for dlay dominant prosss (τ/θ<), but for othr prosss th proportional trm k is most signifiant. For los-to intgrating pross (τ/θ>8), th rul is to inras th intgral trm to avoid poor prforman (slow sttling) to disturban at th plant input ( load disturban ). Ths insights ar usful for th nxt stp whn w want to driv tuning ruls basd on th losd-loop stpoint rspons. y s. Swith th ontrollr to P-only mod (for xampl, inrass th intgral tim τ I to its maximum valu or st K I los to zro). In an industrial systm, with bumplss transfr, th swith should not upst th pross.. Mak a stpoint hang with an ovrshoot btwn. and.6 (about.3 is a good valu) Most likly, unlss th original ontrollr was quit tightly tund, on will nd to adjust (inras) th ontrollr gain to gt a suffiintly larg ovrshoot. From th losd-loop stpoint rspons, s Figur, rord th following valus : Stpoint hang y s y p y s t p t Fig.. Stpoint rspons with P-ontrol. y t

3 4. CORRELATION BETWEEN SETPOINT RESPONSE AND -SETTINGS On ould us th losd-loop stpoint rspons data to first dtrmin th opn-loop modl paramtrs (k, τ, θ) and thn us th -ruls (or othrs) to driv PI-sttings. A mor dirt approah is to dirtly omput from th data th PIsttings as proposd in this study. Th goal is thn to driv a orrlation, prfrably as simpl as possibl, btwn th stpoint rspons data (Figur ) and th PI-sttings in Eq. (3) and (4). For this purpos, w onsidrd 5 first-ordr with dlay modls paramtrizd to ovr a rang of prosss; from tim dlay dominant to lag-dominant (intgrating). Not that A is almost indpndnt of th valu of τθ. This is illustratd in Figur 3 whr w plot kk () as a funtion of kk for th 9 stpoint rsponss. A is th slop of th lin for ah ovrshoot, and is plottd in Figur 4 as a funtion of th ovrshoot. Th following quation (solid lin in Figur 4) fits th data vry wll, A.5(ovrshoot) -.67(ovrshoot) +. () whr th orrlation is basd on data with frational ovrshoot btwn. and.6. Not that a good fit of k is not so important for dlay-dominant prosss (τ/θ<), in th lowr lft ornr in Figur 3, whr th intgral ontribution is th most important. τθ=.,.,.4,.8,.,.5,.,.5,3., 5., 7.5,.,., 5.,. For ah of th 5 pross w obtaind th valu of k and τ I using th -stting in Eq. (3) and (4) for τ =θ. Furthrmor, for ah of th 5 prosss w gnratd 6 stp stpoint rsponss (Figur ) using P-ontrollrs that giv diffrnt frational ovrshoots. Ovrshoot=.,.,.3,.4,.5 and.6 A A =.58(ovrshoot) -.67(ovrshoot) +. In total w thn hav 9 stpoint rsponss. Not that small ovrshoots, lss than., wr not usd. On rason is that it is diffiult in prati to obtain from xprimntal data aurat valus of th ovrshoot and pak tim if th ovrshoot is too small kk kk =.8649kk kk =.79kk kk =.659kk kk =.5546kk kk =.4986kk kk =.456kk. ovrshoot. ovrshoot.3 ovrshoot.4 ovrshoot.5 ovrshoot.6 ovrshoot 3 kk 6 9 Fig. 3. kk vs. kk for diffrnt ovrshoot. W first sk a rlationship for th ontrollr gain k. Intrstingly, for a fixd valu of th ovrshoot, th ratio k /k is approximatly onstant, k A () k ovrshoot(frational) Fig. 4. Variation of A with ovrshoot Nxt, w want to find a orrlation for th intgral tim. Sin th tuning formula in Eq. (4) uss th minimum of two valus, it sms rasonabl to look for a similar rlationship, that is, to find on that maths prosss with a rlativly larg dlay (τ I =τ) and on that works wll for intgrating pross (τ I =8θ), and thn tak th minimum. First, onsidr prosss with rlativly larg dlay (τ/θ<8 or θ>.5τ), whr th -rul is to us τ I = τ. From Figur, it is lar that for a dlay-dominant pross (θ>τ) th intgral trm (K I ) is most important. This mans that it is partiularly important to obtain a good valu of KI k I in this rgion. In othr words, it is not so important that k and τ I ar orrt individually, but rathr that thir ratio K I is los to th -valu. Insrting τ = τ I in th rul for k in Eq. (6) and solving for τ I givs I kk () To gt K I orrt, w hr must us th atual valu for th ontrollr gain k. From () w hav obtaind th orrlation k /k =A, whr A is givn as a funtion of th ovrshoot in Eq. (). Howvr, w also nd th valu of th pross gain k, and to this fft, writ kk kk k k (3) Hr from Eq. (9), kk b b whr b is obtaind from th stady-stat valu of th stpoint rspons. In summary, w hav following quation for τ I for a dlay dominant pross

4 b I A (4a) b whr θ is th fftiv tim dlay. Similarly, for a lagdominant (intgrating) pross (τ>8θ) th rul givs τ I =8θ (4b) Equations (4a) and (4b) for th intgral tim hav all known paramtrs xpt th fftiv tim dlay θ. On ould obtain th fftiv tim dlay dirtly from th losdloop stpoint rspons, but this may b diffiult. Fortunatly, as shown in Tabl, thr is a good orrlation btwn θ and th pak tim t p whih is asir to obsrv. Cas-a: For prosss with a rlativly larg tim dlay (θ>τ/8), th ratio θ/t p varis btwn.7 and.5 (dpnding on th ovrshoot and valu of τ/θ). W slt to us th valu θ=.43t p, (not that a larg valu is mor onsrvativ as it inrass th intgral tim). This givs Pross with rlativly larg tim dlay: b τ I =.86 A t (5a) p b Tabl. Variation of θ/t p with τ/θ and ovrshoot. θ/t p θ/t p θ/t p τ/θ Ovrshoot=. Ovrshoot=.3 Ovrshoot= Cas-b: For a lag-dominant pross (τ>8θ) w find that θ/t p varis btwn.5 and.36 (dpnding on th ovrshoot and valu of τ/θ). W slt to us th avrag valu θ=.35t p and gt Intgrating pross: τ I =.44t p (5b) In onlusion, th intgral tim τ I is obtaind from th minimum of th abov two valus and boms min I.86 A b tp,.44 tp (6) b 5. ANALYSIS AND SIMULATION Simulations hav bn ondutd for diffrnt typs of pross and th proposd tuning produr provids rasonabl ontrollr sttings with rspt to both prforman and robustnss. This stion prsnts only twlv typial ass to show th fftivnss of th proposd tuning rul. Th rsults of th stp tst for th ontrollr tuning and orrsponding PI stting with M s valu ar listd in Tabl. Th pak of maximum snsitivity (M s ) is a masur of robustnss and is dfind as Ms max/[ gpg( i)] ; a small M s valu indiats that th stability margin of th ontrol systm is larg. Th rsulting losd-loop PI-rspons for as E5, E7, E8, E and E ar shown in Figur 5, 6, 7, 8 and 9. A unit stp stpoint hang is mad at t= and a unit stp hang for a load disturban at th pross input is mad at t=,, and 5 for E7, E8 and E, rsptivly. For E5 and E, th prformans ar valuatd by giving a unit stp hang at t= and a stp input of magnitud and.5 in th load disturban at t=5 and. Th rommndd PI sttings vary somwhat with th ovrshoot as sn in Tabl. From Figur 5-9, it is lar that th rsponss ar los to thos with th sttings. Th rsulting PI tunings dpnd on th ovrshoot usd in th xprimnt and basd on th fitting and th rsults in xampl prosss (Tabl ) w rommnd that an ovrshoot around.3 is usd in pratis. 6. CONCLUSION A simpl and nw approah for PI ontrollr tuning has bn dvlopd. From a singl losd-loop stpoint stp tst, using a P-ontrollr with gain k, on obtains thr haratristi numbrs: Th ovrshoot (typially around.3), th tim to th first pak t p and th rlativ stady stat hang b. Th proposd PI-sttings for th Shams s stpoint mthod mthod ar: k k A min I.86 A b tp,.44 t p b whr A.5(ovrshoot) -.67(ovrshoot) +. Th abov sttings giv a robust and rasonabl fast rspons (orrsponding to a losd-loop tim onstant τ = θ in th original mthod). If on wants to dtun th ontrollr to gt a smoothr rspons with mor robustnss and lss input usag thn on may introdu th dtuning paramtr F>: k k A/F; min I.86 A b tp,.44 tpf b On may in som ass hoos F< to spd up th rspons but th systm will thn b lss robust. Th nw mthod works for a wid varity of th prosss, xpt unstabl and highly osillating systm. Th novlty of th proposd mthod is that only on xprimnt in losdloop is suffiint for gtting signifiant information for ontrollr tuning. W bliv that it ould b th simplst and asist approah for PI ontrollr tuning to us in pross industris.

5 Tabl : PI ontrollr stting for proposd and Cas Pross modl k ovrshoot t p b k τ I M s E s s E s s E3.3s.8s ss.4s.s.5s 3 E4 s.s E5.3s 6s3s s8ss E6 s ss E7 s s 6sss E8 9 ss s9 E9.7s ss.8s E 7.4s E. s s.5s E s s s.5s

6 ovrshoot=.9 ovrshoot=.344 ovrshoot= Tim Fig. 6. Rspons for as E7 pross ovrshoot=. ovrshoot=.3 ovrshoot= Tim Fig. 9. Rspons for as E pross ovrshoot=.9 ovrshoot=.344 ovrshoot= Tim Fig. 7. Rspons for as E8 pross ovrshoot=. ovrshoot=.35 ovrshoot= Tim Fig. 8. Rspons for as E pross..4. ovrshoot=.8 ovrshoot=.38 ovrshoot= Tim Fig. 5. Rspons for as E5 pross. 7. REFERENCES Ziglr, J. G., Nihols, N. B. (94). Optimum Sttings for Automati Controllrs. Trans. ASME, (65), Rivra, D. E., Morari, M. and Skogstad, S. (986). Intrnal Modl Control. 4. PID Controllr Dsign, Ind. Eng. Chm. Pross Ds. Dv., (5), Sborg, D. E., Edgar, T. F. and Mllihamp, D. A. (4). Pross Dynamis and Control, nd d., John Wily & Sons, Nw York, U.S.A. Skogstad, S. (3). Simpl Analyti Ruls for Modl Rdution and PID Controllr Tuning, J. Pross Control, (3), Åström, K. J., Hägglund, T. (984). Automati Tuning of Simpl Rgulators with Spifiations on Phas and Amplitud Margins, Automatia, (),

A Simple Method of Tuning PI Controllers for Interval Plant of Cold Rolling Mill

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