Publiction [P5] L. Eriksson T. Oksnen PID Controller Tuning for Integrting Processes: Anlysis nd New Design Approch in Proc. ourth Interntionl Symposium on Mechtronics nd its Applictions (ISMA7 Shrjh UAE Mrch 69 7. 6 p. 7 ISMA7. Reprinted with permission.
Proceeding of the 4 th t Interntionl Symposium on Mechtronics nd its Applictions (ISM7 Shrjh U.A.E. Mrch 69 7 PID CONTROLLER TUNING OR INTEGRATING PROCESSES: ANALYSIS AND NEW DESIGN APPROACH Lsse Eriksson Helsinki Uniersity of Technology Control Engineering Lbortory P.O.Box 55 I5 TKK inlnd lsse.eriksson@tkk.fi ABSTRACT This pper discusses PID controller tuning for integrting processes with time-dely nd first order lg. Most of the existing tuning rules for this kind of processes he the sme generl structure nd the properties of these rules re discussed especilly in connection with rying time-dely systems. The pper proposes noel tuning method tht optimizes the closed-loop performnce with respect to certin robustness constrint while considering the dely rince i jitter mrgin imiztion. urther we deelop new PID controller tuning rules bsed on the tuning method. The pper discusses the new tuning rules in detil nd compres them with some of the recently published results. The work ws originlly motited by the need for robust but simultneously well performing PID controller tuning prmeters in n griculturl mchine cse process. We lso demonstrte the superiority of the proposed tuning rules with this cse process.. INTRODUCTION The tuning of the PID controller hs been discussed in numerous rticles nd books nd there exists riety of tuning methods. Mybe the best known tuning rules re those proposed by Ziegler nd Nichols lredy in 94. Still tody the Z-N methods re populr in process control. It is obious tht the Z-N tuning methods do not meet the requirements of ll the processes in tody s industry. An exmple of this is networked control system where rying time-delys might endnger the stbility. PID tuning is not completely soled problem despite of the decdes of reserch. On the contrry its reserch seems to grow []. Some of the recent tuning methods re presented in []. The reserch described in this pper ws motited by the need for PID tuning rules for integrting processes where rible trnsport delys nd gin prmeters ffect the system stbility nd performnce. The present tuning rules re inestigted in this frmework nd new tuning method nd rules re lso proposed. The need for tuning rule deelopment ws experienced when prototyping some of the most recent tuning rules [3] in cse process tht is presented in Section. The section lso presents the preliminries required to understnd the proposed tuning pproch nd reiews the current tuning rules for the OLIPD (first order lg plus integrl plus dely process model tht is considered throughout the pper. Section 3 nlyzes the properties of PID controlled OLIPD systems. Section 4 presents the Timo Oksnen Helsinki Uniersity of Technology Automtion Technology Lbortory P.O.Box 55 I5 TKK inlnd timo.oksnen@tkk.fi new tuning pproch nd rules tht re compred with other tuning rules in Section 5. Section 6 sttes the conclusions.. CONTROL SYSTEM The generl lyout of the control system nd its components re discussed in this section. In ddition the cse process is described. The PID controller tuning rules currently found in the literture re lso reiewed. The control system goodness mesures used in this pper such s the distnce from the robustness circle nd the jitter mrgin re presented... Process model The generl lyout of the control system is seen in igure. We consider n integrting process in connection with low-pss mesurement filter. Alterntiely the low-pss filter cn be prt of the process (integrtor + first order lg. In both cses the process model is gien s K Ps ( = e. ( s + st ( This is lso known s the OLIPD model [] where K is the elocity gin T the filter or lg time constnt L the time-dely nd s the Lplce rible. The tuning rules re lter deeloped by ssuming tht T is fixed by the process mesurement setup such tht dequte noise compenstion is chieed... Cse process In the experiments rel integrting process is used. The process is prt of n griculturl trctor nd it consists of hydrulic system n electroniclly controllble hydrulic le hydrulic cylinder ctutor connected to weight nd position sensor. The position of the weight is controlled. The mss of the weight ries nd lso occsionl counterforces by ground contct re eident. The hydrulic le is controlled i CAN bus nd this limits the control cycle to ms. The dely of the complete system is identified to ry between nd 3 ms consisting of communiction bus dely le dynmics oil pressure nd flow in the hydrulic pipes nd position mesurement delys. The integrting cse process hs rible trnsport dely nd -gin nd with mesurement filter it cn be modeled s (. ISMA7-
Proceeding of the 4 th t Interntionl Symposium on Mechtronics nd its Applictions (ISM7 Shrjh U.A.E. Mrch 69 7 y r C( s u K e s + Ts y m This tuning is clled AMIGO tuning (pproximte M-constrined integrl gin optimiztion..5. PID tuning rules for OLIPD igure. The generl lyout of the control system..3. PID controller We consider the continuous-time PID controller of the form [4] ( = ( r ( m( t dyr ( t dym( t ki ( e( τ dτ kd c u t k by t y t + + dt dt ( where e(τ = y r (τ - y m (τ is the error signl between reference signl nd mesured (filtered output. The prmeters k k i nd k d re proportionl- integrtion- nd deritie gins respectiely. The set-point weighting prmeters b nd c re fixed in dnce here b = nd c =. The trnsfer function of ( is C( s = k+ ki + kds. (3 s.4. AMIGO tuning The AMIGO tuning rules (see [3] [5] were recently deeloped both for non-integrting nd integrting processes. The good experiences with these tuning rules for non-integrting processes encourged the uthors to prototype the AMIGO pproch for integrting processes in the cse process. Neertheless the results from the cse process indicted tht the performnce of the control system could be improed by replcing the AMIGO tuning with some other method. The tuning method proposed in this pper tkes the similr pproch s the AMIGO tuning where the process is modeled with simple first order liner model or the integrtor model. The model is deeloped in the spirit of Ziegler-Nichols i step response experiments. The tuning rules re then deried bsed on the few process prmeters (gin time constnt dely. The AMIGO tuning for integrting processes is bsed on chrcterizing the process using the IPD (integrl plus dely model structure [] K Ps ( = e. (4 s After numerous nlyses of prmeter reltionships nd extensie studies of robustness nd performnce the following tuning rules re proposed in [3] nd [5] for integrting processes. k =.45 / K ki =.565 / ( KL (5 k =.5 L/ K. d As the AMIGO tuning rules were originlly tested with the cse process with unstisfctory results other PID tuning rules for integrting processes were inestigted. or the pure integrtor process there re seerl tuning methods but for the OLIPD there re not so mny. Numerous tuning rules re collected into Hndbook of PI nd PID Controller Tuning rules [] lso for OLIPD process model. Most of the tuning rules for OLIPD re gien in the form T k = ki kd K L = = K L (6 where is tuned with rious methods. Vitečko et l. [6] he tuned bsed on oershoot criterion nd O'Dwyer [] hs deried from gin nd phse mrgins. Numericl lues of ry roughly from.3 to.. By writing out the open-loop eqution ( ( C s P s = + s e KL KL s( + st = e Ls T it cn be seen tht the open-loop nd thus lso the closed-loop is independent of the lues of K nd T. In other words the controller structure elimintes those known process prmeters. Other tuning rules for OLIPD re presented by Rier nd Jun [7] nd these rules re conerted in [] into form K ( L+ λ ( L+ T + λ T k = k = k = K L K L K L i d ( + λ ( + λ + λ where λ is n djustble prmeter lue tht should correspond pproximtely to the closed-loop response speed. If the lg T in ( is rther smll compred to the dely L the tuning rules for IPD cn lso be used. Åström nd Hägglund [4] propose the following rules bsed on Ziegler-Nichols ultimte cycle equilent method.94.94.47 k = ki k d K L = KL = K. (9 All the tuning rules collected in [] for IPD nd OLIPD were tested with the cse process using simultion. Some rules seemed to work only for certin rnge of process prmeters L nd T nd some rules ge unstisfctory performnce ersus robustness rtio tht is n importnt fctor in the cse process..6. Robustness for disturbnces: The M-circle The deelopment of the AMIGO rules ws bsed on the following robustness criterion: if the Nyquist cure of the loop trnsfer function does not intersect circle with center c R nd rdius r R defined s (7 (8 ISMA7
Proceeding of the 4 th t Interntionl Symposium on Mechtronics nd its Applictions (ISM7 Shrjh U.A.E. Mrch 69 7 c R M M + M = r = ( M M R ( M ( M the sensitiity function nd the complementry sensitiity functions re less thn M for ll frequencies [8]. The robustness is thus cptured by one prmeter only M. The lue M =.4 ws used in the AMIGO rule deelopment lthough finlly the rules did not quite stisfy the constrint. or the test process btch 5 % increse of M ws reported resulting in M.6..7. Robustness for dely rince: The jitter mrgin Wheres the robustness prmeter M concerns the disturbnces such s mesurement noise in the cse process lso other type of robustness is required. The process suffers from rying timedelys s mentioned in Section.. Often such time-delys re known to be bounded nd it might be tempting to design worstcse controller using the imum dely. Unfortuntely controller designed for the imum dely does not gurntee tht the closed-loop system would be stble s the dely ries in the rnge from the minimum to the imum lue. [9] Recently proposed stbility criteri for systems with rying time-delys [] re suitble for our usge since they cn be formulted s objectie functions in the optimiztion of PID controller prmeters. The jitter mrgin is n upper bound for dditionl dely tht cn be dded into closed-loop control system while mintining stbility. The dely cn be of ny type (constnt time-dependent rndom but the jitter mrgin determines the bound for the imum lue of the dely. A continuoustime SISO system is stble for ny time-rying delys defined by if ( δ δ δ Δ ( = t ( t ( t ( P( jω C( jω < ω [ [ ( + P( jω C( jω δ ω where δ is the imum dditionl dely (the jitter mrgin. This criterion hs been successfully pplied in the derition of PID tuning rules for non-integrting processes in rying timedely systems []. δ + Gol ( jω L L < = sin( ωl + jωgol ( jω ω ω. (4 In order to clculte the nlyticl expression for the jitter mrgin in the OLIPD cse with the controller tuning (6 the expression (4 should be minimized with respect to frequency ω. This turns out to be hrd problem nlyticlly but rther esy using numericl methods. or the nlysis of the jitter mrgin we concentrte on prmeter rnges.368.8 nd. L. The rnge for is chosen similrly s in [6] where determines the oershoot of the closed-loop response ( 5 %. The rnge for dely L is simply chosen to be ery wide. The jitter mrgin for the OLIPD process model with the PD controller (6 is shown in igure with respect to prmeters nd L. A closer look t the jitter mrgin surfce reels tht for fixed the jitter mrgin is nerly liner function of dely L. or prcticl use of this nlysis it would be conenient to he n expression for (the closed-loop performnce prmeter s function of L nd δ. Often the minimum dely (L nd the possible dditionl dely (δ of the system re known but the problem is how to select between robustness nd performnce (. Thus we clculte n pproximtion for the jitter mrgin nd sole it for. The jitter mrgin surfce cn be pproximted by.9485 δ =.6356 L (5 which gies.9485l =. (6 δ +.6356L This is the imum lue for gin tht cn be used with certin jitter mrgin requirement. or exmple if the process minimum time-dely is L =.5 nd the required jitter mrgin is 5 % of L it is possible to use gin =.835 which gies pproximtely 3 % oershoot for the closed-loop system with good performnce. In order to he less oershoot the gin cn be decresed without endngering the stbility since the smller lues of increse the jitter nd gin mrgins. In the chosen rnge of prmeters the estimte (5 gies imum error of ±.5 % of the true jitter mrgin (4. 3. ANALYSIS O OLIPD TUNING RULES or the OLIPD process model most of the PID tuning rules he the sme generl structure (6 i.e. PD controller. According to (7 when these tuning rules re pplied the open-loop system becomes independent of T nd K. The Nyquist cure of the open-loop trnsfer function is Gol ( jω = e = ( sin( ωl + jcos( ωl. (3 Ls ωl s= jω This indictes tht with high frequencies the Nyquist cure conerges to the origin. The prmeters nd L determine the distnce from (- such tht with higher lues of the gin mrgin decreses. The jitter mrgin of the system becomes 4. NEW TUNING APPROACH In this section we propose new PID controller tuning pproch tht explicitly tkes into ccount the robustness criteri presented in Section. We introduce two objectie functions to be optimized simultneously nd use simultion bsed constrined optimiztion to sole the optiml prmeters for the PID controller in connection with the OLIPD process. The tuning rules re then deried bsed on the prmeter surfces produced in the optimiztion phse. Multi-objectie optimiztion is used for soling the problem since the optiml controller prmeters should minimize more thn one conflicting objecties simultneously. ISMA7
Proceeding of the 4 th t Interntionl Symposium on Mechtronics nd its Applictions (ISM7 Shrjh U.A.E. Mrch 69 7 Jitter mrgin δ 5 5 5 Dely L Jitter mrgin δ (L..4.6.8 Tuning prmeter igure. The jitter mrgin (OLIPD + PD controller. 4.. Multi-objectie optimiztion In order to sole the controller tuning problem we use multiobjectie constrined optimiztion. A generl multi-objectie optimiztion (here minimiztion problem is gien s Min st.. { } ( x = f( x fk ( x x Ω gi ( x i =... m Ω= xhj( x = j=... m T x = [ x xn ]. (7 where f l (x l = k re nonliner objectie functions tht re to be minimized simultneously x i re the decision ribles nd g i (x nd h j (x re the nonliner inequlity nd equlity constrints respectiely (see e.g. []. There re numerous lgorithms for soling the boe problem of which the gol ttinment method will lter be used for deriing the tuning rules. The gol ttinment problem is defined s Min γ s.t. ( x α γ g (8 x Ω where γ is n uxiliry rible α is ector of weights nd g is ector of gols i.e. the objectie function lues tht should be ttined. 4.. Problem formultion In order to pply multi-objectie optimiztion the objectie functions must be set. We use the ITAE cost criterion to mesure the performnce of the closed-loop system. The other criterion is the jitter mrgin tht should be imized. The robustness with respect to disturbnces is lso tken into ccount by introducing n optimiztion constrint tht keeps the open-loop Nyquist cure outside the robustness circle ( similrly s in the AMIGO rules. Here M =.5 is used s the robustness prmeter which pproximtely corresponds to the obtined lue tht ws reported with AMIGO tuning rules. Note tht we use this lue s hrd constrint wheres in AMIGO rules M =.4 ws rther soft constrint or n objectie. As mentioned before the AMIGO rules finlly hd to relx (increse this lue up to 5 %. The optimiztion problem is formulted in (9-( where f (x re the objectie functions to be minimized nd g(x is the constrint function tht the decision ribles x must stisfy. where f ( x = t e( t dτ = t y ( t y( t dτ r (9 f ( x = = δ + H ( jω min ω [ [ jωh( jω ( ( H c ( R H rr ( gx ( = d = min Re( + Im( ( ω [ ] T x = k k k nd H ( jω = C( jω P( jω. ( 4.3. Tuning results i d The controller tuning problem ws soled using MATLAB s Optimiztion Toolbox nd fgolttin function. The weights of the gol ttinment method were chosen such tht both performnce (9 nd jitter mrgin ( were eqully weighted. The gol for ITAE criterion ws set equl to the ITAE criterion for the Vitečko et l. [6] tuning (6 with =.4 corresponding to oershoot of % for the closed-loop system. The gol for the jitter mrgin ws set to T + L corresponding to the effectie ded-time of the process. This mrgin would llow the dely to increse by % from the effectie ded-time while gurnteeing the stbility. Note tht this is in some cses quite high objectie but s it is hndled s n objectie rther thn constrint it is resonble. The initil lues of the controller prmeters for the optimiztion were chosen ccording to the Vitečko et l. [6] tuning. The rnge for prmeters T nd L ws chosen from. to but only lues for which the rtio T / L remins in the rnge [. ] were considered. This restriction ws motited on one hnd by simultion ccurcy since the system tends to become stiff for lues outside of this rnge nd on the other hnd by resoning. One of the prmeters esily becomes negligible outside this rnge. The optimiztion results re presented in igure 3 where controller gins (k k i nd k d nd the ITAE cost re presented with respect to the process nominl dely L nd process time constnt T. The controller gin k increses s L nd T decrese. The integrl gin k i remins in zero. The deritie gin k d decreses s L nd T decrese. The ITAE cost increses nturlly s the dely increses but lso s T increses. igure 4 shows the open-loop Nyquist cures when pplying the optiml tuning to the OLIPD process (. ISMA7-4
Proceeding of the 4 th t Interntionl Symposium on Mechtronics nd its Applictions (ISM7 Shrjh U.A.E. Mrch 69 7 Controller gin k Controller gin k i.5 The loop trnsfer function Nyquist cures nd the robustness circle with M =.5 k 4 ki (- log (T log (L - log (T log (L Im -.5 Controller gin k d ITAE cost (log -scle - kd.5 log (T log (L log (ITAE 5-5 log (T igure 3. Optimized PID controller prmeters. 4.4. Tuning rules log (L Bsed on the controller prmeter surfces the tuning rules were deeloped. This phse included both determining the rule structure nd estimting the coefficients. The rule identifiction ws done similrly s in []. As with the other OLIPD tuning rules the integrl gin is zero nd the proportionl nd the deritie gins re inersely proportionl to the process elocity gin K. The proposed tuning rules for the OLIPD processes re T k = k = k = (3 f ( L T g( L T h( L i d KL K where f( L T =.7( T / L.794 T / L.34.5 glt ( =. + (.5.76log ( T L.5 hl ( =.97.48 L. 5. EXPERIMENTS (4 The deeloped tuning rules were utilized in the cse process. The step response of the process ws recorded in rying cses. Identifiction of the process prmeters ws utomted with MATLAB script. It ws found out tht the process gin K ries between.3 nd. nd the trnsport dely L between.5 nd.35. Resonble lues for the mesurement filter time constnt T re.-.. Bsed on the set of identifiction results the process ws fixed with prmeters K =.8 L =.5 T =.5. 5.. Comprison of tuning rules by simultion All the tuning rules collected in [] for IPD nd OLIPD were tested with the cse process model using simultion. The new tuning rules were compred to these. igure 6 presents the robustness properties nd the simulted step responses of the cse process when using different tuning rules. On the left there re the Nyquist cures nd on the right the unit step responses. -.5.5 -.5 - -.5.5 Re igure 4. Optimized open-loop systems Nyquist cures. The circles in igure 6 represent the robustness circle ( with M =.5. Only the proposed tuning nd Vitečko et l. [6] tuning gie good performnce with dequte robustness properties tht fulfill the sero control requirements presented boe. Neertheless the proposed tuning gies better settling time nd robustness properties simultneously nd there is no oershoot. The lowest plots of igure 6 compre the step responses nd robustness properties between the proposed rules nd the Vitečko et l. rules with oershoot of %. It cn be concluded tht the new rules re superior to the other rules in mny respects. 5.. Experiments with the rel process In the cse process the response of the hydrulic le is not liner to the control signl in the whole rnge but in the experiments only prt of the full rnge ws used (-4 % where the le is pproximtely liner to the control signl. This control signl limittion ws lso tken into ccount in the simultions. The comprison of the simulted nd rel process step responses using the new PID tuning rules is presented in the igure 5. 6. CONCLUSIONS In this pper the PID controller tuning for integrting processes ws considered. The few existing tuning rules for OLIPD process model were nlyzed with respect to robustness for disturbnces nd for rying time-dely. The common structure of these tuning rules ws nlyzed nd the dependency of tuning rule performnce prmeter nd time-dely robustness criterion ws shown. In ddition noel tuning method for the PID controller ws proposed nd bsed on the design concept new tuning rules were deeloped. The new tuning rules were tested by simultion nd in cse process. The new tuning rules were shown to outperform the other known tuning rules for OLIPD process..8.6.4. 3 4 5 6 time (s igure 5. with simultor nd rel process using the new rules. sim rel ISMA7-5
Proceeding of the 4 th t Interntionl Symposium on Mechtronics nd its Applictions (ISM7 Shrjh U.A.E. Mrch 69 7 Åström nd Hägglund (995 Hy (998 method 3 3 Åström nd Hägglund (995 Hy (998 method 3 - - O'Dwyer Am=3 phsm=6 ( - - AMIGO (4 - - Vitecko OS= ( - - - - Vitecko OS= ( - - Rier & Jun lmbd= ( - - The New Rules - - 4 6 8 4 6 8 time (s.5.5 O'Dwyer Am=3 phsm=6 ( Vitecko OS= ( 4 6 8 4 6 8 time (s.5.5 AMIGO (4 Rier & Jun lmbd= ( 4 6 8 4 6 8 time (s.5.5 Vitecko OS= ( The New Rules 4 6 8 4 6 8 time (s igure 6. Comprison of loop trnsfer function Nyquist cures (robustness circle with M =.5 nd unit step responses. 7. REERENCES [] A. O Dwyer Hndbook of PI nd PID Controller Tuning Rules Imperil College Press London 3. [] IEEE Control Systems Mgzine ol. 6( 6. [3] K. J. Åström nd T. Hägglund Adnced PID Control ISA- The Instrumenttion Systems nd Automtion Society 6. [4] K. J. Åström nd T. Hägglund PID Controllers: Theory Design nd Tuning nd ed. Instr. Soc. of Americ 995. [5] K. J. Åström nd T. Hägglund "Reisiting the Ziegler- Nichols step response method for PID control" Journl of Process Control ol. 4 pp. 635-65 4. [6] M. Vitečko A. Vitecek nd L. Smutny "Controller Tuning for Controlled Plnts with Time Dely" in Proc. IAC Workshop: Digitl Control pp. 8388. [7] D. E. Rier nd K. S. Jun "An integrted identifiction nd control design methodology for multirible process system pplictions" IEEE Control Systems Mgzine ol. (3 pp. 57. [8] K. J. Åström H. Pngopoulos nd T. Hägglund "Design of PI controllers bsed on non-conex optimiztion" Automtic ol. 34 pp. 585-6 My 998. [9] K. Hiri nd Y. Stoh "Stbility of System with Vrible Time Dely" IEEE Trns. Automtic Control ol. c5(3 pp. 55-554 98. [] C.-Y. Ko nd B. Lincoln "Simple stbility criteri for systems with time-rying delys" Automtic ol. 4(8 pp. 49-434 Aug. 4. [] L. M. Eriksson nd M. Johnsson "PID Controller Tuning Rules for Vrying Time-Dely Systems" to Apper in Proc. 7 Americn Control Conference New York USA 7. [] G. P. Liu J. B. Yng nd J.. Whidborne Multiobjectie optimistion nd control Reserch Studies Press Ltd. 3. ISMA7-6