Dublin Institute of TechnologyARROW@DITConference papers School of Electrical Engineering Systems2002-01-01A non-linear PID controller for CSTR using localmodel networksRuiyao GaoDublin Institute of TechnologyAidan O'DwyerDublin Institute of Technology, aidan.odwyer@dit.ieEugene CoyleDublin Institute of TechnologyThis Conference Paper is brought to you for free and open access by theSchool of Electrical Engineering Systems at ARROW@DIT. It has beenaccepted for inclusion in Conference papers by an authorizedadministrator of ARROW@DIT. For more information, please contactyvonne.desmond@dit.ie, arrow.admin@dit.ie.Recommended CitationGao, Ruiyao and O'Dwyer, Aidan and Coyle, Eugene : A non-linear PID controller for CSTR using local model networks.Proceedings of the IEEE 4th World Congress on Intelligent Control and Automation (WCICA 2002), Shanghai, China, 10-14 June.A Non-linear PID Controller for CSTR Using Local Model Networks Ruiyao Gao Aidan OdywerEugene Coyle School of Control Systems and Electrical Engineering Dublin Institute of Technology Kevin Street, Dublin 8. IRELAND

Abstract-ThebasicPIDcontrollers have difficulty in dealing with problemsthatappearincomplexnon-linearprocesses.Thispaper presentsapracticalnon-linearPIDcontrollerthatdealswiththese non-linear difficulties. It utilises a local model (LM) network, which combinesasetoflocalmodelswithinanartificialneuralnetwork (ANN)structure,toadaptivelycharacterisetheprocessnon-linearity.Thenalocalcontrollernetworkisformulatedthrougha gating system deduced from the LMN to handle the non-linearity. A continuousstirredtankreaction(CSTR)casestudyillustratesthe practicality of this method in the modelling and control of non-linear processes.PIDcontrollersarestillaliveandappropriateforthe control of non-linear processes. I.INTRODUCTION Proportional, integral and derivative (PID) controllers have existedformorethan50years.ThereasonwhyPID controllers have gained such popularity is that the controller can be tuned by means of simple rules of thumb, and detailed knowledge about the system is not necessary.ThebasicPIDcontrollershavedifficultyincontrolling processeswithcomplexnonlinearity.Todate,many sophisticatedalgorithmshavebeenusedtohelpthePID controllerworkundersuchdifficulties([1],[2]).Talking aboutneuralnetworks,generally,therearetwodifferent waysofapplyinganeuralnetworktosolvecontrol-engineeringproblems.OneistouseArtificialNeural networks (ANN) to adjust the parameters of a conventional controller.TheothermethodistheuseoftheANNasa directcontroller.HowarethePIDparametervalues connectedwithneuralnetworkcontrollers?Likethetwo ways to build ANN controllers, there are two ways to make the connection. One method is to adjust the PID parameters byANN([3],[4]);theothermethodistocreatetheANN based on the systems output error signal ([5], [6], [7]). The firstmethodinvolvesemulatingthethoughtsofanexpert controlengineerbytweakingthetuningparameters accordingtotheempiricalrules.Itsapplicationislimited and the computation load is normally extensive. The second method is an application of adaptive control, which has been usedinindustrialapplicationandbringssomepromising results. However, the selection of the network training data setsisnotatrivialproblem,andthecomputationalloadis highlyintensive.Sofar,mostoftheapproacheslosethe simplicityofimplementation,whichisthemostattractive feature of the original PID control approach. Recent advances in LM networks give a neat extension of thebasicPIDstructuretohandlenon-linearsystemsusing neural networks. This approach decomposes the system into a numberofsmoothlyoverlappinglocaloperatingregimesin whichalocalmodeldescribesthelocalregionalproperties. Theglobalnon-linearmodeliscreatedbycombiningthe local models through an ANN structure.In this paper, we have discussed the issue of PID controller applicationusingneuralnetworks.Insections2and3,the constructionoflocalmodel(LM)networksandlocal controller (LC) networks are discussed. Then the CSTR case studyispresentedandperformanceresultsaregivenin section4.Thepaperendswithsomeconclusionsand suggestions for future work in section 5. II.LOCAL MODEL NETWORKS Localmodel(LM)networkswerefirstintroducedby JohansenandFossin([8],[9])asameansofdecomposing non-linearauto-regressivemovingaveragewithexogenous inputs(NARMAX)modelsintoaninsightfulstructurefor systemidentificationandcontrol.Murray-Smith([10],[11]) presentedfurtherreportsonLMnetwork,whichpresented thisapproachasoneofthestandardtechniquestocombine linearmodelsandANNtocharacterisethenon-linearity. Figure 1 shows the general structure of this scheme. Fig.1. Local Model Networks Action 1NiMMMM( )1 1 1 1, , c( )i i i ic , ,( )n n n nc , , Norma-lisation( ) ~1f( ) ~fi( ) ~nfi 1 n ~We assume that at each time instant, the process behaves insomeuniquelycharacterisablewaywitheachlocal operatingregime i ,ofwhichweuseafunction ifto describetheproperty.Thenweassociateavalidity functionitodeterminethevalidityoftheoperating regimes given the current operating point ~. The modelling problemistorobustlyestimatethefunction iffrom observationdataandexistinga-prioriinformationsoasto pre-structureandparameterisethemodelstructure if. Please refer to ([8] & [9]) for detailed information.One straightforward and simple approach to the modelling problemistouseasetoflinearlocalmodels,whichis appealingformodellingcomplexnon-linear systems due to itsintrinsicsimplicityandtheweakassumptionsrequired. The linear models can be obtained in several different ways: fittingtheparametersofaspecifiedmodelstructureto input/output data obtained from the physical process, fitting theparameterstothesimulatedresponsefromadetailed fundamentalmodel,orcalculatingtheseparametersusing differential linearization. Weshallconsiderthegeneralnon-linearstatespace system, with state vectorx and inputu : ( )( ) u x g yu x f x,, & (1) Linearizationofnon-lineardynamicsystemsoftheform of(1) is a standard procedure ([12]).Considerthelinearizationof( ) u x f , withrespecttoN designedoperatingregimes,theselinearisedmodelsare created andindexed by i together with an operating point vector~and N validity functions ias follows ([9]). ( ) ( ) ( ) ~] [ ,1iNidi iei iu B x x A u x f+ ( ) ( ) ( ) ~] [ , 1iNidi iei ieiu D x x C y u x g+ + (2) inwhich, eidu u ui ,thesuperscriptedenotesthe equillibriumrelatedvariable.Thestateandoutputofthe nonlinearsystem(1)canbeapproximatelyrecreatedfrom the N linear systems of (2). III. LOCAL CONTROLLER NETWORK Localcontroller(LC)networks,thecontrolversionof LMNwasintroducedin[13]andfurtherextendedin ([14],[15]). In general, the global control signal isdefined by ( ) ( ) ( ) ( ) ( ) niicit t C t u1~ iC denotesthelocalcontrollerforeachlocalmodelif . The n local controllers thus obtained are blended using the samevalidityfunctioni whichareusedintheLMN.The controllerinformationvector c consistsofpastcontrol inputs,currentandpastplantoutputs,andthecurrentand past values of the reference signalrefy . Figure 2 shows a LC network with a gating system. Its basic idea is to adaptively blend various controllers at different operating regions of the process in a proper way through a gating system. The gating system i resultsfromtheapproachformulatingtheLM network. IV.CASE STUDY A.CSTR PROCESS Continuousstirredtankreactor(CSTR)isahighlynon-linear process. A shematic of the CSTR system is shown in Fig.3.Asingleirreversible,exothermicreactionisassumed to occur in the reactor. Theprocessmodelconsistsoftwononlinearordinary differential equations ([16]): M Fig. 2.Local controller network MMM( ) t qc( )cfT t qc ,C(t),T(t)Feed in f f fT q C , ,Coolant Fig.3. CSTRplant model ( ) ( ) ( ) ( )( )( )( )( ) ( ) t T Tt qKt q Kt RTEt C K t T TVqt Tcfccff11]1

,_

+

,_

+ 321exp 1exp& ( ) ( ) ( )( )

,_

t RTEt C K t C CVqt Cffexp ) (0& where( ) t qcis the coolant flow rate, T(t) is the temperature ofthesolutionsandC(t)istheeffluentconcentration.The modelparametersdefinedandnominaloperatingconditions areshownintable1.TheobjectiveistocontrolC(t)by manipulating( ) t qc. Table 1.Nominal CSTR Operating Conditions fq = 100 l/min, product flow rate fC =1 mol/l,input concentration fT =350 K,input tempraturecfT =350 K,temprature of coolant K1=1.44*1013 Kl/min/mol,V =100 l , container volume RE=104 K,activation energy01 . 02 K /l, constant K3=700 l/min. constant 10010 * 2 . 7 Kmin-1 , constant TheCSTRprocessiswithexponentialtermsandproduct terms.Itsopen-loopsteptestsshowthattheoutput concentrationresponsesvaryfromover-dampedtounder-damped,indicatingthevariabledynamicsintheCSTR process.Fig.4isthestepresponseofconcentrationoutput ) (t Cwhenthecoolantflow rate) (t qcvariesfrom85l/min to 110 l/min. The CSTR exhibits highly non-linear dynamical behaviour. 0 10 20 30 40 50 600.040.060.080.10.120.140.160.18 Fig 4. Dynamic response of the CSTR plant Eigenvalueanalysisshowsthatthestableequilibrium regimeoftheCSTRliesin( ) ( ) 13566 . 0 , 0 t C mol/l& ( ) t qc( ) 8 . 110 , 0 l/min, which is shown in Figure 5. 0 20 40 60 80 100 12000.020.040.060.080. 10.120.14 Fig.5 steady-state concentration output from CSTRB.MODELLING THE CSTR PROCESS The difficulty of modelling using LM networks lies in that it requirescarefulconsiderationofthefollowingoptions:the number of the regimes, the variables to be used to define the regimes, and the size and shape of the regimes. We manually decomposedtheworkregimeoftheCSTRintoNsmall regimes based on a-priori information, each of which linearly approximatesthelocalpropertyoftheassignedregime. NormalisedGaussianfunctionisusedastheweighting functionintheANNstructuretoblendthelocalmodels . Simulations were carried out to model the system with from 3 up to 10 models. The global model with 5 local models meets the best tradeoff between the number of the local models and the quality of the performance. The selected 5 local operating regimes are as follows: min / 2 10 . 1 , 95 . 432 , / 1 2980 . 11 1 1l e q K T l mol e Cco o o min / 1 8899 . 9 , 442 , / 2 5060 . 82 2 2l e q K T l mol e Cco o o min / 1 8291 . 8 , 450 , / 2 8541 . 53 3 3l e q K T l mol e Cco o o min / 1 8788 . 6 , 465 , / 2 9468 . 24 4 4l e q K T l mol e Cco o o min / 1 0438 . 5 , 481 , / 2 4630 . 11 2 5l e q K T l mol e Cco o o in which(icoioioq T C , , ) denotes the linearization point of the ith local model.Wechooseasetofstepinputsignal) (t qc,whichvaries from50l/minto110l/min,asshowninFig.6.Itcoversthe highlydynamicoperatingareaofCSTRprocess.TheLM networkoutputsaregiveninFig.7,whichcomparesthe effluentconcentrationouputsC(t)andthetemperature outputsT(t)fromtheCSTRprocesswiththecorresponding outputs from the LM network, when the input signal (coolant flowrate)( ) t qcvaries.Wecanseethegoodnessofthe matchingbetweentheLMnetworkmodeloutputsandthe process CSTR outputs. It is whorth while to note that, when thecoolantflowrate( ) t qc=110l/min,theCSTRouputis highly under-damped, however, the LM network output still Time(min) Concentration C(t)(mol/l) Coolant flow-rate) (t qc (l/min) Concentration C(t)(mol/l) matchestheCSTRouputwithhighaccuracy,whichproves the effectivity ofthe proposed the LC network.0 10 20 30 40 50 60 70 805060708090100110 Fig.6. Step changes in coolant flow-rate( ) t qc

0 10 20 30 40 50 60 70 80420430440450460470480490

0 10 20 30 40 50 60 70 8000.020.040.060.080.10.120.140.16 Fig. 6: Comparison of the CSTR process output and the LM network output Solid line is from the plant, dash-dotted line is from the LM networks. C.PI/PID CONTROLLER FOR CSTR ThelocalPI/PIDcontrollerparametersweredesigned firstly using the Zeigler-Nichols ultimate cycle tuning rule basedonthemodelsdevelopedabove,andthenempirically adjustingthePIDparameterstogetoptimalperformance. The control signal is deduced from ( ) ( ) ( )( )

,_

+ + ttdipdtt deT dt t eTt e K t u01 The PID parameters we use in this paper are as follows: 1 pK=2.55e+003,1 iT =9.59e-001, 1 dT =0.22 2 pK =3.31e+002, 2 iT =5.12e-001, 2 dT =0 3 pK =4.042e+002, 3 iT =2.67e-001, 3 dT =0 4 pK =2.08e+003,4 iT = 4.34e-001, 4 dT =0 5 pK =2.01e+004,5 iT = 5.35e-001, 5 dT =0Theglobalnon-linearPIDcontrollerisformulatedby blendingthelocalcontrollersthroughthegatingsystem resulted from the LMN structure. The behaviour of the non-linearPIDcontrollerisshowninFig.8,fromwhichwecan seethesmoothtransientresponsewhentheset-pointC(t) changesfrom0.02mol/lupto0.13mol/l.It should be noted thattheset-pointC(t)=0.13isveryclosetotheinstable regionofCSTRprocess,inwhichthereisonlysmall overshoot.ItvalidatestheLCnetworkdevelopedbasedon the LM network.Moreover,theglobalperformanceoftheLCnetwork highlydependsontheperformanceofthelocalcontrollers. Thegatingsystem,asaweightingfunction,smoothesthe transient response when the set-point changes.0 10 20 30 40 50 60430435440445450455460465470475 Time (min.) Temperature T(t) (k) Time (min.) Effluent concentration C(t) (mol/l)Time (min.) Coolant flow-rate qc(t) (l/min) Temperature T(t) (k) Time (min.) 0 10 20 30 40 50 6000.020.040.060.080.10.120.14

Fig. 7. CSTRstep responses of the temperature T(t) and of the effluent concentration C(t) in closed loop. Solid line is from the LN controller; dashed line is the input. V.CONCLUSION Wepresentanon-linearPIDcontrollerusingLMNand LCNinthispaper.Thispaperillustratesthesimplicityand practicality of the method in the identification and control of non-linearsystemCSTR(Continuous Stirred Tank Reactor). ItprovesthatPIDcontrollersarestillaliveandappropriate undernon-lineardifficulties.Concerningthedifficultyof manuallytuningPIDparameters,furtherworkwillfocuson auto-tuning PID controller strategies as well as stability issues for non-linear process control based on the LM Network.VI. REFERENCES [1]Taylor, J.H.,m o str A & &&K.J., A non-linear PID auto tuning algorithm,AmericanAutomaticcontrolconference, Seattle, W.A., pp. 1-6, 1986. [2]UnarM.A.,Murray-SmithD.J.andSyedFarmanAli Shah,DesignandtuningforfixedstructurePID controllersAsurvey,reportCSC-96016,1996. Centreforsystemsandcontrol&departmentof mechanical Engineering, university of Glaslow. [3]RuanoA.E.B.,FlemingP.J.,.JonesD.I,Connectionist approachtoPIDautotuning,IEEproceedings-D, Vol.139, No.3, pp. 279-285, May 1992. [4]YuWen-ShyongandLuTien-Ching,PIDcontroller designusingdynamicalneuralnetworks,0-7803-4859-1/1998 IEEE,1998, pp. 2131-2135. [5]ChanK.C.,LeongS.S.andLinG.C.I,Aneural networkPIcontrollertuner,ArtificialIntelligencein Engineering 9, pp. 167-176,1995 [6]ChenC.L.andChangF.Y.,Designandanalysisof neural/fuzzy variable structural PID control systems, IEEProceedingsControlTheoryApplication., Vol.143, No.2, March 1996, pp. 200-208.[7]Vance VanDoren, Model free adaptive control, Control engineering, Europe, Feb/Mar, 2001, pp.25-31. [8]JohansenT.A.andFossB.A.,ANARMAXmodel representationforadaptivecontrolbasedonlocal models,Modelling,Identification,andControl, Vol.13, No.1, 1992,pp.25-39. [9]Johansen, T.A and Foss B.A., Constructing NARMAX models using ARMAX models, International Journal of Control, Vol.58, 1993, pp.1125-1153. [10]Murray-SmithR,LocalModelnetworksandlocal learning,inFuzzyDuisburg,94,pp.p404-409, Feb.1994(onlineversionat ftp://eivind.imm.dtu.dk/pub/rod-ocal_learning.ps.gz). [11] Murray-Smith R. and Johansen T. A., Multiple Model ApproachestoModellingandControl,Taylorand Francis, 1997. [12]T.Kailath, Linear Systems, Prentice Hall, 1980. [13] Middleton R.H., Goodwin, G.C., Hill D.J., and Mayne, D.Q,Designissuesinadaptivecontrol,IEEE transaction on automatic control, Vol.33, No.1, 1988, pp.50-58. [14]MorseA.S,Towardaunifiedtheoryofparameter adaptivecontroltunability,IEEEtransactionon automaticcontrol,Vol.35,No.9,1990,pp.1002-1012. [15] Weller S.R. and Goodwin G.C, Hysteresis Switching adaptivecontroloflinearmultivariablesystems, IEEE transaction on automatic control, Vol.39, No.7, 1994, pp.1360-1375. [16]HensonM.AandSeborgD.E.,Input-Output linearisation of general non-linear processes, AIChE Journal Vol.36, 1990, pp.1753-1757. Time (min.) Effluent concentration C(t) (mol/l)