Neural Network Predictive Control of a Chemical Reactor

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1 Ata Chimia Slovaa, ol., No., 009, 1-36 Neural Network Preditive Control of a Chemial Reator Anna asičkaninová*, Monika Bakošová Institute of Information Engineering, Automation and Mathematis, Faulty of Chemial and Food ehnology, Slovak University of ehnology, Radlinského 9, 8137 Bratislava, Slovakia monika.bakosova@stuba.sk Abstrat Model Preditive Control (MPC) refers to a lass of algorithms that omute a seuene of maniulated variable adjustments in order to otimize the future behaviour of a lant. MPC tehnology an now be found in a wide variety of aliation areas. he neural network reditive ontroller that is disussed in this aer uses a neural network model of a nonlinear lant to redit future lant erformane. he ontroller alulates the ontrol inut that will otimize lant erformane over a seified future time horizon. In the aer, simulation of the neural network based reditive ontrol of the ontinuous stirred tank reator is resented. he simulation results are omared with fuzzy and PID ontrol. Keywords: model reditive ontrol, fuzzy ontrol, PID ontrol, neural network, ontinuous stirred tank reator Introdution Conventional roess ontrol systems utilize linear dynami models. For highly nonlinear systems, ontrol tehniues diretly based on nonlinear models an be exeted to rovide signifiantly imroved erformane. Model Preditive Control (MPC) onet has been extensively studied and widely aeted in industrial aliations. he main reasons for suh oularity of the reditive ontrol strategies are the intuitiveness and the exliit onstraint handling. he reditive ontrollers are used in many areas, where high-uality ontrol is reuired, see e.g. Qin and Badgwell (1996), Qin and Badgwell (000), Rawlings (000). Model-based reditive ontrol refers to a lass of ontrol algorithms, whih are based on a roess model. MBPC an be alied to suh systems, as e.g. multivariable, nonminimum-hase, oen-loo unstable, non-linear, or systems with long time delays.

2 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator Constrained model reditive ontrol beomes the standard algorithm for advaned ontrol in roess industries.several versions of MPC tehniues are Model Algorithmi Control (MAC), see e.g. Rihalet et al. (1978), Dynami Matrix Control (DMC), see e.g. Cutler et al. (1980), and Internal Model Control (IMC), see e.g. Garia et al. (198). Although the above tehniues differ from eah other in some details, they are fundamentally the same, beause all of them are based on linear roess modelling. If the nonlinear model is available, the omutational reuirements are very high, see e.g. Garia et al. (1989), eseially for nonlinear MIMO roesses. It is estimated that, in a tyial ommissioning rojet, modelling efforts an take u to 90% of the ost and time in imlementing a model reditive ontroller by Morari and Lee (1999). he Neural Network Model Preditive Control (NNMPC) is another tyial and straightforward aliation of neural networks to nonlinear ontrol. When a neural network is ombined with MPC aroah, it is used as a forward roess model for the redition of roess outut, see e.g. Hunt et al. (199), Nørgaard et al. (000). Control of hemial reators is one of the most studied areas of roess ontrol. In this aer, a neural network based reditive ontrol strategy is alied to a ontinuous-time stirred reator with two arallel first-order irreversible exothermi reations. Simulation results show that the neural network based reditive ontrol gives romising results. heoretial Model-based reditive ontrol MBPC is a name of several different ontrol tehniues. All are assoiated with the same idea. he redition is based on the model of the roess, as it is shown in Figure 1. Fig. 1. Classial model-based reditive ontrol sheme Ata Chimia Slovaa, ol., No., 009, 1-36

3 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator 3 he target of the model-based reditive ontrol is to redit the future behaviour of the roess over a ertain horizon using the dynami model and obtaining the ontrol ations to minimize a ertain riterion, generally J N ( k,u( k) ) = ( ym ( k + j) yr ( k + j) ) +λ ( u( k + j )) j=n 1 N u j= 1 1 (1) Signals y m (k+j), y r (k+j), u(k+j) are the j-ste ahead reditions of the roess outut, the referene trajetory and the ontrol inut, resetively. he values N 1 and N are the minimal and maximal redition horizon of the ontrolled outut, and N u is the redition horizon of the ontrol inut. he value of N should over the imortant art of the ste resonse urve. he use of the ontrol horizon N u redues the omutational load of the method. he arameter λ reresents the weight of the ontrol signal. At eah samling eriod only the first ontrol signal of the alulated seuene is alied to the ontrolled roess. At the next samling time the roedure is reeated. his is known as the reeding horizon onet. he ontroller onsists of the lant model and the otimization blok. E. (1) is used in ombination with the inut and outut onstraints: u y u min min y min min u u y y u y max u max y max max he ability to handle onstraints is one of the key roerties of MBPC and also auses its sread, use, and oularity in industry. MBPC algorithms are reorted to be very versatile and robust in roess ontrol aliations. () Neural network reditive ontrol Neural networks have been alied very suessfully in the identifiation and ontrol of dynami systems. he universal aroximation aabilities of the multilayer eretron make it a oular hoie for modelling of nonlinear systems and for imlementing of nonlinear ontrollers. he use of a neural network for roess modelling is shown in Figure. he unknown funtion may orresond to a ontrolled system, and the neural network is the identified lant model. wo-layer networks, with sigmoid transfer funtions in the hidden layer and linear transfer funtions in the outut layer, are universal aroximators. Ata Chimia Slovaa, ol., No., 009, 1-36

4 4 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator Fig.. Neural network as a funtion aroximator he redition error between the lant outut and the neural network outut is used as the neural network training signal. he neural network lant model uses revious inuts and revious lant oututs to redit future values of the lant outut. he struture of the neural network lant model is given in the Figure 3, where u(t) is the system inut, y (t) is the lant outut, y m (t) is the neural network model lant outut, the bloks labelled DL are taed delay lines that store revious values of the inut signal, IW i,j is the weight matrix from the inut j to the layer i. LW i,j is the weight matrix from the layer j to the layer i. Fig. 3. Struture of the neural network lant model his network an be trained off-line in bath mode, using data olleted from the oeration of the lant. he roedure for seleting the network arameters is alled training the network. he Levenberg-Maruardt (LM) algorithm is very effiient for training. he LM algorithm is an iterative tehniue that loates the minimum of a funtion that is exressed as the sum of suares of nonlinear funtions. It has beome a standard tehniue for nonlinear least-suares roblems and an be thought of as a ombination of steeest desent and the Ata Chimia Slovaa, ol., No., 009, 1-36

5 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator 5 Gauss-Newton method, see e. g. Kelley (1999), Levenberg (1944), Madsen et al. (004), Maruardt (1963), Mittelmann (004). When the urrent solution is far from the orret one, the algorithm behaves like a steeest desent method: slow, but guaranteed to onverge. When the urrent solution is lose to the orret solution, it beomes a Gauss-Newton method. Let f be an assumed funtional relation whih mas a arameter vetor n estimated measurement vetor x = f ( ),x R ˆ ˆ m R to an. An initial arameter estimate 0 and a measured vetor x are rovided, and it is desired to find the vetor + that best satisfies the funtional relation f, i.e. minimizes the suared distane e e with e= x xˆ. he basis of the LM algorithm is a linear aroximation to f in the neighbourhood of. For a small δ, a f +δ f + Jδ where J is the aylor series exansion leads to the aroximation ( ) ( ) Jaobi matrix ( ) f. Like all non-linear otimization methods, LM is iterative: initiated at the starting oint 0, the method rodues a series of vetors 1,,..., that onverge towards a loal minimizer + for f. Hene, at eah ste, it is reuired to find the δ that minimizes the uantity e Jδ. he sought δ is thus the solution of a linear least-suare roblem: the minimum is attained when Jδ e is orthogonal to the olumn sae of J. his leads to J ( Jδ e) = 0, whih yields δ as the solution of the normal euations: J Jδ = J e. (3) he matrix J J in the left hand side of E. (3) is the aroximate Hessian, i.e. an aroximation to the matrix of seond order derivatives. he LM method atually solves a slight variation of E. (3), known as the augmented normal euations Nδ = J e, where the off-diagonal elements of N are idential to the orresonding elements of J J and the diagonal elements are given by [ ] ii N =µ + J J for some µ > 0. he strategy of altering the diagonal ii elements of J J is daming and µ is referred to the daming term. If the udated arameter vetor +δ with δ omuted from E. (3) leads to a redution of the error e, the udate is aeted and the roess reeats with a dereased daming term. Otherwise, the daming term is inreased, the augmented normal euations are solved again and the roess iterates until a value of δ that dereases error is found. Ata Chimia Slovaa, ol., No., 009, 1-36

6 6 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator In LM, the daming term is adjusted at eah iteration to assure a redution in the error e. he LM algorithm terminates when at least one of the following onditions is met: 1. he magnitude of the gradient of e e, i.e. J e in the right hand side of E. (3), dros below a threshold ε 1.. he relative hange in the magnitude of δ dros below a threshold ε. 3. he error e e dros below a threshold ε A maximum number of iterations k max is omleted. If a ovariane matrix for the measured vetor x is available, the minimum is found by solving a weighted least suares roblem defined by the weighted normal euations J Jδ = J e (4) Fuzzy Control Classi ontrol theory is usually based on mathematial models whih desribe the behaviour of the ontrolled roess. he main aim of fuzzy ontrol is to simulate a human exert (oerator), who is able to ontrol the roess by translating the linguisti ontrol rules into a fuzzy set theory. In 1965, Lotfi A. Zadeh introdued fuzzy sets, where a more flexible sense of membershi is ossible. he ast few years have witnessed a raid growth in the use of fuzzy logi ontrollers for the ontrol of roesses that are omlex and badly defined. Most fuzzy ontrollers develoed till now have been of the rule-based tye by Driankov et al. (1993), where the rules in the ontroller attemt to model the oerator s resonse to artiular roess situations. An alternative aroah uses fuzzy or inverse fuzzy model in roess ontrol, see e.g. Babuška et al. (1995), Jang (1995), beause it is often muh easier to obtain information on how a roess resonds to artiular inuts than to reord how, and why, an oerator resonds to artiular situations. A review of the work on fuzzy ontrol has been resented by Lee (1990). Design of a simle fuzzy ontroller an be based on a three ste design roedure, that builds on PID ontrol: start with a PID ontroller; insert an euivalent, linear fuzzy ontroller; make the ontroller gradually nonlinear. he fuzzy ontroller an inlude emirial rules. his roerty is eseially useful in oerator ontrolled lants. Let us onsider e.g. a tyial fuzzy ontroller: Ata Chimia Slovaa, ol., No., 009, 1-36

7 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator 7 if error is negative and hange in error is negative then outut is negative big, if error is negative and hange in error is zero then outut is negative medium. he olletion of rules is alled a rule base. he omuter is able to exeute the rules and omute a ontrol signal deending on the measured inuts error and hange in error. he inuts are most often hard or ris measurements from some measuring euiment. A dynami ontroller would have additional inuts, for examle derivatives, integrals, or revious values of measurements bakwards in time. he blok fuzzifiation onverts eah iee of inut data to degrees of membershi by a looku in one or several membershi funtions. he rules may use several variables, both in the ondition and the onlusion of the rules. Basially, a linguisti ontroller ontains rules in the if-then format, but they an be resented in different formats. he resulting fuzzy set must be onverted to a number that an be sent to the roess as a ontrol signal. his oeration is alled defuzzifiation. here are several defuzzifiation methods. Outut saling is also relevant. In ase the outut is defined on a standard universe this must be saled to engineering units. Fig. 4. Fuzzy ontroller akagi-sugeno tye ontroller he outut sets an often be linear ombinations of the inuts, or even a funtion of the inuts. he develoed Fuzzy Logi oolbox for the software akage Matlab imlements one of the hybrid shemes known as the Adative Network based Fuzzy Inferene System (Anfis). Anfis reresents a Sugeno-tye fuzzy system in the seial five-layer feed forward Ata Chimia Slovaa, ol., No., 009, 1-36

8 8 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator network arhiteture and uses a hybrid learning algorithm to identify the membershi funtion arameters of single-outut, Sugeno tye fuzzy inferene systems. Suose the rule base of a Sugeno - akagi fuzzy system is as follows, see e.g. Nauk et al. (1977), akagi et al. (1985), Kvasnia et al. (009): if x 1 is A i and x is B i then y = i x 1 + i x +r i, i=1,..n (5) he if-arts (anteedents) of the rules desribe fuzzy regions in the sae of inut variables error e, its derivative de. he then-arts (onseuents) are funtions of the inuts, usually linear with onseuent arameters i, i, r i. Further, y is an outut variable, A i, B i are fuzzy sets haraterized by three linguisti variables (small, middle, large). Exerimental Consider a ontinuous stirred tank reator (CSR) by asičkaninová et al. (005), asičkaninová et al. (006) with two arallel first-order irreversible reations aording to the sheme A k k 1 B,A C, where B is the main rodut and C is the side rodut. he measured and ontrolled outut is the temerature of the reation mixture. he ontrol inut is the volumetri flow rate of the ooling medium. Possible disturbanes inlude hanges in the feed temerature and the oolant temerature. he only maniulated variable is the oolant flow rate. he simlified non-linear dynami mathematial model of the hemial reator onsists of five differential euations: d dt A d dt B d dt C = = = Av A k1 A k A (6) Bv B + k1 A (7) + k Cv C A (8) Ata Chimia Slovaa, ol., No., 009, 1-36

9 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator 9 d dt = v Ak ρc [ ] Q& r + ρc (9) d dt = v Ak + ρ C [ ] (10) he reation rate oeffiients are non-linear funtions of the reation temerature being defined by the Arrhenius relations k 1 = k 10 e E 1 R k = k 0 e E R (11) he heat generated by hemial reations is exressed as Q& = k1 ( H1 )+k ( H ) r A r A r (1) Here, are onentrations, are temeratures, are volumes, ρ are densities, C P are seifi heat aaities, are volumetri flow rates, r H are reation enthalies, A is the heat transfer area, k is the heat transfer oeffiient. he subsrit denotes the oolant, r the reating mixture and the suersrit s denotes the steady-state values in the main oerating oint. Parameters and inuts of the reator are enumerated in able 1. able 1: Reator arameters and inuts ariable Unit alue ariable Unit alue C ρ ρ C C C A k k 10 k 0 E 1 /R E /R m 3 min -1 m 3 m 3 kg m -3 kg m -3 kj kg -1 K -1 kj kg -1 K -1 m kj m - min -1 K -1 min -1 min -1 K K r H 1 r H Av Bv Cv s v s v s s s s A s B s C kj kmol -1 kj kmol -1 kmol m -3 kmol m -3 kmol m -3 K K m 3 min -1 K K kmol m -3 kmol m -3 kmol m Ata Chimia Slovaa, ol., No., 009, 1-36

10 30 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator he reations in the desribed reator are exothermi ones and the heat generated by the hemial reations is removed by the oolant in the jaket of the tank. he ontrol objetive is to kee the temerature of the reating mixture lose to a desired value. Results and disussion Neural Network Model Preditive Control of the CSR he designed ontroller uses a neural network model to redit future CSR resonses to otential ontrol signals. An otimization algorithm then omutes the ontrol signals that otimize future lant erformane. he neural network lant model was trained using the Levenberg-Maruardt algorithm. he training data were obtained from the nonlinear model of the CSR (6)-(10). he used model reditive ontrol method was based on the reeding horizon tehniue. he neural network model redited the lant resonse over a seified time horizon. he reditions were used by a numerial otimization rogram to determine the ontrol signal that minimizes erformane riterion (1) over the seified horizon. he ontroller blok was imlemented in Simulink. Constraints and arameters values: 0 u 0.0, 354 y 365, N 1 = 1, N = 7, N u =3, λ= 0.5. akagi-sugeno ontroller for the CSR Sugeno-tye fuzzy inferene system was generated using subtrative lustering in the form: if e is A i and de is B i then u = i e + i de + r i, i=1,... 3 (14) where e is the ontrol error, de is the derivation of the ontrol error, u is the alulated ontrol inut (t) and i, i, r i are onseuent arameters. he symmetri Gaussian funtion (gaussmf in MALAB) was hosen as the membershi funtion and it deends on two arameters σ and as it is seen in (14) f ( x;σ,) = e ( x ) σ he arameters σ and for gaussmf are listed in the able. For obtaining of these arameters, it was neessary to have the data sets of e, de and u at first. hese data were obtained by simulation of PID ontrol of the CSR. he onseuent arameters in the ontrol (15) Ata Chimia Slovaa, ol., No., 009, 1-36

11 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator 31 inut rule (14) are listed in able 3 and the resulting lot of the outut surfae of a desribed fuzzy inferene system is resented in Figure 5. able : Parameters of the Gaussian urve membershi funtions e de σ i i σ i i able 3: Conseuent arameters i i r i Fig. 5. akagi-sugeno ontroller - ontrol signal u as funtion of ontrol error e and its derivation de. Ata Chimia Slovaa, ol., No., 009, 1-36

12 3 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator PID ontrol For feedbak ontroller tuning, the aroximate model of a system with omlex dynamis an have the form of a first-order-lus-time-delay transfer funtion (16). he roess is haraterised by a steady-state gain K, an effetive time onstant and an effetive time delay D. G P K s + 1 Ds ( s) = e he transfer funtion desribing the ontrolled hemial reator was identified from ste resonse data in the form (16) with arameters: K = -157, = 14 min, D = min. hese arameters were used for feedbak ontroller tuning. he feedbak PID ontrollers were tuned by various methods, see e.g. Ogunnaike and Ray (1994). he best simulation results were obtained with PID ontroller (17) tuned using Chien-Hrones-Reswik method. he ontroller arameters are K C = , I = 16.8, D = he transfer funtion of the used PID ontroller is following G C 1 ( s) = K + s C 1+ D I s Figure 6 resents the simulation results of the reditive ontrol of the CSR. hese results are omared with those obtained by fuzzy ontrol and PID ontrol of the CSR (16) (17) Fig. 6. Comarison of the reating mixture temerature ontrol: reditive ontrol (... ), fuzzy ontrol (- - - ), PID ontrol ( ), referene trajetory ( ) Ata Chimia Slovaa, ol., No., 009, 1-36

13 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator 33 he ste hanges of the referene y r were generated and the MBP, fuzzy and PID ontrollers were omared using iae and ise riteria desribed as follows: iae = ise = e dt 0 (18) 0 e dt he iae and ise values are given in able 4. able 4: Comarison of the simulation results by integrated absolute error iae and integrated suare error ise ontrol method iae ise reditive ontrol fuzzy ontrol PID ontrol (19) Figure 7 resents the simulation results of the reditive ontrol, fuzzy ontrol and PID ontrol of the CSR in the ase when disturbanes affet the ontrolled roess. Disturbanes were reresented by oolant temerature hanges from 98 K to 37 K at t=100 min, from 37 K to 91 K at t=300 min and from 91 K to 310 K at t=500 min. he iae and ise values are given in able 5. Fig. 7. Comarison of the reating mixture temerature ontrol in ase when disturbanes affet the ontrolled roess: reditive ontrol (... ), fuzzy ontrol (- - - ), PID ontrol (-.-.-), referene trajetory ( ) Ata Chimia Slovaa, ol., No., 009, 1-36

14 34 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator able 5: Comarison of the simulation results by integrated absolute error and integrated suare error in ase when disturbanes affet the ontrolled roess. ontrol method iae ise reditive ontrol fuzzy ontrol 3 71 PID ontrol Used fuzzy ontroller is simle, and it offers lesser value ise than the reditive ontroller in the ase when the reator is not affeted by disturbanes. he disadvantage of the fuzzy ontroller is, that using the ontroller leads to nonzero steady-state errors. he steady-state errors vary from 0.05 K to 0.17 K, when the reator without disturbanes is ontrolled. In the ase of the reator ontrol in the resene of disturbanes, the steady-state errors vary from 0.17 K to 1.4 K. he advantage of the fuzzy ontrol is that the ontrol resonses do not show any overshoots and undershoots. he worst simulation results were obtained using the PID ontroller. he ontrol resonses are most osillating, and the PID ontroller used in a simle feedbak ontrol loo is not able to attenuate disturbanes. he best simulation results were obtained using the neural network reditive ontroller. Although the ontrol resonses are osillating, the maximum overshoot is smaller than the one with the fuzzy ontroller. Simultaneously, the steady state errors are very small, the maximum steady state error is 0.38 K in the resene of disturbanes. he followed integral riteria also onfirm that the best of three ontrollers in the neural network reditive ontroller. Conlusions In this aer, an aliation of a neural network based reditive ontrol strategy to a CSR is resented. he simulation results onfirm that the neural network based reditive ontrol is one of the ossibilities for suessful ontrol of CSRs. he advantage of this aroah is that it is not linear-model-based strategy and the ontrol inut onstraints are diretly inluded to the synthesis. Comarison of the MBPC simulation results with fuzzy ontrol and lassial PID ontrol demonstrates the effetiveness and sueriority of the roosed aroah. hese roerties are aarent, eseially in the ase, when the ontrolled roess is affeted by disturbanes. Ata Chimia Slovaa, ol., No., 009, 1-36

15 A.asičkanová, M.Bakošová, Neural Network Preditive Control of a Chemial Reator 35 Aknowledgments he work has been suorted by the Sientifi Grant Ageny of the Slovak Reubli under grants 1/4055/07, 1/0071/09 and by the Slovak Researh and Develoment Ageny under the rojet AP Referenes Babuška R, Sousa J, erbruggen HB (1995) Model-Based Design of Fuzzy Control Systems. Proeedings of the EUFI 95: Cutler CR, Ramaker BL (1980) Dynami matrix ontrol - a omuter ontrol algorithm. Pro. Amer. Control Conf., WP5-B. Demuth H. Beale M (00) Neural Network oolbox For Use with MALAB. Control Systems: URL htt://www. image. ee. ntua. gr /ourses_stati/ nn/ matlab/ nnet.df. Driankov D, Hellendom H (1993) An Introdution to Fuzzy Control. Sringer erlag, London. Garia CE, Morari M (198) Internal model ontrol-i. A unifying review and some new results. Ind. Eng. Chem. Proess Des. Dev. 1: Garia CE, Prett DM, Morari M (1989) Model reditive ontrol, theory and ratie-a survey. Automatia 5(3): Hunt KJ, Sbarbaro D, Zbikowski R, Gawthro PJ (199) Neural networks for ontrol systems-a survey. Automatia 8(6): 1083-I 11. Jang JSR, Sun C (1995) Neuro-Fuzzy Modeling and Control. he Proeedings of the IEEE May 83: Kelley C (1999) Iterative Methods for Otimization. SIAM Press, Philadelhia. Kvasnia M, Hereg M, Čirka Ľ, Fikar M (009) ime-otimal Control of akagi-sugeno Fuzzy Systems. Proeedings of the 10th Euroean Control Conferene: Lee CC (1990) Fuzzy Logi in Control Systems: Fuzzy Logi Controllers, IEEE rans. Syst. Mn. Cybern., SMC 0: 404. Levenberg K (1944) A Method for the Solution of Certain Non-linear Problems in Least Suares, Quarterly of Alied Mathematis, (): Madsen K, Nielsen HB, ingleff O (004) Methods for Non-Linear Least Suares Problems. ehnial University of Denmark, Leture notes, Available: htt://www. imm. dtu. dk /ourses /0611/nlls. df. Maruardt W (1963) An Algorithm for the Least-Suares Estimation of Nonlinear Parameters. SIAM Journal of Alied Mathematis, 11(): Mittelmann D (004) he Least Suares Problem. Available: htt:// lato. asu. edu /tois /roblems/ nlols. html. Morari M, Lee JH (1999) Model reditive ontrol: ast, resent and future, Comuters and Chemial Engineering, 3: Ata Chimia Slovaa, ol., No., 009, 1-36

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