grey-box modelling and control of chemical processes

Chemical Engineering Science 57 (2002) 1027–1039www.elsevier.com/locate/ces

Grey-box modelling and control of chemical processes

Qiang Xiong, Arthur Jutan∗

Department of Chemical and Biochemical Engineering, The University of Western Ontario,London, Ont., Canada N6A 5B9

Received 5 January 2001; accepted 17 September 2001

Abstract

The success of model based control of chemical processes is dependent on good process models. Many of these processes exhibitstrong nonlinearity and time varying parameters and are often di1cult to model accurately. The ‘grey-box’ model which combines partialknowledge of the process, with a neural network to capture the remaining dynamics, is a promising modelling tool for nonlinear processes.This modelling methodology maximizes the use of a priori process knowledge. This, in turn, reduces the size of the neural networkrequired to capture the remaining dynamics, hence, less data for training and faster convergence can be achieved. The grey-box model iscombined with a generic model control structure and applied to a number of simulations as well as a real-time process. ? 2002 ElsevierScience Ltd. All rights reserved.

Keywords: Grey-box modelling; Neural networks; Nonlinear process control; Generic model control

1. Introduction

Strongly nonlinear behavior is often found in chemicalengineering processes, such as highly exothermic batch reac-tors and pH processes. Another characteristic which makeschemical processes di1cult to control is their time-varyingnature due to many factors, for example, fouling of heatexchangers and deactivation of catalysts. Model-basednonlinear adaptive control techniques are most often usedto control these nonlinear time-varying processes. Thesetechniques require good nonlinear models which are anessential part of advanced control strategies. Model-basedcontrol strategies, such as model predictive control (MPC)and internal model control (IMC), are well developed, butthe stumbling block is often a poor model.

Many nonlinear input–output representations are avail-able to model and control nonlinear processes, such asVolterra series (Doyle III, Ogunnaike, & Pearson, 1995),Hammerstein (Zhu & Seborg 1994), Wiener (Norquay,Palazoglu, & Romagnoli, 1996) and bilinear (Bruni,DiPillo, & Koch, 1974) models, to name a few. These mod-els share a common drawback, in that they can describe only

∗ Corresponding author. Present address: University of Newcastle,CIDAC, Dept. of Electrical Engineering, Centre Integrated Dynamics &Control, Callaghan, 2308, NSW, Australia. Tel.: +1-519-661-2111-88322;fax: +1-519-661-3498.

E-mail address: ajutan@uwo.ca (A. Jutan).

a speciLc class of nonlinearity. This limitation restricts theimplementation of these models in practice. Recently, neu-ral networks have gained much attention because of theircapabilities of nonlinear function approximation. Theoreti-cally, neural networks with three layers can approximate anynonlinear function with arbitrary accuracy (Hornik, Stinch-combe, & White, 1989; Park & Sandberg, 1991), and thisfeature makes the neural networks a promising nonlinearmodelling tool. There are already many successful appli-cations using neural networks in chemical process control(Bhat & McAvoy, 1990; Hussain & Mohamed Azlan, 1995;te Braake, van Can, Scherpen, & Verbruggen, 1998). Onecommon problem associated with these studies, is that in or-der to capture as much as possible of the nonlinear dynamicsof the process, the neural network models used often con-tain many layers and neurons which leads to a large numberof neural parameters. Training the neural network modelswith such a large number of parameters requires large setsof data and long training times. In addition, over-Lt causedby a large number of parameters can lead to poor predictionperformance.

In this paper, we use the ‘grey-box’ modelling approach tohelp alleviate some of these di1culties. A ‘grey-box’ modelis also called a ‘hybrid’ model in the literature and it has beenused successfully to model and control some chemical pro-cesses (Cubillos, Alvarez, Pinto, & Lima, 1996; Psichogios& Ungar, 1992; ZbiciMnski, Strumi l lo, & KamiMnski, 1996).

1028 Q. Xiong, A. Jutan / Chemical Engineering Science 57 (2002) 1027–1039

This paper is organized as follows. Section 2 presentsthe architecture of the grey-box model and its advantages.Section 3 gives a brief introduction to generic model control(GMC) design. Simulation and experimental examples areused to test the grey-box model-based controller in Section4. Section 5 concludes this work.

2. Grey-box model

A process model can be either a Lrst-principles model,which is constructed using physical knowledge of the pro-cess, or an empirically identiLed black-box model. Bothhave their disadvantages: Lrst-principles models are rigor-ous but often expensive or even impossible to obtain, whileblack-box models do not always capture the physical realityand they are usually valid only in a limited range, howeverthese models are much less expensive to obtain. The combi-nation of partially known process knowledge and black-boxmodelling results in a so-called grey-box model. From apractical point of view, the grey-box modelling is a very con-venient way to model nonlinear processes, since, the modelstructure can be derived from the Lrst principles of mass andenergy balances and the nonlinear characteristics of the pro-cess can be modelled as an empirical additive component.

Tompson and Kramer (1994) give a detailed summaryand comparison of diQerent approaches of modelling chem-ical processes using prior knowledge and neural networks.We focus our attention on the semiparametric design ap-proaches, which combine a neural network with a Lxed formparametric model, either in series or in parallel. In Cubilloset al. (1996), the Lrst-principles model is used to describea particulate solid drying process, while the unknown pa-rameters, p; in the Lrst-principles model are approximatedby neural networks. Fig. 1 shows the architecture of a serialgrey-box model structure. Psichogios and Ungar (1992) alsoused the serial grey-box structure to model a fermentationprocess. This serial grey-box modelling technique with neu-ral networks oQers substantial advantages over a black-boxneural network modelling approach. Unlike the neural net-work model, which has a fully interconnected network withmany neural parameters, the Lrst principles model speciLesprocess variable interactions from physical consideration.Thus, the hybrid model can be more easily trained and up-dated and is more reliable for prediction purposes.

Sometimes, we are faced with the situation, where a com-plete description of the entire process is not available. Forinstance, for an exothermic reactor, the heating=cooling sys-tem may have been well studied but the kinetics of the reac-tion system cannot be easily modeled. The serial approachcannot be applied here, but an alternate parallel approachcan be used. The parallel structure of a hybrid neural modelis shown in Fig. 2. The output of the grey-box model is thesum of two separate model outputs

y = y1 + y2; (1)

First Principles Model

Neural Network

Fig. 1. Serial structure of a grey-box model with neural network.

ApproximateModel

Neural Network

Fig. 2. Parallel structure of a grey-box model with neural network.

where y1 is the output of an approximate model and y2 isthe output of a neural network. The approximate model canbe a simpliLed Lrst-principle model or an identiLed simplelinear=nonlinear model, which is the best available approxi-mation of the process and can approximate the overall pro-cess dynamics. This is the same philosophy employed bycontrol engineers who use a Lrst-order plus dead time struc-ture to approximate higher order dynamics. The neural net-work is placed in parallel and captures both model mismatchand process disturbances. In this architecture, the load on theneural network is much lower compared with a black-boxneural model which tries to model the entire process. Thisoccurs because the approximate model captures most of themain process dynamics, leaving only the remainder for theneural network. Thus, the size of the neural network can besubstantially reduced. The structure of the parallel grey-boxmodel can also be considered as an overall neural network,but with a certain number of pre-Lxed weights, from theapproximate model.

3. Controller design

3.1. Generic model control (GMC)

GMC (Lee & Sullivan, 1988) proposed by Lee and Sul-livan has strong advantages for developing nonlinear con-trollers and has been used successfully as mentioned in Cottand Macchietto (1989). A brief introduction to GMC algo-rithm is given below.

Consider a process described by the following stateequations:

x = f(x; u; t); (2)

Q. Xiong, A. Jutan / Chemical Engineering Science 57 (2002) 1027–1039 1029

y = g(x; u; t); (3)

where x is the state vector; u is the process input; y is theprocess output; t is time; f and g are nonlinear functions.DeLne a reference trajectory y∗ and equate its derivative y ∗

to a proportional integral (PI) type function as follows:

y ∗ = Kp(ysp − y) + Ki

0(ysp − y) dt; (4)

where Kp and Ki are tuning parameters and relate y∗ to theprocess output with

y = y ∗: (5)

By letting the derivative of the process output equal theproportional integral term which operates on the diQerencebetween the current output y and its set-point value ysp; theGMC can be viewed as an extension of the PID controller.However, it allows the use of nonlinear models because ofthe implicit relationship between the control signal u andprocess output y in Eq. (5).

The following tuning parameters Kp and Ki:

Kp =2��; (6)

Ki =1�2 ; (7)

are suggested by Lee and Sullivan (1988), where � and �characterize the target proLle of the controlled variable y∗. ALgure outlines the relative control performances of diQerentcombinations of � and � is presented in Lee and Sullivan(1988).

3.2. GMC for discrete systems

Since, we want to implement GMC on discrete systems,Eq. (5) should be expressed in a discrete form

D[y(t)] = D[y∗(t)]; (8)

where D denotes the derivative operator. By discretizingEq. (4) at t=nTs, the right-hand side of Eq. (8) is written as

D[y∗(n)] =Kp(ysp(n) − y(n))

+Kin∑k=0

(ysp(k) − y(k))Ts; (9)

where Ts is the sampling time. The derivative of processoutput D[y(t)] can be approximated by a Lrst order forwarddiQerence

D[y(n)] =y(n+ 1) − y(n)

Ts: (10)

From Eqs. (9) and (10), the desired output of the nextsample time is then obtained as

y(n+ 1) = y(n) + Ts

{Kp(ysp(n) − y(n))

+Kin∑k=0

(ysp(k) − y(k))Ts

}: (11)

Knowing y(n + 1), the control action at current time u(n)is obtained by inverting the process model (3).

3.3. GMC on systems with time delay

When applying the GMC to systems with time delay, theinversion of the process model for the calculation of the con-trol action requires that the controllers know in advance thefuture outputs of the process. Therefore, some sort of pre-diction is required. One way around this, is to increase thesampling time of the discrete model, which, in eQect, de-creases the number of time delays. Dunia and Edgar (1996)proposed a modiLed version of GMC which is termed thepredictive generic model control (PGMC). Since, the currentcontrol action only aQects the output in the future, PGMCsolves the current control action by requiring system out-put d steps ahead to be equal to some reference output. Aslight modiLcation of Eq. (8) for processes with time delayd results in

D[y(n+ d)] = D[y∗(n+ d)]: (12)

Substituting Eqs. (9) and (10) into the above equation gives

y(n+ d+ 1) − y(n+ d)Ts

=Kp(ysp(n+ d) − y(n+ d))

+Kin+d∑k=0

(ysp(k) − y(k))Ts (13)

in which y(n+1) to y(n+d) are unknown. Replacing themby their predictions from the model gives

y(n+ d+ 1) − ym(n+ d=n)Ts

=Kp(ysp(n+ d) − ym(n+ d=n))

{n∑k=0

(ysp(k) − y(k))

+n+d∑k=n+1

(ysp(k) − ym(k=n))

}; (14)

where ym(k=n) denotes the predicted output at time step kgiven information up to time n. The predictions ym(k=n),k = n + 1; : : : ; n + d, can be calculated from the availableinformation of the process input and output up to currenttime n and y(n+d+1) can thus be obtained from Eq. (14).The controller output u(n) is then calculated by invertingthe process model.

ApproximateModel

Controller Processr u y

Neural Network

Fig. 3. Diagram of the closed-loop system with GMC controller.

3.4. GMC using parallel grey-box model

For the parallel grey-box model given in Eq. (1), we sumthe two separate parts of the model to provide the modeloutput y

y(n+ 1) = y1(n+ 1) + y2(b+ 1)

=f1(u(n)) + f2(u(n)): (15)

The nonlinear equation in Eq. (11) can then be solvediteratively to Lnd the control action using Newton’s method

uk+1 = uk − f1(uk) + f2(uk)f′

1(uk) + f′2(uk)

: (16)

The derivative f′1(uk) can be easily obtained from the equa-

tions from the known part of the model. The derivative ofthe unknown part, which is described by a neural network,is calculated using the backpropagation method.

The overall closed-loop system with the controller is setup as shown in Fig. 3. The neural network is adapted on-line,to model the diQerence between the outputs of the true pro-cess and the approximate model.

4. Examples

4.1. Simulation

The following nonlinear system was used to evaluate thecontrol performance of the grey-box GMC controller.

y(k) =0:5y(k − 1)2

1 + y(k − 1)2 + y(k − 2)2 + 0:2y(k − 2)

+ 0:3u(k − d) + 0:1u(k − d− 1); (17)

where d is the time delay of the system. An approximatethird-order linear model for this system was used in thegrey-box model,

y(k) = a1y(k − 1) + a2y(k − 2)

+ b1u(t − d) + b2u(t − d− 1); (18)

where the parameters were identiLed as: a1 = −0:0319,a2=−0:1862, b1=0:2998 and b2=0:0896. This linear modelwas Lt to input–output data generated from the nonlinearmodel (17) and was able to capture the main process dy-namics. As a result, the load for the neural network, whichcaptured the remaining dynamics, was signiLcantly lowered,which allowed us to choose a single layer neural network.

4.1.1. Case 1. Time delay d= 1The tuning parameters of the GMC were calculated from

Eqs. (6) and (7) as Kp= 0:3 and Ki = 0:0004 using trial anderror. The number of neurons in the hidden layer is set ata (small) value of 2. The reference input in this simulationis a square wave with amplitude 1:5 and period 200 s. Thecontrol result is shown in Fig. 4. Some oscillation occurs atthe beginning of the simulation run, which is expected be-cause of the adaptation of the neural network in the grey-boxmodel. After that, the controller is able to track the referenceinput well.

For comparison, the simulation is run with only the linearmodel used and the same tuning parameters for the GMCcontroller. The result is shown in Fig. 5. It can be seenfrom this Lgure, that, with only the linear model, the GMCcontroller fails to track the set-point because of the stronglynonlinearity of the process represented by Eq. (17).

4.1.2. Case 2. Time delay d= 5This system has a time delay larger than 1, so the PGMC

algorithm has to be used. The rest of the parameters are sameas in Case 1. Fig. 6 shows, the PGMC can be successfullyapplied to processes with large time delay. There is somedrop in response speed due to the large delay but the controlquality, nevertheless, remains good.

4.2. Simulated exothermic batch reactor

The grey-box GMC controller is tested on an exothermicbatch reactor which was studied by Cott and Macchi-etto (1989). The schematic diagram of the batch reactoris given in Fig. 7. The reactor is cylindrically shaped,with a heating=cooling jacket. Two cascaded tempera-ture controllers, TC1 and TC2 are used to control the

0 100 200 300 400 500 600_2

time(s)

set_pointy

0 100 200 300 400 500 600_4

time(s)

Fig. 4. Simulation result using GMC with grey-box model.

0 100 200 300 400 500 600_2.5

time(s)

set_pointy

Fig. 5. Control result using GMC controller with linear model.

temperature inside the reactor. Control of the temperatureof the heating=cooling Vuid is handled by temperature con-troller TC2 for the jacket inlet stream. In this simulationstudy, we assume that perfect temperature control of the

jacket feed temperature can be achieved by TC2 and ourattention is focused on TC1.

A well-mixed, liquid-phase reaction system inside the re-actor with a Lxed volume of reactants is considered. Two

0 50 100 150 200 250 300 350 400 450_2

time(s)

set_pointy

0 50 100 150 200 250 300 350 400 450_4

time(s)

Fig. 6. Control result on a system with large time delay.

TC1 TC2

HotCold

Reactor Heat Exchanger

Fig. 7. Schematic diagram of a batch reactor.

reactions of four components A, B, C and D are describedby

reaction1: A + B → C;reaction2: A + C → D;

with reaction rates R1 and R2 given as

R1 = k1MAMB; (19)

R2 = k2MAMC; (20)

where k1 and k2 are reaction rate constants; MA; MB,and MC are the number of moles of components A, Band C. The rate constants k1 and k2 are dependent onthe reactor temperature through the Arrhenius relation.Both reactions have a large exothermic heat of reaction(WH1 = −41 840 kJ=kmol; WH2 = −25 105 kJ=kmol).

The heat balance of the batch reactor is given by Cott andMacchietto (1989)

�VCpdTrdt

= Qr + UA(Tj − Tr); (21)

�jVjCpjdTjdt

= −UA(Tj − Tr) + �jFjCpj(Tw − Tj); (22)

where � is the density of the reactor contents, V is the vol-ume of the reactor contents, Cp is the heat capacity of thereaction contents, Tr is the reactor temperature, Tj is thejacket temperature, Tw is the temperature of the Vuid fedinto the jacket, Qr is the reaction heat, U is the heat-transfercoe1cient, A is the heat-transfer area, �j is the density ofthe jacket contents, Vj is the volume of the jacket, Cpjis the heat capacity of the jacket contents and Fj is theVow-rate of the heating=cooling Vuid. A list of values ofall these variables can be found in Cott and Macchietto(1989).

Though Eq. (21) gives a nearly exact description of thereactor, the reaction heat Qr cannot be obtained directly us-ing detailed kinetic models because of the complexity ofthe chemical reactions. Jutan and Uppal (1984) proposeda promising approach to estimate the reaction heat releasewhich used the energy balance of the reactor. However, theassumptions they made were restrictive in that they assumenegligible heat capacity for the reactor walls, which is not al-ways the case in practice. Another technique for estimatingQr is the extended nonlinear Kalman Llter and its eQective-ness was demonstrated by de ValliXere and Bonvin (1989).

0 1000 2000 3000 4000 5000 6000 7000 800020

set_point controlled output

0 1000 2000 3000 4000 5000 6000 7000 800040

time(s)

Fig. 8. Control result using the grey-box GMC controller.

Here, the batch reactor is modeled by a parallel grey-boxmodel. The approximate model used in the grey-box modelis a simpliLed version of Eqs. (21) and (22).

�VCpdTrdt

= UA(Tj − Tr); (23)

�jVjCpjdTjdt

= −UA(Tj − Tr) + �jFjCpj(Tw − Tj) (24)

in which the reaction heat Qr is omitted since a detaileddescription of the kinetics (and hence Qr) is unavailable. Aneural network is then used to estimate the nonlinear eQectof the reaction heat Qr on the reactor temperature Tr , andany model inaccuracies of the approximate model.

After the discretization of the approximate model de-scribed by Eqs. (21) and (22) and substituting the parameter

0 1000 2000 3000 4000 5000 6000 7000 8000_1

time(s)

Fig. 9. Output of the neural network for the grey-box model.

values in Cott and Macchietto (1989), the following discretetransfer function is obtained:

Tr(q)Tw(q)

=0:0172q−1 + 0:0132q−2

1 − 1:425q−1 + 0:4553q−2 ; (25)

where q−1 is the backward shift operator. The sampling timechosen here is 60 s rather than 12 s as in Cott and Macchietto(1989). The reason for this choice is that a sampling timeas small as 12 s can destabilize the known part of the modelsince it has a pole very close to the unit circle.

A set of random numbers between−1 and 1 was generatedto initialize the weights of the neural network. The neuralnetwork then updates its weights as new input and outputmeasurements of the process become available. The numberof training cycles in each sampling interval can be adjustedto achieve a satisfactory performance. The tuning parametersKp and Ki are chosen as in Cott and Macchietto (1989), i.e.,to produce a small overshoot of the process output and arise time of about 20 min.

The manipulated variable, Tw, is the temperature of theVuid fed into the jacket for heating and cooling and is con-Lned to the range 20–120◦C. The control results for the re-actor temperature Tr and the manipulated variable Tw for areference input of 80◦C are shown in Fig. 8. The output ofthe parallel neural network is shown in Fig. 9. The adapta-tion of the neural network from the initial random weightsexplains the oscillation at the beginning of the run. Afterthis, an estimate of proLle of the heat release Qr in the reac-tor can be seen. It Lrst rises as the reactor temperature riseswhich then increases the reaction rate; it then decreases be-cause of the accelerated consumption of the reactants. Thisresult is consistent with a physical understanding of theprocess.

I/P Transducer

Mixer / Jet Pump

I/O Board (DT2805 series)

ControllerSplitterComputerT

Temperaturemeasurement

SteamTrap

Fig. 10. Experimental set up of a CST.

_1 _0.8 _0.6 _0.4 _0.2 0 0.2 0.4 0.6 0.8 1_1

u1(steam)u2(water)

Fig. 11. Parametric control.

4.3. Real-time application to a CST

A more stringent test of the grey-box GMC controller is toapply it to a real-time chemical process, such as a continuousstirred tank (CST). Katende and Jutan (1993) also used thisprocess to test an adaptive self-tuning PID (STPID) and gen-eralized minimum variance (GMV) (Clarke & Gawthrop,1975) controller.

4.3.1. Experimental set upThe schematic diagram of this lab-scale CST is shown in

Fig. 10. The CST is a 2 l reactor Llled with water, which isheated or cooled by Vuid Vowing through the jacket. Steam

and cold water are mixed in a jet pumper mixer and Vowsthrough the jacket and enables control of the temperature in-side the CST. Since there are two manipulated variables, i.e.,signals to the steam and water valves and only one processoutput, parametric control (Jutan & Uppal, 1984) is usedto combine the two manipulated variables into one, using apiecewise linear transformation. A typical transformation isshown in Fig. 11 . The control range of the steam and watervalves is normalized to lie between −1 and 1, correspond-ing to fully closed and open positions. The control signalz is mapped to two values using the two lines in Fig. 11,e.g., at full heating (z= 1) the steam valve is fully open andwater valve is 20% open.

0 0.5 1 1.5 2 2.5

time(s)

set_point process output

Fig. 12. CST temperature under GMC controller with linear model.

0 0.5 1 1.5 2 2.5 3

time(s)

Fig. 13. Control variable z using GMC controller with linear model.

4.3.2. Experimental runA linear model was used as the approximate model

in the grey-box structure. To identify the linear model,

a series of PRBS signal was used to excite the CSTand a set of data was collected with a samplingtime of 100 s. The following linear model was

0 0.5 1 1.5 2 2.5 3

time(s)

set_point process output

Fig. 14. Controlled CST temperature proLle using grey-box model.

0 0.5 1 1.5 2 2.5 3

time(s)

Fig. 15. Control variable z using grey-box model.

0 0.5 1 1.5 2 2.5 3

time(s)

CST temperatureset_point

Fig. 16. Controlled CST temperature using self-tuning PID controller.

obtained

y(t) =6:0067q−1 + 1:9047q−2

1 − 0:6832q−1 u(t − 1): (26)

This linear model approximates the CST reasonablywell within a limited range around the operating pointof 40◦C. The CST was operated in a range from 30◦Cto 70◦C which is well beyond the valid region of thelinear model. This can be seen clearly in the test runabove.

First, only the linear model given in Eq. (26) was usedin the GMC controller. The controlled reactor temperatureis shown in Fig. 12 and the corresponding control variableis given in Fig. 13. It can be clearly seen that the controllerloses its ability gradually as the process moves away fromthe 40◦C operating point. The controller is able to controlthe temperature of the reactor at 40◦C and 50◦C with fairlygood performance. Its performance deteriorates when theset-point is 60◦C and it loses control of the temperature at70◦C at which point severe oscillations occur as shown inFig. 12. This conLrms the severe nonlinearities found in thisprocess (Katende & Jutan, 1993).

In the next experimental run, the grey-box model is usedwith the GMC controller to control the CST temperature.The neural network in the grey-box model contains Lveneurons in one hidden layer and it is trained 5 times duringeach sampling interval. Fig. 14 shows the proLle of the

controlled temperature and Fig. 15 shows the control vari-able z. As in the previous run, using only the linear model,the grey-box GMC controller is also able to track the set-point change of the reactor temperature at 40◦C and 50◦Cfairly quickly except for a high spike at the beginning of theexperiment caused by the choice of random weights at thestart of the neural network adaptation. Here, we can con-trol reactor temperature at higher set-point values (60◦C and70◦C), showing that the grey-box model can compensate forthe model mismatch when the operating point moves awayfrom the valid region of the linear approximate model. Be-cause of the simple neural network used in this grey-boxmodel, its capability to model all the nonlinearities in theCST is limited. This can be seen from the higher overshootof the controlled reactor temperature when the set-point is70◦C compared to 60◦C.

Katende and Jutan (1993) used a STPID controller andGMV controller to control the same CST process. A changein set point from 30◦C to 62◦C and back down to 40◦C wasused to test these two algorithms. Fig. 16 gives the con-trol result of the STPID controller and Fig. 17 shows theresult using a GMV controller. Both algorithms can con-trol the reactor temperature at 30◦C and 40◦C fairly well.When the set point changes from 30◦C to 62◦C, both con-trollers sustain a large overshoot of 20◦C and the STPIDcontroller has an oscillatory behavior which dies out veryslowly. The grey-box GMC controller shows an overshootof 4◦C at setpoint 60◦C and an overshoot of 10◦C when the

0 0.5 1 1.5 2 2.5 3

time(s)

CST temperatureset_point

Fig. 17. Controlled CST temperature using GMV controller.

set point is at the high value of 70◦C. The grey-box GMCcontroller is thus able to outperform the STPID and GMVcontrollers.

5. Conclusions

A grey-box model based control strategy using GMC al-gorithm is developed and successfully applied to a simulatedexothermic batch reactor and a real-time CST process.

Using a parallel structure for the grey-box model, thebroad process dynamics are Lrst modelled using an approxi-mate model which is either a Lrst-principle model or an iden-tiLed linear model. The neural network component is thenused to compensate for any model mismatch. Because of areduced load this neural model can be trained more quicklyand can be simpler in structure. The control of a simulatedexothermic batch reactor shows that the grey-box model canalso be used to estimate heat release inside the reactor.

The grey-box based GMC controller is applied to the con-trol the real-time CST process with improved performance.The grey-box model based GMC controller is shown to out-perform a STPID controller and a GMV controller used inprevious studies.

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