recursive data-based prediction and control of product quality for a pmma batch process

Chemical Engineering Science 58 (2003) 3215–3221www.elsevier.com/locate/ces

Recursive data-based prediction and control of product qualityfor a PMMA batch process

Yangdong Pana, Jay H. Leeb;∗

aSchool of Chemical Engineering, Purdue University, West Lafayette, IN 47907-1283, USAbSchool of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA

Received 16 May 2002; received in revised form 24 January 2003; accepted 25 February 2003

Abstract

In many batch processes, frequent process/feedstock disturbances and unavailability of direct on-line quality measurements make it verydi5cult to achieve tight control of product quality. Motivated by this, we present a simple data-based method in which measurementsof other process variables are related to end product quality using a historical data base. The developed correlation model is used tomake on-line predictions of end quality, which can serve as a basis for adjusting the batch condition/time so that desired product qualitymay be achieved. This strategy is applied to a methyl methacrylate (MMA) polymerization process. Important end quality variables, theweight average molecular weight and the polydispersity, are predicted recursively based on the measurements of reactor cooling rate.Subsequently, a shrinking-horizon model predictive control approach is used to manipulate the reaction temperature. The results in thisstudy show promise for the proposed inferential control method.? 2003 Published by Elsevier Ltd.

Keywords: Data-based control; Quality control; Recursive prediction; Batch reactor control; MMA polymerization

1. Introduction

Batch operation is often the preferred choice for producinglow-volume/high value-added materials, including certainpolymers, <ne chemicals, bio-chemicals, semi-conductors,etc. In order to derive the maximum bene<t from batch op-erations, product quality must be controlled tightly. Tradi-tionally, engineers have attempted to achieve this indirectly,that is, by controlling some secondary variables, such astemperatures, pressures, and >owrates, in the hope that con-sistent operating condition would lead to consistent prod-uct quality. While this approach is undoubtedly helpful foreliminating some of the variabilities, it is not entirely ef-fective when the properties of feedstock vary signi<cantlyfrom batch to batch. In such cases, maintaining operatingvariables over <xed trajectories does not render consistentproduct qualities in general.To combat the problem, the aforementioned strategy

is typically complemented by a statistical process con-trol (SPC) method that monitors the product quality

∗ Corresponding author. Tel.: +1-404-385-2148;fax: +1-404-894-2866.

E-mail address: [email protected] (J. H. Lee).

measurements made in the laboratory to see if any statis-tically signi<cant change has occurred to the process. Ofcourse, the “after-the-fact” nature of the analysis means lim-ited eGectiveness: It is helpful only in dealing with dis-turbances of signi<cant magnitudes and sustained nature.Nomikos and MacGregor (1994) introduced a new SPCtechnique for batch and semi-batch processes based on theconcept of multi-way PCA. In their paper, on-line mea-surements of process variables are used to detect variousabnormalities, thus removing the major drawback of thetraditional SPC approach. However, there can still be sig-ni<cant quality variations, even in the absence of abnor-malities. Therefore, it is desired to predict and control endquality in order to reduce the natural variability seen in nor-mal batches.In order to build an empirical regression model for on-line

quality prediction, certain problems associated with the na-ture of the batch data must be addressed. The primary con-cern is related to the conditioning problems resulted fromthe extremely large number of highly correlated variablesinvolved in building data-based regression model. Anotherconcern is to deal with the missing measurement whenthe <nal quality is predicted during the batch. Multivari-ate statistical modeling techniques like principle component

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3216 Y. Pan, J. H. Lee / Chemical Engineering Science 58 (2003) 3215–3221

regression (PCR) and partial least squares (PLS) are oftenused for this purpose. Nomikos and MacGregor (1995) in-troduced multi-way PLS technique for batch and semi-batchprocesses. In their paper, the missing measurements areregenerated either based on some ad hoc assumptions oraccording to statistical correlations.The area of batch quality control is less developed than

monitoring. Although many researchers (Kaspar & Ray,1993; Piovoso & Kosanovich, 1994; Chen & McAvoy,1996; Chen, McAvoy, & Piovoso, 1998) have used mul-tivariate statistical modeling techniques in the context ofquality control, most of them were concerned with con-tinuous processes and the techniques cannot be applieddirectly to batch processes. Dong, McAvoy, and Za<riou(1996) used nonlinear neural network PLS models forbatch-to-batch optimization of input pro<les. In an eGort tomake more frequent controls within the batch, Yabuki andMacGregor (1997) suggest to use on-line as well as infre-quent oG-line measurements at some mid-course point of abatch to predict the <nal product quality. If the predictedquality deviates signi<cantly from the target, the correctionmove needed is then calculated and implemented. Josephand Hanratty (1993) also proposed a similar approach thatrelies on neural network models. However, the assumptionof known initial conditions restricts the applicability of thisprocedure.Most recently, Russell, Kesavan, Lee, and Ogunnaike

(1998) generalized these approaches to develop a method topredict and control the end quality on a recursive basis. Thesuggested approach use a single-regression model to makerecursive prediction of <nal product quality at any pointduring the batch using only on-line measurements availableup to that point. Unlike previous approaches that treatedmissing measurements in an ad hoc manner, the Kalman <l-ter theory is used to produce statistically optimal recursivequality predictions.In this paper, we adopt the basic method by Russell

et al. (1998) and tailor it to control the product qualitiesfor a PMMA process. Relevant secondary measurementsare identi<ed and their eGectiveness in predicting the <nalproduct qualities in the presence of disturbances and noisesare examined. Finally, the bene<ts from manipulating thereaction temperature based on the on-line predictions aredemonstrated.

2. Data-based procedure introduction

The main di5culty in quality control for batch processesstems from the fact that the operator has to rely upon sec-ondary measurements to monitor and control the processsince the measurements of end product quality do not be-come available until the batch is <nished. Therefore, forachieving tight quality control, the main issue is how to de-velop a model that enables prediction of end quality basedon the input and output measurements available during the

batch. The end-quality prediction is to be formed as

qtf|i = f(Yi ;U); (1)

qtf|i represents the end-quality predicted using Yi, the mea-surements, collected up to the ith sample time) as well asU,the input trajectory for the entire batch. tf is the total num-ber of sample time steps during a whole batch, and thereforerepresents the batch cycle time. The challenge is to identifythis relationship from existing data.To simplify the identi<cation task, one may choose to

develop a linear predictor that corresponds to the linearizedform of the above general predictor with respect to some“nominal” (mean or reference) trajectories Y0

i , U0:

q′tf|i = AY′i + BU

′; (2)

where Y′i =Yi −Y0

i , etc. From here on, the superscript {}′will be suppressed for the convenience of exposition.In order to build the suggested empirical model for on-line

quality prediction, certain problems associated with the na-ture of batch processes must be addressed. First, to detectquality deviations and make necessary corrections in time,we need to be able to predict the end-quality accuratelybased on partial batch information. Hence, the predictionsare developed over the progression of a batch, preferablyat a number of sample steps located in diGerent phases ofthe operation. This may require several models to be devel-oped, one for each sample time at which the prediction isdesired. Alternatively, one may develop a single model thatuses all the measurements collected during a whole batch,but in order to use this model in the middle of a batch, onemust somehow <ll in the missing future measurements.This problem was addressed in Russell et al. (1998). In

their work, a statistically optimal linear recursive predictorof qtf is formulated based on a Kalman <lter with stateZ=[YT qTtf ]

T. The states of the Kalman <lter consist of allthe on-line measurements collected during a batch from t=0to t = tf. The state-space model used for the <lter design is

Zi+1 =Zi ; (3)

yi = [0 · · · 0 Iny 0 · · · 0]︸︷︷︸Ci

Zi ; (4)

Ci is a time-varying matrix that picks out yi, the measure-ments collected at the ith sample step, fromZi. The Kalman<lter equations are given by

Zi+1|i+1 = Zi|i + Ki+1

[yi+1 − Ci+1Zi|i

]; (5)

Ki+1 = Pi|iCTi+1(Ci+1Pi|iCT

i+1)−1; (6)

Pi+1|i+1 = (I − Ki+1Ci+1)Pi|i : (7)

This Kalman <lter is to be initialized with the covariancematrix calculated based on historical batch data (P0|0 =1=N

∑Ni=1 Z(i)ZT(i), where N is the number of batches

used for modeling). As new measurements become avail-able, the predictions of the quality variables are recursively

Y. Pan, J. H. Lee / Chemical Engineering Science 58 (2003) 3215–3221 3217

upgraded based on the correlation between the measuredstates and the quality variables (given by the covariancematrix). Note that the quality predictions are simply the lastnq elements of the <lter state estimate:

qtf|i = [ 0 · · · 0 Inq ]Zi|i : (8)

Having developed the recursive predictor for the purelystochastic case, the eGects of deterministic inputs can beadded in a straightforward manner. First, the input eGectson the measurement and end quality can be modeled asZ=BuU+E or more generally Z=f(U)+E. The deter-ministic model BuU (or f(U) may come from an alreadyavailable empirical model or by performing identi<cationexperiments (if the historical data do not contain su5cientinput adjustments for the model identi<cation). Then theinput eGects are subtracted from the original data for Z,leaving the residual data, Z = Z − BuU, that contain thecorrelation between Y and qtf that is irrespective of the de-terministic input moves. The Kalman <lter is then formu-lated just as before based on the residual state Z.Another di5culty in developing the prediction model

from historical data stems from the large dimension of in-puts and outputs and strong correlation that exist amongthem. This not only results in a high demand for compu-tation and storage, but also makes the initial covariancematrix P0|0 ill-conditioned and very di5cult to identify. Toovercome both the high dimensionality and the condition-ing problem, multivariate statistical methods such as theprincipal component analysis (PCA) can be used to projectthe <lter state Z onto a subspace of much lower dimensionto create reduced-order state Z. For the reduced-order <lterdesign, the model is therefore given by

Zi+1 = Z i ; (9)

yi = C iZ i + [b 0 · · · b i]

u0

...

ui

+ �i: (10)

Note that residual error term � has been included in the mea-surement equation. This modi<cation is necessary, becausesome information is inevitably lost when projecting the statevector down to the lower-dimensional space. A Kalman <l-ter can be designed straightforwardly based on the abovemodel. The quality prediction is given by

qtf|i = [ 0 Inq ]

ˆZ i|i + [b 0 · · · b tf−1]

u0

...

utf−1

:

(11)

From a practical standpoint, it is convenient to divide thetotal batch cycle time into several diGerent phases and thenapply the control algorithm at the beginning of each phase.In addition, the dimension of control input can be reduced

through some appropriate parameterization. That is, withineach interval, the input trajectory may be parameterized withonly a few parameters.Having developed the on-line quality prediction, the fu-

ture inputs calculation can now be formulated as a quadraticoptimization problem,

min�+iqTtf|i�qqtf|i + [U+

i (�+i )]

T�uU+i (�

+i ); (12)

whereU+i contain the future input values (with respect to the

ith sample step) and �+i represent the associated parameters.�q and �u are weighting matrices. Although all the futureinput moves are calculated, only the moves for the presentphase are implemented and the rest are recomputed at thebeginning of next phase. The number of input parameterscalculated decreases as the batch proceeds, giving rise to thename “shrinking horizon control”.

3. MMA process introduction

Methyl methacrylate (MMA) is an important industrialmaterial. Modeling and control of MMA batch polymer-ization system has received much attention. Soroush andKravaris (1992) designed a nonlinear controller to tracka pre-determined optimal temperature pro<le in order toachieve the desired value of weight-average molecularweight. Peterson, Hernandez, Arkun, and Schork (1992)developed a nonlinear DMC algorithm to control the reactortemperature. More recently, Crowley and Choi (1996) usedthe extended Kalman <lter (EKF) algorithm to estimate theeGective reactor wall heat transfer coe5cient, which is usedto bring the reactor temperature to its target as rapidly aspossible with minimal overshoot. All these work focus onthe problem of tracking a pre-assigned temperature pro<leand hence the control of end-quality could suGer when initialcharge disturbances occur. Ellis, Taylor, and Jensen (1994)implemented on-line estimation and control of molecularweight distribution on an experimental MMA solution poly-merization system, based on a two-time scale EKF. Butthe fundamental model needed to design the EKF is gen-erally very di5cult and expensive to develop in practice.The data-based procedure adopted in this paper oGers theadvantage of easy implementation and should, therefore, bequite attractive from the viewpoint of industrial application.MMA polymerization is a typical free radical chain

growth polymerization. It is also a diGusion-controlled re-action, which exhibits gel eGects and glass eGects near thehigh conversion stage of batch. In this study, the modelgenerated by Seth and Gupta (1996) is selected for simula-tion. This model overcomes some important shortcomingsof other models that previously existed and has alreadybeen tested and found to be suitable for an industrial use.Solution polymerization with initial volume fraction of

solvent f0s=0:3 is chosen for this study. The nominal opera-

tion is isothermal with reaction temperature, T =70◦C. Therelevant product quality in this process is characterized by


the weight-average molecular weight (Mw) and the polydis-persity (Pd), which are critical for end-use polymer charac-teristics. The control objective is to drive the batch processsuch that products of desired end-quality, Mw = 400; 000,and Pd = 2:6, be obtained.In addition to the lack of on-line quality measurements,

the quality control problem is further complicated by thepresence of frequent feed disturbances. The two major feeddisturbances are initiator charge >uctuation and monomerimpurity. The initiator charge >uctuation is assumed to bein the range of ±10%, while the monomer impurity is sim-ulated as the existence of inhibitors. To simulate the eGectof residual inhibitor on the polymerization, the initiator e5-ciency was assumed to be reduced to 0.1 for the <rst 20 min.After that, the inhibitor was assumed to have been consumedin reaction with the initiator, and the initiator e5ciency wasset back to 0.5 for the rest of the polymerization.Although measurements of Mw and Pd are made only

after the completion of batch, on-line measurements of var-ious secondary process variables are available for controlpurposes. In this study, the cooling rate needed to maintain aconstant temperature pro<le, is measured every minute andused for prediction of end quality. When a disturbance oc-currs, the heat generated from the polymerization reactionchanges and results in a deviation of the cooling rate pro<lefrom the nominal trajectory. This information was found tobe strongly correlated with the deviation of end-quality vari-ables and, therefore, useful for predicting the end-quality forcontrol purposes.In this paper, we also assumed that the reaction tempera-

ture was controlled perfectly through a lower-level temper-ature loop. So the reaction temperature was treated as aninput variable that can bemanipulated for control purposes—within some constraints. The batch cycle time (6 h) was

Fig. 1. Measurement/input pro<les and recursive predictions of end qualities for a typical batch.

divided into <ve intervals, which begin at 0, 50, 100, 150,and 250 min marks, respectively. The reactor temperaturewas assumed to be held constant within each interval. Thedata-based control procedure introduced earlier was used tocalculate the reactor temperatures during these intervals.

4. Results

The very <rst step in applying the data-driven control pro-cedure is generation of model building data. It is important torecognize that the prediction capabilities of data-based mod-els are heavily dependent upon the nature of data suppliedfor building the model. The batch data should contain theeGects of various disturbances as well as the potential inputmoves one may make for the purpose of control. This willallow the correlation among the disturbances, input moves,measurement process outputs, and <nal product quality to becaptured in the model. In our simulation study, 60 batchesdata were generated by randomly changing the initial chargeand reactor temperature values during each interval.Having generated the model-building data, we next pro-

ceeded to pre-process the data so that they are in a suitableform for model building. First, the nominal operation data,which represent the mean values of various measurementtrajectories, are subtracted from the raw data, to create datafor the deviation variables. Second, we scaled the data sothat every variable has the same variance. This is necessarysince various input and output variables are available in dif-ferent units and, therefore, their variations are typically ofdrastically diGerent magnitudes. Without the scaling, the re-gression model would put more weight on the variables withlarger variations. However, for a <xed variable, the samescaling must be applied to all samples. If diGerent scaling is


predictionactual

0 2 4 6 8 10 12 14 16 18 20−2

−1

0

1

2

3

4

Batch #

Mw

(sc

aled

val

ue)

Fig. 2. Prediction results for Mw at the end of batch for the 20 tested batches.

predictionactual

0 2 4 6 8 10 12 14 16 18 20−3

−2

−1

0

1

2

3

4

Batch #

Mw

(sc

aled

val

ue)

Recursive prediction of Mw at t=100 min

Fig. 3. Prediction results for Mw at t = 100 min for the 20 tested batches.

used for samples at diGerent time instants, it would be equiv-alent to assuming noise of diGerent magnitudes for diGerenttimes.The <rst objective of the simulation study was to assess

the prediction capability of the proposed recursive method.For this, another 20 batches with randomly varying distur-bances were simulated. Fig. 1 shows the results for onetested batch. Fig. 1c–d show the trajectories for the measuredvariable (cooling rate) and manipulated variable (reactor

temperature), respectively. The peaks in Fig. 1c indicate theeGects of inputs (setpoint changes for the reaction temper-ature) at beginning of each control interval. Fig. 1a and bgive an indication of how the prediction of the end quali-ties from the recursive <lter evolves throughout the batch asmore and more measurements become available. As can beseen from the <gures, the end-quality predictions eventuallyconverge to the actual values as more and more on-line mea-surements are taken in. The predictions of weight average


Fig. 4. Prediction results for Mw for the 20 batches in the presence of strong measurement noises.

0 2 4 6 8 10 12 14 16 18 203.6

3.7

3.8

3.9

4

4.1

4.2

4.3

4.4

4.5

4.6x 10

5

batch #

Mw

comparison of data-based control and isothermal operation

Fig. 5. Comparison between operation with the data-based control and isothermal operation: dash line—isothermal operation, solid line—data-based control.

molecular weight for the entire 20 tested batches are shownin Fig. 2. These predictions are those given by the recursive<lter at the end of the batches. From this, we conclude thatthe prediction capability of the data-based correlation modelis quite satisfactory.Since an accurate prediction of the end-quality at an early

stage of reaction is the key to making the corrective adjust-ments needed to suppress the disturbance eGects, the predic-tion capability was also tested at various mid-points duringa batch. Fig. 3 plots the predictions at the 100 min mark forthe same 20 batches. The predictions are still very good.

Since the recursive prediction adopted in this paper isbased on on-line measurements, the accuracy of measure-ments should certainly aGect the prediction capability.Several levels of measurement noises were added to testthe sensitivity of the prediction. One tested case with anextremely high level of noises is illustrated in Fig. 4. InFig. 4, we can see that the noise was so large that the inputeGect could not be discriminated. Despite this, the predic-tions for the 20 batches still follow the right trend. Thisdemonstrates the robustness of the recursive method withrespect to measurement noises.


Table 1Comparison between operation with the data-based control and isothermaloperation

Deviation Min Max Mean

Data-based control 5.970e2 1.1840e4 4.019e3Isothermal 2.2297e3 5.1782e4 3.2971e4

The second objective of this simulation study was toassess the performance of the quality control strategy. Atbeginning of each interval, future input parameters were cal-culated according to Eq. (12). To test the eGectiveness ofthe control strategy, another 20 batches with randomly vary-ing disturbances were simulated. The comparison betweenthe operation with the data-based control and the conven-tional isothermal operation is shown in Fig. 5 and Table 1.It shows that much tighter quality control can be achievedwith the data-based control approach.

5. Conclusion

In this paper, a general data-based approach developedin Russell et al. (1998) is examined and applied to an in-dustrially relevant polymerization process, a PMMA batchreactor. A main advantage of the approach is that the mod-eling process requires only a historical data base for on-lineprocess measurements and relevant end-quality. Other ad-vantages include the conceptual simplicity and low on-linecomputational demand. However, it is shown to be im-portant to use an appropriate model-building data set (thatinclude the eGects of various disturbances as well as poten-tial control moves) for the success of this strategy. For thePMMA process, the prediction capability of the recursivemethod was found to be robust with respect to signi<cantmeasurement noises. The ultimate bene<t of the on-line pre-diction was the reduction in the eGect of batch-to-batch feedvariation on the <nal product quality.

Acknowledgements

The Authors gratefully acknowledge the <nancial supportfrom the National Science Foundation (CTS-#0096326).

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recursive data-based prediction and control of product quality for a pmma batch process

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