Statistical Prediction of Product Quality in Batch Processes with

Article pubs.acs.org/IECR

Statistical Prediction of Product Quality in Batch Processes with Limited Batch-Cycle Data Zhiqiang Ge,*,† Zhihuan Song,† and Furong Gao‡,§ †

State Key Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou 310027, Zhejiang, P. R. China ‡ Department of Chemical and Biomolecular Engineering and §Center for Polymer Processing and Systems, Fok Ying Tung Graduate School, The Hong Kong University of Science and Technology, Hong Kong, China ABSTRACT: This paper develops a modeling approach to address the end-of-batch product quality prediction problem for batch processes with limited batch-cycle data. Generally, those batch processes that have multiple phases are the focus of the present paper. Different from the traditional multiway/phase-based partial least-squares (PLS) method, which unfolds the threeway data set through the batch direction, the proposed method unfolds the data set through the variable direction, in order to generate more training data samples. Reproducing the product quality data with the noise injection method allows a statistical model to be developed in each phase of the batch process. This, however, does not remove the nonlinearity of the batch process data, as practically addressed by the typical batch normalization. Therefore, a nonlinear regression model is subsequently introduced to handle this problem for product quality prediction modeling. To compare the performance of linear and nonlinear statistical models, phase-based PLS and relevance vector machine models have both been developed for prediction of product quality in an industrial injection molding process.

1. INTRODUCTION

Generally, with the three-way batch data, which corresponding to the batch, variable, and time directions, the typical statistical modeling approach first unfolds the data through the batch direction, to result in a two-dimensional data set. In this case, each batch is considered as a training sample in the multiway PLS model; thus, the number of modeling data samples is equal to the number of batches, which is assumed to be limited in the present paper. However, if we could unfold the three-way process data set through the variable direction, each of the sampling intervals during the batch is treated as a training data sample in the two-dimensional data set. As a result, more training data samples will be generated. However, since each batch has only one quality data, it is infeasible to implement the statistical modeling method with unequal numbers of input and output data samples. Besides, without the batch normalization step (which is always carried out when the three-way data set is unfolded through the batch direction), a significant nonlinearity of the process data may be exhibited.13 Therefore, a linear PLS model may be not sufficient to characterize the relationship between process variables. Recently, for batch process monitoring wirh limited reference batches, Lu et al.14 has proposed a moving-window-based method, in which the local covariance structure of the data was extracted for phase division and the monitoring model was developed in each phase. To address the same problem, Zhao et al.15 developed an adaptive monitoring scheme for batch processes, which was based on the independent component analysis method. However, neither of them has considered the

For producing low-volume and high-value added products, batch and semibatch processes have played an important role in modern industrial manufacturing. To keep the quality of the final product inside a predefined specification zone, product quality monitoring and control is needed, which have already received great attention. However, for most batch processes, online product quality control is difficult to implement, which is mainly due to the lack of online measurements of quality variables. Therefore, in the past years, significant efforts have been made for the development of quality prediction methods, among which the multivariate statistical analysis method has received particular interest. Since the work of Nomikas and MacGregor,1 a large number of research papers have been published on the statistical method for monitoring and predicting product quality in batch processes.2−12 Most existing methods have assumed that the reference batches are sufficient for modeling. While it is easy to run trials for certain batch processes with short batch duration and inexpensive precusors, for some other batch processes it may not possible to collect sufficient batches for modeling, due to long batch duration or the expense of running trials. Besides, when the operation condition is changed frequently in some batch processes, the quality prediction model should also be updated or rebuilt in time. In those cases, the statistical product quality prediction model needs to be developed on the basis of a limited number of batch data. In such a case, a problem may arise with the traditional methods, such as multiway partial least-squares (PLS) and stage-PLS, that are used directly, that is, with only limited modeling data (for example, less than the variable number), the developed model may be overfitted and have a generalization performance problem. © 2012 American Chemical Society

Received: Revised: Accepted: Published: 11409

November 8, 2011 July 7, 2012 July 19, 2012 July 19, 2012 dx.doi.org/10.1021/ie202554r | Ind. Eng. Chem. Res. 2012, 51, 11409−11416

Industrial & Engineering Chemistry Research

Article

Figure 1. Illustration of the data unfolding and noise injection methods.

compared in the case study part. The following subsections give detailed descriptions of these data treatments and modeling procedures. 2.1. Data Unfolding and Noise Injection. Given a threeway batch process data set X(I × J × K), it is first unfolded into a two-dimensional data matrix X(KI × J); suppose the whole batch process is divided into S phases, then X(KI × J) can be further partitioned into

quality prediction problem; i.e., the quality information has not been incorporated for modeling, and the nonlinearity of the process data also remains unexplored. In the present paper, we intend to build the statistical model for end-of-batch product quality prediction based on limited training batches for multiphase batch processes. First, the threeway batch data set X(I × J × K) is unfolded through the variable direction X(KI × J), where I is the number of batches, J is the numbers of process variables, and the duration of each batch is K. In the second step, each quality data is copied K times, in order to provide the same number of the process data samples KI. To obtain a reasonable diversity between different quality data, a noise injection method is used. Then, the twodimensional data sets X(KI × J) and Y(KI × Jy) are divided into several sub data sets, corresponding to different phases of the batch process. Finally, a phase-based statistical prediction model (linear and nonlinear) is constructed for each of the sub data sets. An illustration of the developed method will be shown later in Figure 1. The remainder of this paper is structured as follows. A detailed description of the method is provided in section 2, including the noise injection method, the phase-based PLS model, and the nonlinear phase-based model, and performance evaluation and further discussions of the proposed method are included. In section 3, an industrial case study of the injection molding process is provided to show the feasibility and efficiency of the proposed method. Finally, conclusions are made.

X(KI × J ) = [X1(K1I × J )X 2(K 2I × J )...XS(KSI × J )] (1)

where K1, K2, ..., KS are the numbers of time intervals in different phases. Correspondingly, the quality data set Y(I × Jy) is copied K times, where Jy is the number of quality variables, in order to generate the new quality data set Y(KI × Jy), which has the same row dimension as X. In order to generate a reasonable diversity between different data samples, the noise injection method is used. Much experimental evidence has been explored in the past years supporting that noise injection into the original training data set can improve the generalization performance of the model.19,20 An important issue of the noise injection method is how to determine the noise level. While a high noise level may produce garbage data, a too low level noise may not be able to introduce diversity into the copied data set. In this paper, the simple Gaussian noise distribution with zero mean and variance σ2 is added to the copied data set and given as follows ⎡ Y(I × J )⎤ y ⎥ ⎢ ⎢ Y(I × J )⎥ y ⎥ Ỹ (KI × Jy ) = ⎢ + N (0, σ 2) ⎢ ⎥ ⋮ ⎢ ⎥ ⎢ Y(I × J )⎥ y ⎦ ⎣ KI

2. METHODOLOGY To divide the batch process into several phases, many phase division methods have been developed, such as expert knowledge based methods, process analysis techniques, and data clustering based method.8,16−18 In the present paper, however, it is assumed that the batch process has already been divided into several phases. In each phase, the data is first cumulated through the variable direction, and the quality variable is coped with a noise injection step. On the basis of the new modeling data set in each phase, a phase-based statistical model can be developed. Here, two kinds of models (linear and nonlinear) are used and the performance of which will be

(2)

where N(0,σ2) is a Gaussian distribution data set with appropriate dimensions. In most cases, a small value of σ2 is sufficient to provide a reasonable diversity among different data. Similarly, the noise injected data set Ỹ (KI × Jy) can be divided into different phases 11410

dx.doi.org/10.1021/ie202554r | Ind. Eng. Chem. Res. 2012, 51, 11409−11416


Article

Ỹ (KI × Jy ) = [Y1̃ (K1I × J )Y2̃ (K 2I × J )...YS̃ (KSI × J )]

By introducing a prior distribution over each parameter weight, RVM incorporates the Bayesian method into the kernel model structure. If the training data set in each phase is denoted as {x,ys}i,i=1,2,...,KsI, the relationship between the process and quality variables is given as follows

(3)

On the basis of the data sets {Xs(KsI × J),Ys(KsI × J)}s=1,2,...,S, the statistical model can be developed in each phase for product quality prediction purpose. A PLS-based linear model and a relevance vector machine (RVM) based nonlinear model are developed respectively in the next two subsections, the performance of which will be compared in section 3. An illustration of the data unfolding and noise injection method is provided in Figure 1. 2.2. Phase-Based PLS Model. On the basis of the data set of each phase, a corresponding PLS model can be developed as follows

ys = f (x , ws) + e

where s = 1, 2, ..., S and e is the random error, which is assumed to be an independent zero-mean Gaussian distributed with variance σe2; thus, e ∼ N(0,σe2). On the basis of the kernel algorithm, the nonlinear function f(x) can be expressed as a linearly weighted sum of different basis functions, given as K sI

f (x , ws) =

T X s(KsI × J ) = TP s s + Es

where ws = [ws,0,ws,1,ws,2,...,ws,KsI] is the weighted parameter vector of the basis functions ψs(x) = [1,K(x,x1),K(x,x2),...,K(x,xKsI)]T. On the basis of the prior probability of the weighted parameter25

(4)

where s = 1, 2, ..., S; PsT ∈ RJ×As and Qs are the loading matrices; Ts ∈ RKsI×As is the principal component matrix; Es and Fs are the residual matrices; As is the retained number of principal components; Ws is the weighting matrix; Rs is the regression matrix for the corresponding PLS model, and ∑Ss = 1Ks = K. For product quality prediction of a new batch at some specific time interval, representing the data sample as xnewk, where k = 1, 2, ..., K, the prediction results can be calculated as follows

ynew ̂ k

(7)

T

Ys̃ (KsI × J ) = TQ s s + Fs

⎧ R Tx k , 1 ≤ k ≤ K 1 ⎪ 1 new ⎪ R Tx k , K + 1 ≤ k ≤ K + K 1 1 2 ⎪ 2 new =⎨ ⋮ ⎪ S−1 S ⎪ T k R x , K + 1 ≤ k ≤ ∑ Ks ⎪ ⎪ S new ∑ s ⎩ s=1 s=1

∑ ws ,iK (x, x i) + ws ,0 = wsT ψs(x) i=1

T

R s = Ws(PsT Ws)−1Q sT

(6)

K sI

p(ws|αs) =

∏ N(ws ,i|0, αs ,i−1) i=0

=

⎛ α w 2⎞ s,i s,i ⎟ 2 ⎟⎠ ⎝

K sI

1

∏ αs ,i1/2 exp⎜⎜−

(2π )(KsI + 1)/2

i=0

(8)

where αs is the hyperparameter set of the distribution function. The RVM model can be developed by maximizing the following likelihood function max L[ys(αs , σe 2)] = max log{p(ys|X, ws , σe 2)}

{∫ p(y |X, w , σ ) p(w |α ) dw } 2

= max log

s

s

e

s

s

s

(9)

where ys = (y1,y2,...,yKsI), X = (x1,x2,...,xKsI). The probability of each quality data is given as follows:

(5)

2.3. Phase-Based RVM Model. As mentioned in the Introduction , without the batch normalization step, the nonlinearity of the batch process may remain in the unfolded data set. In this case, a nonlinear model can be used to improve the modeling efficiency for product quality prediction. Conventionally, there are nonlinear regression modeling methods, such as nonlinear PLS, artificial neural network (ANN), and support vector regression (SVR).21−24 Compared to the nonlinear PLS- and ANN-based regression methods, SVR seems to have better generalization performance for new data samples. However, as has been reported, SVR has some significant drawbacks. First, SVR is subjected to the Mercer condition and thus cannot use arbitrary kernel function in the model. Second, SVM needs to determine the trade-off and insensitivity parameters, which may need a cross-validation step. Recently, another kernel learning method named relevance vector machine (RVM) has been proposed,25 which shares a similar model structure with SVR, but has no type limitation of the kernel function. Another advantage of the RVM model is that RVR has a sparser model while maintaining comparable generalization error compared to SVR, which leads to a faster prediction step on test data samples. Therefore, the RVM method is used for nonlinear quality prediction modeling in the present paper.

p(ys|x , ws , σe 2) =

⎧ 1 ⎫ 1 ⎨ ⎬ exp y ( x ) w − || − ψ || s s 2 s (2πσe 2)KsI /2 ⎩ 2σe ⎭ (10)

As a result, the optimal values of σe and α can be obtained, on the basis of which the posterior distribution of the weighted parameter ws can be determined as follows, which is also Gaussian: 2

p(ws|ys , αs , σe 2) = =

p(ys|ws , σe 2) p(ws|αs) p(ys|αs , σe 2) 1 (K sI + 1)/2

(2π )

|Σs|−1/2

1 × exp − (ws − μs )T Σ−s 1(ws − μs ) 2

{

} (11)

the mean value of which is given as follows

11411

ws̅ = μs = σe−2 Σsψ T (x)ys

(12)

Σs = (σe−2 ψs T (x) ψs(x) + A s)−1

(13)



Article

where ψ s (x) = [ψ 1 (x),ψ 2 (x),...,ψ K s I (x)], A s = diag(αs,0,αs,1,...,αs,KsI). As a result, many elements of w̅ s become zero, and only a few nonzero elements are effective in the RVM model. Those data samples whose corresponding weighted parameters are nonzero are termed as “relevance vector”, similar to “support vector” in the SVR model. Therefore, each phase-based RVM model can be represented as follows:

quality prediction can be changed to the traditional form, in which case the modeling performance of the traditional method is better. Compared to the linear regression model, the nonlinear model may be more appropriate to characterize the relationship among the data. This is because by unfolding the three-way batch data matrix through the variable direction, the nonlinearity of the process data cannot be ignored. In this case, a linear model may not be sufficient to build the relationship between the process and quality variables. RVM is an efficient kernel learning method; by selecting some important training data samples (relevance vectors), it directly uses these data samples for prediction of new data samples. Since the data patter of training samples in the same phase of the batch process is similar, RVM can significantly reduce the number of relevance vectors. Therefore, only a very small portion of the training data samples is selected as the relevance vectors in each RVM model, which will greatly improve the online product quality prediction efficiency in the batch process.

RVs

f (x , ws̅ ) =

∑ ws̅ ,iK (x, x i) + ws̅ ,0 = ws̅ T ψs RV(x)

(14)

i=1

where s = 1, 2, ..., S, and RVs is the number of relevance vectors in each phase RVM model. To examine how well each of the weighted parameter has been determined, the following γ quantity can be defined:25 γs , i = 1 − αs , i Σs , ii

(15)

where i = 1, 2, ..., RVs. It is noticed that the value of γs,i is between zero and one. When αs,i has a large value, the corresponding weighted parameter ws,i is highly constrained by the priority, thus γs,i will approach zero. On the other hand, when the αs,i value is small and the γ value is close to one, the weighted parameter ws,i will fit the data well. Similarly, for product quality prediction of a new batch at a specific time interval xnewk, the prediction results can be calculated as follows:

ynew ̂ k

3. APPLICATION TO AN INJECTION MOLDING PROCESS In this section, an industrial injection molding process is used as an example for performance evaluation of the proposed method. The weight of the final product is selected as the quality variable in this process. For prediction of this quality variable, several important measured process variables are used, such as temperatures, pressures, and the screw velocity, all of which are described in Table 1. A data set that contains 55

⎧ w T ψ RV(x k), 1 ≤ k ≤ K new 1 ⎪ 1̅ 1 ⎪ T RV k w ψ (x new ), K1 + 1 ≤ k ≤ K1 + K 2 ⎪ ⎪ ̅2 2 =⎨ ⋮ ⎪ S 1 S − ⎪ ⎪ wS̅ T ψS RV(x new k), ∑ Ks + 1 ≤ k ≤ ∑ Ks ⎪ ⎩ s=1 s=1

Table 1. Select Variables for Quality Prediction

(16)

where ψs = [1,K(xnew ,x1),K(xnew ,x2),···,K(xnew ,xRVs)]T, s = 1, 2, ..., S is the basis function set based on each phase RVM model. 2.4. Performance Evaluation and Further Discussions. For performance evaluation of the proposed method, the rootmean-square error (RMSE) criterion can be used, which is defined as follows RV

(xnewk)

k

k

∑i =te 1 || yi − yî k || Ite

variables

unit

1 2 3 4 5 6

screw stroke screw velocity injection press barrel temperature 1 barrel temperature 2 temperature 1

mm mm/s bar °C °C °C

batches has been collected in the injection molding process for modeling training and testing. To simulate the case of limited batches, only five of the training batches are selected for modeling, and the remaining 50 batches are used for performance evaluation. For simplicity, the batch duration is assumed to be constant, which is 635 sampling intervals. Therefore, the three-way training and testing data sets can be represented as Xtr(5 × 6 × 635) and Xte(50 × 6 × 635). A detailed description and experimental design of the injection molding process can be found in refs 4, 26, and 27. Suppose the whole batch process has been divided into different phases (three steady phases and four transition phases), the sampling intervals of the these phases are 1−18, 19−117, 118−119, 120−367, 368−369, 370−609, and 610− 635. Therefore, after the original three-way training data set has been rearranged into the two-dimensional matrix Xtr(3175 × 6), it can be further divided into seven submatrices, which are X1tr(90 × 6), X2tr(495 × 6), X3tr(10 × 6), X4tr(1240 × 6), X5tr(10 × 6), X6tr(1200 × 6), and X7tr(130 × 6). On the basis of the noise injection method, the original quality data set ytr(5 × 1) is copied 635 times and divided into seven submatrices, which correspond to the seven phases. Here, the Gaussian noise has been used, with zero mean and variance 0.01. It is noted that

2

I

RMSE(k) =

k

no.

(17)

where k = 1, 2, ..., K; yi, i = 1, 2, ..., Ite is the measured quality variable of each testing batch; and ŷik is the predicted value of the testing batch i at time point k. The main contribution of the present work is to propose a general modeling idea for batch processes with limited modeling data. In batch processes, the limited data problem can widely exist, especially for those processes which have long durations. Another common case is that when a batch process is changed to produce a new product grade, the modeling batches will be extremely limited at the initial stage. Therefore, the question of how to provide an efficient quality prediction model by using only limited data is important and deserves research investigation. With the growth of modeling batches, the product quality prediction model can adapt or be updated periodically. When there are sufficient modeling batches, the 11412



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Figure 2. RMSE values during the whole batch duration: (a) PLS and (b) RVM.

Figure 3. End-of-batch prediction results of testing batches.

Figure 4. Product quality prediction results of testing the 10th batch: (a) PLS and (b) RVM.

the value of the noise variance should not be selected as too significant, or the main data information could be obscured. Then, for each subdata set, phase-based PLS and RVM models are developed, respectively. Through cross-validation, the number of latent variables in each of the PLS model is determined as 4, and the kernel parameter of the RVM model is selected as 3. The calibrations of both PLS and RVM models

are made to obtain the best result of each. As a result, at the end-of-batch sampling point of the batch process, the calibrated RMSE value of the training data set by the PLS model is 0.0044, while the value provided by the RVM model is 0.0045. First, an overall performance view of the two methods is given in Figure 2, which exhibits the RMSE values during the whole batch duration. It can be seen that the quality prediction 11413



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Figure 5. Product quality prediction results of testing the 50th batch: (a) PLS and (b) RVM.

Figure 6. Relevance vectors of the phase RVM models in the (a) first transition period, (b) second transition period, (c) third transition period, and (d) fourth transition period.

In each of the phase RVM models, only a small portion of training data samples have been determined as the relevance vectors, which are used for prediction of new data samples. While the weighted parameters of other training data samples have been eliminated, it has a significant value for each relevance vector. To examine the detailed information of relevance vectors and their corresponding weighted parameters values, Figure 6 shows the results of the phase RVM models of four transition phases in this process. As can be seen, the selected numbers of relevance vectors in each phase RVM

performance has been significantly improved by the introduction of the nonlinear regression model. Particularly, the end-ofbatch quality prediction values of each testing batch are shown in Figure 3 for the two prediction models. It has shown that the prediction accuracy of the RVM method is much higher than that based on the PLS model. Therefore, it can be inferred that in this process the nonlinearity of the process data cannot be ignored. Detailed results of two selected testing batches are provided in Figures 4 and 5, both of which have also shown the superiority of the nonlinear method. 11414



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batch process. Therefore, the nonlinearity of the data is significant, which indicates that the incorporation of the nonlinear modeling approach is necessary.

model are 12, 2, 2, 13, with detailed sample labels provided in Table 2. The relevance vector rate of each RVM model is also Table 2. Sample Labels of Relevance Vectors in Each Phase RVM Model phase RVM RVM RVM RVM RVM

1 3 5 7

sample labels

RV rates, %

9, 18, 21, 30, 31, 38, 40, 46, 49, 59, 77, 90 2, 8 2, 5 6, 7, 13, 16, 18, 20, 30, 31, 38, 69, 92, 96, 99, 103

13.33 20.00 20.00 10.77

4. CONCLUDING REMARKS In the present paper, the product quality prediction problem of multiphase batch processes under the limited batch case has been studied. Different from existing methods, which unfold the process data set through the batch direction, the data set is unfolded through the variable direction in the proposed method. In order to generate an equal number of quality data as the process variables, the noise injection method has been introduced. Two modeling methods, PLS and RVM, have been used for linear and nonlinear quality modeling. The feasibility and efficiency of both methods have been evaluated through an industrial application case study. As a result, the nonlinear method performs much better than the linear method. In practice, with the increase of available reference batches, an adaptive form of the proposed method can be easily developed. However, the advantages and disadvantages of the multiway, phase/stage-based, and proposed methods need further investigation. For example, how many batches should be considered as a sufficient modeling data set? When does the proposed method perform better than the traditional methods or vice versa? In our opinion, it is difficult to give general guidance; thus, different methods may be appropriate for

shown in Table 2. Hence, only a very small part of the training data samples have been selected for quality prediction. This is reasonable because we have cumulated the process data from the same batch and copied the quality data. In this case, some overlapped data information may be generated, which will enlarge the sparseness of the RVM model. Furthermore, the calculated γ value for each relevance vector is given in Figure 7 for different phase RVM models. It can be found that most weighted parameters have been well determined, since their γ values are close to one. In contrast, by examining the data explanation rates of both input and output data in the linear PLS model, it can be found that the linear PLS model cannot efficiently extract the main data information from the batch process. Actually, data explanation rates of the linear PLS model are around 25−35% in each of the seven phases in the

Figure 7. γ values of relevance vectors of the phase RVM models in the (a) first transition period, (b) second transition period, (c) third transition period, and (d) fourth transition period. 11415



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different batch processes. However, the combination of these methods may be a possible way to enhance performance of batch processes, since different methods may complement with each other for product quality prediction.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported in part by the National Natural Science Foundation of China (NSFC) (61004134), Project National 973 (2012CB720500), Guangzhou scientific and technological project (11F11140010), Zhejiang Provincial Natural Science Foundation of China (LY12F03008), and the Fundamental Research Funds for the Central Universities.

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Statistical Prediction of Product Quality in Batch Processes with

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