PCA-Based Modeling and On-line Monitoring Strategy for Uneven

This paper extends the stage-based sub-PCA modeling method originally proposed by ... information necessary for the development of PCA monitoring mode...
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Ind. Eng. Chem. Res. 2004, 43, 3343-3352

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PROCESS DESIGN AND CONTROL PCA-Based Modeling and On-line Monitoring Strategy for Uneven-Length Batch Processes N. Lu,†,‡ F. Gao,*,† Y. Yang,† and F. Wang‡ Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, and School of Information Science and Engineering, Northeastern University, Shenyang, Liaoning (110004), P. R. China

This paper extends the stage-based sub-PCA modeling method originally proposed by the authors to the monitoring of batch processes with durations of uneven lengths. Two models for each stage are developed, one for the stage division and the other for process monitoring. The purposes of the stage division are two-fold, to enhance process understanding and to provide stage-division information necessary for the development of PCA monitoring models. With the proposed method, batch processes with durations of uneven lengths can be effectively monitored for fault detection and diagnosis. 1. Introduction The pioneering works of Nomikos and MacGregor1-4 have spurred the development of PCA- or PLS-based batch process monitoring methods5-14 in the past decade. These methods have been shown to be effective in both off-line and on-line process analysis and monitoring and fault diagnosis. The data used for modeling, however, should be of equal length to be arranged in a three-way matrix, X(I × J × K), where I is the number of reference batches, J is the number of selected process variables, and K is the number of samples in each batch. There are, however, many industrial processes15 in which the total batch duration and the durations of various operating stages within the batch are not fixed in length because of changes in operating conditions or control objectives. In such cases, as shown in Figure 1a, the reference batches, Xi(J × Ki) (i ) 1, ..., I) have different trajectory lengths, Ki. Figure 1b shows an example process consisting of an uneven-length stage (stage I) and an even-length stage (stage II). The existing modeling and on-line monitoring methods cannot be applied unless the trajectories are of the same length. Two simple methods can be used to deal with cases in which the historical successful batches have different lengths but the trajectories overlap in a common time part.16 One is to cut the batches to the minimum length, provided that the uneven-length problem is not serious and that the main events have occurred in the common time part.17 The other is to derive a model using the data from long batches and to treat the absent parts of the trajectories of the shorter batches as missing data.16 In a general case, uneven-length phenomena can occur in one or more specific operating stages, resulting in the * To whom correspondence should be addressed. Tel.: +8522358-7139. Fax: +852-2358-0054. E-mail: [email protected]. † Hong Kong University of Science and Technology. ‡ Northeastern University.

Figure 1. Data representation for uneven-length batch process: (a) three-way representation, (b) illustration for a two-stage uneven-length process.

total batch duration and the durations of operating substages both being unequal. The two above-mentioned methods are not applicable to such a general case. Nomikos and MacGregor2 suggested replacing time by an appropriate indicator variable that progresses monotonically in time and has the same starting and ending value for each batch, for example, the cumulative monomer feed7 or the the reaction extent.9,10 Interpolation is often needed to find measurements at regular intervals of the selected indicator variable. This has to

10.1021/ie030736f CCC: $27.50 © 2004 American Chemical Society Published on Web 05/29/2004

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be done properly; otherwise, the auto- and cross-correlations might be seriously affected.16 The indicator variable method can perform well, provided that there exists a priori knowledge for determining such an indicator variable for the process. Attempts have also been made to use dynamic time warping (DTW)18,19 to resolve the uneven-length problem.15 DTW is used to synchronize two trajectories by appropriately translating, expanding, and contracting localized segments within the trajectories to achieve a minimum distance between the two trajectories. As DTW works with pairs of patterns, it needs to synchronize separately each batch trajectory, Xi(J × Ki) (i ) 1, 2, ..., I) with a predefined reference batch trajectory, XREF(J × KREF). This approach can be effective as a pretreatment tool in aligning batches of different durations. In on-line monitoring, the evolving batch trajectories have to be warped to have a minimum distance with the reference batch trajectories, which can distort the fault patterns and, hence, reduce the detection ability. Furthermore, the distorted fault patterns can introduce difficulties for fault diagnosis. In a comparison work, Rothwell et al.17 showed that the DTW-based method is less successful in fault detection than the methods of cutting to minimum length or using an indicator variable. In addition, the PARAFAC2 model20 can also handle batches of unequal lengths,21 but it is computationally complex and inefficient.22 In general, uneven lengths are likely to arise in some specific operating stages of a batch process. This paper extends a stage-oriented sub-PCA modeling and monitoring strategy23 developed by the authors to such uneven-length batch processes. The original stage-based sub-PCA modeling method is based on the recognition that the process correlation remains largely the same within the same “operating” stage, so that the process can be divided into stages according to the changes of process correlation. As noted in the original paper, the stages referred to here might differ from the actual operating stages. For an evenlength process, each batch, Xi(J × K) (i ) 1, ..., I), has the same data length K. The time-slice matrices, X ˜ k(I × J) (k ) 1, ..., K), generate K loading matrices, P ˜ k(I × J), containing process correlation information at each sampling interval k. A k-means clustering algorithm has been developed for dividing the loading matrices into groups and for defining process stages, as illustrated in Figure 2.23 When the original stage-based sub-PCA modeling method is applied to a process with uneven-length stages, several issues arise: (i) How should the reference batch data, Xi(J × Ki), be normalized for stage division and process monitoring? (ii) How should the batch be divided into substages according to the change of its process correlation? Note that the covariance structure might be distorted because of the variation in stage durations. For instance, the time-slice covariance matrix for the data in span B of Figure 1b is meaningless. (iii) How should one determine the stage to which the current data belong in on-line monitoring? Process time alone might be not sufficient for stage division because of the variation in stage durations. (iv) How should process abnormalities be differentiated from stage changes? The next section of this paper answers those questions to extend the stage-oriented sub-PCA modeling method

Figure 2. Scheme of stage-oriented sub-PCA modeling method.

to batch processes with uneven-length stages. An application example to injection-molding process is given in section 3, followed by the conclusions in the last section. 2. PCA-Based Modeling and Monitoring for Uneven-Length Batch Processes 2.1. Data Normalization. The most commonly used normalization method for pretreating three-way batch data in the existing literature is the method presented by Nomikos and MacGregor,1 abbreviated as NM normalization here. Focusing on batch-to-batch variation, NM normalization removes the average trajectories from each batch and scales the variables to unit variance by dividing by their standard deviations. Mathematically

x˜ ijk )

xijk - xjjk sjk

(1)

where

xjjk )

sjk )

1

1

I

∑xijk I i)1

x∑

I-1

I

(xijk - xjjk)2

i)1

For uneven-length batch processes, particularly for processes with uneven-length stages, the NM method cannot be used because it has difficulties in obtaining average trajectories, xjjk, and standard deviations, sjk. Taking the process illustrated in Figure 1b as an example, the process consists of two operating stages, and without loss of generality, the first stage is assumed having varying duration. Note that the above information on process stages might be unknown a priori. Two main problems result from applying NM normalization to such a process: (i) How should the data in the irregular tail of span D in Figure 1b be normalized? (ii)

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Figure 3. Two separate stages of the process illustrated in Figure 1b.

Subtracting the average trajectories and dividing by the standard deviations for the data in span B will become meaningless. Kourti16 attempted to solve the first problem by adding zeros for the unavailable data, resulting in the difficulty that the “average trajectories” are not the true average trajectories, because they have been altered by the added zeros. This paper proposes that the average trajectories and standard deviations in the irregular tails of uneven-length stages be calculated from the actual available data rather than these data with added zeros. Although the confidence of the estimated mean and standard deviation is reduced statistically because fewer data are available, this can be compensated by increasing the corresponding control limits. The second problem can be solved by separating the data according to the corresponding process stages, as shown in Figure 3, where Lc stands for the data length in stage c. Thus, the NM normalization technique can only be used for data pretreatment after the stagedivision information becomes available. Another normalization for modeling batch processes was proposed by Wold et al.,8 abbreviated as WKFH normalization in this paper. It is conducted on the twoway matrix X(KI × J), unfolded in such a way as to retain the variable direction. Each point of the trajectory of a batch is considered as an object; WKFH normalization involves subtraction of a constant, the average of each variable over all batches and over the entire batch durations, and division of each variable at each time by the overall standard deviation. Mathematically

x˜ ijk )

xijk - xjj sj

(2)

where

1

xjj )

sj )

1

K

I

∑ ∑xijk

KIk)1i)1

x∑∑

KI - 1

K

stage division. After the stage-division information becomes available, the NM method is used to normalize the stage data for development of the monitoring model. 2.2. Stage-Oriented Sub-PCA Modeling. 2.2.1. PCA Model for Stage Division. The purposes of dividing the stages and finding the stage durations of each batch are two-fold. One is to enhance the understanding of a batch process, and the other is to develop a stage PCA model for process monitoring. For the proposed method, we assume that information on only successful historical batches is available, without any other prior knowledge. We also assume that the reference database contains I normal batches, Xi(J × Ki) (i ) 1, ..., I), and that the shortest batch length is Ks. Thus, Ks time-slice matrices, X ˜ k(I × J) (k ) 1, ..., Ks), can be obtained from the reference data for the calculation of the the time-slice loading matrices, P ˜ k (k ) 1, ..., Ks). As stated in the development of the original stage-oriented PCA-based modeling method,23 (1) a batch process can be divided into several “operating” stages reflecting its inherent process correlation nature; (2) even though the process might be time-varying, the correlation of its variables remains largely the same within the same “operating” stage. We further show here that the process correlation remains essentially the same within the same stage for normal batches, notwithstanding variations in duration. That is, the process correlation will not change significantly until the process enters the next stage. On the basis of this conclusion, one can find the duration of each stage by checking whether the data can be explained by the current-stage PCA model, using the corresponding squared prediction error (SPE) statistics. For the process illustrated in Figure 1b, the process correlation for the data in span A, the common part belonging to stage I for all reference batches, will be largely unchanging according to the aforementioned assertions. With the clustering algorithm,23 the loading matrices in span A should form an independent and stable cluster, representing the inherent process characteristics of the first stage. A PCA model can be developed on the basis of the clustering result, as described later. The data in the first stage should be explained well by this first-stage PCA model, with relatively small SPE values. In contrast, for data not belonging to the first stage, this model will result in large SPE values, as the first-stage PCA model can no longer explain correlations among variables beyond the first stage. The stage PCA model derived from the common part of the first stage can be used to find the duration of the first stage for all reference batches, given a properly selected threshold, θ. Clustering of Loading Matrices for the Development of the First-Stage PCA Model. For time-slice matrices, X ˜ k (k ) 1, ..., Ks), the conventional two-way PCA can be applied directly, generating Ks loading matrices, P ˜ k (k ) 1, ..., Ks)

I

(xijk - xjj)2

k)1i)1

WKFH normalization can be applied directly to uneven-length data, although it is ill-suited for process monitoring because it focuses on the variation along the time direction, not the batch direction. Considering the merits and deficiencies of these two normalization techniques, WKFH normalization is proposed to be used for the pretreatment of the data for

X ˜k ) T ˜ k(P ˜ k)T

(k ) 1, 2, ..., Ks)

(3)

As in ref 23, the clustering algorithm can be applied to the set of weighted loading matrices, {P ˇ 1, P ˇ 2, ..., P ˇ Ks}. In ref 23, the clustering algorithm is used to partition the loading matrices into C groups, representing C different process characteristics. Process stages are then associated with process operating times. For the unevenlength case, the clustering algorithm can obtain clear

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first stage. The subscript s indicates that the information is used for stage division, and the superscript 1 stands for the first stage. h /s and P h˜ /s , for the P/s can be divided into two parts, P principal component and residual subspaces, respectively, where the retained principal components are selected by the method in ref 23. For data vector x, the principal score and SPE can be calculated by the following stage-division PCA model Figure 4. Illustration for an extreme multistage process.

and stable clusters only for the common parts of various stages, such as spans A and C in Figure 1b. From the clustering results, a two-way PCA model can be developed for the first stage, but not for other stages. For example, for the process illustrated in Figure 4, where the difference in durations of the first-stage processes is greater than the length of the second stage, no common part will exist for the second stage; hence, it will not be possible to derive the second-stage model using the above clustering results. For this reason, the PCA submodels developed for stage division will have to be derived one by one in accordance with the sequence of process “stages”, that is, determination of the process stages will have to be conducted in series, not in parallel as in ref 23. The clustering algorithm can be summarized as follows: ˇ 2, The inputs are the patterns to be partitioned, {P ˇ 1, P ..., P ˇ Ks}, and the threshold, θ, for cluster elimination. The outputs are the cluster centers, {W1,W2, ..., WC}; the cluster membership of P ˇ km(k): P ˇ k f {1, 2, ..., C}; and the number of clusters, C. The index variables are the iteration index, i, and the pattern index, k. (i) Choose C0 (i ) 0) cluster centers W0c (c ) 1, 2, ..., 0 C ) from the Ks loading matrices along the time series. In practice, the initial cluster centers can be assumed to be uniformly distributed in the pattern set. (ii) Merge pairs of clusters whose intercenter distance, i-1 i-1 dist(Wc1 , Wc2 ), is below the threshold θ. (iii) Calculate the distances from each pattern P ˇ k to k to the closest all of the centers, dist(P ˇ k, Wi-1 ), assign P ˇ c center Wi-1 c* , and denote its membership as m(k) ) c*. (iv) Eliminate the centers that catch few patterns after a set number of iterations i > I•num to avoid singular clusters. (v) Update the number of cluster centers to be Ci; recompute the new cluster centers, Wic (c ) 1, 2, ..., Ci), using the current cluster membership, m(k). (vi) Return to step 2 if a convergence criterion is not met. Typical convergence criteria are minimal changes in the cluster centers and/or a minimal rate of decrease in the squared errors. (vii) Retain the information on the first stage, {k|m(k) ) 1, k ) 1, ..., Ks}. Development of the First-Stage PCA Model for Stage Division. Define P/s |1, the first-stage representative loading matrix, as

1 P/s |1 ) min dist(P P ˜k ˜ k, P/s |1) ) 1 k k L {k|m(k) ) 1, k ) 1, 2, ..., Ks} (4)

∑

where L1 is the number of process data belonging to the

ts ) xP h /s es ) x - xP h /s (P h /s )T

(5)

SPEs ) esTes If the correlations among the variables can be explained by the PCA model, the SPE statistics should be within the nominal control region. On the other hand, significant SPE statistics will result from the currentstage PCA model if the data belong to a different stage. By checking whether the SPE statistic is beyond a predetermined threshold, SPE*, one can readily determine stage durations for each reference batch. With the information for the first stage, the first-stage data of each reference batch can be removed and the remaining data used to form a new reference data set. The same procedure can readily be applied to determine the duration of the second stage, and so. The stage division can hence be conducted iteratively until all stage information becomes available. The two parameters θ in clustering and SPE* in stage division are important for the success of the proposed stage-division algorithm. The threshold θ determines the accuracy and complexity of the developed stagebased sub-PCA model. Obviously, large θ values result in coarse clustering and less accurate modeling, whereas small θ values can improve modeling accuracy but need more submodels and increase the modeling complexity. The parameter SPE* is recommended to be assigned a value that is slightly larger than the maximum SPE values of the data belonging to the common part of the first stage. 2.2.2. PCA Model for Process Monitoring. The PCA model derived in the preceding section can successfully be used to divide the process into stages and to determine the duration of each stage. It is, however, ill-suited for process monitoring because of the WKFH normalization method adopted. WKFH normalization focuses on the variation of the trajectories of the variables, rather than the variation among the batches.24 For this reason, the data in each stage are renormalized by the NM method to develop a stage PCA monitoring model for fault detection and diagnosis. The time-slice loading matrices for the data in the irregular tail region are calculated from the time-slice matrices, X ˜ k, of the available data of the longer batches. The contribution to the stage PCA monitoring model should be reduced by a weight wk ) Ika/I (0 < wk e 1), where Ika is the number of available batches for calculating the time-slice loading matrix and I is the total number of reference batches. The stage PCA monitoring model is finally defined as a weighted mean of the loading matrices

Ind. Eng. Chem. Res., Vol. 43, No. 13, 2004 3347

P/m|c )

∑k wkP˜ k ∑k w

(k ) 1, ..., Lc)

(6)

k

where the subscript m indicates that the derived stage representative loading matrix is used for process monitoring, c is the stage index, and k is the data index in stage c. The principal score and SPE of data vector x are calculated by the following stage PCA monitoring model

tm ) xP h /m em ) x - xP h /m(P h /m)T

(7)

SPEm ) emTem Determination of the control limits can follow the same procedure as in ref 23. 2.3. Stage-Oriented Sub-PCA Monitoring. Two stage models are developed in section 2.2 above. The stage-division model (eq 5) is used to divide the process into stages, find the uneven-length stages, and determine the duration of each stage. With the stage-division information, that is, the length of each stage, Lci (i ) 1, ..., I; c ) 1, ..., C) and the shortest and longest lengths c , reference of an uneven-length stage c, Lcmin and Lmax batch data are processed by NM normalization to develop the stage-monitoring model (eq 7) for fault detection and diagnosis. With the proposed modeling method, an issue in process monitoring is how to differentiate an abnormality from a change of process stage. In process monitoring, there are three possible results: normal conditions, abnormal conditions, and stage change. (i) For the normal case, the two statistics T 2 and SPE, calculated by the current-stage model (eq 7), should be below their corresponding control limits. (ii) For the stage-change case, the process enters the new stage, and there is no abnormality. However, the current-stage model will give significant T 2 and SPE statistics, indicating that the corresponding PCA model can no longer explain the correlation of the current data. The data, however, should be well explained by the monitoring model of the next stage, generating incontrol T 2 and SPE statistics. The stage change, therefore, can readily be identified by applying the two consecutive stage-monitoring models and checking their corresponding T 2 and SPE values. (iii) For the abnormal case, both the current and the next stage-monitoring models should give large T 2 or SPE values. As mentioned before, process time is insufficient for defining process stages because of the problem of uneven lengths. The above knowledge is used to determine the stage to which new data belong in on-line process monitoring. For even-length stages, on-line monitoring is similar to that in the original stage-oriented sub-PCA method. For uneven-length stages, data are partitioned into two parts for on-line monitoring. The data before the shortest stage length, Lcmin, can be directly monitored by the current-stage model. Significant SPE or T 2 statistics can be used with confidence to indicate a process abnormality. For the data between Lcmin and c , out-of-control statistics represent two possibiliLmax

ties: a process abnormality or a stage change. The above-stated knowledge can be used to associate the change with a proper cause. As with any statistical modeling and monitoring method, the reference data used for the modeling are assumed to cover statistically all normal operating conditions. Any case in which the uneven-length stage of a new batch is shorter than the minimum stage length, Lcmin, or longer than the maxic , will be considered an abnormality mum length, Lmax by the monitoring procedure. Dividing a batch process into stages and developing a two-way PCA monitoring model for each stage can enhance process understanding and allow for rapid fault diagnosis after detection of an abnormality. The contribution plot,25 a commonly used diagnostic tool, can be used to indicate the variables seriously impacted by the abnormality that has occurred. The modeling and on-line monitoring strategies for uneven-length batch processes can be summarized as follows. Modeling Procedure. The available information includes the number of successful historical batches Xi(J × Ki), I, and the shortest batch length, Ks. The purposes are to divide the process into stages according to the change of process correlation and to develop stage PCA monitoring models. The main steps are as follows: (i) Normalize the reference batches by the WKFH normalization method. (ii) Generate Ks time-slice matrices and calculate the corresponding loading matrices. (iii) Derive the stage-division PCA model (eq 5) by the clustering algorithm for the first stage. (iv) Determine the duration of the first stage, L1i (i ) 1, ..., I), by checking SPE values for each reference batch, and find the longest and shortest durations in 1 and L1min, respectively, if the the reference data, Lmax first stage has a different duration. (v) Extract the first-stage data, normalize them by the NM method, derive the corresponding stage PCA monitoring model (eq 7), and compute the corresponding control limits for the T 2 and SPE statistics. (vi) Remove the first-stage data to form a new reference data set, and repeat steps 2-5 for the next stage. Monitoring Procedure. The available information includes the stage-division information, Lci (i ) 1, ..., I; c , and Lcmin; the average trajectories c ) 1, ..., C), Lmax and standard deviation trajectories for each stage; and the stage-monitoring PCA models. The purposes are to monitor the process conditions, to detect any abnormality, and to provide possible the cause for the detected abnormality. Case I applies to data with sampling times belonging to range [1, Lcmin]. (i) Normalize the data by the NM method using the average value and standard deviation at the corresponding time. (ii) Call the current stage-monitoring model and calculate the two statistics T 2 and SPE. (iii) If either of the two statistics exceeds its control limit, the monitoring procedure gives an alarm indicating an abnormality in the process. If the two statistics are both within the normal operating regions, the current data are normal; return to step 1 for the next data.

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Figure 5. Simplified illustration of an injection-molding machine.

Figure 6. First loading vector of each time-slice loading matrix. Table 1. Descriptions of the Process Variables

Case II applies to data with sampling times belonging c ]. to range [Lcmin, Lmax (i) Apply the same procedures as in case I until either of the statistics is beyond the normal control limit. (ii) Normalize the data by regarding them as the first data of the next stage, and use the monitoring PCA model of the next stage to calculate new T 2 and SPE values. (iii) If both of the statistics are below their control limits, the process enters the next stage. Otherwise, a process abnormality has occurred. Use a contribution plot to analyze the possible cause for the fault. 3. Application Illustration 3.1. Injection-Molding Process. Injection molding,26,27 an important polymer processing technique, transforms polymer granules into various shapes and types of products. Figure 5 shows a simplified diagram of a typical instrumented reciprocating-screw injectionmolding machine. As a multistage process, injection molding operates in stages, among which injection (or filling), packing-holding, and cooling are the most important phases. During filling, the screw moves forward and pushes melt into the mold cavity. Once the mold is completely filled, the process then switches to the packing-holding phase, during which additional polymer is “packed” at a pressure to compensate for material shrinkage associated with cooling and solidification of the material. The packing-holding stage continues until the gate freezes off, which isolates the material in the mold from that in the injection unit. The process then enters the cooling phase, during which the polymer in the mold continues to solidify until it is rigid enough to be ejected from the mold without damage. Concurrently with the early cooling phase, plastication takes place in the barrel where polymer is melted and conveyed to the front of the barrel by screw rotation, preparing for the next cycle. As mentioned above, the filling phase stops when the mold is completely filled. The filling time is therefore not fixed, but rather depends on the injection velocity. Obviously, a lower injection velocity requires a longer filling time, resulting in a longer batch with more process data than a batch operation with a higher injection velocity. Although it is possible for the real injection-molding machine to repeat each cycle rigidly with a high degree of automation, this process is intentionally simulated as an uneven-length batch

no.

description

units

1 2 3 4 5 6 7 8 9 10 11

nozzle temperature nozzle pressure stroke injection velocity hydraulic pressure plastication pressure cavity pressure screw rotation speed SV1 opening SV2 opening cavity temperature

°C bar mm mm/s bar bar bar rpm % % °C

process to demonstrate the proposed modeling and monitoring method. In this paper, the filling phase is designed to have a varying duration as the injection velocity is changed from 22 to 26 mm/s. The maximum difference in number of data points in the injection stage is 13, about 15% of the average filling length. The other stages are controlled to have exactly the same data length. The material used in this work is high-density polyethylene (HDPE). The process variables selected for modeling are listed in Table 1. The operating conditions are set as follows: the injection velocity changes from 22 to 26 mm/s; the nozzle packing pressure is set at 200 bar; the mold temperature is set at 25 °C; the sevenband barrel temperatures are set at 200, 200, 200, 200, 180, 160, and 120 °C, respectively; the packing-holding time is fixed at 6 s; the cooling time is set at 15 s; and the sampling interval is 20 ms. In total, 35 normal batch runs are conducted, where the shortest and longest batch lengths are 1180 and 1193 samples, respectively. Three abnormal batches, with a check-ring problem, a material disturbance, and an SV1 valve-stick fault, are introduced to test the proposed process monitoring and diagnosis scheme. 3.2. Illustration of Stage Division. The reference data are first normalized by the WKFH method for stage division. Assuming that no information is available in advance on the number of stages and the length of each stage, the clustering algorithm has to partition Ks (Ks ) 1180) loading matrices for the determination of the duration of the first stage. The first column vector of each time-slice loading matrix, P ˜ k (k ) 1, ..., Ks), is plotted in Figure 6, where each line stands for the elements in the first loading vectors over the batch duration. Note that, in Figure 6 and the remaining figures, the unit of the horizontal axis is 20 ms, equal to the sampling interval. From Figure 6, it is obvious

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Figure 7. Clustering results for the determination of the firststage PCA model.

Figure 9. Clustering results for the determination of other stages.

Figure 10. SPE monitoring chart for a normal batch. Figure 8. Determination of the duration of the first stage for three normal batches.

that the loading vectors indeed remain similar across several operating periods and show significant differences for different periods. The corresponding clustering results are shown in Figure 7. It is also easy to find the four major clusters, corresponding to the four operating stages injection, packing-holding, plastication, and cooling. The odd points between major stages can be caused by the uneven-length filling stage. Figure 8 shows the stage-division results for three test batches (batches 1-3), from which the determination of the durations of their filling stages can be done easily and accurately. The injection velocities of the three test batches are 22, 24, and 26 mm/s, respectively. According to process knowledge, test batch 1 is the longest batch, and test batch 3 has the shortest filling time. The SPE trajectories, obtained after application of the stagedivision PCA model to these test batches, exactly agree with the true durations of the filling phase. The longest filling stage has 93 data points, whereas the shortest one has 80. The data remaining after the removal of the fillingstage data have almost the same length and overlapping trajectories. The same stage-division method can be applied to the remaining data to determine the durations of the packing-holding, plastication, and cooling stages. The clustering results are illustrated in Figure 9. A short period between the packing-holding and

plastication stages forms a separate stage, which was specially designed for retraction of the nozzle into the molding. 3.3. Illustration of Process Monitoring and Fault Diagnosis. 3.3.1. Monitoring a Normal Batch. This section focuses on illustrating the proposed on-line process monitoring and the method of differentiating a stage change from a process abnormality. The SPE monitoring chart for a normal batch is shown in Figure 10. For Figure 10 and all remaining figures, points marked with squares are SPE values calculated by applying the filling-stage PCA model, whereas points marked with triangles are SPE values calculated with the packing-holding-stage model. The solid line represents the control limits corresponding to the confidence level R ) 0.01, and the dashed line is the control limits for R ) 0.05. After the stage-division procedure described in section 3.2 has been applied, the following stage information has been obtained from the experimental results. (1) The filling stage has a varying length, and the other stages have fixed operating durations. (2) The longest and shortest filling stages have 93 and 80 data points, respectively, which implies that the filling time of any normal batch should fall within the range [80, 93], provided that the reference database has covered all normal possibilities. For on-line monitoring, the data before sample 80, the shortest length of the filling phase, are normalized using the corresponding mean and standard deviation trajectories as presented in section 2.3. The filling-stage PCA

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Figure 11. Monitoring of an abnormal batch with a check-ring problem: (a) SPE monitoring chart, (b) contribution plot.

Figure 12. Monitoring of an abnormal batch with material disturbance: (a) SPE monitoring chart, (b) contribution plot.

model is then applied to these data to compute T 2 and SPE statistics. If both statistics are below the control limits, the data are considered normal. Otherwise, any significantly large statistic indicates an abnormality. For the data within the period [80-93], to differentiate a process abnormality from a stage change, those data that exhibit significant statistics using the fillingstage monitoring model are regarded as the beginning of the packing-holding stage and renormalized accordingly. The packing-holding-stage PCA model is then applied to the normalized data to compute the T 2 and SPE statistics. For the case in which process has actually entered into packing-holding stage, the evolving data should have in-control statistics according to packing-holding-stage PCA model, as illustrated in Figure 10. For the data after sample 93, the longest filling period of the reference data, all normal processes should finish the first operating stage and enter the packing-holding phase. The packing-holding-stage PCA model is applied to the evolving data to monitor the process conditions. For the process monitored in Figure 10, the actual operation switches from the filling stage to the packingholding stage at sampling interval 90. The continued use of the filling-stage monitoring model after this transition over the period 90-93, as indicated in a dashed-line box of Figure 10, gives higher statistics than the corresponding control limits, even though these data

are well explained by the second-stage model, i.e., the packing-holding-stage monitoring model. This can be clearly seen in Figure 10. 3.3.2. Monitoring Abnormal Batches. Check-ring problems, material disturbances, and valve sticking are typical faults in injection molding. Check-ring problems affect the filling and packing-holding stages; material disturbances can change process correlations over all operating stages. Valve-sticking faults, simulated in the middle of the packing-holding stage, will have normal process measurements in the filling stage and the early part of packing stage. Figures 11-13 show the monitoring charts and diagnosis results for these three faults in the filling and packing-holding stages. The monitoring and diagnosis results for the other stages are not shown, as this paper focuses on stage-division and monitoring for uneven-length batch processes. The check-ring valve, a device that allows the polymer melt to flow from the screw channel to the nozzle during plastication, closes during the injection and packing stages to prevent polymer backflow from the nozzle to the screw channel. A batch with a check-ring problem means that the check-ring valve fails to close completely during the injection and packing stages, resulting in a smaller amount of material being injected into the mold at the same given injection velocity. This fault will typically result in a lower nozzle pressure (2), hydraulic pressure (5), and cavity pressure (7). According to the correlation model, the pressures and injection velocity

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statistics. When the SV1 valve fails to work in the packing stage, the nozzle pressure (2) increases. The cavity pressure (7) also increases following the change of nozzle pressure. The correlation between these two variables is not altered by this fault, resulting in insignificant SPE statistics for the faulty data. The T 2 statistics, however, increase beyond the control limit because of the increased nozzle and cavity pressures as shown in Figure 13a. The contributions of the variables to the first principal component are shown in Figure 13b, where two pressure variables (2 and 7) contribute most to the significant T 2 statistics. The above-described experimental results indicate that the proposed modeling and monitoring scheme can successfully obtain accurate stage information, detect process faults, and provide reasonable explanations for the detected faults for a batch process with durations of uneven lengths. 4. Conclusion Industrial batch processes are often characterized by durations of uneven lengths. This paper has proposed a modeling and on-line monitoring method for such a process without the need of aligning the data to the same length for modeling and monitoring. Two models for the two stages have been developed: one for stage division and the other for process monitoring. The problem of batch durations of uneven lengths is solved by dividing the process into stages, and fault detection and diagnosis are conducted using a PCA monitoring model developed for each stage. The application to the injection-molding process shows the feasibility of the proposed method, illustrating the potential for application to other industrial batch processes with durations of uneven lengths. Figure 13. Monitoring of an abnormal batch with valve-stick fault: (a) T 2 monitoring chart, (b) contribution plot.

Literature Cited have a negative correlation; thus, the same pressures will result in a higher injection velocity (4). The SPE monitoring chart, shown in Figure 11a, can clearly indicate the abnormality during the filling and packing stages as the SPE values are clearly beyond the control limits. The contribution plot, Figure 11b, shows that the process variables that are seriously impacted by the abnormality agree well with the fault pattern of checkring leakage described earlier. The abnormal batch with a material disturbance fault is carried out by adding a small amount of polypropylene (PP) to the processing of HDPE. Because PP is more viscous than PE, to keep the same injection velocity, the injection pressure (i.e., hydraulic pressure, 5) must be larger than the normal case, which also results in increasing nozzle (2) and cavity (7) pressures. In addition, PP has higher crystallization rate than PE for the same chilling water temperature, so the mold temperature (11) is lower than in the normal case. The monitoring chart in Figure 12a can clearly detect the fault, and the diagnosis chart in Figure 12b can reveal the impacts of this fault on the aforementioned process variables. The SV1 valve-sticking fault is introduced in the middle of packing stage, resulting in an uncontrolled nozzle pressure. Prior to this point, the process measurements in the filling stage and the early part of the packing stage should yield in-control SPE and T 2

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Received for review September 29, 2003 Revised manuscript received March 23, 2004 Accepted March 23, 2004 IE030736F