Pseudo time-slice construction using variable moving window-k

Article pubs.acs.org/IECR

Pseudo Time-Slice Construction Using a Variable Moving Window k Nearest Neighbor Rule for Sequential Uneven Phase Division and Batch Process Monitoring Shumei Zhang,†,‡ Chunhui Zhao,*,† Shu Wang,*,‡,§ and Fuli Wang‡,§ †

State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, China ‡ State Key Laboratory of Integrated Automation of Process Industry Technology and Research Center of National Metallurgial Automation, and §College of Information Science & Engineering, Northeastern University, Shenyang, 110819, China ABSTRACT: Multiphase characteristics and uneven-length batch duration have been two critical issues to be addressed for batch process monitoring. To handle these issues, a variable moving window-k nearest neighbor (VMW-kNN) based local modeling, irregular phase division, and monitoring strategy is proposed for uneven batch processes in the present paper. First, a pseudo time-slice is constructed for each sample by searching samples that are closely similar to the concerned sample in which the variable moving window (VMW) strategy is adopted to vary the searching range and the k nearest neighbor (kNN) rule is used to find the similar samples. Second, a novel automatic sequential phase division procedure is proposed by similarity evaluation for local models derived from pseudo time-slices to get different irregular phases and ensure their time sequence. Third, the affiliation of each new sample is real-time judged to determine the proper phase model and fault status can be distinguished from phase shift event. The proposed strategy can be readily extended to the case with limited batches. To illustrate the feasibility and effectiveness, the proposed algorithm is applied to a typical multiphase batch process, i.e., injection molding process, with an uneveness problem.

1. INTRODUCTION To meet rapidly changing market demand for high-value-added products, batch and semibatch processes play a significant role in modern industrial processes such as in the specialty chemical, semiconductor, food, and biological industries. It is very important to monitor these batch processes to ensure safety, consistency, and reliability. However, batch processes have many characteristics that differ from those of a continuous process. It is more difficult to establish a monitoring model for batch processes. A three-dimensional data structure is a significant characteristic of batch processes. Multivariate statistical techniques,1,2 such as multiway principal component analysis (MPCA) and multiway partial least-squares (MPLS), have been widely applied in batch process analysis and monitoring to handle these problems. But these methods cannot reveal the changes of process correlations along the time direction because they take the entire batch data as a single object.3 Besides, the future status has to be estimated in the online monitoring since the data from the current time to the end of the batch are not available. Batch processes are usually conducted in a series of steps, resulting in different process segments, which are called stages or phases. Each phase has its own variable correlations and data characteristics. Many approaches have been developed for multiphase batch processes, which can be roughly summarized into two different categories: multiblock and phase-separated methods. © 2016 American Chemical Society

Multiblock methods use a single model with data grouped in several blocks to capture the relations between complex process variables and reflect local behaviors of a process. MacGregor et al.4 developed a multiblock partial least-squares (PLS) method to monitor the subsections as well as the entire process. Westerhuis et al.5 compared multiblock and hierarchical PCA and PLS methods from a theoretical or algorithmic viewpoint. Qin et al.6 extended the multiblock approaches based on a combined index for fault detection and diagnosis. Zhao et al.7 proposed an improved multiblock-based spectral calibration and statistical analysis approach. Although multiblock methods have been successfully applied in many industrial areas, most of these traditional multiblock methods assume that prior process knowledge is known. However, prior process knowledge is unavailable in many industrial processes, which limits the application of the multiblock methods. Besides, many batch processes may have significant transition behaviors from phase to phase, while most of the multiblock methods have not well considered the transitions between any two adjacent phases. Phase-based methods build a separate statistical model for each phase of a batch process. Since the prior process knowledge Received: Revised: Accepted: Published: 728

September 26, 2016 December 11, 2016 December 22, 2016 December 22, 2016 DOI: 10.1021/acs.iecr.6b03743 Ind. Eng. Chem. Res. 2017, 56, 728−740

Article

Industrial & Engineering Chemistry Research

Figure 1. Illustration of the VMW-kNN rule for searching similar samples.

time part. Another common used method is called “warping technique”, such as dynamic time warping (DTW) and correlation optimization warping (COW), which has always been used as a method of pattern matching. Recently, they have been applied to resolve the uneven-length problem in batch monitoring.22,23 However, this kind of monitoring method is complicated to achieve and the result may not be accurate because the relationships between process variables may be distorted when the trajectories are stretched or compressed to the reference. Also, warping technique can be only used for offline modeling but cannot be used online because online batch cannot be synchronized to the reference until the end of the batch. Many other methods have been proposed to deal with the uneven length in the recent years. Wang24 proposed a novel local neighborhood standardization (LNS) strategy as a data preprocessing method and constructed a global PCA model for process monitoring. LNS-PCA method can only eliminate influence caused by diversity of mean in different phases but cannot closely describe the covariance structures of each phase, so it would fail to represent each operating phase precisely due to statistical averaging. The Gaussian mixture model (GMM)25 is another common used method which can tackle the uneven problem in batch process monitoring. Since the duration time of transitional phases is much shorter than the stable time, the information contained in transitional phases may be overwhelmed when all the historical data are used to train a GMM. GMM may not efficiently capture the local features of transitions. Besides, both LNS-PCA and GMM methods did not consider the phase information. Zhao et al.26 proposed a sequential uneven phase identification method regarding the influences on monitoring performance. However, they simply assume that the consecutive samples within a certain time-region have the similar characteristic. In fact, owing to uneven problem and multiphase characteristic, the samples from uneven batches may have different process characteristic. On the basis of the above analysis, three main issues should be addressed for process monitoring which are common in batch processes: multiple phases, transition characteristics from phase to phase, and uneven-length problems. To handle these issues, a variable moving window k nearest neighbor (VMWkNN) based local modeling strategy has been proposed in

is hard to obtain in many processes, an important issue for phase-separated methods is how to divide the batch process into different phases. According to the fact that the changes of the process correlations may relate to the phase shift in multiphase batch processes, many clustering-based phase division methods8−11 have been well developed. Lu et al.12 proposed phase-based sub-PCA/PLS models for multiphase batch processes using improved K-means clustering algorithm. Yu and Qin13 proposed a phase partition method based on the Gaussian mixture model (GMM) which uses the posterior probability to classify the operation phases. However, these methods do not consider the time sequence of operation phases. Thus, the points in one phase may be assigned to different phases, and time segments at different phases may be mixed as a single phase. Besides, they neglect the transition characteristics between two adjacent phases, which may compromise the accuracy of phase-representative monitoring models. To solve the time sequence problem, Camacho14 divided the process into multiple segments according to the percentage of sum of squares explained by the principal component. Sun et al.15 put forward a phase partition method in which the results are in time sequence. Zhao et al.16,17 proposed a stepwise sequential phase partition (SSPP) algorithm which considered the effects of phase partitioning on monitoring performance. Considering the transitions, Zhao et al.18 proposed a soft-transition multiple PCA (STMPCA) method to detect and model transitions for online process monitoring. Other related works for dealing with the transition problem include those of Kourti,19 Ng and Srinivasan,20 etc. However, most of these methods assume that each batch has the same duration and the sampling intervals are equal without considering the uneven-length problem. In fact, uneven-length problems widely exist in batch processes. The length and the intervals may vary from batch to batch in terms of practical operation conditions. To solve this problem, many methods have been developed in the past years. The simplest method is cutting the batches to the minimum length21 or deriving a model with long batches by treating the absent part of short batches as missing data.19 However, these two methods are suitable only when the uneven length problem is not serious and the trajectories overlap in a common 729

DOI: 10.1021/acs.iecr.6b03743 Ind. Eng. Chem. Res. 2017, 56, 728−740

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Figure 2. Illustration of offline phase division.

window is used around the phase shift. Then the similar samples are selected from the historical data set within the MW by using kNN rule. An online judgment rule is defined to determine the affiliation of each new sample and distinguish the fault status from phase shift. In comparison with other methods for uneven batch processes, the method show more sensitive and reliable fault detection results even when the batches are limited.

which a pseudo time-slice is constructed for each sample and the irregular phase duration is determined for each batch. The kNN rule27 is used to find similar data, which can avoid synchronization problems caused by uneven length. In order to reduce the computational burden of kNN as well as preserve the local neighborhood information, the VMW strategy is adopted in the paper to determine the searching range. A small window is used for the stable phase while the large 730


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Figure 3. Illustration of online identification of phase affiliation and monitoring.

The paper is organized as follows. In section 2, some preliminaries about PCA and kNN rules are briefly revisited. In section 3, the methodology of sequential phase division and offline modeling by constructing pseudo time-slice is first described in detail. Online phase identification and monitoring algorithm is then introduced in subsection 3.3. The new proposed approach is demonstrated for the injection molding process in section 4. Finally, section 5 outlines the concluding remarks of this paper.

d(x t , x t ,1) ≤ d(x t , x t ,2) ≤ ... ≤ d(x t , x t , k)

2.2. Principal Component Analysis. As a commonly used dimensionality reduction technique, principal component analysis (PCA) has been widely used for process monitoring. Consider a data matrix X = [x1, x2, ..., xn]T ∈ Rn×J, where J is the number of process variables with n samples. PCA is carried out upon the covariance matrix of X. Traditionally, the singular value decomposition (SVD) method can be employed for construction of the PCA model. Suppose first q principal components have been selected in the PCA model, X can be decomposed as

2. PRELIMINARIES 2.1. kNN Rule. The kNN rule was first developed by Fix and Hodges,27 and then it was widely used in pattern classification, where the unlabeled samples are classified by examining its distances to the nearest neighboring training samples in the feature space. Recently, the kNN rule has been widely used to determine the local model in process monitoring.28 For a data matrix X = [x1, x2, ..., xn]T ∈ Rn×J, where J is the number of process variables with n samples, k nearest neighbors of xt (1, 2, ..., n) should be found out according to Euclidean distance between xt: d(x t , x t , k) = ∥x t − x t , k∥

(2)

q

X = X̂ + E = TPT + E =

∑ tjpTj + E j=1

(3)

where E is the residual matrix, T ∈ Rn×q is score matrix with score vectors tj (j = 1, 2, ..., q), P ∈ RJ×q is loading matrix with PC loadings pj (j = 1, 2, ..., q). The prediction of the PCA q model is given as X̂ = XPPT = TPT = ∑ j = 1 tjp Tj . To monitor the process, the statistics used for PCA are T2 and SPE, which are calculated in different PCA subspaces, respectively:

(1)

Then the neighborhood set of xt can be recognized as N(xt) = {xt,1, xt,2, ..., xt,k}, where the following condition should be satisfied:

q

T2 =

∑ j=1

731

t2j λj

(4) DOI: 10.1021/acs.iecr.6b03743 Ind. Eng. Chem. Res. 2017, 56, 728−740

Article

Industrial & Engineering Chemistry Research SPE = e Te = (x − x̂)T (x − x̂)

running in one steady state, in other words, the characteristics of variables such as mean and covariance are constant. Thus, the same stable phase in different batches is highly repeatable, which means that the information is sufficient even within a small searching range. (2) In contrast to stable phase, the transition between two stable phases is a dynamic process with a very short duration time. Thus, a large searching range is required to obtain enough similar samples to construct the reliable local models for the concerned transition sample, especially when the batches are uneven. In the paper, variable moving window (VMW) is used to determine the range of searching the relevant data samples. And kNN rule is used to select the similar data set from the searching range. Two window sizes are used to define the searching range, which are L1 and L2 satisfying L1 < L2. Here, L1 denotes the length of initial searching range, which is set to be 10. In contrast, L2 is not fixed which is varying to enlarge the initial search range with more candidate samples available. For the tth sample xit in the ith batch, the searching range can be described as

(5)

where λj (j = 1, 2, ..., q) are the eigenvalues of the covariance matrix of X, which are arranged in descending order to determine the principal components (PCs). The control limits of two statistics can be determined by the sampling distributions of T2 and SPE, respectively: Tα2 ∼

q(n − 1) Fα(q , n − q) n−q

SPEα ∼ gχh,2 α

(6) (7)

where Fα(q, n − q) is an F-distribution with the level of significance α. The degrees of freedom are q and n − q. The control limit of SPE can be calculatedfrom an χ2-distribution with the level of significance (α). g and h are calculated by g = b/2a, h = 2a2/b, where a and b are the estimated mean and variance of the SPE.29

3. METHODOLOGY 3.1. Pseudo Time-Slice Construction. Considering an uneven-length batch with J process variables measured over sampling points Ki, the historical data collected in normal working condition are composed of a three-dimension array X(I × J × Ki), where I is the number of batches. The ith batch can be described as Xi(J × Ki) (i = 1, 2, ..., I). Since the batches are not even-length indicated by Ki, the conventional time-slice cannot be constructed for statistical modeling to reveal similar process characteristics at the same time. Despite the uneven operation duration, for each sample, some samples may present similar process characteristics which may be located in different batches and in different time. The key point is how to find these similar samples so that a “pseudo” time-slice can be constructed from which the variable correlations can be derived to describe the process variations around the concerned sample. Here, a query-based search approach is developed. The kNN rule is a common method to determine the similar data set from the historical data at all sample time. However, it will be difficult and time-consuming when the historical database is large. Besides, the sample in a local model may be close in space but far away in time, which is unreasonable. To cope with the issue, the moving window (MW) strategy is used by Hu et al.30 to determine the range of searching the relevant data samples. A series of moving time windows are set according to the current sample time because a new data sample is just most likely relevant to the historical samples at around the current sample time. The window size tunes the speed and range for searching the relevant data samples. When the window size is large, the searching range is wide and the information is enough to reflect the current process operating condition sufficiently. But a large window size means large computational burden and scarce sensitivity to changes. When the window size is small, the model may capture process changes quickly. But the information may be not adequate to construct a reliable local model, especially when the batches have uneven-length phase durations. So it is important to select an appropriate window size that determines the searching range. The proposed method is based on the following recognition: (1) The stable phase plays the primary role in the batch processes and it occupies the most production time to yield high productivity. The process variables in stable phase are

X t ,se = [X 1t ,se, X t2,se, ..., X tI ,se]

(8)

where

X it ,se

⎧ Xi(1: t + L ) 1 ≤ t ≤ L1 1 ⎪ ⎪ i = ⎨ X (t − L1: t + L1) L1 < t ≤ K i − L1 ⎪ ⎪ X i (t − L : K ) K i − L1 < t ≤ K i ⎩ 1 i

(9)

and Ki denotes duration of the ith batch. The kNN rule is utilized to find the similar data samples of xit from the searching range Xt,se. The neighborhood set of xit can i be recognized as N(xit) = {xit,1, xit,2, ..., xt,k }, which can be regarded as the pseudo time-slice around the concerned sample xit. Then the set distance is defined as following: k

dist(x it ) =

∑ d(x it , x it ,f )2 /k f =1

(10)

If the set distance dist(xit) satisfies the following condition,

dist(x it ) < δ

(11)

it means the truly similar samples have been found to construct the reliable local model and the searching should be stopped. Otherwise, the collected samples by kNN within the initial searching range are not truly similar so that the information in “pseudo” time-slice may be not adequate to construct a reliable local model. Therefore, the searching range should be extended. The length of searching range (L2) will be enlarged step by step (i.e., one sampling time across all batches) until enough similar samples are founded satisfying eq 11. Here, the threshold δ can be determined simply in the following step: (1) the set distance dist(xit) of the data sample xit(1 ≤ i ≤ I, 1 ≤ t ≤ h) is calculated, where h is the shortest duration of the phases that has been obtained by priori knowledge; (2) the threshold δ can be determined based on the maximum value in the data set {dist(xit)}(1 ≤ i ≤ I, 1 ≤ t ≤ h). In the extended searching range, the kNN rule is used again to find the relevant data samples. The illustration of the VMWkNN rule is shown in Figure 1. In case 1, the sample x1t1 satisfies 1 ≤ t1 ≤ L1, thus the searching range should be from time 1 to time t1 + L1. Then the kNN rule is used to find the neighborhood 732


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Step 3: PCA is used to extract the main feature of N(xit), and the loading matrix Pit ∈ Rm×k can represent the variable correlation for each sampling time (xit). Step 4: The mean matrix of the first h samples of all the batches can be calculated according to the following equation:

1 P*c = Ih

I

h

∑ ∑ Pit i=1 t=1

(13)

where Pc* is the representative model of the cth phase, and the value of h should be the shortest duration of the phases as mentioned before. Step 5: The similarity between each sample up to the hth time and the representative model Pc* is defined as ⎛ || Pi − P* || ⎞ c ⎟ Sti = exp⎜ − t 2 J ⎠ ⎝

Figure 4. Simplified schematic diagram of injection molding machine.

where J is the number of the variables. {Sit} (1 ≤ i ≤ I, 1 ≤ t ≤ h) can determine the variation of similarity between each sample and the represent model, from which, the minimum similarity value is denoted as γ. Step 6: The similarity between the tth (t = h + 1, h + 2, ...) sample of each batch and the representative model P*c is calculated and compared with γ. If Sit ≥ γ, it can be considered that the tth sample of the ith batch still belongs to phase c and continues to calculate the similarity between the next sample and P*c ; otherwise, it can be inferred that phase c of the ith batch is over. Step 7: Remove the corresponding data of phase c in each training batch and the remaining data are aligned in the head. The analysis data are updated as the new input and go back to step 4. The phase division is conducted iteratively by repeating steps 4−7 until all irregular phases are separated. The illustration of the proposed approach is shown in Figure 2. Compared to the traditional methods, the proposed method is more sensitive and reliable even when the batches are limited since similar samples are collected around each sample to construct a pseudo timeslice using the proposed searching method so that a reliable PCA can be derived. Then based on phase separation result, all the data samples which belong to the same phase are collected together to build a PCA model. The control limits of two statistics (i.e., T2 and SPE) can be calculated according to eqs 6 and 7. 3.3. Online phAse Identification and Monitoring Algorithm. For online monitoring in uneven batch processes, when a new data sample xt has been obtained, the current phase affiliation should be determined to choose the appropriate phase-representative PCA model to monitor the online process data. According to the characteristic of the time sequence, the batch process certainly begins from the Phase 1. Thus, for the first sample, i.e. when t = 1, the online phase affiliation information can be described as u(1) = 1. VMW-kNN rule is used to find the relevant data samples of x1 from the history data X(I × J × Ki), and the data sample x1 is normalized using the neighborhood set according to the eq 12. Then the PCA model of the first phase is used to monitor the first sample. If the statistics are under the control limits, the process is deemed to be operating normally. Otherwise, the process is assumed to be abnormal. Assume that the current phase information at time (t − 1) has been known and its phase information is described as

Table 1. Monitoring Variables in the Injection Molding no.

variable description

units

no.

variable description

units

1 2 3 4 5

nozzle temperature nozzle pressure screw stroke injection velocity hydraulic pressure

°C bar Mm mm/s bar

6 7 8 9

plastication pressure cavity pressure SV1 opening SV2 opening

bar bar % %

set. Because all the samples in the searching range are from the stable phase 1, the samples in the neighborhood set N(x1t1) are the real similar data. In case 2, the sample x1t2 is in the transition between two stable phases. When the window size is small, many samples in the searching range are from another phase due to the uneven lengths. Thus, although the samples in the neighborhood set N(x1t2) are the nearest neighboring training samples to x1t2, they may not the real similar data resembling to x1t2. The local model constructed by these data may not be reliable. So the searching range should be enlarged within a large window size L2 to gain the enough information for constructing the reliable local model. In case 3, when the sample x1t3 satisfies L1 < t3 ≤ Ki − L1, the searching range should be from time t3 − L1 to time t3 + L1. 3.2. Sequential Uneven Phase Division and Offline Modeling. In phase division of batch processes, the time sequence of process phases which is overlooked by the conventional clustering-based division algorithm should be considered. For the three-dimension array X(I × J × Ki) with different batch duration, the proposed automatic sequential phase division method is described as follows: Step 1: For the tth sample xit (i = 1, 2, ..., I, t = 1, 2, ..., Li) in the ith batch, the neighborhood set N(xit), i.e. the pseudo timeslice around the data sample xit, can be determined using the proposed method shown in subsection 3.1. Step 2: xit should be normalized using the neighborhood set according to the following equation: x̅ it =

x it − mean(N(x it )) std(N(x it ))

(14)

(12)

mean(N(xit))

where, and std(N(xit)) represent the mean and the standard deviation of N(xit), respectively. Thus, the normalized data of the ith batch is described as X̅ i(J × Ki) = [xi1̅ xi2̅ ... xiK̅ i] (i = 1, 2, ..., I). 733


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Figure 5. Length of each phase for 50 batches (mean ± STD) using the proposed phase division algorithm in injection molding process.

Figure 6. Similarity between the sample and the corresponding phase-representative model (the red vertical line denotes the phase boundary).

u(t − 1) = c, where t is the current sampling time. For the tth sample xt (t ≥ 2), there are two possible cases for the current sample: (1) The sample xt belongs to phase c. (2) The sample xt belongs to phase c + 1. It needs further analysis to identify the current phase information. The monitoring model of phase c is used to monitor the current sample. If the statistics are under the control limits, the process is deemed to be operating normally and the phase information does not change, and thus u(t) = c. Otherwise, the PCA model of phase c + 1 is used to monitor the current sample xt and compared with the control limits of phase c + 1. In-control statistics reveal that the current sample is normal and the process has turned into the next phase with u(t) = c + 1. Otherwise, it indicates that both two phase models cannot well describe the variation of the current sample, revealing that the current process is abnormal. Besides, the phase affiliation of the current fault sample should be determined to indicate which phase model should be adopted for the

monitoring of next sample. Here, the phase affiliation information on similar samples that have been found for the current fault sample is evaluated from which, the phase affiliation is determined by finding the specific phase to which the most number of samples are affiliated. The illustration of the online identification and monitoring are shown in Figure 3. It is worth noting that in practical application, the process is assumed to be abnormal if and only if consecutive d (d equals 10 in this paper) samples go beyond the control limits rather than a single sample, and this can effectively avoid false alarms. In comparison to the traditional methods based on just-in-time-learning strategy, the proposed method can provide the phase information on the current fault samples which can be used for fault diagnosis. Here, the proposed method focuses on analysis of timewise dynamics, i.e., multiphase characteristics. The multimode problem in fact reveals the batchwise dynamics which is common in 734


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Figure 7. Set distance of pseudo time-slice for each sample constructed within (a) a small fixed searching range and (b) the proposed variable searching range.

batch processes owing to alterations of feed stock, fluctuations in the external environment, and various product specifications. Although the proposed method is presented for single-mode batch processes, it can be extended to multimode and multiphase batch processes by analyzing the batchwise dynamics and timewise variations simultaneously. The present work provides the basis for multimode and multiphase process monitoring which deserves further devotion.

shown in Table 1. In this work, different operation recipes of injection are adopted by setting the injection velocity at different values, resulting in different uneven groups regarding the injection phase. 50 normal training batches are conducted, where the shortest and longest batch lengths are 1152 and 1356 samples, respectively. The number of the nearest neighbors equals to 50. 4.2. Illustration of Phase Division. A pseudo time-slice is constructed for each training sample using the VMW-kNN rule proposed in subsection 3.1. Then the steps of sequential phase division are used to divide the injection molding process into eight phases. The four long phases correspond to the four main physical operation phases, i.e., the injection, packing-holding, plastication, and cooling phases. A few short transitional phases emerge between the four main phases, corresponding to the transitional phases from one major phase to the next. A detailed division of a batch process into steady and transient phases can improve process monitoring and diagnosis performance as well as enhance process analysis and understanding. Figure 5 shows the phase division result for 50 batches in injection molding process. From the figure, it can be found that that the uneven problem in injection molding trajectory process mainly existed in the first phase, i.e. the injection phase. The duration of first phase varies greatly in 50 batches in which, the shortest phase covers 45 samples and the longest phase cover 217 samples. The durations of all the other phases vary in a very small range. Figure 6 shows the similarity between the sample of one random batch in the training group and the corresponding phase

4. APPLICATION AND RESULTS 4.1. Process Description. Injection molding31,32 is a typical multiphase batch process, which transforms polymer materials into various shapes and types of products. There are three major operation phases in a typical injection molding process: (1) injection of molten plastic into the mold; (2) packing− holding of the material under pressure; (3) cooling of the plastic in the mold until the part becomes sufficiently rigid for ejection. Besides, plastication takes place in the barrel in the early cooling phase, where polymer is melted and conveyed to the barrel front by screw rotation, preparing for next cycle. A simplified schematic diagram of a reciprocating-screw injection molding process is shown in Figure 4. For process analysis, some important variables such as the temperature, pressure, displacement, and velocity can be measured online by their corresponding transducers, providing abundant process information. The material used in this work is high-density polyethylene (HDPE). Nine process variables are selected for modeling as 735


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Figure 8. Process monitoring of a normal batch using the (a) LNS-PCA; (b) GMM; and (c) the proposed methods.

representative model. From the figure, it can be seen that the similarity between the data sample and the representative model of its corresponding phase is high, while the similarity between the first data sample of each phase and the representative model of the previous phase is very small. It again illustrates that the phase division result is correct and reliable. Figure 7 shows the set distance defined in eq 10 for the pseudo time-slice of each sample in a random batch. In Figure 7a, the set distance is calculated when the small searching range is fixed for each sample. From the plot, it can be seen that when the process is operating in the stable phase, the set distance is very small even within the small searching range, illustrating that the samples in the pseudo time-slice are truly similar to the concerned sample. But when the process is operating within the transitional phases, the set distance is very large for the pseudo time-slice constructed within a small searching range, indicating that the small searching range has not covered

enough similar samples. Figure 7b shows the set distance for the pseudo time-slice constructed using the proposed VMWkNN rule. From the plot, it can be seen that the set distance is small since the searching range has been enlarged for the concerned sample in the transitional phase to cover enough similar samples. 4.3. Illustration of Process Monitoring. After phase division, the monitoring models for each phase are established. To illustrate the proposed algorithm, three test batches, including a normal batch process and two abnormal batches with a check-ring problem and a sensor fault respectively, are introduced to test the proposed process monitoring method. To illustrate the feasibility and effectiveness of the proposed algorithm, two methods including local neighborhood standardization-principle component analysis (LNS-PCA)29 and Gaussian mixture model (GMM)13 are compared with the proposed methods. For GMM, a probability monitoring index 736


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Figure 9. Process monitoring of an abnormal batch with a check-ring problem using the (a) LNS-PCA; (b) GMM; and (c) the proposed methods.

(BIP) proposed by Yu and Qin13 is the most commonly used statistic which is employed for comparison. Besides, the Mahalanobis distance based monitoring index proposed by Xie, etc.,33 is also used for monitoring, termed BID here. The monitoring results for the normal batch are presented in Figure 8. In the T2 plot of Figure 8a, many normal samples jump above the corresponding control limit with false alarms triggered with the overall false alarm rate as high as 8.38%. That is because LNS-PCA can only eliminate influence caused by diversity of mean in different phases, but cannot deal with the differences of the covariance structures of each phase. Therefore, owing to the fact that a global PCA model is actually the statistical average of all operation modes, a global PCA model may lose important local information and may lead to low resolution for some phases when the phases have different

covariance structures. The monitoring results of GMM for the normal batch are presented in Figure 8b. Most of the statistics are under the control limit, while both BIP and BID statistics go beyond the control limit for the transitional phase. It is because that the information contained in transitional phases may be overwhelmed when all historical data are selected to train a GMM. Figure 8c shows the monitoring chart of the proposed method for a normal batch. Most of the statistics are under the control limit in the whole batch process. The check-ring valve is a device that allows the polymer melt to flow from the screw channel to the nozzle during plastication. It should be closed 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 737


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Figure 10. Process monitoring of an abnormal batch with a sensor fault using (a) LNS-PCA; (b) GMM; and (c) the proposed methods.

Table 2. Monitoring Performance Comparison for the Injection Molding Process LNS-PCA type I error (%) type II error (%)

case 1 case 3 case 2 Case3

GMM

proposed method

T2

SPE

BIP

BID

T2

SPE

8.38% 8.32% 2.19% 0

1.34% 0.37% 54.58% 98.29%

6.03% 5.29% 49.54% 0.40%

5.98% 5.18% 48.19% 0.40%

2.43% 0.37% 2.87% 2.66%

0.84% 1.47% 0.42% 0

and 2.19%, respectively. Figure 9b shows the monitoring results of GMM for an abnormal batch with a check-ring problem. The missing alarm rate of BIP statistics and BID statistics are as high as 49.54% and 48.19% respectively with large numbers of faulty samples undetected. Figure 9c shows the monitoring chart of the proposed method for the abnormal batch. The check-ring valve is closed at the beginning and the two statistics are under

and packing stages, resulting in a smaller amount of material being injected into the mold at the same given injection velocity. Figure 9a shows the monitoring results of LNS-PCA for an abnormal batch with a check-ring problem. From Figure 9a, it can be seen that SPE statistics of LNS-PCA perform poorly while T2 statistics performs better on avoiding false negative rates for abnormal samples and their false alarm rate are 54.28% 738



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the control limit in the filling stage. Then the check-ring valve cannot be closed completely and the statistics are beyond the control limit in the rest of the batch which indicated the abnormality. For the sensor fault, the thermocouple sensor is damaged from the 813th data and the measured value of the temperature is zero. Figure 10a shows the monitoring results of LNS-PCA for this abnormality. The T2 statistic in Figure 10a detects this disturbance timely with no time delay while the SPE statistics perform poorly with a high missing alarm of 98.29%. Figure 10b shows the monitoring results by GMM. It is observed that both BIP and BID statistics jump above the corresponding control limit as soon as the fault occurs. However, false alarms are observed particularly in transitional phases. Figure 10c shows the monitoring result by the proposed method. The two statistics are under the control limit before the fault occurs. The two statistics immediately go beyond the control limit in response to the occurrence of sensor fault. Table 2 compares the monitoring performance for three methods regarding normal batches and two fault cases. Compared with LNS-PCA and GMM, the proposed method based on VMW-kNN rule obtained the best performance, with significantly reduced false alarm (type I error) and missing alarm (type II error).

5. CONCLUSIONS In the present paper, a novel monitoring method based on VMW-kNN rule has been proposed for uneven multiphase batch processes. The variable moving window (VMW) strategy can constrain the searching range with reduced computational burden in comparison with the traditional kNN and preserve the local neighborhood information. The similar observations are selected from the historical data within the VMW to resemble each sample by using the kNN rule so that the local model can well describe the underlying characteristics at each time. It avoids the synchronizing problem caused by uneven length. Different phases are thus sequentially obtained for phase model development which can be adopted for online application by properly determining the phase affiliation for each sample. The feasibility and efficiency of the proposed method have been verified for both normal and fault cases in the injection molding process.

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Article

NOMENCLATURE h = shortest duration of the phases I = number of batches J = number of variables k = number of the nearest neighbors in pseudo time-slice Ki = duration in the ith batch L1 = initial window size L2 = enlarged window size n = number of samples q = number of remained principal components E = residual matrix e = residual vector for each data sample N(xit) = neighborhood set for the tth sample in ith batch P = loading matrix P*c = representative model of the cth phase Pit = loading matrix of the pseudo time-slice pj = loading vector T = score matrix tj = score vector X = data matrix Xi = data matrix of ith batch Xt,se = searching range for the tth sample in ith batch X̂ = prediction of the PCA model xit = tth sample in ith batch α = confidence level δ = threshold of set distance

Subcripts

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c = phase index i = batch index j = variable index t = sample index

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (C.Z.). *E-mail: [email protected] (S.W.). ORCID

Chunhui Zhao: 0000-0002-0254-5763 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the support from the following foundations: the National Natural Science Foundation of China (nos. 61422306, 61273166, and 61433005), the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (no. ICT1600197), and the Fundamental Research Funds for the Central Universities (N140404020). 739


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Pseudo time-slice construction using variable moving window-k

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