Fault monitoring of batch process based on multi-stage optimization regularized neighborhood preserving embedding algorithm

Abstract

Batch process is an important type of industrial production process, and the process mechanism is complex. It is difficult to accurately describe the dynamic changes of the production process of multi-stage time-varying batch process. In addition, the data of batch process contain not only global information but also local information. The traditional neighborhood preserving embedded algorithm is used to maintain the local geometric structure of data while ignoring the global information, and the extracted latent variables cannot fully characterize batch process. Therefore, we propose a multi-stage optimization regularized neighborhood preserving embedding (ORNPE) algorithm. First, the multiple process stages are separated by affinity propagation (AP) algorithm. Second, based on maintaining local information of neighborhood preserving embedding algorithm, slow feature analysis algorithm is used to extract dynamic time-varying global information. Then, cross-entropy is used to optimize the global information, and the extraction ability of the global information is improved. Finally, a monitoring index based on support vector data description is constructed to eliminate adverse effects of non-Gaussian data for monitoring performance. The effectiveness and advantages of the proposed algorithm based on monitoring strategy are illustrated by the penicillin fermentation process and a semiconductor industry process.

Keywords

Batch process multi-stage fault detection affinity propagation clustering neighborhood preserving embedding

Introduction

Batch process is widely used in semiconductor, pharmaceutical, injection molding, and other production processes due to its high added value, small production batches, and meeting individual needs. It is a crucial production mode in modern manufacturing (Luo and Bao, 2018; Qin, 2012). At present, as the complexity of the production process increases and the scale continues to expand, the production process also contains vast safety risks and the probability of faults continues to grow (Lavanya et al., 2021; Prasanth, 2021). Therefore, real-time monitoring of process ensures that faults can be detected timely, and accurate fault detection has essential economic value and practical significance (Fu and Zhang, 2017; Jiang et al., 2020; Zhang et al., 2018, 2019, 2020).

With the development of advanced control systems, a large amount of process data is collected and stored, which provides a development basis for data-driven modeling and monitoring technology. Multivariate statistics methods (e.g. Principal Component Analysis (PCA; Abdi and Williams, 2010; Cotrufo and Zmeureanu, 2016)) and Partial Least Squares (PLS; Helland, 2014; Li, 2010) have been widely used in process monitoring. These methods have good performances for dimension reduction and process monitoring. A strategy of multi-directional expansion data processing is used to obtain Multiway Principal Component Analysis (MPCA; Jeffy et al., 2018; Majid et al., 2011) and Multiway Partial Least Squares (MPLS; Camarrone and Van Hulle, 2018; Wang et al., 2016), which are applied in fault monitoring of the process. Besides, some experts and scholars have done a lot of researches (Jiang and Yin, 2018; Peng et al., 2020; Zhang et al., 2021a, 2021b). Although these researches have achieved good results in fault monitoring, they usually consider the global structure of data and ignore the local feature information.

However, He et al. (2005) proposed Neighborhood Preserving Embedding (NPE) algorithm based on Locally Linear Embedding (LLE; Roweis and Saul, 2000). Compared with LLE algorithm, NPE algorithm can more accurately obtain the mapping matrix and has better practicability. In process monitoring, unlike PCA, PLS, and other global structure preserving algorithms, NPE maintains the neighborhood structure by projecting neighboring points in high-dimensional space to low-dimensional space, and pays more attention to the local structure information.

Generally speaking, the occurrence of faults often leads to changes of the global and local structure information. In this sense, it is essential to consider both the global structure information and the local structure information. Therefore, only focusing on the global structure or the local structure can hardly reflect the actual working conditions of batch process. To extract the global and local information of batch process simultaneously, Zhang et al. (2011) proposed a Global-Local Structure Analysis (GLSA) algorithm that considered the extraction of global structure and local structure features. On this basis, Yu (2016) proposed a monitoring model of local and global principal component analysis (LGPCA) algorithm, which could better extract the global and local structure. Zhao et al. (2016) proposed a Global Neighborhood Preserving Embedding (GNPE) algorithm in fault detection of batch process, which combined PCA and NPE algorithm. Xu and Ding (2021) proposed a Manifold Regularized Slow Feature Analysis (MRSFA) algorithm, which gave sufficient consideration to the global time change and local structure information of original data: so that the extracted latent variables can more truly represent the process data. These algorithms preserve the global and local information of the data through different strategies and also achieve better fault monitoring results. However, the lack of process optimization for preserving global information mediately loses part of the data information. As a result, some faults are not detected, and the effect of fault monitoring is directly affected. Because each stage of batch process has its own process mechanism, different stages show different process characteristics. Therefore, batch process has a multi-stage characteristic. However, traditional algorithms such as MPCA, MPLS, Multiway Neighborhood Preserving Embedding (MNPE), and their improved algorithms regard batch process as a single stage in the monitoring process. They do not consider the multi-stage information of batch process. In order to improve the monitoring performance of batch process under the influence of multi-stage characteristic, some researchers have proposed different modeling methods to strengthen process monitoring. Ningyun et al. (2010) used K-Means (KM) algorithm to divide the process into multiple stages through clustering. Zhao and Sun (2013) proposed a step-wise sequential phase division method. Liu et al. (2016) proposed a step-wise sequential phase division method based on windows. Gao et al. (2014) proposed a multi-stage modeling method by using fuzzy clustering, and process data could be divided into multiple categories simultaneously to complete the stage division. Guo et al. (2017) divided the stages according to the characteristics of different stages of batch process. Ge et al. (2012) proposed a new staged method based on a defined repeatability factor. However, the above clustering methods need to set the clustering model parameters in advance during the stage division process. The manual set parameters affect the accuracy of stage segmentation and cannot accurately reflect the stage characteristic of the process.

To solve the accuracy problem of stage division caused by the manual setting of parameters, affinity propagation (AP) clustering algorithm is used to classify data without knowing the number of classifications in advance, which can divide batch process into different operating stages. We utilize Dynamic Time Warping (DTW) algorithm to process the data of different batches at the same stage with equal length. The divided sub-stages have similar data structures and characteristics, which are convenient for statistical monitoring. In each sub-stage, we use ORNPE algorithm for dimension reduction and feature extraction. Fault detection is performed by constructing statistics of T² and R² based on the multi-stage ORNPE algorithm. Finally, the Penicillin fermentation process and semiconductor etching process are used to verify the effect of the proposed algorithm on batch process monitoring.

Batch process is a typical multi-stage process. The data at the same stage have similar characteristics and correlations which can establish a unified model to ensure that the established monitoring model has the smallest error. In process monitoring, the traditional NPE algorithm only considers the local manifold structure of the data, but ignores the global information of the data, resulting in incomplete information. Therefore, it is necessary to improve NPE algorithm to extract important global information while extracting local information.

According to the aforementioned problems, we propose a multi-stage ORNPE algorithm for batch process monitoring. The contributions of this paper are given as follows:

AP clustering algorithm is used to divide batch process into stages. Due to the difference in the stage division of each batch, DTW algorithm processes the same stage of different batches with equal length.

A new dimensionality reduction algorithm named ORNPE is proposed to extract latent variables in each stage, which considers the global dynamic information and the local geometric structure information of the original data, consequently providing a more faithful low-dimensional representation of the original data.

At each stage, since the original data is non-Gaussian, based on the ORNPE algorithm, SVDD algorithm is used to construct statistical indicator to avoid the adverse effects of the non-Gaussian distribution of process data.

The rest of this paper is organized as follows. The section “Preliminaries” introduces the principle of NPE algorithm, AP clustering algorithm, and slow feature analysis algorithm. The section “Multi-stage monitoring process based on ORNPE algorithm” proposes a multi-stage process monitoring method based on ORNPE algorithm. The proposed algorithm is demonstrated through the penicillin fermentation process and the semiconductor industry process in the section “Case studies.” The section “Conclusion” concludes the work.

Preliminaries

NPE

NPE (He et al., 2005) is a manifold learning algorithm that approximates LLE algorithm. It can mine local topological structures and high-dimensional structure relationships. The basic idea is to linearize the original training data set $X (x_{1}, x_{2}, \dots, x_{n}) \in R^{D}$ in high-dimensional space through the nearest neighbors. The dimensionality is reduced through mapping matrix from high-dimension to low-dimensional space. The obtained data set is as follows: $Y = (y_{1}, y_{2}, \dots, y_{d})$ , where $A (a_{1}, a_{2}, \dots, a_{d})$ (d ≤ D) is a mapping from high-dimension space to low-dimensional space and satisfies Y = A^TX. Specific steps are as follows:

Step 1: Construct a neighborhood graph

The Euclidean distance between samples in the training data set $X (x_{1}, x_{2}, \dots, x_{n}) \in R^{D}$ is arranged by the K Nearest Neighbor (KNN) method in ascending order. The first k points of each sample are selected as neighbor points, and a neighborhood graph is formed by comparing and connecting.

Step 2: Calculate the weight matrix

If the sample point x_j is the neighbor point corresponding to the selected connection, its weight coefficient is w_ij. And if there is no connection, then w_ij = 0. All the weight coefficients form the weight matrix W. The weight matrix W can be obtained by minimizing the reconstruction error

Φ (W) = \min \sum_{i = 1}^{n} | | x_{i} - \sum_{j = 1}^{n} w_{ij} x_{j} | |^{2}

(1)

The normalization constraint is $\sum_{j = 1}^{n} w_{ij} = 1$ that the sum of the weights meets the normalization condition, and the result is 1.

Step 3: Calculate the mapping matrix

The mapping matrix A is obtained by solving equation (2) to minimize the cost function

\begin{matrix} f (A) = \sum_{i = 1}^{n} {(y_{i} - \sum_{j = 1}^{k} w_{ij} y_{j})}^{2} \\ = y^{T} (I - W)^{T} (I - W) y \\ = a^{T} X (I - W)^{T} (I - W) X^{T} a \end{matrix}

(2)

That is $A = \arg \min a^{T} X (I - W)^{T} (I - W) X^{T} a$ , where the constraint is $y^{T} y = a^{T} X X^{T} a = 1$ .

We introduce the Lagrange multiplier method to transform the solution of equation (2) into the generalized eigenvalues of equation (3)

XM X^{T} a = λ X X^{T} a

(3)

where $M = (I - W)^{T} (I - W)$ . We can find the eigenvectors corresponding to the first d smallest eigenvalues $(λ_{1} \leq λ_{2} \leq \dots \leq \dots \leq λ_{d})$ from the mapping matrix $A (α_{1}, α_{2}, \dots, α_{d})$ , which satisfies Y = A^TX.

AP clustering

AP clustering (Frey and Dueck, 2007) was a clustering method proposed by Frey and Dueck in 2007. This algorithm does not need to set the number of clusters in advance, but automatically obtains clusters based on the cluster samples in the iterative process. Clustering centers are clustered according to the similarity between data objects. Before clustering, all sample points are regarded as potential cluster centers, and the similarity calculation performs in a loop. Each sample point converges through iteration to obtain the final representative point set of the best category.

Given a data set $X = {x_{1}, x_{2}, \dots, x_{i}}$ , where x_i represents a sample point and d represents the dimension of the data point. There is $x_{i} = {x_{i 1}, x_{i 2}, \dots, x_{id}}$ . The calculation steps of AP clustering algorithm are as follows:

Step 1: Calculate similarity matrix

The similarity between samples selects different measurement criteria according to different scenarios, such as Euclidean distance. Depending on the similarity criteria, the similarity between samples may be symmetrical or asymmetrical. These similarities form a similarity matrix with N×N dimension similarity matrix S (N is the number of data samples). Here, we choose the square of the negative Euclidean distance to calculate the similarity and construct the similarity matrix, that is, $s (i, k) = - ‖ x_{i} - x_{k} ‖^{2}$ , similarity matrix S is shown as equation (4)

S = [\begin{matrix} s (x_{1}, x_{1}) & s (x_{1}, x_{2}) & \dots & s (x_{1}, x_{N}) \\ s (x_{2}, x_{1}) & ⋱ & \dots & s (x_{2}, x_{N}) \\ ⋮ & ⋮ & s (x_{k}, x_{k}) & ⋮ \\ s (x_{N}, x_{1}) & s (x_{N}, x_{2}) & \dots & s (x_{N}, x_{N}) \end{matrix}]

(4)

Step 2: Divide clusters based on cluster centers

To select the appropriate cluster centers, the degree of attribution a(i, k) and the degree of attraction r(i, k) are defined. The degree of attribution a(i, k) indicates that the data point x_i belongs to one of the categories represented by the data point x_k. The stronger the numerical information expressed by a(i, k) and r(i, k), the greater the probability that the point is the center of the cluster. The degree of attribution a(i, k) and the degree of attraction r(i, k) are shown in equations (5) and (6), and the initial value is 0

a (i, k) = {\begin{matrix} \min {0, r (k, k) + \sum_{i^{'} \notin ∣ i, k]} \max {0, r (i^{'}, k)}}, i \neq k \\ \sum_{i^{'} \neq k} \max {0, r (i^{'}, k)}, i = k \end{matrix}

(5)

r (i, k) = s (i, k) - \max_{k^{'} \neq k} {a (i, k^{'}) + s (i, k^{'})}

(6)

The damping coefficient is added to adjust the convergence speed during each iteration update, which prevents oscillations during the iteration process. By repeating the above steps, equations (7) and (8) are updated

r_{t + 1} (i, k) = λ \times r_{t} (i, k) + (1 - λ) \times r_{t + 1} (i, k)

(7)

a_{t + 1} (i, k) = λ \times a_{t} (i, k) + (1 - λ) \times a_{t + 1} (i, k)

(8)

Cluster center point H is obtained. The original data set is divided into H clusters through cluster center point H.

Slow feature analysis

Slow feature analysis (SFA; Shang et al., 2016) is a dimensionality reduction algorithm, which can find a projection matrix to convert time series data into latent variables that change as slowly as possible, which latent variables contain the main information of the original data. The specific steps are as follows:

Given an m-dimensional input signal $x (t) = [x_{1} (t), x_{2} (t), \dots, x_{i} (t), \dots, x_{m} (t)]^{T}$ , where $t \in [t_{0}, t_{1}]$ represents time, SFA seeks a transformation function $g (x) = [g_{1} (x), \dots, g_{J} (x)]^{T}$ to change the output $y (t) = {[y_{1} (t), \dots, y_{J} (t)]}^{T}$ of the function as slowly as possible, where $y_{j} (t) = g_{j} (x (t))$ . By minimizing the variance of the first derivative of the slow feature, the optimization of SFA is derived

\min {〈 {\overset{\cdot}{y}}_{i}^{2} 〉}_{t}

(9)

And, three constraints are derived

{〈 y_{i} 〉}_{t} = 0

(10)

{〈 y_{i}^{2} 〉}_{t} = 1

(11)

\forall i \neq j, 〈 y_{j} y_{i} 〉 = 0

(12)

where ${\overset{\cdot}{y}}_{i}$ represents the first-order derivative of the slow feature and ${〈〉}_{t}$ represents the average over time, which is defined as equation (13)

{〈 f 〉}_{t} \dot{=} \frac{1}{t_{1} - t_{0}} \int_{t_{0}}^{t_{1}} f (t) dt

(13)

The constraint condition of equation (10) ensures that the mean value of the extracted slow feature information is zero. The purpose of equation (11) is to exclude the trivial solution of the output signals. The constraint condition of equation (12) guarantees that the output signals are multiple. The components are uncorrelated so that they carry different aspects of information.

The slow feature of x can be expressed as y. The projection matrix $A = [α_{1}, α_{2}, \dots, α_{l}]$ is obtained by two-step singular value decomposition.

To whiten the slow feature, singular value decomposition of the covariance matrix of the original data is performed, and shown as equation (14)

{〈 x x^{T} 〉}_{t} = U \sum U^{T}

(14)

where U represents the eigenvector and ∑ represents the eigenvalue. The data after whitening is c and meets $cov (c) = {〈 c c^{T} 〉}_{t} = I$ , and the data after whitening eliminates the cross-correlation between variables. The covariance of the first derivative of c is found, and the second step of singular value decomposition is performed, as equation (15)

{〈 \overset{\cdot}{c} {\overset{\cdot}{c}}^{T} 〉}_{t} = Ξ^{T} Ω Ξ

(15)

The slow feature of x is obtained by equation (16)

y = Ξ c = Ξ^{\frac{- 1}{2}} \sum U^{T} x

(16)

where $Ω = diag {w_{1}, w_{2}, \dots, w_{m}}$ is composed of eigenvalues and arranged on its diagonal in ascending order to satisfy $w_{i} = {〈 {\overset{\cdot}{y}}_{i}^{2} 〉}_{t}$ , thus it ensures that the slowest feature has the lowest exponent. The matrix $Ξ$ is composed of eigenvectors corresponding to eigenvalues.

Multi-stage monitoring process based on ORNPE algorithm

This section introduces the process of multi-stage fault monitoring based on ORNPE algorithm: including data preprocessing, stage division, algorithm model and establishment of statistical indicators.

Three-dimensional expansion of batch process data

Compared to continuous process, batch process has three-dimensional data. When fault diagnosis of batch process is performed, it is necessary to unfold the three-dimensional data into the two-dimensional data to establish a fault diagnosis model. Therefore, the three-dimensional data expansion method is used to process the three-dimensional data. First, the three-dimensional data are unfolded as two-dimensional data along the batch direction, as shown in equation (17), then, it is standardized by column, and the standardized two-dimensional matrix is arranged according to the variable direction to form a two-dimensional matrix, as shown in equation (18). The expansion method can eliminate data pre-estimation problems during online monitoring and can monitor process changes more effectively

X (I \times KJ) = [\begin{matrix} x_{1, 1}^{1}, x_{2, 1}^{1} \dots x_{J, 1}^{1} \dots x_{1, K}^{1}, x_{2, K}^{1} \dots x_{J, K}^{1} \\ x_{1, 1}^{2}, x_{2, 1}^{2} \dots x_{J, 1}^{2} \dots x_{1, K}^{2}, x_{2, K}^{2} \dots x_{J, K}^{2} \\ ⋮ ⋱ ⋮ \\ x_{1, 1}^{I}, x_{2, 1}^{I} \dots x_{J, 1}^{I} \dots x_{1, K}^{I}, x_{2, K}^{I} \dots x_{J, K}^{I} \end{matrix}]

(17)

where $x_{j, k}^{i}$ represents the variable x_j of the ith batch at the kth sampling time

X (K I \times J) = [\begin{matrix} [(x_{1, 1}^{1}, x_{1, 1}^{2} \dots x_{1, 1}^{J}) & \dots & (x_{1, 1}^{1}, x_{1, 1}^{2} \dots x_{1, 1}^{J})]^{T} \\ [(x_{2, 1}^{1}, x_{2, 1}^{2} \dots x_{2, 1}^{J}) & \dots & (x_{2, 1}^{1}, x_{2, 1}^{2} \dots x_{2, 1}^{J})]^{T} \\ ⋮ & ⋱ & ⋮ \\ [(x_{K, 1}^{1}, x_{K, 1}^{2} \dots x_{K, 1}^{J}) & \dots & (x_{K, I}^{1}, x_{1, I}^{2} \dots x_{K, I}^{J})]^{T} \end{matrix}]

(18)

where $x_{k, i}^{j}$ represents the variable x_j of the ith batch at the kth sampling time.

The process of data obtained by the expansion method is shown in Figure 1 (I represents the batch of data, J represents the variable of data, K represents the sample time of data).

Figure 1.

The process of three-dimensional variable expansion.

Stage division based on AP clustering

After three-dimensional data are processed, AP clustering algorithm is used to divide the process data into different stages. The steps of AP clustering are as follows:

Step 1: Divide data into the matrix $X = {(x^{1}, \dots, x^{i}, \dots, x^{I})}^{T}$ , where xⁱ represents the ith batch of data.

Step 2: Slice X along time to get K time slices and calculate the similarity matrix S according to equation (4).

Step 3: Take $i (i = 1, 2, \dots, I)$ batch data as the AP clustering input and replace s(k, k) in the similarity matrix s(i, k) with the optimal bias parameter p. We take the average of all similarities as the setting value of p, and the number of clusters obtained at this time is medium. If the value of the minimum similarity is taken as the value of p, the number of clusters obtained is small. The AP clustering bias parameter divides the batch process into stages. The process is initially divided into n stages.

The stage division of each batch is shown in Figure 2.

Figure 2.

Unequal length of batch data.

Sub-stage unequal length processing based on DTW algorithm

DTW is an algorithm of speech recognition technology. The algorithm can find the optimal curve path, and match the coordinate points in one time series with the coordinate points with the most significant feature similarity in another time series.

First, for two unequal length series, the distance between two series as matrix d and the distance between series g_i and h_i are defined as equation (19)

d (g_{i}, h_{j}) = ‖ g_{i} - h_{j} ‖_{p}

(19)

where $‖ ‖_{p}$ represents the norm and p usually takes the value 2.

DTW algorithm synchronizes two unequal length time series by searching for the shortest distance and the optimal path, and optimization performance index of the shortest distance is shown as equations (20) and (21)

D (m, n) = \min_{F} [\sum^{\underset{L}{l = 1}} d (i (l), j (l))]

(20)

W = \arg \min_{F} [D (m, n)]

(21)

where $\max (m, n) \leq L \leq m + n$ , D(m, n) is the sum of the two series of local shortest distances along the optimal path, the optimal path $W = {w_{1}, w_{2}, \dots, w_{l}} w_{l} = (i (l) j (l))$ is a series searched in the m×n grid based on the shortest distance D(m, n).

To avoid excessive data distortion and skipping, the W setting of each step is continuous. There are three options for the points before the grid point: point (i– 1, k), point (i– 1, k– 1), point (i– 1, k– 2); therefore, equation (20) can be obtained by equation (22)

D (i, k) = {\begin{matrix} D (i, k) + d (i, k) \\ D (i - 1, k - 1) + d (i, k) \\ D (i - 1, k - 2) + d (i, k) \end{matrix}}

(22)

After the cumulative distance matrix D(m, n) is determined, the last element is used to find the point where the optimal path passes gradually. Finally, we find the corresponding data to be synchronized through the optimal path value. The newly obtained series is a synchronized series. DTW algorithm is used to regularize the sub-stages of different batches, and the data of each batch after equal length is shown in Figure 3.

Figure 3.

Each batch of data after the equal length.

Monitoring scheme based on ORNPE

Cross-entropy

Cross-entropy represents the similarity of two probability density functions, and it has high efficiency for multi-objective global optimization. It can update the rules through analysis and calculation to make the optimization path more efficient and faster.

Cross-entropy is used to globally optimize the two-dimensional data matrix and feature reduction through multiple iterations. Defining the probability density functions as f and g, the definition of cross-entropy is given in equation (23)

\begin{matrix} D (f (x) ∥ g (x)) = \sum_{x \in X} f (x) \ln \frac{f (x)}{g (x)} \\ = E_{f} (x) \ln \frac{f (x)}{g (x)} \end{matrix}

(23)

where E() is the mathematical expectation.

From equation (23), we know that $D (f (x) ∥ g (x)) \geq 0$ , because cross-entropy is a downward convex function, there is only f(x) = g(x), which meets $D (f (x) ∥ g (x)) = 0$ . Therefore, cross-entropy characterizes the similarity of the two distributions. The more similar the two distributions, the smaller the cross-entropy between them. For matrix $A (M \times N) f (i, j) (\geq 0)$ , the definition of two-dimensional entropy of the matrix is given by equation (24)

H = - \sum^{\underset{M}{i = 1}} \sum^{\underset{N}{j = 1}} p_{ij} \log p_{ij}

(24)

p_{ij} = \frac{f (i, j)}{\sum^{\underset{M}{i = 1}} \sum^{\underset{N}{j = 1}} f (i, j)}

(25)

The iterative optimization of cross-entropy can significantly improve the global structure of the matrix after dimensionality reduction, thereby improving the fault detection rate.

Regularized NPE

When a fault occurs, it usually causes the global and local structure of normal data to change. Only considering local or global information would inevitably lead to the loss of some information. Therefore, in the process of fault diagnosis, global structure information and local structure information have the same importance.

SFA algorithm only reveals the time change of the global information without considering the local geometric structure. NPE algorithm only considers the local structure. Therefore, to consider the global and local information of the process data simultaneously, a global–local feature extraction algorithm is constructed through SFA and NPE, which comprehensively considers the global time change and local structure information of original data.

Equation (9) can be rewritten as equation (26) according to the trace

J_{S F A} = \sum^{\underset{n - 1}{i}} {‖ {\dot{y}}_{i} ‖}^{2} = t r ({\dot{Y}}^{T} \dot{Y}) = t r (K^{T} ({\dot{Z}}^{⊤} \dot{Z}) K)

(26)

where K is the projection matrix, tr() denotes the trace of a matrix, and $\overset{\cdot}{Z} = [{\overset{\cdot}{z}}_{1}, {\overset{\cdot}{z}}_{2}, \dots, {\overset{\cdot}{z}}_{n - 1}]^{T}$ , evidently, $Y^{T} Y = K^{T} (Z^{T} Z) K = K^{T} K = I$ .

We introduce cross-entropy algorithm in the training set and calculate the value of the objective function corresponding to each sample. We sort from small to large and update the probability distribution function by using the sample features with the larger function value. In this way, the sample quality is continuously improved globally, and finally, the optimal value is obtained

J_{SFA} = \sum_{I}^{n - 1} {‖ {\overset{\cdot}{y}}_{i} ‖}^{2} S_{ij}^{p}

(27)

where $S_{ij}^{p}$ is the corresponding cross-entropy probability matrix.

Because the data are sampled at discrete intervals, therefore, the first-order derivative of the collected data can be approximate as the corresponding first-order difference

{\overset{\cdot}{z}}_{j} (t) \approx \frac{z_{j} (t) - z_{j} (t - Δ t)}{Δ t}

(28)

Therefore, the objective function of ORNPE algorithm is given by equation (29)

\begin{matrix} argmin (\sum_{i}^{n - 1} {‖ \overset{\cdot}{y} ‖}^{2} S_{ij}^{p} + η \sum_{i}^{n - 1} {‖ y_{i} - \sum_{j}^{n - 1} w_{ij} y_{j} ‖}^{2}) \\ = tr ({\overset{\cdot}{Y}}_{S}^{T} {\overset{\cdot}{Y}}_{S}) + η tr (Y^{T} {(I - W)}^{T} (I - W) Y) \\ = tr (K^{T} ({\overset{\cdot}{Z}}^{T} \overset{\cdot}{Z}) K) + η tr (K^{T} (Z^{T} MZ) K) \\ = tr (K^{T} HK) \end{matrix}

(29)

where η is the tuning parameter, $H = {\overset{\cdot}{Z}}^{T} \overset{\cdot}{Z} + η Z^{T} MZ$ , and $M = (I - W)^{T} (I - W)$ . Equation (29) can be obtained by seeking the generalized characteristic solution of equation (30)

HK = K Ψ

(30)

According to the reference (Xu and Ding, 2021), the tuning parameters η can be determined by the spectral radius, shown as equation (31)

η = \frac{ρ (Φ)}{ρ (G) + ρ (Φ)}

(31)

where $G = {\overset{\cdot}{Z}}^{T} \overset{\cdot}{Z}$ , $Φ = Z^{T} MZ$ , and ρ() denotes the spectral radius of a matrix.

Monitoring statistics based on Support Vector Data Description algorithm

The goal of Support Vector Data Description (SVDD) algorithm is to find a hypersphere with the smallest volume that contains all or most of the training data. A relaxation factor ξ_i and a penalty factor C are introduced to successfully solve outliers’ influence caused by measurement errors and noise interference. Here, we apply SVDD algorithm to monitor the data distribution in the projection matrix $Y = [y_{1}, y_{2}, \dots, y_{i}, \dots, y_{n}]$ , and monitoring statistics are constructed. For the projection matrix Y of the normal working condition data, the SVDD optimization problem is given in equation (32)

\begin{matrix} \min_{R, a, ξ_{i}} (R^{2} + C \sum_{i} ξ_{i}) \\ s . t . ‖ ϕ (x_{i}) - ε^{2} ‖ \leq R^{2} + ξ_{i} \end{matrix}

(32)

where ε is the center of the sphere and R is the radius of the hypersphere $ξ_{i} \geq 0, i = 1, 2, 3, \dots, n$ .

The optimization problem of equation (33) is transformed into a dual form, shown as equation (34)

\begin{matrix} \min_{α_{i}} \sum^{\underset{n}{i = 1}} α_{i} 〈 ϕ (x_{i}) \times ϕ (x_{i}) 〉 - \sum^{\underset{n}{i = 1}} \sum^{\underset{n}{j = 1}} α_{i} α_{j} 〈 ϕ (x_{i}) \times ϕ (x_{j}) 〉 \\ s . t . \sum_{i} α_{i} = 1, 0 \leq α_{i} \leq C \end{matrix}

(33)

where α is the Lagrangian factor. The kernel function K(x_i×x_j) is used to replace the inner product $〈 ϕ (x_{i}) \times ϕ (x_{j}) 〉$ to realize the linear conversion of the nonlinear problem in the high-dimensional kernel space and obtain the dual equation (34)

\begin{matrix} \min_{α_{i}} \sum_{i} α_{i} K (x_{i} \times x_{i}) - \sum^{\underset{n}{i = 1}} \sum^{\underset{n}{j = 1}} α_{i} α_{j} K (x_{i} \times x_{j}) \\ s . t . \sum_{i} α_{i} = 1, 0 \leq α_{i} \leq C \end{matrix}

(34)

The radius R and the ε core of the hypersphere are shown as equations (35) and (36)

ε = \sum^{\underset{n}{i = 1}} α_{i} φ (x_{i})

(35)

R^{2} = K (x_{k} \times x_{k}) - 2 \sum^{\underset{n}{i = 1}} α_{i} K (x_{k} \times x_{i}) + \sum^{\underset{n}{i = 1}} \sum^{\underset{n}{j = 1}} α_{i} α_{j} K (x_{i} \times x_{j})

(36)

where x_k is the support vector of SVDD.

The distance between the new sample and the center of the hypersphere can be expressed by equation (37)

\begin{matrix} R_{new}^{2} = ‖ φ (x_{new}) - ε ‖^{2} \\ = K (x_{new} \times x_{new}) - 2 \sum^{\binom{i = 1}{n}} α_{i} K (x_{new} \times x_{i}) \\ + \sum^{\binom{i = 1}{n}} \sum^{\binom{j = 1}{n}} α_{i} α_{j} K (x_{i} \times x_{j}) \leq R^{2} \end{matrix}

(37)

If $R_{new}^{2} \leq R^{2}$ , the sample is normal, otherwise, it is abnormal.

Monitoring process

Offline training

Step 1: Under normal working conditions, obtain training sample set $X \in R^{m \times n}$ , and the three-dimensional data are unfolded according to batch-variable method and standardized. The method for training sample X(I×J×K), first, along the batch direction, according to equation (17), a two-dimensional matrix X(I×JK) is obtained, and then it is arranged as X(IK×J) along the variable path according to equation (18).

Step 2: According to section “Stage division based on AP clustering,” AP clustering algorithm is used to divide multiple batch processes into stages, and DTW algorithm is used to perform equal length processing on the divided sub-stages.

Step 3: According to section “Monitoring scheme based on ORNPE,” ORNPE algorithm is used for feature extraction for each division stage to obtain the low-dimensional structure $Y = [y_{1}, y_{2}, \dots, y_{d}]$ .

Step 4: The kernel density estimation method is used to obtain the control limits of offline training samples.

Online monitoring

Step 1: Process data are collected online, unfolded, and standardized by using the batch-variable method. The preprocessed test data are obtained.

Step 2: The preprocessed data are divided into stages according to the stage division in the offline modeling process.

Step 3: For each stage, ORNPE algorithm is used to obtain the low-dimensional structure y_new, in which the R² and T² statistics of the online data are obtained.

Step 4: The online statistics are compared with the control limits to determine whether the limits are exceeded. If the control limit is exceeded, the fault occurs, otherwise, go back to Step 1.

The fault detection of batch process based on multi-stage ORNPE algorithm includes offline modeling and online detection process. The flow chart is shown in Figure 4.

Figure 4.

Flow chart of multi-stage ORNPE algorithm.

Case studies

In this section, the effectiveness of the multi-stage ORNPE algorithm is verified through two examples of batch processes. The two examples are the penicillin fermentation simulation process and the actual semiconductor industry process—the AI reactor corrosion process. They are typical batch processes. A modified multi-stage GNPE method (Yao et al., 2021), Multiway Slow Feature Analysis (MSFA; Shumei Zhang and Zhao, 2018), MNPE (Sun et al., 2018), and MPCA (Zhaomin et al., 2014) are used for comparison.

Penicillin fermentation simulation process

Penicillin is an antibiotic that has been widely used in clinical medicine, and its fermentation process is a typical batch process—the schematic diagram of the penicillin fermentation process is shown in Figure 5. To facilitate data monitoring analysis, the Pensim2.0 simulation platform designed by Birol et al. (2002), Chicago Illinois Institute of Technology, USA, is selected to simulate batch process. The simulation platform generates batch data of different conditions by setting the initial conditions. The Pensim2.0 simulation platform can set three types of faults: Aeration rate fault, agitator power fault, and substrate feeding rate fault. The introduced fault signal types include step signal and ramp signal, which can effectively simulate various variables in many different types.

Figure 5.

Schematic diagram of the penicillin fermentation process.

We select 10 process variables of 18 variables in the platform as experimental test variables, as shown in Table 1.

Table 1.

Detection variables.

No.	Process variables
1	Aeration rate (L/h)
2	Agitator power (r/min)
3	Substrate feed flow rate (L/h)
4	Substrate feed flow temperature (K)
5	Substrate concentration (L/h)
6	Dissolved oxygen concentration (%)
7	Reactor volume (L)
8	Fermentation temperature (K)
9	PH
10	CO₂ (%)

We collect 30 batches of data under normal operating conditions on this platform as training samples. The reaction time of each batch is set to 400h, and the sampling interval is set to 1h. To get the closer actual fermentation process, we add Gaussian white noise to the process data. A training data set X(30 × 400 × 10) is obtained. The 30 batches of obtained training data are unfolded into a two-dimensional matrix and divided into stages. The results of the stage division are shown in Figure 6. The penicillin fermentation process is divided into three stages: the first stage is $1 ~ 43 h$ , the second stage is $44 ~ 169 h$ , and the third stage is $170 ~ 400 h$ .

Figure 6.

Results of stage division of penicillin fermentation process.

The corresponding fault batches are formed by adding step and ramp disturbances into the three variables of aeration rate, agitator power, and substrate feeding rate, as shown in Table 2. To make the process data closer to the actual fermentation process, all data variables are added Gaussian white noise.

Table 2.

Fault batch samples.

Fault no.	Fault variable name	Disturbance type	Amplitude (%)	Fault introduction period
F1	Aeration rate	Step	4	100–300 h
F2	Aeration rate	Ramp	2	300–400 h
F3	Agitator rate	Step	2	250–350 h
F4	Agitator rate	Ramp	4	300–00 h
F5	Substrate feeding rate	Step	3	200–300 h
F6	Substrate feeding rate	Ramp	0.8	250–400 h

The detection rates of MPCA, MSFA, MNPE, multi-stage GNPE, and multi-stage ORNPE algorithms for the faults F1 and F6 in Table 2 are shown in Table 3. We can see that fault F1 and fault F2 are the easiest to be detected, and all algorithms can fully detect them. The proposed algorithm in this paper has the highest detection rates for faults F3–F6. Ramp fault disturbances are more challenging to be detected due to the slow changes of the variables. The fault caused by the substrate feeding rate variable is more difficult to be detected, this is because the propagation speed of the reaction substrate is slow.

Table 3.

Fault detection rate of each fault batch by different methods.

Fault no.	MPCA		MNPE		MSFA		multi-stage GNPE		multi-stage ORNPE
Fault no.	T ²	SPE	T ²	SPE	S ²	SPE	T ²	SPE	T ²	R ²
F1	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
F2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
F3	0.556	0.621	0.596	0.684	0.699	0.732	0.745	0.809	0.911	0.932
F4	0.780	0.910	0.550	0.770	0.640	0.780	0.920	0.940	0.980	0.990
F5	0.357	0.383	0.543	0.497	0.552	0.649	0.675	0.773	0.717	0.803
F6	0.903	0.908	0.931	0.944	0.933	0.940	0.951	0.955	0.971	0.972

Figure 7 shows the fault monitoring results of fault F4 by using MPCA, MNPE, MSFA, multi-stage GNPE, and the multi-stage ORNPE algorithm. Fault F4 is agitation rate, and it added a ramp disturbance with a fault amplitude of 4% between $300 ~ 400 h$ , Figure 7(a) is the statistics T² and SPE monitoring diagram of MPCA under fault F4, and it can be seen that the fault is detected at the 322h and 309h, respectively, and the fault false alarm is serious under normal operating conditions. Figure 7(b) is the statistics T² and SPE monitoring diagram of MNPE algorithm under fault F4, and it can be seen that the fault is detected at the 345h and 323h, respectively, and compared with MPCA algorithm, the fault false alarm is more miniature. But when the fault is detected, the time is delayed, which means that the fault cannot be detected in time. Figure 7(c) is the statistics S² and SPE monitoring diagram of MSFA algorithm under fault F4, and it can be seen that the fault is detected at the 336h and 322h respectively, which the fault false alarms are fewer than MPCA algorithm, and the fault is detected earlier than MNPE algorithm. Figure 7(d) is the statistics T² and SPE monitoring diagram of multi-stage GNPE algorithm under fault F4, and it can be seen that the fault is detected at the 308h and 306h, respectively, and due to the multi-stage characteristic of the penicillin fermentation process, multi-stage GNPE algorithm performs modeling and monitoring in stages to reduce fault false alarms under normal operating conditions. Multi-stage algorithm can detect the occurrence of faults in a timely manner than single-stage algorithms. Figure 7(e) is the statistics T² and R² monitoring diagram of the proposed algorithm in this paper under fault F4, and it can be seen that the fault is detected at the 302h and 301h, respectively. There are few fault false alarms under normal operating conditions, and compared to other algorithms, the proposed algorithm can detect the fault earlier and have a better monitoring effect.

Figure 7.

Monitoring results for the penicillin fermentation process of the fault F4: (a) MPCA, (b) MNPE, (c) MSFA, (d) multi-stage GNPE, and (e) multi-stage ORNPE.

For process data, fault detection is delayed due to the dynamic characteristic of the data. The proposed algorithm uses SFA to extract dynamic characteristic and effectively deal with the influence of dynamic characteristic on detection delay. In addition, the proposed algorithm also considers the stage characteristic of the data. For the stage of fault occurrence, the calculation of statistics is more in line with the actual situation. Therefore, the real-time performance of fault detection is better.

Figure 8 is a bar graph that compares the average fault detection rates of selected fault batches. From Figure 8, we can see the superiority of the proposed algorithm for the fault detection of the penicillin fermentation process.

Figure 8.

Graph of average fault detection rate of fault batches during penicillin fermentation.

Semiconductor industry process

The semiconductor industry process—AI reactor corrosion process is applied to compare the performances of different fault detection algorithms. The data are derived from the actual data of the semiconductor production process of Texas Instruments (Azamfar et al., 2020). It is a typical complex multi-stage batch process. The data set is composed of 107 normal batches and 20 fault batches, and the fault batch contains Tape Carrier Package (TCP) power fault, Radio Frequency (RF) power fault, pressure fault, CI₂ fault or BCI₃ flow fault. We select 17 variables from 40 measured variables as test variables, as shown in Table 4.

Table 4.

Test variables.

No.	Measured variables	No.	Measured variables
1	BCI₃ Flow	10	RF power
2	CI₂ Flow	11	RF impedance
3	RF bottom power	12	TCP tuner
4	Endpoint A Detector	13	TCP phase error
5	Helium pressure	14	TCP impedance
6	Chamber pressure	15	TCP top power
7	RF tuner	16	TCP load
8	RF load	17	Vat valve position
9	Phase error	—	—

In this experiment, we select 50 normal batches for modeling, and each batch is 95 hours in length. We select 17 variables to monitor process status. The three-dimensional normal sample modeling data matrix is X(50 × 17 × 95). The new batch of test data is $X_{test} (95 \times 17)$ . The semiconductor industry process is divided into four stages: $0 ~ 5 h$ , $6 ~ 43 h$ , $44 ~ 60 h$ , and $61 ~ 95 h$ . The results of the stage division are shown in Figure 9.

Figure 9.

Results of stage division of semiconductor industry process.

This experiment selects TCP+50, RF-12, BCl₃+5, Pr-2, Cl₂-5, and He chuck pressure fault batches to introduce faults in all periods of each batch. Table 5 shows the fault detection rates of six types of faults under different algorithms. It can be seen from Table 5 that although the fault detection rate from MPCA, MNPE to MSFA has increased in turn. Because they are single-stage detection algorithms, the overall detection effects of various faults are poor. Multi-stage GNPE algorithm has a higher detection rate than the single-stage algorithms, multi-stage ORNPE algorithm proposed in this paper has a higher detection rate than multi-stage GNPE algorithm.

Table 5.

Fault detection rates of each fault type by different algorithms.

Fault type	MPCA		MNPE		MSFA		Multi-stage GNPE		Multi-stage ORNPE
Fault type	T ²	SPE	T ²	SPE	S ²	SPE	T ²	SPE	T ²	R ²
TCP+50	0.323	0.351	0.379	0.382	0.401	0.442	0.698	0.833	0.794	0.820
RF-12	0.310	0.368	0.336	0.402	0.399	0.437	0.599	0.779	0.801	0.812
BCl₃+5	0.322	0.372	0.411	0.454	0.405	0.532	0.671	0.831	0.779	0.802
Pr-2	0.347	0.389	0.421	0.436	0.368	0.452	0.757	0.884	0.852	0.915
Cl₂-5	0.331	0.441	0.401	0.427	0.399	0.411	0.597	0.644	0.799	0.852
He Chuck	0.348	0.339	0.397	0.422	0.402	0.541	0.661	0.689	0.833	0.901

Figure 10 shows the fault monitoring results of the fault batch Pr-2 by using MPCA, MNPE, MSFA, multi-stage GNPE, and multi-stage ORNPE algorithm. It can be seen from Figure 10(a)–(c), three algorithms have seriously missed fault alarms during the fault detection process, and more faults are not detected. It can be seen from Figure 10(a)–(c) that the fault detection effect is gradually getting better. The detection effect of statistic SPE is better than the statistics T²/S². Still, MPCA, MNPE, and MSFA belong to single-stage detection algorithms, so their detection effects for batch process are poor. The detection algorithms used in Figure 10(d) and (e) are multi-stage detection algorithms. Figure 10(d) is the statistical monitoring chart of multi-stage GNPE algorithm. Figure 10(e) is the statistical monitoring chart of multi-stage ORNPE algorithm proposed in this paper, which has better detection results than the multi-stage GNPE algorithm of Figure 10(d). The main reason is that the proposed algorithm preserves the dynamic global details of the data while considering the local information of the process data.

Figure 10.

Monitoring results for semiconductor industry process of the Pr-2: (a) MPCA, (b) MNPE, (c) MSFA, (d) multi-stage GNPE, and (e) multistage ORNPE.

Figure 11 is a bar graph that compares the average fault detection rates of selected semiconductor industrial process for detection algorithms under multiple fault batches. The fault detection effect of the multi-stage ORNPE algorithm is better than other four algorithms. The proposed algorithm can better monitor the process.

Figure 11.

Graph of average fault detection rate of fault batches during semiconductor etching.

Conclusion

In this paper, we propose a fault monitoring strategy based on multi-stage ORNPE for batch process. First, after the initial division of each batch of data into stages, the same stages of different batches are processed in equal length. Second, on the basis of NPE which can reveal the local structure information and lose global information of data, a global objective function is established by cross-entropy optimized SFA to extract both essential features and dynamic global information of the process data. Finally, the statistics are built to monitor batch process. The penicillin fermentation process and a real semiconductor process are adopted to verify the effectiveness and superiority of the proposed algorithm. For the penicillin fermentation process, the average fault detection rates of statistics T² and R² of the proposed algorithm are 0.95 and 0.93, the average detection rates of statistics T² and R² of the semiconductor process are 0.85 and 0.81, respectively. The detection rates are higher than other comparison algorithms. Compared with the comparison algorithms, the proposed algorithm in this paper increases the calculation amount, but the computational complexity still is O(n²) since the proposed algorithm only considers fault detection and does not consider the identification of fault variables. In the future, based on improving the effect of fault detection, the further research on fault diagnosis and fault prediction will be carried out.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by funding received from the National Nature Science Foundation of China (No. 61763029), the Science and Technology Project of Gansu Province (21YF5GA072, 21JR7RA206), Open Fund project of Gansu Provincial Key Laboratory of Advanced Control for Industrial Process (2022KX07), and the National Key Research and Development Plan (2020YFB1713600).

ORCID iDs

Xiaoqiang Zhao

Kai Liu

Yonyong Hui

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