Recursive transformed component statistical analysis for incipient fault detection

doi:10.1016/j.automatica.2017.02.028

Automatica

Volume 80, June 2017, Pages 313-327

https://doi.org/10.1016/j.automatica.2017.02.028 Get rights and content

Abstract

This paper presents a new data-driven process monitoring method called recursive transformed component statistical analysis (RTCSA) for the purpose of incipient fault detection. Without space partition, RTCSA processes data in sliding windows to obtain orthogonal transformed components (TCs) recursively using rank-one modification. The statistical information of TCs can reveal some important process features, implying that faults can be detected by monitoring the statistics of TCs. With second-order statistics, the detection index reduces to relative changes of ordered eigenvalues of the sample covariance matrix. Fault detectability is analyzed in a statistical sense, leading to the analysis of the eigenvalues of stochastic matrices, including the closed-form expressions for the probability distribution function of the arbitrary $l$ th largest eigenvalue of a class of real uncorrelated Wishart matrices. It indicates that a scaled ordered eigenvalue is sensitive to small changes. The structure of the detection index ensures that RTCSA is sensitive to incipient faults. Compared with existing multivariate statistical process monitoring approaches such as principal component analysis (PCA) and its variants, the superior detectability of RTCSA is illustrated by a numerical example and the Tennessee Eastman process.

Introduction

Recent years have witnessed an increase in the demand for safety and reliability of modern industrial processes. Along these lines, data-driven process monitoring has attracted considerable interest owing to the merit that neither system models nor a priori fault information is required (Ge et al., 2013, Qin, 2012, Yin et al., 2014, Yin et al., 2015). As an important branch of data-driven process monitoring techniques, multivariate statistical process monitoring (MSPM) has been successfully applied in various industrial processes, including chemicals, polymers, and microelectronics manufacturing (Qin, 2003). Principal component analysis (PCA), which is an important method of multivariate analysis, has been widely used in various fields such as data compression, feature extraction, pattern recognition, and process monitoring (Ding, 2014, Duan et al., 2012, Lloyd et al., 2014, Price et al., 2006, Ringnér, 2008, Sun et al., 2008). As a basic technique of MSPM, PCA plays an important role in numerous industrial processes for fault detection and diagnosis (Chiang et al., 2000, Kruger et al., 2007). Its variants such as recursive PCA (RPCA) (Li, Yue, Valle-Cervantes, & Qin, 2000), dynamic PCA (DPCA) (Ku et al., 1995, Russell et al., 2000), and kernel PCA (Choi, Lee, Lee, Park, & Lee, 2005) are usually used for advanced process monitoring such as adaptive process monitoring, dynamic process monitoring, and nonlinear process monitoring.

In practical cases, numerous abnormal conditions gradually evolve from incipient faults (Watanabe, Matsuura, Abe, Kubota, & Himmelblau, 1989). This implies that, if faults are detected in their incipient stages, abnormal conditions may be effectively avoided. However, compared with serious faults, incipient faults are easily masked by normal variation or measurement noise owing to their small magnitudes; as a result, incipient fault detection is a challenging task. Recently, some approaches have been proposed in the literature to address the problems associated with incipient fault detection. Kiasi, Prakash, and Shah (2015) presented a modified GLR-based approach to detect and diagnose incipient sensor faults of an LTI system. For a class of nonlinear distributed processes with incipient component and actuator faults, Armaou and Demetriou (2008) presented a robust detection and accommodation scheme. Alwi, Edwards, and Tan (2009) proposed sliding mode estimation schemes for incipient sensor faults. Ge, Wang, Zhou, Wu, and Jin (2015) proposed a two-step incipient fault detection method combining wavelet analysis with residual evaluation. Harmouche, Delpha, and Diallo (2014) presented an incipient fault detection method based on Kullback–Leibler divergence using PCA.

For most MSPM methods, process variables are jointly monitored to detect faults. Based on correlations, statistical models are built based on sufficient training data, leading to the decomposition of the original measurement space (Qin, 2003). During online monitoring, sample vectors are directly projected onto corresponding subspaces in sequence. This implies that the latest sample is projected separately without considering statistical information among samples. When detecting incipient faults, samples belonging to normal and abnormal conditions usually overlap to a large extent owing to their small fault magnitudes. As a result, conventional sample-wise MSPM methods easily lead to a significant number of missed detections.

One possible solution to reduce the missed detection rate is to utilize statistical information among measurements. Window-based monitoring methods can partially alleviate data overlap. He and Wang (2011) proposed statistics pattern analysis (SPA) to address the challenges encountered in semiconductor processes, which was also extended to continuous process monitoring (Wang & He, 2010). Instead of monitoring process variables, SPA monitors the statistics of process variables in sliding windows, demonstrating a superior performance over PCA and DPCA. However, SPA may not effectively detect some incipient faults with small magnitudes. Kano, Hasebe, Hashimoto, and Ohno (2002) proposed DISSIM to monitor the dissimilarity of process data. DISSIM monitors data distribution in sliding windows, and uses the dissimilarity index to differentiate between normal and abnormal conditions. It is sensitive to incipient faults occurring in some processes but may lack the portability for others.

Considering the problem of incipient fault detection, we propose a new MSPM method called recursive transformed component statistical analysis (RTCSA). It obtains orthogonal vectors called transformed components (TCs) by transforming the axes in the original measurement space. This transformation represents a rigid rotation of axes such that the scores in the transformed coordinates are orthogonal with maximum sample variance under constraints. TCs extracted in sliding windows are linear combinations of normalized process measurement vectors. The statistical information of TCs can reveal some important process features, which implies that condition changes can be detected by monitoring the statistics of TCs. We also use rank-one modification to update the sample covariance matrix and its eigenpairs recursively to improve the algorithm efficiency.

The main contributions of this paper are summarized as follows. (i) A new MSPM method RTCSA is proposed. Different from existing methods such as PCA and SPA, RTCSA extracts orthogonal TCs without space partition. Statistical information including higher-order statistics of TCs is extracted for process monitoring. (ii) The detection index is well-designed to ensure that RTCSA is sensitive to incipient faults. With second-order statistics, the detection index reduces to relative changes of ordered eigenvalues of the sample covariance matrix. Its structure ensures a wide spectrum of fault detection, because a scaled ordered eigenvalue is sensitive to faults with small magnitudes. (iii) The fault detectability of RTCSA is analyzed in a statistical sense for a general multivariate process (Alcala & Qin, 2009). For multivariate Gaussian distribution, the sample covariance matrix is decomposed into five parts considering the small magnitude of the incipient fault. This leads to the analysis of the eigenvalues of stochastic matrices, including the closed-form expressions for the probability distribution function (p.d.f.) of the arbitrary $l$ th largest eigenvalue of a class of real uncorrelated Wishart matrices. (iv) A numerical example and the benchmark Tennessee Eastman process both illustrate the superior detectability of RTCSA, compared with the existing MSPM methods, such as PCA, RPCA, DPCA, SPA, and DISSIM.

The remainder of this paper is organized as follows. In Section 2, the algorithm of RTCSA is introduced in detail, including transformed components, statistical analysis, recursive computation, and the corresponding analysis of computational complexity. In Section 3, the detection indices of RTCSA and window width selection are demonstrated. The detectability of RTCSA for additive sensor fault and process fault is analyzed in Section 4, including the closed-form expectations for the arbitrary $l$ th largest eigenvalue of a class of real uncorrelated Wishart matrices and the lower bounds on the expectations of eigenvalues of other stochastic matrices. In Section 5, both a numerical example and the Tennessee Eastman process are used to examine the detectability of RTCSA. Conclusions are given in Section 6.

Section snippets

Transformed components (TCs)

Consider the original measurement matrix $X \in R^{n \times m}$ , where $n$ and $m$ denote the number of samples and measured variables, respectively. We construct a one-step sliding window to stack process measurements: $X_{k} = [\begin{matrix} x_{k - w + 1, 1} & x_{k - w + 1, 2} & \dots & x_{k - w + 1, m} \\ x_{k - w + 2, 1} & x_{k - w + 2, 2} & \dots & x_{k - w + 2, m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{k, 1} & x_{k, 2} & \dots & x_{k, m} \end{matrix}]$ where $k$ is the time index of the latest sample in the sliding window, and $w$ denotes the window width. The original measurements are normalized as ${\bar{X}}_{k} = (X_{k} - 1 μ_{0}^{T}) Σ_{0}^{- 1}$ where $μ_{0} \in R^{m}$ denotes the reference mean, and the diagonal

Detection index

Once TCs and their statistics are calculated from training datasets, the next step is to quantify the dissimilarities of statistics under normal and abnormal conditions, and to determine the upper control limit.

In brief, the detection index at the $k$ th sampling instant can be calculated by $D_{k} = {‖ ς^{- 1} (Θ_{k} - Θ_{0}) ‖}_{p}$ where $Θ_{k} \in R^{s m}$ denotes the statistics of TCs in the $k$ th sliding window, $Θ_{0} \in R^{s m}$ represents the reference mean of $Θ$ trained from historical dataset under normal conditions, $s$ is the type number of

Data description

Consider a general multivariate process (Alcala & Qin, 2009) represented by $x = A s + e$ where the measurement vector $x \in R^{m}$ has $m$ variables, the coefficient matrix $A \in R^{m \times r}$ is assumed to be column full rank, $s \in R^{r}$ denotes $r$ independent data sources $(r < m)$ , with each sample i.i.d., $e \in R^{m}$ denotes Gaussian white noises with variances satisfying $σ_{e_{i}}^{2} = {(κ_{i} + 1)}^{- 1} E {x_{i}^{2}}$ , and $κ_{i}$ denotes the signal-to-noise ratio (SNR) in the $i$ th measurement channel. Denote $γ_{i} = κ_{i} - E {x_{i}}^{2} σ_{e_{i}}^{- 2}$ . Then, $σ_{x_{i}}^{2} σ_{e_{i}}^{- 2} = γ_{i} + 1$ . Note that $s$ can be

A numerical example

Consider a multivariate process (14) generated by the following equation: $[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}] = [\begin{matrix} 0.2183 & - 0.1693 & 0.2063 \\ - 0.1972 & 0.2376 & 0.1736 \\ 0.9037 & - 0.1530 & 0.6373 \\ 0.1146 & 0.9528 & - 0.2624 \\ 0.4173 & - 0.2458 & 0.8325 \end{matrix}] [\begin{matrix} s_{1} \\ s_{2} \\ s_{3} \end{matrix}] + [\begin{matrix} e_{1} \\ e_{2} \\ e_{3} \\ e_{4} \\ e_{5} \end{matrix}]$ where $s$ denotes independent Gaussian distributed data sources with mean ${[2.3, 1.7, 3.1]}^{T}$ and unit standard deviation, respectively, and $e$ denotes Gaussian white noises with standard deviation ${[0.061, 0.063, 0.198, 0.176, 0.170]}^{T}$ . Four types of incipient faults are given as follows:

Conclusion

In this paper, a new data-driven process monitoring method called RTCSA is proposed for incipient fault detection. In the proposed approach, process measurement vectors are transformed into orthogonal TCs in each sliding window. Instead of monitoring the original measurements, statistical information of TCs is extracted. Various statistics including higher-order statistics are utilized to capture process features. The algorithm extracts TCs in each sliding window recursively using rank-one

Jun Shang received the B.Eng. degree from the Department of Control Science and Engineering, Harbin Institute of Technology, China, in 2013. He is currently a Ph.D. candidate in the Department of Automation, Tsinghua University, China. His research interests include industrial process monitoring, fault diagnosis, data analytics, and statistical learning.

References (50)

C.F. Alcala et al.
Reconstruction-based contribution for process monitoring
Automatica
(2009)
H. Alwi et al.
Sliding mode estimation schemes for incipient sensor faults
Automatica
(2009)
L.H. Chiang et al.
Fault diagnosis in chemical processes using Fisher discriminant analysis, discriminant partial least squares, and principal component analysis
Chemometrics and Intelligent Laboratory Systems
(2000)
M. Chiani
Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution
Journal of Multivariate Analysis
(2014)
S.W. Choi et al.
Fault detection and identification of nonlinear processes based on kernel PCA
Chemometrics and Intelligent Laboratory Systems
(2005)
J.J. Downs et al.
A plant-wide industrial process control problem
Computers and Chemical Engineering
(1993)
R. Dunia et al.
A unified geometric approach to process and sensor fault identification and reconstruction: the unidimensional fault case
Computers and Chemical Engineering
(1998)
J. Harmouche et al.
Incipient fault detection and diagnosis based on Kullback–Leibler divergence using principal component analysis: Part I
Signal Processing
(2014)
M. Kano et al.
Comparison of statistical process monitoring methods: application to the Eastman challenge problem
Computers and Chemical Engineering
(2000)
F. Kiasi et al.
Detection and diagnosis of incipient faults in sensors of an LTI system using a modified GLR-based approach
Journal of Process Control
(2015)

U. Kruger et al.

Improved principal component monitoring using the local approach

Automatica

(2007)

W. Ku et al.

Disturbance detection and isolation by dynamic principal component analysis

Chemometrics and Intelligent Laboratory Systems

(1995)

W.H. Li et al.

Recursive PCA for adaptive process monitoring

Journal of Process Control

(2000)

S.J. Qin

Survey on data-driven industrial process monitoring and diagnosis

Annual Reviews in Control

(2012)

E.L. Russell et al.

Fault detection in industrial processes using canonical variate analysis and dynamic principal component analysis

Chemometrics and Intelligent Laboratory Systems

(2000)

S. Yin et al.

A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process

Journal of Process Control

(2012)

A. Armaou et al.

Robust detection and accommodation of incipient component and actuator faults in nonlinear distributed processes

AIChE Journal

(2008)

J.R. Bunch et al.

Rank-one modification of the symmetric eigenproblem

Numerische Mathematik

(1978)

L.H. Chiang et al.

Fault detection and diagnosis in industrial systems

(2001)

M.S. Choudhury et al.

Diagnosis of poor control-loop performance using higher-order statistics

Automatica

(2004)

S.X. Ding

Data-driven design of fault diagnosis and fault-tolerant control systems

(2014)

D.H. Duan et al.

The changes of cultivated land area and its driving mechanism in Qingdao based on the method of principal component analysis

Journal of Shandong University of Science and Technology

(2012)

R. Dunia et al.

Subspace approach to multidimensional fault identification and reconstruction

AIChE Journal

(1998)

S.C. Eisenstat et al.

Relative perturbation techniques for singular value problems

SIAM Journal on Numerical Analysis

(1995)

Z.Q. Ge et al.

Review of recent research on data-based process monitoring

Industrial and Engineering Chemistry Research

(2013)

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Maoyin Chen received the B.S. degree in mathematics and the M.S. degree in control theory and control engineering from Qufu Normal University, Shandong, China, in 1997 and 2000, respectively, and the Ph.D. degree in control theory and control engineering from Shanghai Jiaotong University, Shanghai, China, in 2003.

From 2003 to 2005, he was a Postdoctoral Researcher with the Department of Automation, Tsinghua University, Beijing, China. From 2006 to 2008, he visited Potsdam University, Potsdam, Germany, as an Alexander von Humboldt Research Fellow. Since October 2008, he has been an Associated Professor with the Department of Automation, Tsinghua University. His research interests include fault prognosis and complex systems.

Hongquan Ji received the B.Eng. degree in automation from Shandong University, Jinan, China, in 2012. He is currently working toward the Ph.D. degree in control science and engineering with the Department of Automation, Tsinghua University, Beijing, China. His research interests include data-driven industrial process monitoring, and fault diagnosis with application in high-speed trains.

Donghua Zhou received the B.Eng., M.Sci., and Ph.D. degrees in electrical engineering from Shanghai Jiaotong University, China, in 1985, 1988, and 1990, respectively.

He was an Alexander von Humboldt Research Fellow with the University of Duisburg, Germany from 1995 to 1996, and a Visiting Scholar with Yale university, USA from 2001 to 2002. He joined Tsinghua university, China in 1997, and was a Professor and the Head of the Department of Automation from 2008 to 2015. He is now the vice president of Shandong University of Science and Technology. He has authored and coauthored over 150 peer-reviewed international journal papers and 6 monographs in the areas of process identification, fault diagnosis, fault-tolerant control, reliability prediction, and predictive maintenance.

Dr. Zhou is a member of the IFAC Technical Committee on Fault Diagnosis and Safety of Technical Processes, a senior member of IEEE, an associate editor of the Journal of Process Control, and the vice Chairman of Chinese Association of Automation (CAA). He was also the NOC Chair of the 6th IFAC Symposium on SAFEPROCESS 2006.

^☆: This work was supported by National Natural Science Foundation of China under Grants 61490701, 61210012, 61290324, 61473164, Tsinghua University Initiative Scientific Research Program under Grant 20131089240, and Research Fund for the Taishan Scholar Project of Shandong Province of China under Grant LZB2015-162. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Antonio Vicino under the direction of Editor Torsten Söderström.

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Recursive transformed component statistical analysis for incipient fault detection☆

Abstract

Introduction

Section snippets

Transformed components (TCs)

Detection index

Data description

A numerical example

Conclusion

Automatica

Automatica

Chemometrics and Intelligent Laboratory Systems

Journal of Multivariate Analysis

Chemometrics and Intelligent Laboratory Systems

Computers and Chemical Engineering

Computers and Chemical Engineering

Signal Processing

Computers and Chemical Engineering

Journal of Process Control

Automatica

Chemometrics and Intelligent Laboratory Systems

Journal of Process Control

Annual Reviews in Control

Chemometrics and Intelligent Laboratory Systems

Journal of Process Control

Robust detection and accommodation of incipient component and actuator faults in nonlinear distributed processes

AIChE Journal

Rank-one modification of the symmetric eigenproblem

Numerische Mathematik

Fault detection and diagnosis in industrial systems

Diagnosis of poor control-loop performance using higher-order statistics

Automatica

Data-driven design of fault diagnosis and fault-tolerant control systems

The changes of cultivated land area and its driving mechanism in Qingdao based on the method of principal component analysis

Journal of Shandong University of Science and Technology

Subspace approach to multidimensional fault identification and reconstruction

AIChE Journal

Relative perturbation techniques for singular value problems

SIAM Journal on Numerical Analysis

Review of recent research on data-based process monitoring

Industrial and Engineering Chemistry Research