Multivariate/minor fault diagnosis with severity level based on Bayesian decision theory and multidimensional RBC
Introduction
In industrial manufacturing processes, fault diagnosis can ensure the safety of the process and improve the quality of products. With the rapid development of data technology, more and more attention has been paid to the data-driven fault diagnosis methods [1].
As a typical data-driven fault detection technique, multivariate statistical process monitoring (MSPM) is widely applied in industries [2]. Principal component analysis(PCA) is a representative of MSPM methods, and its core objective is to extract main characteristic information from the process data to calculate monitoring statistics such as SPE, T, or their combined Index [3].
After the fault is detected, fault diagnosis is indispensable to judge the fault source variables. Miller et al. proposed a contribution plot method to identify a fault, which selects the variable with largest contribution among all the variables as the main cause of the fault [4]. This method is simple so that it is widely applied in industries. Dunia et al. identified sensor faults through fault reconstruction method on basis of PCA model [5]. Yue et al. proposed a combined index for fault reconstruction, and extracted the fault direction from the historical fault data through Singular Value Decomposition (SVD) to form a fault subspace [6]. Liu and Chen improved the contribution plot method, which defined a new statistical indicator called Reduction of Combined Index(RCI) to locate the fault source variables [7].
In 2009, Alcala and Qin proposed the Reconstruction based Contribution (RBC) method [8]. Compared with traditional contribution plot method, RBC can ensure correct diagnosis under the condition of single fault with large magnitude. Westerhuis et al. thought each variable contributed equally to the detection index under normal conditions, and presented the concept of relative RBC [9].
Due to the correlation between variables, fault variables tend to affect normal variables, and the contribution of normal variables even possibly exceeds that of fault variables, which is called ”smearing effect”[10]. Wong et al. [11] and Liu et al. [12] introduced a Bayesian filter on basis of RBC to reduce the smearing effect and filter out secondary fault variables. Considering the difference between the losses caused by type I error (misdiagnosis) and type II error (missed diagnosis), Zheng et al. proposed normalized relative RBC-based minimum risk Bayesian decision method to make the fault diagnosis result more reliable [13].
Although RBC performs satisfactorily on the diagnosis of a single fault with large magnitude, it cannot handle multiple faults or minor faults (which has secondary influence on the process compared with the primary faults). Mnassri et al. proved that the RBC of normal variable is likely to be larger than the fault variable in the case of multiple fault, and proposed the concept of multi-dimensional RBC and RBC ratio [14]. Li et al. proposed the concept of generalized RBC for output-related fault diagnosis [15]. In addition, they combined multi-directional RBC with dynamic PCA to diagnose the root causes of the process faults [16]. Kariwala et al. combined missing data analysis method with Branch and Bound(BAB) to search fault source variables step by step [17], while its computational cost and time is high. He et al. proposed a reconstruction based multivariate contribution analysis (RBMCA) method [18]. On basis of this, penalized RBMCA framework was established by introducing a penalty function to further improve the diagnostic efficiency [19]. Kuang et al. analyzed the similarity between fisher discriminant analysis(FDA) and regression analysis, and transformed the multivariate fault diagnosis problem into LASSO penalty regression problem [20], which reduced the high computational cost caused by BAB to a certain extent. Further, based on this method, the Bayesian theory and LASSO method was combined for fault diagnosis in [21].
In this paper, a multivariate fault diagnosis method based on Bayesian decision theory and multi-dimensional RBC is proposed. Firstly, the probability density function(PDF) of normal samples is estimated. Then, a deviation factor is defined and taken as the characteristic of the sample; the conditional PDF of the deviation factor under normal/faulty condition is calculated. To make use of the historical diagnosis results, Bayesian decision theory is adopted to choose the suspicious fault source variables by accumulating their fault probability. Finally, multi-dimensional RBC determine the fault source variables with the severity level which is associated with the fault occurrence probability. A numerical example and Tennessee Eastman process shows that the proposed method not only can isolate multivariate fault and minor fault with severity level, but also can greatly improve the diagnostic efficiency with less computation time.
The remaining sections of this paper are organized as follows. Section 2 provides a general description of the theoretical basis. The proposed approach is presented in detail in Section 3. Section 4 provides a numerical example and the Tennessee Eastman (TE) process to validate the effectiveness of the proposed method. And the last section gives conclusion.
Section snippets
PCA-based fault detection
PCA-based monitoring model is extensively applied in practical processes. Assume that the historical data is described as , where is the sample with variables. After is standardized, PCA decomposes it into principal subspace and residual subspace as: where , and , are the scores and load matrices for the principle and residual subspaces respectively; and is the number of principal components.
For
The proposed approach
In this section, the contribution of the fault variables is considered, and a multivariate fault diagnosis method based on Bayesian decision theory and multi-dimensional RBC is proposed to solve multiple/minor fault identification problem.
Numerical examples
In this section, the Monte Carlo simulation is adopted to verify the proposed method. The constructed process model is where is the normal sample, are three latent variables with zero mean and standard deviation of 1, 0.8 and 0.6 respectively; and . The training set contains a total of 3000 normal samples, and the
Conclusions
In this paper, a fault diagnosis method based on Bayesian theory and multidimensional reconstruction-based contribution is proposed. Firstly, the PDF of the variables is estimated based on the normal dataset. Then the characteristic of the normal/fault samples, i.e., the deviation factor, is defined and calculated. The PDF of the deviation factor is deduced based on the PDF of the variables. Considering historical diagnosis results as prior knowledge, Bayesian theory is adopted to obtain the
CRediT authorship contribution statement
Ying Zheng: Conceptualization, Methodology, Investigation, Formal analysis, Writing - original draft, Visualization. Wei Zhou: Conceptualization, Methodology, Formal analysis, Software, Data curation, Writing - original draft, visualization. Weidong Yang: Conceptualization, Methodology, Formal analysis, Resources, Writing - review & editing, Supervision. Lang Liu: Software. Yuanle Liu: Writing - review & editing. Yong Zhang: Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant 61873102, 61873197) and key Natural Science Foundation of Hubei (Grant 2019CFA047).
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