(615g) Branch and Bound Method for Fault Isolation through Missing Variable Analysis
In recent years, multivariate statistical process monitoring (MSPM) has received considerable attention for fault detection and isolation [1,2]. Although the use of monitoring statistics like Hotelling's T2 and squared prediction error (SPE) is well established for fault detection, the diagnosis of the source or cause of the detected fault is relatively more difficult. The primary tool used for fault diagnosis in MSPM is contribution analysis, which quantifies the contribution of individual variables to T2 and SPE . However, the contribution plots based on T2 and SPE can identify different sets of faulty variables making the decision subjective to operators' experience. Furthermore, contribution analysis investigates individual variables one by one and is ineffective in isolating multiple variables which jointly contribute to the occurrence of fault.
To obtain a unified monitoring statistic, as opposed to using T2 and SPE individually, different approaches have been proposed including methods to combine T2 and SPE algorithmically  and the use of a fully probabilistic model . These studies have also suggested an alternate approach to quantifying the contribution using the idea of missing variables. However, the major difficulty is the large number of possible variable combinations that are required to be evaluated. Therefore, previous studies made the restrictive assumption that the combinations of faulty variables are specified a priori according to known faulty modes  or focussed on the analyzing the contribution of individual variables towards the faults .
This paper extends the missing-variable based contribution analysis to consider the joint effect of multiple variables, namely multivariate contribution analysis. In this study, we choose probabilistic PCA (PPCA)  for modelling normal operating data and on-line process monitoring. Based on the PPCA, a statistical criterion is derived to quantify contribution of multiple missing variables. Using the criterion, when a fault condition is identified through on-line process monitoring, the fault diagnosis can then be conducted by solving a series of subset selection problems. The bBranch and bound (BAB) technique has recently been shown to be a promising approach for handling the combinatorial difficulty of the subset selection problems [7,8]. In this paper, a numerically efficient BAB algorithm is developed to isolate the set of faulty variables based on the criterion derived. The efficiency of the proposed method is demonstrated by its application to the Tennessee Eastman Process .
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