Publication: IIE Transactions
Publication Date: 01-MAR-07
Delivery: Immediate Online Access
Author: Li, Zhiguo ; Zhou, Shiyu ; Ding, Yu
Article Excerpt
1. Introduction
Process monitoring and control technology, which focuses on the detection, identification, diagnosis, and elimination of process faults, can help a company maintain on edge in today's highly competative market place. This is because it can help to reduce process downtime, and hence, the operation costs. The rapid advances in sensing and information technology that are currently being made mean that a large amount of data is readily available that requires process control methodologies to be developed for its interpretation.
Statistical Process Control (SPC) (Montgomery (2005) and the references therein) is the primary tool used in practice to improve the quality of manufacturing processes. SPC methods compare the statistical distribution of a process output at normal (in control) working conditions with that at current working conditions. If a large disparity is found, then an alarm is signaled to indicate an abnormal (out of control) condition. However, SPC is purely a statistical technique that is able to detect a departure from normal conditions but is unable to pin down the process fault that caused the alarm. This process fault is often called the root cause of the alarm by practitioners. The job of root-cause identification is actually left to plant operators or quality engineers.
In light of this limitation of SPC methods, considerable research efforts have been expended on developing methodologies for root-cause identification. The basic approach among the reported methodologies is to use a diagnostic fault-quality model, which connects the measured product quality characteristic to process faults. For example, in a machining operation, dimensional features such as the position, orientation, and size of a machined feature are affected by deviations in process variables such as fixturing errors and/or machine tool errors. In this example, the dimensional features are the product quality characteristics and are treated as the outputs of a fault-quality model, while the errors in the process variables are the process faults and thus are the inputs to the model.
A wide variety of approaches can lead to the development of a diagnostic fault-quality model. Ceglarek and Shi (1996) linked process faults and the product quality of assembly processes using principal components. A linear fault-quality model of an explicit input-output format was presented for assembly processes by Apley and Shi (1998). Their work was extended to multistage assembly processes by Jin and Shi (1999) and Ding et al. (2002a). Djurdjanovic and Ni (2001) and Zhou et al. (2003b) developed a linear-deviation propagation model for multistage machining processes.
All of the above models were developed on the basis of governing physical laws in the corresponding processes. They bear the same linear-model structure because the process faults are assumed to be of a smaller magnitude as compared to nominal product dimensions. A linear fault-quality model can be generally expressed as:
y = Af + [epsilon], (1)
where y is a n x 1 vector of product quality measurements, A is a n x p constant system matrix determined by process/product designs, f is a p x 1 random vector representing the process faults, and [epsilon] is a n x 1 random vector representing the influence of measurement noise, un-modeled faults, and high-order nonlinear terms. Without loss of generality, we can further assume that the columns of the A matrix are of unit length. This can be achieved through a simple scaling process. In this paper, we focus on variation-source identification. Therefore, the process fault vector f is modeled as a random vector to describe the variation errors, as opposed to a mean-shift type of error in the process. Since the mean of quality measurements can always be subtracted, it is also assumed that the means of f and [epsilon] are zero.
After the fault-quality model is established, either statistical estimation methods or pattern matching methods can be used to identify which process fault has occurred based on product quality measurements.
In estimation methods, the fault-quality model of equation (1) is treated as a linear mixed model. The variances of the process faults ([[sigma].sub.[f.sub.l].sup.2],..., [[sigma].sub.[f.sub.p].sup.2]) are the variance components to be estimated in this mixed model (Searle et al., 1992; McCulloch and Searle, 2001). Based on the mixed model, Apley and Shi (1998) and Chang and Gossard (1998) used ordinary least-squares to estimate the random input [^.f] and then calculate its variance as if [^.f] was directly measured. Zhou et al. (2004) used a maximum-likelihood estimator and also provided the confidence intervals of the estimated variance off. Ding et al. (2005) compared different variance estimation methods and provided guidelines on method selection under different circumstances. Furthermore, Ding et al. (2002b) and Zhou et al. (2003a) performed diagnos-ability studies to identify the necessary conditions that need to be satisfied for an estimation method to be applicable.
Although based on the same fault-quality model, the pattern matching method is different from statistical estimation methods. The basic idea of the pattern matching technique is illustrated in Fig. 1. First, based on the fault-quality model, we can obtain the signatures of potential faults. Then the pattern of an occurring fault is extracted from measured data. Finally, the occurring fault can be identified if there is a match between the patterns of the fault symptom and the fault signature. In Equation (1), the column vectors of A determine how a specific process fault affects the product quality characteristics and thus they are the fault signature vectors. Principal Component Analysis (PCA) (Johnson and Wichern, 1998) is the major statistical tool used to extract the fault symptom from measured data in the pattern matching method.
[FIGURE 1 OMITTED]
Ceglarek and Shi (1996) developed a pattern matching method for fixture fault diagnosis in automotive body assembly processes. The fault signatures were defined based on design information on the assembled part under faulty fixture conditions. After that, the pattern matching technique was used to map the extracted fault symptoms from measured data through PCA to the predefined fault signatures. The authors assumed a structured noise (the covariance matrix of [epsilon], [[SIGMA].sub.[epsilon]] is diagonal) and a large sample size (namely, the covariance matrix of measurements y, [[SIGMA].sub.y], is known). This method was extended by Rong et al. (2000) to the compliant beam structure assembly model by considering the sample properties of the principal eigenvector of [[SIGMA].sub.y] but keeping the structured noise requirement. Ding et al. (2002a) studied the impact of an unstructured noise under the large sample condition. They used a state space model for multistage manufacturing processes and generated the fault signatures based on the model. A PCA-based methodology was adopted to identify the single variation source in the process.
The statistical estimation method and the pattern matching method have different strengths. From a practicability point of view, the pattern matching method is very intuitive and possesses a clear geometric interpretation, which may help practitioners to understand and eliminate the variation source. Thus, the pattern matching method is easy for practitioners to implement and execute as compared with statistical estimation methods. On the other hand, the statistical estimation method allows people to analyze the test performance analytically because the testing statistics are tractable. The available statistical estimation methods for variation-source identification assume that the covariance matrix of the noise term [epsilon] is in the simple form [[sigma].sub.[epsilon].sup.2]I. Unstructured noise is not considered.
This paper focuses on an extension of the pattern matching technique by considering both unstructured noise and sample uncertainty in the matching. The proposed method is robust and maintains a good identification probability. Using a machining process as an illustrative example, the paper demonstrates that current pattern matching procedures can have a remarkably low identification probability when the assumptions are not strictly satisfied. By contrast, our proposed method is more robust, maintaining a much higher identification probability, and is a preferable tool for variation-source identification in manufacturing quality improvement.
This paper is organized as follows. In Section 2, we present a robust pattern matching procedure for fault diagnosis. Section 3 uses a case study to illustrate the effectiveness of the proposed technique. Finally, we conclude the paper in Section 4.
2. Pattern matching techniques for root-cause identification
2.1. Problem formulation and assumptions
The linear fault-quality model as defined in Equation (1) is adopted in this paper. Assuming that the process fault f is independent of system noise [epsilon], we can obtain the following from Equation (1):
[[SIGMA].sub.y] = A[[SIGMA].sub.f][A.sup.T] + [[SIGMA].sub.[epsilon]], (2)
where [[SIGMA].sub.f] and [[SIGMA].sub.[epsilon]] are the covariance matrices of f and [epsilon], respectively. Furthermore, it is usually assumed that process faults (i.e., the elements in f) are independent of one another, and hence, [[SIGMA].sub.f] is a diagonal matrix.
Without loss of generality, let us assume that the ith fault occurs and only a single diagonal element of [[SIGMA].sub.f], [[sigma].sub.i.sup.2], is nonzero. Then, we have:
[[SIGMA].sub.y] = [[sigma].sub.i.sup.2][a.sub.i][a.sub.i.sup.T] + [[SIGMA].sub.[epsilon]], (3)
where [a.sub.i] is the ith column vector of A. If we multiply both sides of Equation (3) by [a.sub.i], we have [[SIGMA].sub.y][a.sub.i] = [[sigma].sub.i.sup.2]([a.sub.i.sup.T][a.sub.i])[a.sub.i] + [[SIGMA].sub.[epsilon]][a.sub.i]. If the components in [epsilon] are independent of one another and have the same variances, then [[SIGMA].sub.[epsilon]] will have the simple form [[sigma].sub.[epsilon].sup.2]I, where [[sigma].sub.[epsilon].sup.2] is the variance of the noise and I is an identity matrix with appropriate dimension. Under these conditions, we have:
[[SIGMA].sub.y][a.sub.i] = ([[sigma].sub.i.sup.2]([a.sub.i.sup.T][a.sub.i]) + [[sigma].sub.[epsilon].sup.2])[a.sub.i]. (4)
Since [[sigma].sub.i.sup.2]([a.sub.i.sup.T][a.sub.i]) +...
NOTE: All illustrations and photos have been removed from this article.
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