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Article Excerpt
1. Introduction
Statistical Process Control (SPC) plays a very important role in quality and productivity improvement of a manufacturing or service enterprise (Montgomery, 2005). Following SPC methodology, measurements are taken from the process or the product, they are then treated as random variables and their distributions are compared with the distributions under normal working conditions. If the measurements show that some characteristics are "out of control" (e.g., deviation from the target or variability is too high), an alarm is generated to indicate that changes have occurred in the process. For most of the available SPC techniques, the measurements are often either explicitly or implicitly assumed to be independent and identically distributed (i.i.d.).
Due to the rapid development of information and sensing technologies in recent years, a large amount of data is now readily available in many processes. Multi-dimensional measurements for discrete manufacturing processes with 100% inspection rate and very high sampling rates for continuous processes are no longer rare in practice. For example, in autobody assembly processes, 100% dimensional inspections have been achieved by the use of inline optical coordinate measurement machines (Ceglarek and Shi, 1995). The extensive datasets provide significant opportunities for more sophisticated analysis for process monitoring, however, on the other hand, it also poses great challenges for SPC. Because of the very high sampling frequency and system inertia, the measurements often exhibit significant autocorrelation (Montgomery, 2005). Many researchers have demonstrated that the performance of SPC methods developed with i.i.d. assumptions will degrade when there exists dependence between successive samples (e.g., Harris and Ross (1991) and Montgomery and Mastrangelo (1991)). These works indicate that, for measurements with significant autocorrelation, dynamic models rather than static models should be used to fully utilize the measurements for process monitoring purposes.
The most commonly used dynamic models in SPC are time series models such as the AR(p) model (Alwan and Roberts, 1989; Montgomery and Mastrangelo, 1991) and the IMA (1, 1) model (MacGregor, 1988; Box and Kramer, 1992; Vander Wiel, 1996). People often fit the auto-correlated measurements using these models and because the model residuals are considered to be i.i.d., the conventional SPC techniques can then be applied to the residuals. Most of the available SPC methods that are based on time series model approaches only handle univariate cases. To monitor multivariate dynamic processes a state space model is often used to model the process dynamics. Negiz and Cinar (1997) used subspace identification methods to establish a state space model for the measurement data under normal conditions, and then the prediction of the state vector is monitored through a [T.sup.2] control chart. Simoglou et al. (2002) used a similar approach and compared different techniques for the fitting of state space models. In addition to model-based methods, various model-free methods have also been developed for multivariate dynamic process monitoring, for example, methods based on asymptotical optimal sequential testing (Basseville and Nikiforov, 1993; Lai, 1995) and various Principal Component Analysis (PCA)-related methods, such as recursive PCA (Li et al., 2000), multiscale PCA (Bakshi, 1998) and dynamic PCA (Ku et al., 1995).
Although various works exist on the statistical monitoring of univariate and multivariate dynamic processes, a fundamental issue regarding process monitoring has not been thoroughly studied in past work. The issue is whether the process measurements contain sufficient information for the detection of process changes. This is referred to as detectability analysis and it is very important, especially in the design phase, because if a process is not detectable due to a problematic system structure, then no matter how much effort we put into the creation of a monitoring algorithm, we will not be able to detect process faults. Thus, if we blindly use statistical monitoring methods without any consideration of the process detectability, we may miss the changes that we want to detect. For example, Harris and Ross (1991) and Wardell et al. (1994) noticed that the one-step-ahead residual approach lacks the ability to detect changes in a time series with AR poles close to the unit circle. However, despite its importance, few papers have been published on the topic of exploring general detectability issues.
Since process changes are always caused by the occurrence of process faults, we treat "change detection" and "fault detection" as being interchangeable in this paper. Basseville (2001) provided a good review on various definitions of fault detectability for different types of faults. In that paper, the process faults are classified as being either: (i) additive faults on a linear system, which are manifested as a mean change of the process inputs and process outputs, or (ii) component (or system) faults, which are manifested as process structure changes such as the variance change of the multivariate process inputs. For each type of fault, the detectability definitions can be put into one of two categories: (i) an intrinsic definition that defines the detectability in an intrinsic manner as a system property, without any reference to any specific fault detection algorithms; or (ii) a performance-based definition that defines the detectability with explicit reference to a specific algorithm, taking into account its performance. Using the above classification, Zhou et al. (2003) and Apley and Ding (2005) investigated the properties of an intrinsic fault detection for both additive (i.e., mean shift of the input) and component faults (i.e., variance change of the input) in a static system. In their work, the process measurements are viewed as being i.i.d. and no autocorrelation and system dynamics are considered. On the other hand, Liu and Si (1997) and Gustafsson (2002) studied the properties of a performance-based detection definition only for additive faults in general dynamic systems. In their work, a state space model is adopted to model the system dynamics and the system observability is utilized to derive the system fault detectability.
In this paper, we will provide an intrinsic fault detectability definition for both additive and component faults in multivariate dynamic processes. A state space model is used to describe the system dynamics. A random vector is added to the system input to represent the faults. In this way, both the mean shift fault (i.e., additive fault) and variance change fault (i.e., a type of component fault) can be modeled. This fault representation can describe a wide range of practical situations and has been adopted in several previous works such as in Negiz and Cinar (1997), Chen et al. (1998) and Zhou et al. (2003). With the fault representation, an intrinsic detectability definition is proposed and the relationship between the defined detectability and the system structure is further investigated in this paper. The conditions under which the system detectability is guaranteed are derived. The results developed in this paper can be used to analyze the process structure to check if certain process faults are detectable and consequently provide quantitative guidelines on system design to improve the performance of the statistical monitoring scheme.
The paper is organized as follows: In Section 2, we will give a formal definition of the detectability. Then, the conditions of mean detectability and variance detectability will be explored, respectively. We also provide a corollary to easily check a system's detectability. In Section 3, a case study will be given to show the effectiveness of our method,
and some possible applications of the result will also be pointed out. The paper is concluded in Section 4.
2. Intrinsic detectability of multivariate dynamic processes
2.1. Problem formulation
Linear Time Invariant (LTI) state space models are widely used because this approach can closely approximate many real systems...
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