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Residual-life estimation for components with non-symmetric priors.

Article Excerpt
1. Introduction

Condition monitoring uses sensory information from a functioning device to evaluate its health. The evolution of this sensory information often exhibits characteristic patterns that reflect the underlying physical transitions that occur during degradation. These patterns are known as degradation signals (Nelson, 1990). Degradation models mathematically characterize these degradation signals and are useful for estimating the residual life of a partially degraded component. Degradation models typically have stochastic parameters that follow some distribution across the population of components. In Gebraeel et al (2005), we develop a Bayesian updating technique that uses degradation signal information to update the distributions of stochastic parameters for linear and exponential degradation models. We also derive expressions for the residual-life distribution assuming normality of the stochastic parameters. The motivation of the present work is to study the practical applicability of these expressions when the normality assumption is not met. We specifically focus on the linear model developed by Gebraeel et al. (2005) and consider the case where the stochastic parameter follows gamma distributions with various degrees of skewness. The gamma distribution provides more flexibility in capturing the characteristics of the real-world sensory data. Based on this assumption, we propose a simulation-based algorithm to estimate residual-life distributions. Finally, we examine the performance of the proposed method for various degrees of skewness in the prior and obtain a threshold beyond which it performs better than that presented in Gebraeel et al. (2005).

2. Literature review

The purpose of real-time condition monitoring is to evaluate the health of a device in order to reduce unnecessary maintenance and unexpected downtime. Significant research exists in areas such as power transformers (Feser et al., 1995), cutting tools (Dimla, 1999), high-voltage induction motors (Thorsen and Dalva, 1999), railway equipment (Fararooy and Allan, 1995) and machine tools (Martin, 1994). Most of this work is diagnostic in nature.

Some researchers use degradation information to estimate life characteristics. Lu and Meeker (1993) derive life distributions for populations of devices using degradation information from randomly selected sets of devices. Wang (2000) uses random coefficient models to characterize degradation and provides a concise listing of the most common assumptions in degradation modeling research. These include (i) the condition of the device deteriorates with operating time and the level of deterioration can be observed at any time; (ii) the mean and variance of device deterioration can be increasing in time; (iii) device failure occurs when the degradation signal reaches a well-defined threshold; (iv) the device being monitored comes from a population of devices, each of which exhibits the same degradation form; and (v) the distribution of the stochastic parameters across the population of devices is known. Both Lu and Meeker (1993) and Wang (2000) assume independent and identically distributed (iid) N(0, [[sigma].sup.2]) error in the degradation signal across the population of devices.

Yang and Yang (1998) develop a random-coefficient-based approach to obtain better estimates of life parameters using life information of failed components and degradation information from partially degraded ones. They experimentally verify that this approach provides better estimators than traditional life testing. Yang and Jeang (1994) use a random coefficients model to study the effect of cutting tool flankwear on surface roughness in metal cutting. Tseng et al. (1995) apply random coefficient models for luminosity with experimental design to improve the reliability of fluorescent lamps. They use a degradation model to determine the combination of manufacturing settings that provide the slowest rate of luminous degradation. Goode et al. (1998) use an exponential degradation model to predict the condition of a hot strip mill. They conclude that the degradation model provides better life predictions than a reliability model.

Doksum and Hoyland (1991) develop inverse Gaussian life models and maximum likelihood estimators for devices subjected to accelerated stress testing. They model the accumulated decay as a Wiener process with drift and diffusion dependent on the stress level. Whitmore (1995) models degradation as a Wiener process and explains how to account for measurement errors. Whitmore and Schenkelberg (1997) use Wiener processes to model degradation data collected from accelerated testing and develop methods for estimating the parameters of time and stress transformations. Lu et al. (2001) suggest methods for forecasting system performance reliability for systems with multiple failure modes. The authors use time series forecasting to develop a joint density function for performance measures such as system reliability. They also develop a model for estimating the conditional performance reliability in real-time for an individual component in operation.

In Gebraeel et al. (2005), we propose a Bayesian technique to update the stochastic parameters for linear and exponential degradation models. Figure 1 shows an example real-time vibration-based degradation signal for a thrust bearing from Gebraeel (2006). Note that the vibration amplitude increases with operating time as the bearing degrades. When the amplitude reaches a pre-specified threshold (based on industry standards), the bearing is assumed to fail.

[FIGURE 1 OMITTED]

Figures 2 and 3 illustrate our updating scheme. Figure 2(a) and Fig. 2(b) show an exponential degradation model of the form [S.sub.i] = [phi] + [theta] exp [[beta][t.sub.i] + [[epsilon].sub.i] for a specific device and the residual life distributions at time [t.sub.i] and [t.sub.i+n] respectively. Here both the parameters [theta] and [beta] are stochastic. In both Fig. 2(a) and Fig. 2(b), we obtain a posterior distribution of the stochastic parameter using the prior distribution and condition monitoring data. Then we use this posterior to compute the residual-life distribution. The small circle in Fig. 2(a) indicates the information obtained up to time [t.sub.i]. For this case we have one specific residual-life distribution. In Fig. 2(b) the small circle shows the information obtained up to time [t.sub.i+n]; in this case, the residual-life distribution has different mean and variance to...

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