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Degradation-model-based remaining useful life (RUL) estimation is essential for effective prognostic and health management; this method can provide information for enabling effective maintenance decisions pertaining to degrading systems to avoid or mitigate loss due to impending failures. According to existing studies that estimate degradation-model-based RUL, stochastic-process-based methods are widely advocated by many researchers because they can capture stochastic dynamics within the degradation processes of systems. However, most existing studies primarily utilize the first passage time (FPT) to define the lifetime/RUL. The definition is generally conservative; thus, the lifetime/RUL estimation may be underestimated. This is particularly true for nonmonotonic stochastic degradation processes. In some extreme cases, with strong fluctuations within the degradation processes, the estimated lifetime/RUL under the FPT can be significantly less than the actual lifetime/RUL. To address this limitation, this study investigates prognostic issues, based on the stochastic degradation process, from the last exit time (LET) perspective. In contrast to the FPT, the lifetime/RUL of the degrading system is defined as the LET of its degradation process, i.e., the instant at which the degradation process passes the failure threshold for the last time. Given the new definition, we consider the most widely used degradation process model (i.e., Wiener-process-based model) as an example to demonstrate how the lifetime/RUL is estimated. Two general methods of lifetime estimation for the Wiener-process-based model are given, and some examples with associated exact and closed-form solutions are also provided for better illustration. Finally, numerical examples and a practical case study are presented to substantiate the theoretical results and illustrate the superiority of the proposed method. The results imply that the proposed method exhibits the potential to prevent premature maintenance and resource wastage because the lifetime/RUL estimation from the LET perspective can help avoid conservative results from being obtained.

Background: Remaining Useful Life (RUL) estimation is the central mission to the complex systems’ prognostics and health management. During the last decades, numbers of developments and applications of the RUL estimation have proliferated. Objective: As one of the most popular approaches, stochastic process-based approach has been widely used for characterizing the degradation trajectories and estimating RULs. This paper aimed at reviewing the latest methods and patents on this topic. Methods: The review is concentrated on four common stochastic processes for degradation modelling and RUL estimation, i.e., Gamma process, Wiener process, inverse Gaussian process and Markov chain. Results: After a brief review of these four models, we pointed out the pros and cons of them, as well as the improvement direction of each method. Conclusion: For better implementation, the applications of these four approaches on maintenance and decision-making are systematically introduced. Finally, possible future trends are concluded tentatively.

Remaining useful life (RUL) prediction is extremely significant to ensure the safe and reliable operation for bearing suffering from the deterioration. The main focus of the RUL prediction is to accurately predict the future failure event, and thus, how to quantify the prediction uncertainty will be a major concern. However, current deep learning based RUL prediction methods are difficult to reflect the uncertainty of the RUL prediction results. Toward this end, we propose a RUL prediction model based on the deep belief network (DBN) and diffusion process (DP) in this article. The proposed method consists of two parts: feature extraction combining DBN and locally linear embedding (LLE), DP-based RUL prediction. In the first part, DBN is used to extract deep hidden features behind the monitoring signals, and then the features with higher tendency are screened as the input of LLE. The health index that can truly reflect the bearing health condition is further determined through LLE. In the second part, a health index evolving model based on DP is presented and the probability density function (PDF) of the predicted RUL is accordingly derived in the sense of the first hitting time (FHT). As such, the proposed method holds promise to improve the prediction accuracy and facilitate the prognostic uncertainty. Finally, experimental studies on the bearing degradation data and the associated comparative analysis verify the effectiveness and superiority of the proposed method.

In the absence of sufficient degradation data of long-lifetime and highly reliable products, a step-stress accelerated degradation model based on nonlinear diffusion process is proposed to estimate the remaining useful life (RUL), with the advantage of requiring small sample size and short test time. For multiple uncertainties caused by inherent properties, individual differences, measurement equipment performance and artificial deviations in the degradation process, this model considers the temporal variability, unit-to-unit variability and measurement errors in both performance degradation and covariates. To estimate the RUL, we derive an analytical approximation to the first hitting time (FHT) of the nonlinear diffusion process crossing a predetermined threshold with the consideration of measurement errors. Based on maximum likelihood estimation (MLE), a modified simulation and extrapolation (SIMEX) method, called MLE-SIMEX, is developed for estimating unknown parameters in the established model. The usefulness of the proposed model is demonstrated by a simulation case and a real-world example. The results reveal that considering nonlinearity and multiple sources of variability in the step-stress accelerated degradation process can improve the model fitting and the accuracy of RUL estimation.

For the condition-based maintenance decision of the degrading equipment, the existing methods usually only consider the impact of maintenance activities on equipment degradation rate or degradation level, ignoring the impact of shocks existing in the life cycle. Toward this end, this paper concentrates on how to make optimal maintenance decision policy for degraded equipment subjected to random shocks under imperfect maintenance. Firstly, we propose a multi-stage degradation model based on the Wiener process and incorporate the influence of shocks and maintenance into the degradation model. The remaining useful life(RUL) is predicted under the concept of the first hitting time (FHT). Secondly, we establish a condition-based maintenance decision model on the basis of the prognostic information and determine the monitoring interval and the preventive maintenance (PM) threshold. Finally, a numerical simulation is provided to substantiate the condition-based maintenance decision model proposed in the paper.

Lifetime prognostic based on the degradation data has been widely investigated and adopted for reliability assessment and maintenance policy. However, the measurement error (ME) is usually inevitable, which leads to the bias of lifetime estimation and erroneous evaluation of the safety risk. In this paper, we mainly focus on an inverse issue: how to specify the sensor's performance (i.e., the ME range) for satisfying a given requirement of the lifetime estimation. Under this consideration, we first analyze the probability distribution functions of the lifetime estimation with/without the ME based on Wiener process degradation model. Then a distance measure based on the relative entropy is formulated to evaluate the difference between these two lifetime estimations. Furthermore, the permissible ranges of the time-dependent and time-independent ME are attained under a given allowable bias of lifetime estimation according to the proposed distance measure. In addition, the influence of the ME on maintenance policy is discussed. Finally, numerical examples and a case study are provided to illustrate.

As an effective way to enhance the system reliability, the standby redundancy technique has been widely applied in many industrial systems. Finding methods to determine the maintenance policy and spare-part inventory management is an interesting practical issue. However, due to the imperfect storage and inner mechanism, the spare parts usually deteriorate over time, which not only degrades their performance but also may lead to storage failure. In addition, the immediate burst failure caused by the external shock should not be neglected in storage. These two aspects make the joint optimization of preventive maintenance and inventory policies for a standby system more challenging. Thus, in this study, we first propose a general iterative approach for the lifetime estimation of standby systems with a hybrid spare-part degradation process, which includes both stochastic degradation and immediate burst failure. We then take an example of the Wiener-process-based model to explain how to obtain the analytical results of the lifetime estimation. On this basis, we further establish a joint optimization model, in which the preventive block replacement interval and the inventory number are treated as two decision variables to minimize the expected cost per unit time. Finally, a numerical example and practical case are provided for illustration.

This article addresses the problem of estimating the remaining useful life (RUL) of degrading products by fusing the accelerated degradation data and condition monitoring (CM) data. The proposed model differs from the existing models in adopting the non-conjugate prior distributions for random-effect parameters. First, a nonlinear diffusion process model is developed to characterize the degradation process of a product. Next, the relationship between the model parameters and accelerated stress level is established, and the accelerated degradation data are used to determine the prior distribution types and estimate the hyperparameters in the prior distributions. Then, to fuse the accelerated degradation data and CM data, the Bayesian inference is used to update the posterior distributions of model parameters once the new degradation observations are available. In addition, the Markov Chain Monte Carlo (MCMC) method based on Gibbs sampling is used to obtain the Bayesian solution numerically. Finally, the approximate RUL distribution considering the randomness of model parameters is obtained by the MCMC method based on the concept of the first hitting time. The proposed method is verified by the practical case study of accelerometers. Comparison results demonstrate that the proposed method can obtain higher RUL estimation accuracy and less uncertainty.

Owing to operating condition changing, physical mutation, and sudden shocks, degradation trajectories usually exhibit multi-phase features, and the abrupt jump often appears at the changing time, which makes the traditional methods of lifetime estimation unavailable. In this paper, we mainly focus on how to estimate the lifetime of the multi-phase degradation process with abrupt jumps at the change points under the concept of the first passage time (FPT). Firstly, a multi-phase degradation model with jumps based on the Wiener process is formulated to describe the multi-phase degradation pattern. Then, we attain the lifetime&rsquo, s closed-form expression for the two-phase model with fixed jump relying on the distribution of the degradation state at the change point. Furthermore, we continue to investigate the lifetime estimation of the degradation process with random effect caused by unit-to-unit variability and the multi-phase degradation process. We extend the results of the two-phase case with fixed parameters to these two cases. For better implementation, a model identification method with off-line and on-line parts based on Expectation Maximization (EM) algorithm and Bayesian rule is proposed. Finally, a numerical case study and a practical example of gyro are provided for illustration.

Standby redundancy is an effective fault-tolerant technique for enhancing reliability and prolonging the standby system’s operating lifetime. How to estimate the total lifetime of a standby system with a predetermined number of standby components (i.e., spare parts) presents an interesting practical issue. Most existing studies, however, have mainly focused on the lifetime or remaining useful lifetime prediction of a single online product. Moreover, spare parts usually deteriorate in storage, which will worsen their performance and even lead to failure. This makes lifetime estimation for a standby system more challenging. This study, therefore, focused on how to estimate a standby system’s lifetime (SSL) with deteriorating spare parts. Unlike prior work, we fully considered the uncertainty and randomness caused by the spare parts’ storage degradation in the SSL estimation. By establishing the transition probability function of the storage degradation process, we first proposed a general iterative algorithm for SSL estimation under the concept of the first passage time (FPT) and provided the proof based on mathematical induction. Then, we extended this result to a Wiener-process-based model and obtained the iterative result in a single integral form. Moreover, we attained the analytical expressions of SSL mean and variance under a non-storage-failure hypothesis and further provided the requirement for the establishment of the hypothesis. Finally, a numerical case and a practical case of the gyroscope are introduced for illustration.

Due to the environment changing, working state switch, and physical mutation, the degradation process often exhibits multi-phase feather and the random jump usually exists at the change point, which may make traditional lifetime estimation approach for single-stage degradation process unsuitable. In this paper, we concentrate on how to model the multi-phase degradation process and estimate its lifetime. Under this consideration, we firstly formulate a general degradation framework based on the multi-phase Wiener model to describe such the degradation process. Unlike traditional methods, we provide an analytical form of lifetime estimation for two-phase model with fixed jump, and a single-integral form for two-phase model with random jump via taking full advantage of the uncertainty of the degradation state at the change point. In addition, we extend these results from two-phase model to multi-phase model and obtain a general result of lifetime estimation. Finally, a numerical case is provided for illustration.

Remaining useful life (RUL) prediction method based on degradation trajectory has been one of the most important parts in prognostics and health management (PHM). In the conventional model, the degradation data are usually used for degradation modeling directly. In engineering practice, the degradation of many systems presents a volatile situation, that is, fluctuation. In fact, the volatility of degradation data shows the stability of system, so it could be used to reflect the performance of system. As such, this paper proposes a new degradation model for RUL estimation based on the volatility of degradation data. Firstly the degradation data are decomposed into trend items and random items, which are defined as a stochastic process. Then the standard deviation of the stochastic process is defined as another performance variable because standard deviation reflects the system performance. Finally the Wiener process and the normal stochastic process are used to model the trend items and random items separately, and then the probability density function (PDF) of the RUL is obtained via a redefined failure threshold function that combines the trend items and the standard deviation of the random items. Two practical case studies demonstrate that, compared with traditional approaches, the proposed model can deal with the degradation data with many fluctuations better and can get a more reasonable result which is convenient for maintenance decision.

The last decades have witnessed the development of degradation modeling based remaining useful life (RUL) estimation, especially for Wiener process based degradation models. However, most researchers paid their attention to the drift coefficient representing the degradation rate, while much fewer eyes focus on the diffusion coefficient. The under consideration of diffusion process may cause bias in degradation modeling and RUL estimation, when the concerned system has volatile degradation path. Therefore, this paper construct a new kind of age-dependent degradation model based on diffusion process with time-dependent drift and diffusion coefficient. Unlike most existing models, the degradation process' temporal variance is assumed to be proportional to the instantaneous expectation of the degradation process. By doing so, volatile degradation process can be modelled. The exact lifetime and RUL distributions under the concept of first hitting time (FHT) are derived, and normal random-effect are incorporated to depict the unit-to-unit variability within degradation process. The maximum likelihood estimation method based on historical degradation data is presented. Through a specific degradation model and its application to the data of accelerators, the presented degradation model has been illustrated and validated.

This paper focuses on the problem of how to estimate remaining useful life (RUL) for a class of systems with high fluctuating dagradation caused by time-varying mean and variance. In engineering practice, the fluctuation of the degradation data can reflect the system stability, and hence it can serve as an additional indicator for the system's health state in addition to the degradation observations. To estimate the RUL for this class of systems, three issues should be considered jointly: (i) how to model the degradation data with high fluctuation, (ii) how to define an indicator denoting the fluctuation's level, and (iii) how to combine the degradation path and the fluctuation for the RUL estimating task. In responses to the above issues, this paper characterizes the degree of the fluctuation by the standard deviation of the stochastic degradation process and defines the standard deviation as an additional performance variable. Moreover, a stochastic degradation model is proposed to reflect the high fluctuating degradation data, in which the standard deviation is time dependent. To obtain the marginal distribution of the RUL derived by the standard deviation, the failure threshold of the standard deviation is given according to its influence on the degradation process. Another marginal distribution derived by the degradation data is also obtained by the time distribution of the degradation process crossing the degradation threshold. Then, through deriving the correlation between the two marginal distributions based on the probability theory, the joint distribution of the RUL is obtained. Finally, a practical case study for gyro is conducted. The results demonstrate the feasibility and applicability of the proposed model. Copyright © 2014 John Wiley & Sons, Ltd.

Remaining useful life (RUL) prediction is a key link in prognostics and health management. More accurate RUL prediction results will lead to more reasonable decision making on sequential management activities including maintenance, replacement, spare parts ordering, etc. In engineering practice, many products exhibit more volatile degradation paths when they have high degradation rates. By this observation, this paper concerns the problem of RUL prediction for a class of critical products which possess positive correlation between degradation rate and volatility. A Wiener-process-based degradation model with a special volatility parameter form is developed to achieve the aim. In this model, the volatility parameter has been set dependent on the drift parameter reflecting the degradation rate to describe the concerned problem. Both Bayesian updating and expectation maximization (EM) algorithm are used to estimate the unknown parameters in the model depend on the historically-observed degradation data. An exact and closed-form RUL distribution, which incorporates the random-effect capturing the unit-to-unit variability, is derived under the concept on the first passage time. Finally, a practical case study is used to illustrate and demonstrate the effectiveness of the presented method. The results show that the proposed approach can provide a more accurate RUL estimation for degradation system with volatile degradation path.