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Evidential reasoning (ER) rule has been widely used in dealing with uncertainty. As an important parameter to measure the inherent property of evidence, the evidence reliability makes the ER rule constitute a generalized reasoning framework. In current research of the ER rule, the evidence reliability tends to be expressed in the form of quantitative value by certain methods or expert knowledge. The single quantitative value lacks the ability to describe the statistical property of reliability, which leads to unreasonable results. In this article, a new ER rule with continuous probability distribution of reliability denoted by ERr-CR is proposed. The combination of two pieces of evidence is discussed in detail, where the reliability is profiled as random variables with specific probability distribution. To characterize the output of ERr-CR, a novel concept of expectation of the expected utility is proposed. In addition, the ERr-CR is expanded to multiple pieces of evidence to show its universality. Further, the basic performances of the ERr-CR are explored to illustrate the rationality. Moreover, a case study of safety assessment of natural gas storage tanks (NGSTs) is conducted to show the potential applications of ERr-CR, which makes the proposed method more practical. [ABSTRACT FROM AUTHOR]

In current studies of safety assessment for complex systems with the evidential reasoning (ER) rule, the evidence reliability is generally given by experts, which makes the observation data by sensors ignored. However, sensors are inevitably affected by such various uncertainties as perturbations in engineering practice, which can reduce their quality and tracking ability. As such, the observation data may become unreliable, and the modeling accuracy of the ER rule is decreased. In this article, a new ER rule-based safety assessment method with sensor reliability for complex systems is proposed, where sensor reliability and perturbation are considered. The coefficient of the variation-based weighting (CVBW) method is employed to obtain sensor weight. The sensor reliability is calculated by static reliability and dynamic reliability, which are determined by experts and the distance-based method, respectively. The perturbation is quantified as a bounded parameter defined as the perturbation factor, which is used to describe uncertainties and aggregate static reliability and dynamic reliability. The performance analysis of safety assessment is conducted to demonstrate the rationality of perturbation and position poor sensors, followed by a safety assessment algorithm. A case study is carried out to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This article aims to propose an evidential reasoning (ER) rule considering the parameter uncertainty. As the essential parameters, the evidence weight and reliability make the ER rule constitute a generalized reasoning framework. Theoretically, the weight is affected by subjective cognition, while the reliability mainly reflects objective variation. However, most of the recent researches have focused on the quantitative calculation methods, which make the differences in the property of the two parameters ignored. In this article, a relatively different idea from previous studies is provided, in which multisource uncertainty of parameters is fully considered. On the one hand, the weight is profiled by interval variable with lower and upper bounds. On the other hand, the reliability is modeled by random variable with probability distribution function. Then, a unified inference model for evidence aggregation is developed based on the inference process of the ER rule. In addition, some basic properties of the model are clearly presented to illustrate the rationality of parameter uncertainty. Finally, a practical example is given to show the potential applications of the proposed model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The belief rule-base (BRB) model is a new intelligent expert system with the characteristics of both expert system and data-driven model. In BRB there are many if-then rules which use belief degrees to express various types of uncertain information, including fuzziness, randomness, and ignorance. As a semi-quantitative modeling tool for complex systems, BRB has the superiorities of dealing both numerical quantitative data and linguistic qualitative knowledge that are derived from heterogeneous sources. Moreover, it is also a white box approach which can provide direct access and transparency to decision makers and stakeholders. Currently, BRB has been widely applied in many fields, such as decision making, reliability evaluation, network security situation awareness, fault diagnosis, and so on. To fully demonstrate the progress of BRB, the original BRB, and some evolution forms are introduced in this article. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Due to the inner deteriorating mechanism or the mutant environmental stress, the degradation systems with multi-phase features have frequently been encountered in engineering practice. The key issue for prognostics of such systems is to account for the impact of the changing-point variability and the associated degradation state at this point on the progression of the degradation process. However, current studies usually treat the degradation state at the change point as a fixed value rather a random variable. Thus, it is still challenging to predict the lifetime of such multi-phase degrading systems. To this end, we first formulate a general degradation modeling framework based on a two-phase Wiener process. In prognostics, we take into full account the uncertainty of the degradation state at the changing point and then derive the analytical expressions of the lifetime and remaining useful life under the concept of the first passage time. The derived results are distinguished from existing results limited to the fixed state at the changing point. Furthermore, we extend our approach and results to cases with unit-to-unit variability and multiple phases. To facilitate the model implementation, we propose both offline and online methods for parameter identification, which make full use of the historical data and the in-service data. Finally, a numerical simulation and a practical case study are provided for illustration. [ABSTRACT FROM AUTHOR]

It is vital to online assess the safety of a complex dynamic system by taking into account the current state, degradation trend, and historical records together. This paper proposes a new safety assessment model with an online algorithm based on the evidential reasoning (ER) approach. It does not only take into account the relative importance of each safety indicator, but also consider the reliability of each indicator. To obtain the integrated safety level, multiple safety indicators are fused at first and the “history,” “current,” and “future” safety states are then integrated. First, a forecasting model based on the third-order Volterra filter is proposed to predict the safety indicators’ information online. Second, an adaptive weighting model is developed to automatically adjust to various conditions and track the characteristics of the dynamic system, and the reliability of each indicator is considered to diminish the influence of inherent disturbance and/or noise. Finally, a safety assessment aggregation scheme based on the ER approach is presented to fuse the history, current, and future safety indicators to obtain the corresponding safety state, and the safety states are then fused synthetically to obtain a comprehensive safety assessment result of the complex dynamic system. A numerical study is examined to demonstrate the implementation and effectiveness. Moreover, a practical example of the inertial navigation system is studied to show the potential applications of the proposed ER-based safety assessment method. [ABSTRACT FROM AUTHOR]

Many industrial systems inevitably suffer performance degradation. Thus, predicting the remaining useful life (RUL) for such degrading systems has attracted significant attention in the prognostics community. For some systems like batteries, one commonly encountered phenomenon is that the system performance degrades with usage and recovers in storage. However, almost all of the current prognostic studies do not consider such a recovery phenomenon in stochastic degradation modeling. In this paper, we present a prognostic model for deteriorating systems experiencing a switching operating process between usage and storage, where the system degradation state recovers randomly after the storage process. The possible recovery from the current time to the predicted future failure time is incorporated in the prognosis. First, the degradation state evolution of the system is modeled through a diffusion process with piecewise but time-dependent drift coefficient functions. Under the concept of first hitting time, we derived the lifetime and RUL distributions for systems with specific constant working mode. Further, we extended the results of RUL distribution in specific constant working mode to the case of stochastic working mode, which is modeled through a flexible two-state semi-Markov model (SMM) with phase-type distributed interval times. The unknown parameters in the present model are estimated based on the observed condition monitoring data of the system, and the SMM model is identified on the basis of the operating data. A numerical study and a case study of Li-ion batteries are carried out to illustrate and demonstrate the proposed prognostic method. Experimental results indicate that the presented method can improve the accuracy of lifetime and RUL estimation for systems with state recovery. [ABSTRACT FROM PUBLISHER]

Reliability estimation is central to enhance safety, availability, and effectiveness of phased-mission systems (PMSs). With the development of information and sensing technologies, condition monitoring (CM) data are now available in many real-world PMSs, and then a more interesting question: how can we dynamically estimate the reliability of PMSs using the in-situ CM data, is of considerable significance to industrial practitioners. In this paper, using the CM data and degradation data of PMS, we present a novel condition-based approach to resolve this question under dynamic operating scenarios. This paper differs from most existing methods which only consider the static scenario without using real-time information, and estimate the reliability only for a population of PMSs but not for an individual PMS in service. To establish a linkage between the historical data and real-time data of the individual PMS, a stochastic filtering model is first utilized to model the phase duration. As such, the updated estimation of the mission time can be obtained by Bayesian law at each phase. To account for the dependency of the degradation progression of PMS on the mission process, the degradation process of PMS is modeled by a Brownian motion with a mission phase-dependent drift coefficient. The corresponding lifetime is derived and the lifetime distribution of PMS can be updated under Bayesian framework once new information is available. Unique to this paper is the union of the CM data and degradation data of PMS to real-time estimate the mission reliability through the estimated distribution of the mission time in conjunction with the estimated lifetime distribution, in which the estimated lifetime considers the dependency of the degradation rate of PMS on mission phase. The effectiveness of the proposed approach is verified by a numerical simulation and a case study. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a novel reliability prediction technique based on the evidential reasoning (ER) algorithm is developed and applied to forecast reliability in turbocharger engine systems. The focus of this study is to examine the feasibility and validity of the ER algorithm in systems reliability prediction by comparing it with some existing approaches. To determine the parameters of the proposed model accurately, some nonlinear optimization models are investigated to search for the optimal parameters of forecasting model by minimizing the mean square error (MSE) criterion. Finally, a numerical example is provided to demonstrate the detailed implementation procedures. The experimental results show that the prediction performance of the ER-based prediction model outperforms several existing methods in terms of prediction accuracy or speed. [Copyright &y& Elsevier]

Abstract: Remaining useful life estimation (RUL) is an essential part in prognostics and health management. This paper addresses the problem of estimating the RUL from the observed degradation data. A Wiener-process-based degradation model with a recursive filter algorithm is developed to achieve the aim. A novel contribution made in this paper is the use of both a recursive filter to update the drift coefficient in the Wiener process and the expectation maximization (EM) algorithm to update all other parameters. Both updating are done at the time that a new piece of degradation data becomes available. This makes the model depend on the observed degradation data history, which the conventional Wiener-process-based models did not consider. Another contribution is to take into account the distribution in the drift coefficient when updating, rather than using a point estimate as an approximation. An exact RUL distribution considering the distribution of the drift coefficient is obtained based on the concept of the first hitting time. A practical case study for gyros in an inertial navigation system is provided to substantiate the superiority of the proposed model compared with competing models reported in the literature. The results show that our developed model can provide better RUL estimation accuracy. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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. • Both spare-part storage degradation and immediate burst storage failure are considered. • A general iterative approach for the lifetime estimation of the standby system is proposed. • A joint optimization model of the maintenance interval and the inventory number is established. [ABSTRACT FROM AUTHOR]