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LSTM network is an effective RNN model to predict the system RUL for its superior performance in sequential data processing. Usually, networks trained by life-cycle labeled dataset would possess similar RUL predicting accuracies, because the network training algorithm could ensure the network optimality for the whole training dataset. However, for networks trained by non-life-cycle labeled samples, the network uncertainty caused by different training conditions could lead to degradation prediction uncertainty for some local points. Further, the RUL predicting results that are computed by these uncertain local points may shows relatively large differences. Therefore, in order to obtain an accurate RUL prediction with networks trained by non-life-cycle labeled samples, our paper proposes a novel network model averaging method to deal with the network uncertainty. What is more, to learn the temporal correlation information of training samples sufficiently, we adopt the Bi-LSTM network to illustrate the application of the proposed network model averaging method. Finally, degradation values of Graphite/LiCoO2 battery are used to verify the effectiveness of the proposed method. The results show that the proposed method could improve the RUL prediction accuracy and reduce the prediction error.

The growth of the Industrial Internet of Things (IIoT) has generated a renewed emphasis on research of prognostic degradation modeling whereby degradation signals, such as vibration signals, temperature and acoustic emissions, are used to estimate the state-of-health and predict the remaining useful life (RUL). Besides the inherent system state, external operating conditions, such as the rotational speed and load also play a significant role in the behavior of degradation signals. Time-varying operating conditions often cause two major effects on the degradation signals. First, they change the degradation rate of systems. Second, they cause signal jumps at condition change-points. These two factors make RUL prediction more difficult under time-varying operating conditions. This paper proposes a RUL prediction method by introducing these two factors into a state-space model. Changes in the degradation rate are introduced into a state transition function, and jumps in the degradation signals are introduced into a measurement function. The separate analysis of these two factors makes it possible to distinguish their own contributions to RUL prediction, thus avoiding false alarms and improving the prediction accuracy. The effectiveness of the proposed method is demonstrated using both a simulation study and an accelerated degradation test of rolling element bearings.

With advances in sensing and monitoring techniques, real time multi-sensor monitoring data of stochastic degrading devices has become the reality. How to effectively fuse these multi-sensor monitoring data to first construct the composite health index (CHI) and then model its degradation evolving process has become the emerging topic in the field of the remaining useful life (RUL) prediction. However, existing works treat the CHI construction and degradation modeling for prognostics under multi-sensor data as the disjoint problems rather than both though they are closely related in nature. To address this issue, this paper presents a novel data-model interactive RUL prediction method for multi-sensor monitored stochastic degrading devices. In the proposed method, based on the CHI extracted from multi-sensor historical data and the associated lifetime prediction via stochastic degradation modeling, an optimization objective function synthesizing the mean squared error between the predicted life and the actual life as well as the variance of the predicted life is constructed. As such, a closed-loop feedback mechanism is established for the CHI constructing and stochastic degradation modeling. Based on this feedback mechanism, the fusion coefficients for multi-sensor data and the failure threshold of the associated CHI are reversely optimized to realize the collaborative interaction between the CHI constructing and stochastic degradation modeling. To do so, the goal of making the constructed CHI automatically match the adopted stochastic degradation model can be achieved naturally. To make the degradation model accurately reflect the current reality of the in-service device, a sequential Bayesian method is proposed to update the degradation model parameters. Based on the updated model, the RUL distribution can be derived under the concept of the first passage time to achieve the prognosis. Finally, through multi-sensor data of aircraft gas turbine engines, we justify the necessity of applying the proposed method in prognosis and show its advantages in the improvements of the signal quality and prognosis accuracy.

Transforming nonlinear degradation paths into nearly linear ones has been widely used for nonlinear degradation modeling and prognostics. However, types of the current transformation functions are difficult to determine. This paper addresses issues in nonlinear stochastic degradation modeling and prognostics from a Box-Cox transformation (BCT) perspective. Specifically, the BCT is first used to transform the nonlinear degradation data into nearly linear data, and then the Wiener process with random drift is utilized to model the evolving process of the transformed data. To determine the model parameters, a two-stage estimation procedure is developed including offline stage and online stage. In the offline stage, the parameters are determined via maximum likelihood estimation method based on the historical degradation data and such estimated values are used to initialize the online stage. During the online stage, the Bayesian method is adopted to update the model parameters using the data of the degrading system in service, in which the hyperparameters are updated by the expectation maximization algorithm. A closed-form solution to remaining useful life with updated model parameters is further derived for prognostics. Finally, case studies for lithium-ion batteries and liquid coupling devices are provided to demonstrate the proposed approach.

Degradation-based modeling methods have been recognized as an essential and effective approach for lifetime and remaining useful life (RUL) estimations for various health management activities that can be scheduled to ensure reliable, safe, and economical operation of deteriorating systems. As one of the most popular stochastic modeling methods, the previous several decades have witnessed remarkable developments and extensive applications of Wiener-process-based methods. However, there is no systematic review particularly focused on this topic. Therefore, this paper reviews recent modeling developments of the Wiener-process-based methods for degradation data analysis and RUL estimation, as well as their applications in the field of prognostics and health management (PHM). After a brief introduction of conventional Wiener-process-based degradation models, we pay particular attention to variants of the Wiener process by considering nonlinearity, multi-source variability, covariates, and multivariate involved in the degradation processes. In addition, we discuss the applications of the Wiener-process-based models for degradation test design and optimal decision-making activities such as inspection, condition-based maintenance (CBM), and replacement. Finally, we highlight several future challenges deserving further studies.

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 the key part of Prognostics and Health Management (PHM), Remaining Useful Life (RUL) estimation has been extensively investigated in recent years. Current RUL estimation studies considering the intervention of imperfect maintenance activities usually assumed that maintenance activities have a single influence on the degradation level or degradation rate, but not on both. Aimed at this problem, this paper proposes a new degradation modeling and RUL estimation method taking the influence of imperfect maintenance activities on both the degradation level and the degradation rate into account. Toward this end, a stochastic degradation model considering imperfect maintenance activities is firstly constructed based on the diffusion process. Then, the Probability Density Function (PDF) of the RUL is derived by the convolution operator under the concept of First Hitting Time (FHT). To implement the proposed RUL estimation method, the Maximum Likelihood Estimation (MLE) is utilized to estimate the degradation related parameters based on the Condition Monitoring (CM) data, while the Bayesian method is utilized to estimate the maintenance related parameters based on the maintenance data. Finally, a numerical example and a practical case study are provided to demonstrate the superiority of the proposed method. The experimental results show that the proposed method could greatly improve the RUL estimation accuracy for the degrading equipment subjected to imperfect maintenance activities. Keywords: Convolution operator, Diffusion process, First hitting time, Imperfect maintenance, Remaining useful life

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.

In this paper, we propose a jump diffusion process with non-homogeneous compound Poisson process to model the degradation process with randomly occurring jumps, which combines two stochastic processes, i.e., traditional diffusion process to describe the continuous degradation and non-homogeneous compound Poisson process to depict random jumps with a time-varying intensity. The approximated analytical lifetime under the concept of the first passage time (FPT) is obtained by a time–space transformation technique. To identify the model parameters, we first present a general method based on Maximum Likelihood Estimation (MLE) for the proposed model, and then specifically provide a two-step approach for linear jump diffusion process via combining MLE and Expectation Conditional Maximization (ECM) algorithm. Finally, a numerical example and a study on the furnace wall are provided to illustrate the effectiveness of the proposed method.

Models with random-effects are generally used in the field of degradation modeling and remaining useful life (RUL) estimation for describing unit-to-unit variability. The wide employment of parameters, which is assumed to be subjected to normal distribution to capture this variability, may disaccord with actual industrial conditions, and will introduce misspecifications. Such misspecification can affect the accuracy of RUL estimation and the subsequent inference results. In this paper, we propose a degradation model with flexible random-effects, which makes it flexible to choose distributions to portray the unit-to-unit variability according to the available information. To do so, the mixture of normal distributions, as a distribution describing random-effects, is incorporated into a class of diffusion process based degradation models whose drift coefficient is a linear combination of some time-dependent functions with known forms. The combination coefficients of each function are treated as random variables drawn from the mixture of normal distributions. An analytical approximated probability density function (PDF) of the RUL is derived under the concept of first passage time (FPT). To identify the model parameters, a framework for parameter estimation is presented based on stochastic expectation maximization (SEM) algorithm. Finally, simulation studies are provided to demonstrate the superiority of the normal mixture over the individual normal distribution for describing random effects in RUL estimation.

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.

Joint maintenance and spare parts inventory optimization has attracted increasing attention in recent years because of its capability in addressing the maintenance planning and the spare parts provisioning of industrial systems simultaneously. However, imperfect maintenance (IM) actions are either neglected or over-simplified as constant improvements in existing studies, which reduces their practicality in industrial applications. To tackle this limitation, this paper investigates the joint maintenance and spare parts inventory optimization for multi-unit systems with the consideration of IM actions as random improvement factors. First, a two-step approximate derivation method is proposed, which overcomes the derivation difficulties of replacement numbers due to the introduction of random improvement factors and enables the construction of the inventory level transition relationship. Then based on the inventory level transition relationship, an expected total cost model is formulated via the finite horizon stochastic dynamic programming (FHSDP). The decision variables are optimized by the joint use of enumeration and the FHSDP. Finally, a numerical simulation of a wind farm is carried out for illustration. Sensitivity analyses are further conducted to study the influences of critical parameters.

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.

Fault diagnosis methods based on dimensionless indicators have long been studied for rotating machinery. However, traditional dimensionless indicators frequently suffer a low accuracy of fault diagnosis for nonlinear and non-stationary dynamic signals of rotating machinery. In this paper, we propose an effective fault diagnosis method based on multi-scale dimensionless indicator (MSDI) and random forests. In the proposed method, the real-time vibration signals are first processed by the variational mode decomposition and then six types of MSDI are constructed based on the decomposed signals. Through utilizing the Fisher criterion, several top ranked MSDIs are selected as fault features. Based on the selected MSDIs, the random forests model is applied to determine fault types. To verify the superiority of the proposed method, several experiments on fault diagnosis are conducted on a centrifugal multi-level impeller blower. The results demonstrate that the proposed method can successfully identify different fault types and the average accuracy can reach 95.58%. In contrast with traditional dimensionless indicators based methods, the proposed method can improve the fault diagnosis accuracy by 7.25% and outperforms other techniques such as back propagation neural network, support vector machine and extreme learning machine. These results indicate that the MSDI can effectively solve the deficiency of the traditional dimensionless indicator, and has stronger distinguishing ability for the fault types.

Once in the hands of end users, such durable equipment as spacecraft, aircraft, ships, automobiles, computers, etc. are in a state of debugging, working or storage. In either state, availability, reliability and super-efficiency are the ultimate goals, which have been achieved through constant monitoring as well as regular, preventive, routine and corrective maintenance. Although some advanced instruments can visualize certain invisible malfunctioning phenomena into visible ones, deeply hidden troubles cannot be found unless monitoring and testing data are addressed using tools that process the data statistically, analytically and mathematically. Some state-of-the-art trouble-shooting and life-predicting techniques and approaches are introduced in this paper.

Some systems may spend most of their time in storage, but once needed, must be fully functional. Slow degradation occurs when the system is in storage, so to ensure the functionality of these systems, condition monitoring is usually conducted periodically to check the condition of the system. However, taking the condition monitoring data may require putting the system under real testing situation which may accelerate the degradation, and therefore, shorten the storage life of the system. This paper presents a case study of condition-based remaining storage life prediction for gyros in the inertial navigation system on the basis of the condition monitoring data and the influence of the condition monitoring data taking process. A stochastic-filtering-based degradation model is developed to incorporate both into the prediction of the remaining storage life distribution. This makes the predicted remaining storage life depend on not only the condition monitoring data but also the testing process of taking the condition monitoring data, which the existing prognostic techniques and algorithms did not consider. The presented model is fitted to the real condition monitoring data of gyros testing using the maximum likelihood estimation method for parameter estimation. Comparisons are made with the model without considering the process of taking the condition monitoring data, and the results clearly demonstrate the superiority of the newly proposed model.

Inertial navigation systems have been widely used in both civilian and military systems because of their autonomous navigation capability. Nevertheless, due to its autonomous characteristics, the navigation precision of an inertial navigation system is heavily influenced by its drift errors, which results from the performance degradation of the system in use. One of the most effective means of eliminating such adverse effects is to predict the drift error values in advance, and compensate for them subsequently. It is therefore significantly important to accurately predict the degrading trend of the drift errors of an inertial navigation system. We propose a novel degradation modeling method based on a nonlinear random-coefficient regression model to predict the drift errors. The parameters of the model are dynamically updated by the expectation maximization algorithm, in conjunction with the Bayesian inference method at the time when a new drift error data is observed. In doing this, the degrading trend of the drift errors can be predicted in real time. Finally, a batch of drift error data of an inertial navigation system is used to validate the feasibility and effectiveness of the developed prognostic method.

Reliable and accurate lifetime estimates for key engineering assets have long been a hot research topic attracting increasing attention in reliability and operational research communities and practices.

Remaining useful life (RUL) estimation is regarded as one of the most central components in prognostics and health management (PHM). Accurate RUL estimation can enable failure prevention in a more controllable manner in that effective maintenance can be executed in appropriate time to correct impending faults. In this paper we consider the problem of estimating the RUL from observed degradation data for a general system. A degradation path-dependent approach for RUL estimation is presented through the combination of Bayesian updating and expectation maximization (EM) algorithm. The use of both Bayesian updating and EM algorithm to update the model parameters and RUL distribution at the time obtaining a newly observed data is a novel contribution of this paper, which makes the estimated RUL depend on the observed degradation data history. As two specific cases, a linear degradation model and an exponential-based degradation model are considered to illustrate the implementation of our presented approach. A major contribution under these two special cases is that our approach can obtain an exact and closed-form RUL distribution respectively, and the moment of the obtained RUL distribution from our presented approach exists. This contrasts sharply with the approximated results obtained in the literature for the same cases. To our knowledge, the RUL estimation approach presented in this paper for the two special cases is the only one that can provide an exact and closed-form RUL distribution utilizing the monitoring history. Finally, numerical examples for RUL estimation and a practical case study for condition-based replacement decision making with comparison to a previously reported approach are provided to substantiate the superiority of the proposed model.

Over the past few decades, condition-based maintenance (CBM) has attracted many researchers because of its effectiveness and practical significance. This paper deals with mission-oriented systems subject to gradual degradation modeled by a Wiener stochastic process within the context of CBM. For a mission-oriented system, the mission usually has constraints on availability/reliability, the opportunity for maintenance actions, and the monitoring type (continuous or discrete). Furthermore, in practice, a mission-oriented system may undertake some preventive maintenance (PM) and after such PM, the system may return to an intermediate state between an as-good-as new state and an as-bad-as old state, i.e., the PM is not perfect and only partially restores the system. However, very few CBM models integrated these mission constraints together with an imperfect nature of the PM into the course of optimizing the PM policy. This paper develops a model to optimize the PM policy in terms of the maintenance related cost jointly considering the mission constraints and the imperfect PM nature. A numerical example is presented to demonstrate the proposed model. The comparison with the simulated results and the sensitivity analysis show the usefulness of the optimization model for mission-oriented system maintenance presented in this paper.

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.

Remaining useful life (RUL) is the useful life left on an asset at a particular time of operation. Its estimation is central to condition based maintenance and prognostics and health management. RUL is typically random and unknown, and as such it must be estimated from available sources of information such as the information obtained in condition and health monitoring. The research on how to best estimate the RUL has gained popularity recently due to the rapid advances in condition and health monitoring techniques. However, due to its complicated relationship with observable health information, there is no such best approach which can be used universally to achieve the best estimate. As such this paper reviews the recent modeling developments for estimating the RUL. The review is centred on statistical data driven approaches which rely only on available past observed data and statistical models. The approaches are classified into two broad types of models, that is, models that rely on directly observed state information of the asset, and those do not. We systematically review the models and approaches reported in the literature and finally highlight future research challenges.

Research highlights? A dynamic evidential reasoning algorithm is presented for dynamic fusion. ? A fault prognosis model is established based on the dynamic evidential reasoning algorithm. ? The optimization models are presented for estimating the parameters of the prognosis model. ? The developed model has been validated by case studies. In this paper, a new fault prediction model is presented to deal with the fault prediction problems in the presence of both quantitative and qualitative data based on the dynamic evidential reasoning (DER) approach. In engineering practice, system performance is constantly changed with time. As such, there is a need to develop a supporting mechanism that can be used to conduct dynamic fusion with time, and establish a prediction model to trace and predict system performance. In this paper, a DER approach is first developed to realize dynamic fusion. The new approach takes account of time effect by introducing belief decaying factor, which reflects the nature that evidence credibility is decreasing over time. Theoretically, it is show that the new DER aggregation schemes also satisfy the synthesis theorems. Then a fault prediction model based on the DER approach is established and several optimization models are developed for locally training the DER prediction model. The main feature of these optimization models is that only partial input and output information is required, which can be either incomplete or vague, either numerical or judgmental, or mixed. The models can be used to fine tune the DER prediction model whose initial parameters are decided by expert's knowledge or common sense. Finally, two numerical examples are provided to illustrate the detailed implementation procedures of the proposed approach and demonstrate its potential applications in fault prediction.

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.