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Remaining useful life (RUL) prediction has become one of the key technologies for reducing costs and improving safety of lithium-ion batteries. To our knowledge, it is difficult for existing nonlinear degradation models of the Wiener process to describe the complex degradation process of lithium-ion batteries, and there is a problem with low precision in parameter estimation. Therefore, this paper proposes a method for predicting the RUL of lithium-ion batteries based on a cubic polynomial degradation model and envelope extraction. Firstly, based on the degradation characteristics of lithium-ion batteries, a cubic polynomial function is used to fit the degradation trajectory and compared with other nonlinear degradation models for verification. Secondly, a subjective parameter estimation method based on envelope extraction is proposed that estimates the actual degradation trajectory by using the average of the upper and lower envelope curves of the degradation data of lithium-ion batteries and uses the maximum likelihood estimation (MLE) method to estimate the unknown model parameters in two steps. Finally, for comparison with several typical nonlinear models, experiments are carried out based on the practical degradation data of lithium-ion batteries. The effectiveness of the proposed method to improve the accuracy of RUL prediction for lithium-ion batteries was demonstrated in terms of the mean square error (MSE) of the model and MSE of RUL prediction.

Remaining useful life (RUL) prediction plays a crucial role in prognostics and health management (PHM). Recently, the adaptive model-based RUL prediction, which is proven effective and flexible, has gained considerable attention. Most research on adaptive degradation models focuses on the Wiener process. However, since the degradation process of some products is accumulated and irreversible, the inverse Gaussian (IG) process that can describe monotonic degradation paths is a natural choice for degradation modelling. This article proposes a nonlinear adaptive IG process along with the corresponding state space model considering measurement errors. Then, an improved particle filtering algorithm is presented to update the degradation parameter and estimate the underlying degradation state under the nonGaussian assumptions in the state space model. The RUL prediction depending on historical degradation data is derived based on the results of particle methods, which can avoid high-dimensional integration. In addition, the expectation-maximization (EM) algorithm combined with an improved particle smoother is developed to estimate and adaptively update the unknown model parameters once newly monitored degradation data become available. Finally, this article concludes with a simulation study and a case application to demonstrate the applicability and superiority of the proposed method.

Over the last several years, many testability growth models (TGMs) have been developed to greatly facilitate engineers and managers in tracking and measuring the growth of testability as system is being improved. Most TGMs consider only one or two variation patterns of the aspects, such as the testability growth effort (TGE) in testability design limitation (TDL) identification, the rectifying delay and the new TDL introduction in TDL correction. However, the ignorance of such joint consideration may lead to a lower fitting ability to the fault detection/isolation data. Inspired by the counting idea of non-homogeneous Poisson process (NHPP), a NHPP based testability growth model (TGM) considering the recurrence rate function (RRF) of TDL identification, TDL correction and new TDL introduction is proposed for the foundation of TGM. A real data set of a missile control system is used to validate the above TGMs in fitting ability, estimation accuracy and prediction capability. Results show that the bell-shaped curve can fit the identification process and rectifying delay process of TDL well, and the imperfect correction of TDL really exists in the testability growth test (TGT), and the inflected s-shaped and Gamma function based TGM gives good capacity to the real data set.

Wiener processes have been extensively used to model the degradation processes exhibiting a linear trend for predicting the remaining useful life (RUL) of degrading components. To incorporate the real-time degradation monitoring information into degradation modeling, the Bayesian method has been frequently utilized to update the model parameter, particularly for the drift parameter in Wiener process. However, due to the inherent independent increment and Markov properties of Wiener process, the Bayesian updated drift parameter only utilizes the current degradation measurement and cannot incorporate the whole degradation measurements up to now. As such, once the updated degradation model in this way is used to predict the RUL, the obtained result may be dominated by partial degradation observations or lower the prognosis accuracy. In this paper, we propose a sequential Bayesian updated Wiener process model for RUL prediction. First, a Wiener process model with random drift efficient is used to model the degradation process with the linear trend. To estimate the model parameters, the historical degradation measurements are used to determine the initial model parameters based on the maximum likelihood estimation (MLE) method. Then, for the degrading component in service, a sequential Bayesian method is proposed to update the random drift parameter in Wiener process model. Differing from existing studies using the Bayesian method, the proposed sequential method uses the Bayesian estimate for random drift parameter in the last time as the prior of the next time. As such, the Bayesian estimate for random drift parameter in the current time is dependent on the whole degradation measurements up to current time, and thus the problem of depending only on the current degradation measurement is solved. Finally, we derive the analytical expressions of the RUL distribution based on the concept of the first passage time (FPT). Two case studies associated with the gyroscope drift data and lithium-ion battery data are provided to show the effectiveness and superiority of the proposed method. The results indicate that the proposed method can improve the RUL prediction accuracy.

The remaining useful life (RUL) prediction is the core of equipment maintenance and decision-making. Accurate RUL prediction can make effective maintenance before the failure occurs to reduce the probability of equipment failure. However, in industrial practice, we often face the situation that prior information is insufficient or inaccurate, which could influence our prediction of RUL and reduce the prediction accuracy severely. To solve this problem, we study the issue of RUL prediction under imperfect prior degradation data by reasonably fusing the prior information and the field degradation data. Firstly, based on the linear Wiener degradation process, we prove a conclusion that the random parameter estimation results by the joint Bayesian algorithm and the expectation-maximization (EM) algorithm are the same as that by the maximum likelihood estimation (MLE) method. It shows that the joint estimation method completely overcomes the influence of prior information, and the more iterations, the smaller proportion of prior information. Secondly, we also prove that the random parameter joint updating method for the nonlinear Wiener degradation process has the same characteristics. Then, a heuristic algorithm that reasonably fuses the prior information and the field degradation data is proposed, which controls the number of iterations of the joint updating algorithm based on the credibility of the prior information. It is also applied to the nonlinear degradation process by not iterating the nonlinear coefficients. Finally, the correctness of the conclusion of this paper is verified by the simulated degradation data, and the effectiveness of the proposed heuristic updating method is verified by the actual lithium battery degradation data.

When existing methods consider the impact of the imperfect maintenance activities on random degraded equipment, they usually assume that the degraded equipment is linear, and set the number of imperfect maintenance activities in advance. However, in engineering practice, most of the degraded equipment is nonlinear, and the number of imperfect maintenance activities has an impact on the development of maintenance strategies. Therefore, this paper proposes a new joint maintenance strategy considering the uncertainty of the number of imperfect maintenance activities for nonlinear degraded equipment. First, nonlinear degradation data are linearized based on Box-Cox transformation (BCT), and the degradation model under the influence of imperfect activities is constructed by the random coefficient regression model. Accordingly, the remaining useful life (RUL) probability distribution can be derived. Secondly, the number of maintenances is calculated by imperfect maintenance level and success probability. Then, the detection cycle, preventive maintenance threshold and maintenance times are taken as decision variables. The optimization objective is to minimize the expected average cost. However, this is restricted by availability and probability of success for imperfect maintenance activities. A multi-objective joint optimization model of condition-based maintenance and spare parts ordering is constructed. Finally, the analysis results based on numerical examples verify the feasibility of the proposed joint optimization strategy.

Remaining useful life (RUL) estimation is the key of prognostics and health management (PHM) technology and is an effective way to ensure the safe and reliable operation of equipment. Aiming at the lack of historical data and prior information for the newly developed small-sample systems, an adaptive RUL estimation method based on the expectation maximization (EM) algorithm is proposed with three-source variability. First, a degradation model based on a Wiener process is established to incorporate three-source variability and dynamic sampling interval, and the analytical solution of RUL distribution is derived in the sense of the first hitting time. Second, an adaptive parameter estimation method based on the EM algorithm is proposed to update the model parameters by using the condition monitoring (CM) data from one working system running up to the current moment. Finally, a practical example of a gyroscope in an inertial navigation system is provided to substantiate the effectiveness and superiority of the proposed method. The results indicate that the proposed method can efficiently improve the accuracy of the RUL estimation.

Accelerated degradation test (ADT) has become the main method to assess system reliability. In ADT, Bayesian D-optimality criterion is an effective objective function to deal with the model parameter uncertainty in designing an ADT plan. Current optimal ADT designs based on Bayesian D-optimality usually assume a single degradation model. However, this assumption sometimes may not be robust enough because the degradation trajectories of the target systems are usually unknown. Thus, besides the model parameter uncertainty, the degradation model uncertainty is also a question worth considering during designing an optimal ADT plan. Integrating the advantage of Bayesian model averaging to Bayesian D-optimality, this paper proposes a modified Bayesian D-optimality to handle the model parameter uncertainty and the model uncertainty simultaneously. Simulations are provided to analyze the robustness of the proposed objective function. The results show that the modified Bayesian D-optimality could effectively deal with the uncertainties of the degradation model and model parameter.

As the fundamental and prerequisite work of remaining useful life (RUL) prediction, degradation modeling directly affects the accuracy of RUL prediction. Existing degradation models are all developed under a single time scale, and less research has been carried out to consider the impact of multiple time scales on the degradation model. Toward this end, we mainly study a nonlinear degradation modeling and RUL prediction method for nonlinear stochastic degraded equipment with bivariate time scales in this paper. Firstly, a nonlinear degradation model considering the influence of two time scales is constructed based on the diffusion process. At the same time, the relationship between the two scales is quantitatively characterized by random proportional coefficient. Then, the analytical expressions of the life and RUL for nonlinear degraded equipment are derived under the concept of first passage time (FPT). In order to realize the adaptive estimation of parameters, the model parameters estimation method based on maximum likelihood estimation (MLE) and Kalman filtering algorithm is developed in this paper. Finally, numerical simulation and the monitoring data of gyroscope verify the effectiveness and superiority of the proposed method. The experimental results show that the method proposed in this paper can effectively improve the accuracy of RUL prediction, and has a broad engineering application space.

As the central component of prognostic and health management (PHM) field, remaining useful life (RUL) estimation approaches based on degradation modeling have played an extremely significant role in recent years. For the newly developed systems working in complex environments, the associated degradation processes not only lack historical data and prior information but also have strong nonlinearity and three-source variability. Therefore, this paper proposes an adaptive RUL estimation approach for the newly developed system based on a nonlinear model. Specifically, a general nonlinear Wiener-process-based degradation model is established to simultaneously characterize three-source variability and nonlinearity, and the associated RUL distribution is derived with an explicit form. In order to utilize the condition monitoring (CM) data of the service system up to date, we present a parameter estimation method based on the expectation maximization algorithm to adaptively estimate and update the model parameters online. As such, the RUL distribution can be updated once the new CM data are available. Finally, the effectiveness and superiority of the proposed method are demonstrated by the numerical example an empirical study for battery data. The results show that the proposed method can provide accurate and robust RUL prediction for the newly developed system.

The nonlinear Wiener process has been widely used as a model for the degradation process. This note concerns parameters estimation of nonlinear Wiener processes with measurement errors (WPME) by the maximum likelihood estimation method. Firstly, we prove a rule that the estimated results based on the sample likelihood function developed through observations at each point are equal to the results from the first differences of the observations. This rule indicates that for reducing computation complexity the first differences of the observations may develop the sample likelihood function. Then we present a simple method to calculate the determinant and the inverse matrix of the covariance matrix of the WPME. This simple method could avoid the overflow error when calculating the determinant of the covariance matrix and the case that the inverse matrix is close to be singular, which could result in wrong estimation results. Secondly, we highlight the unit-specific assumption, which has a significant impact on parameters estimation but has been neglected in many papers. Then, we propose a modified expectation maximization algorithm for parameters estimation with random effects. Finally, to demonstrate the application and superiority of the proposed method, we provide a numerical example and a case study with comparison to several representative methods in the literature.

State-of-Health (SOH) is an intuitive reflection to the capacity during the degradation of lithium-ion batteries. Accurate SOH estimation can not only grasp the battery performance but also achieve a better balance between the safety and economic benefits for lithium-ion battery application system. Relaxation effect refers to the capacity regeneration phenomenon of lithium-ion batteries during long rest time. This paper mainly studies the impact of relaxation effect on the degradation law of lithium-ion batteries, and proposes a novel SOH estimation method based on the Wiener process. First, the life cycle of a lithium-ion battery is divided into three parts, i.e., the degradation process that eliminates the relaxation effect, the capacity regeneration process during the rest time, and the degradation process of the regenerated capacity. Next, the degradation model after eliminating the relaxation effect is established based on linear Wiener process, the capacity regenerated model is developed by the normal distribution, and the degradation model of regenerated capacity is established based on a nonlinear Wiener process. Then, for the degradation model based on nonlinear Wiener process, a two-step maximum likelihood estimation (MLE) method for prior parameters is proposed, and the random variables representing the personality features are updated online under the Bayesian framework. For the capacity regenerated model, a parameter estimation method based on MLE is proposed. In addition, a one-step and a multi-step SOH prediction method based on piecewise modeling are developed. Finally, the experiments are carried out based on the degradation data of lithium-ion batteries published by the NASA, and the results show that the method proposed in this paper can effectively improve the accuracy of the SOH estimation.

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.

Recent developments in prognostic and health management have been targeted at utilizing the observed degradation signals to estimate residual life distributions. Current degradation models mainly focus on a population of “identical” devices or an individual device with population information, not a single component in the absence of prior degradation knowledge. However, the fast development of science and technology provides us with many kinds of new systems, and we just have the real-time monitoring information to analyze the reliability for them. The fusion algorithm presented herein addresses this challenge by combining the excellent modeling ability of Bayesian updating method for the multilevel data and the prominent estimation ability of ECM algorithm for incomplete data. Residual life distributions and posterior distributions are first calculated through the Bayesian updating method based on random initial a priori distributions. Then the a priori distributions are revised and improved for future predictions by the ECM algorithm. Once a new signal is observed, we can reuse the fusion algorithm to improve the accuracy of residual life distributions. The applicability of this fusion algorithm is validated by a set of simulation experiments.

Remaining useful life (RUL) prediction is central to the prognostics and health management (PHM) of lithium-ion batteries. This paper proposes a novel RUL prediction method for lithium-ion batteries based on the Wiener process with measurement error (WPME). First, we use the truncated normal distribution (TND) based modeling approach for the estimated degradation state and obtain an exact and closed-form RUL distribution by simultaneously considering the measurement uncertainty and the distribution of the estimated drift parameter. Then, the traditional maximum likelihood estimation (MLE) method for population based parameters estimation is remedied to improve the estimation efficiency. Additionally, we analyze the relationship between the classic MLE method and the combination of the Bayesian updating algorithm and the expectation maximization algorithm for the real time RUL prediction. Interestingly, it is found that the result of the combination algorithm is equal to the classic MLE method. Inspired by this observation, a heuristic algorithm for the real time parameters updating is presented. Finally, numerical examples and a case study of lithium-ion batteries are provided to substantiate the superiority of the proposed RUL prediction method.

Remaining useful life (RUL) prediction has great importance in prognostics and health management (PHM). Relaxation effect refers to the capacity regeneration phenomenon of lithium-ion batteries during a long rest time, which can lead to a regenerated useful time (RUT). This paper mainly studies the influence of the relaxation effect on the degradation law of lithium-ion batteries, and proposes a novel RUL prediction method based on Wiener processes. This method can simplify the modeling complexity by using the RUT to model the recovery process. First, the life cycle of a lithium-ion battery is divided into the degradation processes that eliminate the relaxation effect and the recovery processes caused by relaxation effect. Next, the degradation model, after eliminating the relaxation effect, is established based on linear Wiener processes, and the model for RUT is established by using normal distribution. Then, the prior parameters estimation method based on maximum likelihood estimation and online updating method under the Bayesian framework are proposed. Finally, the experiments are carried out according to the degradation data of lithium-ion batteries published by NASA. The results show that the method proposed in this paper can effectively improve the accuracy of RUL prediction and has a strong engineering application value.

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’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.

A lithium-Ion battery is a typical degradation product, and its performance will deteriorate over time. In its degradation process, regeneration phenomena have been frequently encountered, which affect both the degradation state and rate. In this paper, we focus on how to build the degradation model and estimate the lifetime. Toward this end, we first propose a multi-phase stochastic degradation model with random jumps based on the Wiener process, where the multi-phase model and random jumps at the changing point are used to describe the variation of degradation rate and state caused by regeneration phenomena accordingly. Owing to the complex structure and random variables, the traditional Maximum Likelihood Estimation (MLE) is not suitable for the proposed model. In this case, we treat these random variables as latent parameters, and then develop an approach for model identification based on expectation conditional maximum (ECM) algorithm. Moreover, depending on the proposed model, how to estimate the lifetime with fixed changing point is presented via the time-space transformation technique, and the approximate analytical solution is derived. Finally, a numerical simulation and a practical case are provided for illustration.

Rotating machinery is widely used in modern industry. It is one of the most critical components in a variety of machinery and equipment. Along with the continuous development of science and technology, the structures of rotating machinery become of larger scale, of higher speed, and more complicated, which results in higher probability of concurrent failure in practice. It is important to enable reliable, safe, and efficient operation of large-scale and critical rotating machinery, which requires us to achieve accurate diagnosis of concurrent fault, for example, rolling bearing diagnosis, gearbox diagnosis, and compressor diagnosis. In this paper, to achieve concurrent fault diagnosis for rotating machinery, which cannot be accurately diagnosed by existing methods, we develop an integrated method using artificial immune algorithm and evidential theory.