Lifetime prognostics for deteriorating systems with time-varying random jumps
Introduction
With the rapid development of technology, prognostics and health management (PHM) has attracted increasing attention and been applied to a large variety of industrial systems [1], [2]. Health prognostics, as an essential part of PHM, plays an increasingly important role in many engineering practice [3], [4]. Health deterioration of the systems often manifests as performance degradation, which results from physical deterioration, damage, environment erosion, running wear, etc [1], [2], [4]. In order to evaluate and predict the health state of degrading systems accurately, the general methods include two key steps: modeling the degradation paths of performance variables, and then conducting the lifetime or remaining useful life (RUL) estimation [5], [6]. According to Pecht’s research [1], he employs the techniques for modeling under the categories of physics of failure, data driven and fusion, where data-driven approach consists of the machine learning and statistics based approaches. Compared with other approaches, statistical data driven approach relies only on the available observed data and statistical models, and has advantages in mathematic properties of the estimated RUL [7]. As analyzed by Jardine et al. [8], statistical data driven approach is an effective way to achieve a well estimated lifetime and RUL. Nowadays, statistical data-driven approach has gained momentum. Particularly, Si et al. have systematically reviewed RUL estimation method based on statistical data driven approaches [7]. For example, Wiener process [9], Gamma process [10], Markov process [11], and inverse Gaussian (IG) process [12] have been widely used to model the degradation process for RUL estimation.
Due to different types of monitored degradation data, the statistical data-driven approaches could be classified into the methods based on directly observed data and the methods based on indirectly observed data [7]. Because of systems’ complexity and uncertainty, the directly observed data (e.g. fatigue crack, thickness of the furnace wall, bearing abrasion, etc.) are often unavailable [7], [8]. In this case, a widely adopted way is to measure some intermediate performance variables instead, such as the temperature sensors for the thickness of the furnace wall. As a result, the degradation process often has high dynamics, high fluctuations, and outstanding random jumps, differing from the continuous degradation case. On the other hand, in engineering practice, with the acceleration of the system’s deterioration, the stability of the system becomes worse gradually and the system is more vulnerable to the shocks. That may make some random jumps occur and their occurrence frequency increase with the aging of the system [13]. Thus, there are two impacts toward the degradation paths, i.e., time-varying random jumps and continuous degradation.
Considering the above practical desire, we propose a jump diffusion process with non-homogeneous compound Poisson process to describe such a kind of degradation process. In order to estimate the lifetime and RUL of the proposed model under the concept of the FPT and then apply such the results into practical usage, two main works have been completed as follows. Explicit forms of the lifetime and RUL under the concept of the FPT are approximately derived using the time–space transformation method in which the FPT is transferred at the first time that the diffusion process exceeds a time-varying random threshold. For parameter estimation, we first give a general method based on Maximum Likelihood Estimation (MLE). Then a two-step parameters estimation method is presented via combining MLE and expectation conditional maximization (ECM) algorithm for linear jump diffusion process, which can improve the online performance. To illustrate the applicability and rationality of our method, numerical example and a study on the practical degradation data of the furnace wall are provided.
The remainder parts are organized as follows. In Section 2, the recently related literatures are reviewed and discussed. In Section 3, the motivating examples and problem formulation are given. Section 4 includes the main result of RUL estimation. In Section 5, a new stochastic model is proposed to describe the degradation path and the framework of the ECM algorithm for unknown model parameters is presented. Two illustrative examples are presented to illustrate and demonstrate the proposed model in Section 6. This paper is concluded in Section 7.
Section snippets
Literature review
Nowadays, most researches mainly focus on degradation modeling and RUL estimation for continuous degradation data but ignore the random jumps, such as [7], [11], [12], [14], [15]. Only a few works concentrate on the degradation modeling and RUL estimation with the random jumps. Generally, there are two broad streams of literature related to our study: (1) the degradation-threshold-shock (DTS) degradation models and (2) the jump diffusion process.
DTS model, describing the systems subject to both
Motivating example of blast furnace
Example: Blast furnace is a typical large-scale complex system and its health is often influenced by many different factors [30]. In practice, the degradation of furnace wall is inevitable for all blast furnace [31]. If the furnace wall burns out, not only the life of blast furnace expires, but also may lead to major personnel and property losses. Simply speaking, the degradation of furnace wall is caused by the erosion of molten iron directly. However, since the actual thickness of furnace
Derivation of the lifetime under the concept of the FPT
Based on the definition of the lifetime and RUL as shown in (3) and (4), it is difficult to deduce the general analytical form of the lifetime or RUL in concept of the FPT from (2). Therefore, we develop an approximate solution in a closed form. For better expression, the following lemma is first given.
Lemma 1 Based on the definition of non-homogeneous Poisson process, the mean or intensity measure of N(t) is
where λ(τ) denotes the intensity function of the non-homogeneous[33]
Parameter estimation for linear degradation models
Based on the above result, if the parameters of the proposed model are given, the lifetime and RUL estimation can be obtained. In the following, we will discuss how to identify the proposed model through statistical inference.
Case study
In this section, two examples are provided: (1) a numerous simulation is adopted to verify the accuracy of parameter estimation and the PDF of lifetime; (2) the actual degradation data of furnace wall are used to illustrate the feasibility of the proposed model.
Conclusion
In this paper, motivated from the practical data of the furnace wall, we present a time-varying jump diffusion model to describe the degradation path. Unlike the traditional jump diffusion process, we adopt the non-homogeneous compound Poisson process to model these random jumps. Then, in order to derive the PDFs of the RUL under the concept of the FPT, we translate the random jumps into random threshold, and then derive the approximate analytical solution of the RUL based on the diffusion
Acknowledgments
This work was supported by National Natural Science Foundation of China (NSFC) under grant nos. 61210012, 61290324, 61573365, 61374126, 61473094, 61473163, 61522309, 61673311, 61573076, 61573366, Young Engineering and Science Scholars of ChinaAssociation for Science and Technology under grant 2016QNRC001,China Postdoctoral Science Foundation under grant 2016M600546,Qingdao Postdoctoral Applied Research Projects under grant 2016112 and Tsinghua University Initiative Scientific Research Program.
References (39)
- et al.
A new class of wiener process models for degradation analysis
Reliab Eng Syst Saf
(2015) - et al.
Prognostic modelling options for remaining useful life estimation by industry
Mech Syst Signal Process
(2011) - et al.
Remaining useful life estimation based on stochastic deterioration models: a comparative study
Reliab Eng Syst Saf
(2013) - et al.
Remaining useful life estimation—a review on the statistical data driven approaches
Eur J Oper Res
(2011) - et al.
A review on machinery diagnostics and prognostics implementing condition-based maintenance
Mech Syst Signal Pocess
(2006) - et al.
Condition-based maintenance using the inverse Gaussian degradation model
Eur J Oper Res
(2015) Wiener processes with random effects for degradation data
J Multivar Anal
(2010)Option pricing when underlying stock returns are discontinuous
J Financ Econ
(1976)- et al.
Numerical prediction on the erosion in the hearth of a blast furnace during tapping process
Int Commun Heat Mass Transfer
(2009) - et al.
A change-point based reliability prediction model using field return data
Reliab Eng Syst Saf
(2016)
Bayesian hazard modeling based on lifetime data with latent heterogeneity
Reliab Eng Syst Saf
Prognostics and health management of electronics
Prognostics and health management of industrial equipment
Diagnostics and prognostics of engineering systems: methods and techniques
Failure prediction and health prognostics of electronic components: a review
2014 Recent Advances in Engineering and Computational Sciences (RAECS)
Remaining useful life estimation based on a nonlinear diffusion degradation process
IEEE Trans Reliab
Accelerated degradation models for failure based on geometric Brownian motion and gamma processes
Lifetime Data Anal
Degradation modeling and monitoring of machines using operation-specific hidden Markov models
IIE Trans
A multi-objective optimization of imperfect preventive maintenance policy for dependent competing risk systems with hidden failure
IEEE Trans Reliab
The inverse Gaussian process as a degradation model
Technometrics
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2021, Reliability Engineering and System SafetyCitation Excerpt :Therefore, researchers have started to explore the RUL prediction under the FPT concept. Based on the FPT concept, Ke et al.[14] and Zhang et al.[15] derived the approximate analytical expressions of RUL and provided the parameter estimation procedures for failure processes made up of gradual degradation and random jumps. Although their proposed methods are capable of describing gradual failure processes like wear and fatigue, they are unable to model sudden failures like brittle fracture.
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