An improved Wiener process model with adaptive drift and diffusion for online remaining useful life prediction

Mechanical Systems and Signal Processing 127 (2019) 370–387

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

An improved Wiener process model with adaptive drift and diffusion for online remaining useful life prediction Han Wang a, Xiaobing Ma a,b,⇑, Yu Zhao a,b a b

School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China The Key Laboratory on Reliability and Environmental Engineering Technology, Beihang University, Beijing 100191, China

a r t i c l e

i n f o

Article history: Received 8 May 2018 Received in revised form 7 February 2019 Accepted 15 March 2019

Keywords: Adaptive Wiener process Monitoring data eliminating Prediction accuracy Recursive filter Remaining useful life prediction

a b s t r a c t Remaining useful life (RUL) prediction plays an important role in the field of prognostics and health management (PHM). Although several Wiener process models with adaptive drift have been developed for RUL prediction, these models assume the diffusion parameter is fixed and therefore fail to capture the real degradation process. Accordingly, this paper proposes an improved Wiener process model for RUL prediction, in which both drift and diffusion parameters are adaptive with the updating of monitoring data. The proposed model considers the quantitative relationship between degradation rate and degradation variation. When a new monitoring data is available, we update the model parameters and therefore the RUL distribution by applying recursive filter and expectation maximization (EM) algorithm. In addition, a prediction region is constructed based on the 3rinterval criterion to eliminate the abnormal monitoring data, followed by a model selection method developed to compare the prediction accuracy of the proposed model with the existing models. The proposed model’s superiority and the effectiveness of the model selection method are illustrated and validated by an application to the identical thrust ball bearings. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Background Over the last decade, prognostics and health management (PHM) has gained lots of attentions because it performs well for evaluating product’s reliability in its actual operating conditions [1]. ‘‘Prognostics” which is usually characterized by predicting the remaining useful life (RUL) of products based on online monitoring data provides effective input information for ‘‘health management” such as making maintenance decisions, executing preventive actions, replanning missions and so on [2–4]. A great many researches related to online RUL prediction have been investigated and developed for a wide range of products including lithium-ion batteries [5–8], laser generators [9], and rotating elements [10] such as bearings [11–17], gears [18–20], gyros [21] and so on. Actually, the rapid development of online RUL prediction is closely related to the extensive applications of sensors in engineering, based on which the online monitoring data can be increasingly collected. Various kinds of sensors can basically monitor product’s any performance parameters. When the monitoring data are available, an appropriate RUL prediction ⇑ Corresponding author at: School of Reliability and Systems Engineering, Beihang University, Beijing, 100191, China. E-mail address: [email protected] (X. Ma). https://doi.org/10.1016/j.ymssp.2019.03.019 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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model is needed to describe the underlying degradation path in the best possible way. Currently, two kinds of RUL prediction models are commonly used, i.e., degradation path dependent models [14–16,21] and stochastic process models [9,18–20]. Degradation path dependent models assume a common degradation form for a population of products. For a specific unit, this kind of models are powerless to capture the unit-to-unit variation and temporal uncertainty, thus stochastic process models have advantages in this view. In particular, Wiener-process-based models have been successfully used under some circumstances because of a characteristic feature that product’s observed degradation path can increase or decrease. For example, to predict the RUL of the gyros in an inertial navigation system (INS), a linear degradation model and an exponential-based degradation model were proposed based on the Wiener process [22]. Both of the Bayesian updating and expectation maximization (EM) algorithm are used to update the model parameters and RUL distribution at the time obtaining a newly observed data. A relatively general degradation model was also proposed for the gyros in INS based on the Wiener process [23]. It performs better when characterizing the three-source variability into RUL estimation simultaneously. Further, for the case of lithium-ion batteries, a Wiener-process-based model with random drift and diffusion parameter as well as measurement error was proposed to characterize off-line population degradation and then a particle filterbased parameter estimation method was exploited to achieve the on-line RUL prediction [24]. Similarly, a nonlineardrifted Brownian-motion-based model with multiple hidden states was proposed and applied to predict the RUL of rechargeable lithium-ion batteries [25], in which some comparisons are also conducted to highlight the superiority of their proposed prognostic method. Besides these, an improved exponential model which can be transformed to a Wiener-process-based model was proposed for predicting the RUL of rolling element bearings [26]. And a general Wiener-process-based model in which the drift parameter was age- and state-dependent was proposed to describe the fatigue-crack-growth process [27]. Overall, various types of Wiener-process-based models have been developed for RUL prediction. And accompanied with these models, a variety of parameter estimation methods have also been investigated and applied for RUL prediction, such as ensemble learning [7], artificial neural network [13,15,17], relevance vector machine [14,16], expectation maximization algorithm [19,22], recursive filtering techniques [24,28], Bayesian updating [29], as well as Markov chain Monte Carlo [30], etc. 1.2. Problems in RUL prediction Although extensive studies related to RUL prediction have been investigated, there are still some issues worthy of further consideration. Note that the prediction accuracy of the RUL can be affected by a great many factors such as the validity of the degradation monitoring data, the rationality of the RUL prediction model, the complexity of the operating condition, the applicability of the parameter estimation method, multi-source uncertainties, etc. Among them, the two most important factors are the validity of the degradation monitoring data and the rationality of the RUL prediction model. Here, we address the urgent problems in RUL prediction from the two aspects successively. 1.2.1. The validity of the degradation monitoring data Note that the degradation monitoring data collected by using sensors are not usually extremely ideal in practice. Sometimes there exist abundant abnormal monitoring data caused by highly variability, such as temporal variability, measurement variability and so on [23]. Under the circumstance, it is difficult to predict the RUL of products precisely which may cause a significant influence on the latter decision making [31]. Thus, considering the validity of the degradation monitoring data plays an important role in RUL prediction. In other words, we need to find an effective way to eliminate the abnormal monitoring data from the total dataset and use the normal monitoring data for RUL prediction. However, as we know, the existing methods for eliminating abnormal data basically depend on the engineering experience and there lacks of a universal elimination criterion. 1.2.2. The rationality of the RUL prediction model About the rationality of the RUL prediction model, we think that there are two problems needed to be considered. The first one is about model assumption. It is not uncommon to see that when the degradation of a unit is faster, the variation of the degradation over time is also higher. This means that a unit with a larger drift parameter is expected to have a larger diffusion parameter as well. However, the existing Wiener process models usually use a fixed diffusion parameter which actually violates the real degradation process. Therefore, a more reasonable Wiener process model is expected to describe the degradation process. The second one is model selection. It is absolutely important and necessary for us to think about how to measure the prediction accuracy of a newly proposed RUL prediction model and compare it with the existing models. The existing researches ignore this work and there lacks of a selection criterion for RUL prediction models. 1.3. Work and contribution In this paper, we address the above problems by proposing an improved Wiener process model with adaptive drift and diffusion for online RUL prediction. Different from the existing RUL prediction models, the improved Wiener process model relates degradation variation to degradation rate quantitatively. In this way, both drift and diffusion parameters can be adaptive with the updating of monitoring data, which makes it more suitable for describing the real degradation process. To utilize the entire history of the degradation monitoring data, the recursive filter and expectation maximization (EM) algorithm

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are considered to update the model parameters and RUL distribution. After that, the 3r-interval criterion helps to construct a prediction region so that we can judge whether a new degradation monitoring data is abnormal or not before carrying out parameter updating procedure. Further, a model selection method is developed to compare the prediction accuracy of different prediction models. The model selection method has not been considered before. With the rapid development of degradation models, it is beneficial to consider the problem so that an optimal model can be selected for RUL prediction. The remainder of the paper is organized as follows. In Section 2, we review two well-adopted adaptive Wiener process models and propose the improved adaptive Wiener process model. In Section 3, we introduce the parameter updating procedure, the algorithm for abnormal monitoring data eliminating, and the model selection method in detail. In Section 4, an application to the identical thrust ball bearings is used to illustrate the proposed model and methods. Section 5 draws the main conclusions. 2. RUL prediction models In this section, three adaptive Wiener process models for RUL prediction are introduced. The first two models have been investigated in previous researches and perform well in some applications. However, for comparison, we review them briefly and then introduce our proposed RUL prediction model. Notations used in this paper are summarized as follows. 2.1. Notations

X ðt Þ ti T0!i ¼ ft 0 ; t1 ;    ; t i g X0!i ¼ fx0 ; x1 ;    ; xi g D T Ri BðtÞ WðtÞ N ðÞ

l l0 li r ri f

g j uðs; hÞ f DX ðDxÞ f T ðt Þ f Ri ð r i Þ EðÞ VarðÞ Eiji Variji

H lnL

q

PAC tF

Degradation path The i-th monitoring time point Total monitoring time points up to ti Total degradation monitoring data up to ti , where xi ¼ X ðti Þ Threshold level FHT of the unit RUL of the unit at t i Standard Brownian motion Another standard Brownian motion independent of BðtÞ Normal distribution Drift parameter Drift parameter at t0 which follows N ða0 ; b0 Þ Drift parameter at ti Diffusion parameter Diffusion parameter at t i Fixed parameter
 Noise of the adaptive drift which follows N 0; e2 Diffusion coefficient of the adaptive drift Determined nonlinear function of s with parameter vector h PDF of DX ðtÞ, where DX ðt Þ ¼ X ðt þ Dt Þ  X ðt Þ PDF of T PDF of Ri Expectation operator Variance operator The mean of the estimated li given X0!i The variance of the estimated li given X0!i Parameter vector where H ¼ ½a0 ; b0 ; f; j; h The log-likelihood function Precision requirement Prognostic accuracy criterion True failure time

2.2. RUL prediction models For a specific unit, assume X ðt Þ to be the cumulative degradation path up to time t with X ð0Þ ¼ 0. For convenience, denote the history of the monitoring time and corresponding degradation monitoring data as T0!i ¼ ft0 ; t1 ;    ; ti g and

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X0!i ¼ fx0 ; x1 ;    ; xi g, where xi ¼ X ðt i Þ, respectively. Under this circumstance, the first hitting time (FHT) of the unit is usually defined as

T ¼ inf ft : X ðt Þ P DjX0!i g;

ð1Þ

and the RUL of the unit at t i is usually defined as

Ri ¼ inf fr i : X ðt i þ r i Þ P DjX0!i g;

ð2Þ

where D represents the threshold level. According to Eqs. (1) and (2), we find that both of T and Ri are assumed to be conditional on the entire degradation monitoring data to date, which means that they are related to the way how degradation accumulated. 2.2.1. Linear adaptive Wiener process model A basic linear Wiener process model is generally represented as

X ðtÞ ¼ X 0 þ lt þ rBðt Þ;

ð3Þ

where X 0 ¼ 0 is assumed to be the initial value, l is the drift parameter reflecting the degradation rate, r is the diffusion parameter reflecting the degradation variation, BðtÞ is the standard Brownian motion. Using the basic linear Wiener process model, the following results are usually utilized [28], i.e., 1) The degradation increments are independent and identically distributed (i.i.d.), following a normal distribution, i.e.,
 DX ðtÞ ¼ X ðt þ DtÞ  X ðt Þ  N lDt; r2 Dt , and the probability density function (PDF) of DX ðt Þ is given as

! 1 ðDx  lDt Þ2 f DX ðDxjl; rÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  ; 2r2 Dt 2pr2 Dt

ð4Þ

2) The FHT of the unit follows an inverse Gaussian (IG) distribution and the conditional PDF of the FHT is given as

! D ðD  lt Þ2 ; f T ðtjX0!i ; l; rÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  2r 2 t 2pr2 t3

ð5Þ

3) The conditional PDF of the RUL of the unit at t i is given as

! D  xi ð D  xi  l r i Þ 2 ; f Ri ðr i jX0!i ; l; rÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  2r2 ri 2pr2 r 3i

ð6Þ

4) The following state-space model, defined as Model 1, is usually considered for RUL prediction, i.e.,

li ¼ li1 þ g;

ð7Þ

xi ¼ xi1 þ li1 ðt i  t i1 Þ þ rDBi ;

ð8Þ


 where g  N 0; e2 and DBi ¼ Bðti Þ  Bðt i1 Þ. It is easy to find that, for Model 1, the drift parameter is adaptive and the diffusion parameter is fixed with the updating of monitoring data. 2.2.2. Nonlinear adaptive Wiener process model A nonlinear Wiener process model is generally represented as

X ðt Þ ¼ X 0 þ l

Z

t

uðs; hÞds þ rBðtÞ;

ð9Þ

0

where X 0 , l, r as well as BðtÞ have the same meanings as those in the linear Wiener process model, and uðs; hÞ is a determined nonlinear function of s with parameter vector h. Using the nonlinear Wiener process model, the following results are usually utilized [25], i.e., 1) The degradation increments are i.i.d., and the PDF of DX ðtÞ is given as

0  2 1 R tþDt Dx  l t uðs; hÞds C 1 B f DX ðDxjl; r; hÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp@ A; 2 r 2 Dt 2pr2 Dt

ð10Þ

2) The FHT of the unit follows an IG distribution and the conditional PDF of the FHT is given as

!   1 Sðt Þ l S2 ðt Þ ; f T ðtjX0!i ; l; r; hÞ ¼ pffiffiffiffiffiffiffiffi þ uðt; hÞ exp  t 2t r 2pt

ð11Þ

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where

Sðt Þ ¼

1

r

  Z t Dl uðs; hÞds ;

ð12Þ

0

3) The conditional PDF of the RUL of the unit at ti is given as

0  R  2 1 R t þr t þr D  xi  l tii i uðs; hÞds  r i uðti þ ri ; hÞ D  xi  l tii i uðs; hÞds C B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp@ f Ri ðr i jX 0!i ; l; r; hÞ ¼ A; 2r2 r i 2pr2 r 3i

ð13Þ

4) The following state-space model, defined as Model 2, is usually considered for RUL prediction, i.e.,

li ¼ li1 þ g;

ð14Þ Z

xi ¼ xi1 þ li1

ti

uðs; hÞds þ rDBi ;

ð15Þ

t i1


 where g  N 0; e2 and DBi ¼ Bðti Þ  Bðt i1 Þ. Similarly, for Model 2, the drift parameter is adaptive and the diffusion parameter is fixed with the updating of monitoring data. 2.2.3. The proposed adaptive Wiener process model For Model 1 and Model 2, we find that the diffusion parameter which reflects the degradation variation is assumed to be a constant and not adaptive with the updating of monitoring data. Actually, this assumption violates the real degradation process because when the degradation of a unit is faster, the variation of the degradation over time is also higher, which means that a unit with a larger drift parameter is expected to have a larger diffusion parameter as well [32]. To address the problem, Theorem 1 is introduced in the following and then we propose an improved adaptive Wiener process model based on it. Theorem 1. For a specific degradation process which can be modelled by using the Wiener process model, when the unit’s degradation mechanism remains consistent within the whole degradation process, a fixed proportion relationship between the drift parameter and the square of the diffusion parameter should be satisfied, i.e.,

l=r2 ¼ 1=f;

ð16Þ

where f is a fixed parameter reflecting the degradation mechanism equivalence. The fixed proportion relationship is derived based on the acceleration factor invariant principle which has been investigated in accelerated techniques [33–35]. It can also be derived by using the basic assumption of accelerated failure time model which has been proved appreciate for describing the degradation path caused by accumulated damage [36]. In addition, when there exists abundant degradation information collected from different stress levels, we can estimate the drift parameter and the diffusion parameter under each stress level separately and then verify the proportion relationship by using the parameter correlation test method. Details of the acceleration factor invariant principle and proof of Theorem 1 can be found in Appendix A. It should be mentioned that the necessary condition for l=r2 ¼ 1=f; i.e., the degradation mechanism remains consistent, is actually a basic assumption for essential RUL prediction. If the mechanism changed, it is impossible to utilize the entire monitoring history for RUL prediction as the degradation trend may be different from the normal state. Based on Theorem 1, an improved Wiener process model which considers the proportion relationship between degradation rate and degradation variation is constructed as

X ðt Þ ¼ X 0 þ l

Z

t

uðs; hÞds þ

pffiffiffiffiffiffi flBðt Þ;

ð17Þ

0

pffiffiffiffiffiffi where r ¼ fl represents the diffusion parameter. For the improved Wiener process model, degradation increments are still i.i.d., following a normal distribution, i.e.,

 Z

DX ð t Þ  N

l

tþDt



uðs; hÞds; flDt ;

ð18Þ

t

where DX ðtÞ ¼ X ðt þ DtÞ  X ðtÞ ¼ l

R tþDt t

uðs; hÞds þ

pffiffiffiffiffiffi flDBðt Þ, and the PDF of DX ðt Þ is given as

0  2 1 R tþDt D x  l u ð s ; h Þd s t 1 B C f DX ðDxjl; f; hÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp@ A: 2flDt 2pflDt

Similar to the nonlinear Wiener process model, the conditional PDF of the FHT of the unit is given as

ð19Þ

H. Wang et al. / Mechanical Systems and Signal Processing 127 (2019) 370–387

where

375

! rffiffiffiffi   1 Sð t Þ l S2 ð t Þ ; þ f T ðtjX0!i ; l; f; hÞ ¼ pffiffiffiffiffiffiffiffi uðt; hÞ exp  t 2t f 2pt

ð20Þ

  Z t 1 Sðt Þ ¼ pffiffiffiffiffiffi D  l uðs; hÞds : fl 0

ð21Þ

To obtain the distribution of the RUL, a new stochastic process Y ðri Þ is defined by considering the difference between the stochastic process X ðtÞ at ti þ r i and t i , i.e.,

Y ðr i Þ ¼ X ðt i þ r i Þ  X ðt i Þ ¼ l

Z

t i þr i

uðs; hÞds þ

pffiffiffiffiffiffi flðBðt i þ r i Þ  Bðti ÞÞ:

ð22Þ

ti

According to the definition of RUL, the FHT of the new stochastic process Y ðr i Þ is actually the RUL of the unit at ti . Thus, the conditional PDF of the RUL at ti is directly given as

0  R  2 1 R ti þri t þr D  xi  l tii i uðs; hÞds  r i uðti þ ri ; hÞ D  x u ð s ; h Þd s i l t i B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp@ f Ri ðr i jX0!i ; l; f; hÞ ¼ A: 2f l r 3 i 2pflr i

ð23Þ

According to the improved Wiener process model, the following state-space model, defined as Model 3, is constructed for online RUL prediction, i.e.,

li ¼ li1 þ jDW i ; xi ¼ xi1 þ li1

Z

ti

ð24Þ

uðs; hÞds þ

t i1

pffiffiffiffiffiffiffiffiffiffiffi fli1 DBi ;

ð25Þ

where j is a diffusion coefficient of the adaptive drift, WðtÞ is a standard Brownian motion independent of BðtÞ, DW i ¼ W ðti Þ  W ðti1 Þ and DBi ¼ Bðti Þ  Bðt i1 Þ. Note that, for Model 3, the drift parameter is actually described by utilizing a random walk model with a Brownian motion which makes it a time-dependent random variable. This means that the model assumes adaptive drift for observed monitoring data and assumes adaptivity in the prediction of the future degradation [37]. Traditional RUL prediction models utilize only a normal distributed noise for the adaptive drift and thus the drift parameter remains constant from the last monitoring time until the unit fails. In addition, according to the proportion relation between degradation rate and degrapffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 dation variation, the diffusion parameter can be updated by using ri ¼ fli ¼ f li1 þ jDW i . In this way, both of the drift parameter and the diffusion parameter are adaptive with the updating of monitoring data. This addresses the problem of the former two models, in which the diffusion parameter is fixed and violates the real degradation process. 3. Statistical inference In the following, we introduce how to carry out the online RUL prediction by utilizing the state-space models discussed in Section 2. It is not difficult to find that, the conditional PDF of both T and Ri derived based on those models use only the current monitoring data but not the entire history of monitoring data to date. This violates the assumption that they are related to the way how degradation accumulated. Thus, the recursive filter and EM algorithm will be considered in this section to update the model parameters and RUL distribution in which the entire history to date will be used. In the meanwhile, an algorithm for abnormal monitoring data eliminating as well as a model selection method will be proposed along with the parameter updating procedure so as to improve and compare the prediction accuracy. Without loss of generality, the following parameter updating procedure is conducted for Model 3. We assume that the current monitoring time point is ti and the next monitoring time point is tiþ1 , as shown in Fig. 1. 3.1. Parameter updating procedure 3.1.1. Updating of the drift parameter Firstly, we assume that the initial drift coefficient l0 follows a normal distribution with mean a0 and variance b0 as required by the state-space model. In this way, li follows a distribution which can be estimated by a recursive filter once the degradation monitoring data up to ti , i.e., X0!i ¼ fx0 ; x1 ;    ; xi g, is available. Denote its mean and variance by

 Eiji ¼ E li X0!i and Variji ¼ Var li X0!i , respectively.
 To obtain Eiji and Variji , we need to compute the conditional PDF of li given X0!i , denoted by p li X0!i . Recursion solu

 tion of p li X0!i can be estimated from p li1 X0!i1 by using the well-adopted Bayesian rule, i.e.,

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H. Wang et al. / Mechanical Systems and Signal Processing 127 (2019) 370–387

Fig. 1. A graphical description of the degradation monitoring process.


 p li X0!i ¼

Z

þ1

1



 p li li1 p li1 X0!i dli1 ¼



 R þ1
p li li1 p xi jli1 ; X0!i1 p li1 X0!i1 dli1 1 : pðxi jX0!i1 Þ

For the proposed Model 3, if Eqs. (24) and (25) are used, it has been well established that p is normally distributed with mean Eiji and variance Variji , i.e.,


 1 p li X0!i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  2pVariji




li  Eiji 2Variji

ð26Þ


 li X0!i computed by Eq. (26)

2 ! ;

ð27Þ

where both Eiji and Variji can be estimated by using the following Kalman filter algorithm. Kalman filter algorithm: 1) Initialize the mean and variance of the drift parameter at time t 0 : E0j0 ¼ a0 and Var0j0 ¼ b0 ; 2) Estimate the variance of li at time ti1 : Variji1 ¼ Vari1ji1 þ j2 ðti  t i1 Þ; R 2 ti uðs; hÞds Variji1 þ fEi1ji1 ðti  ti1 Þ; 3) Estimate the filter gain: K i ¼ ti1   R ti R ti uðs; hÞdsK 1 xi  xi1  Ei1ji1 ti1 uðs; hÞds ; 4) Update the mean of li : Eiji ¼ Ei1ji1 þ Variji1 ti1 i R 2 ti uðs; hÞds K 1 5) Update the variance of li : Variji ¼ Variji1  Variji1 ti1 i Variji1 . As a result, the entire history of the degradation monitoring data up to date are captured via recursively updating Eiji and
 Variji , based on which the conditional PDF of li , i.e., p li X0!i , is derived.

3.1.2. Updating of the other parameters Next, on the basis of the above-mentioned results, we estimate the other parameters in the proposed Model 3, including the fixed parameter f, the diffusion coefficient of the adaptive drift j, the parameter vector h in the nonlinear function uðÞ, as well as the initial value of the mean and variance of the drift parameter, i.e., a0 and b0 . Define parameter vector H ¼ ½a0 ; b0 ; f; j; h, then the log-likelihood function for X0!i is given as




 lnLi ðHÞ ¼ lnp X0!i ; li H ¼ lnp X0!i jli ; H þ lnp li H i i   X  
 X lnp lj lj1 ; H þ lnp xj lj1 ; H ; ¼ lnp l0 H þ j¼1

ð28Þ

j¼1




where li ¼ l0 ; l1 ;    ; li1 denotes the hidden variables, i.e., the history of the updated drift parameters, and p X0!i ; li H denotes the joint PDF of the monitoring data X0!i .

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377


 According to Eqs. (24) and (25), we know that l0  N ða0 ; b0 Þ, lj lj1  N lj1 ; j2 tj  tj1 and 
 R tj xj lj1  N xj1 þ lj1 tj1 uðs; hÞds; flj1 tj  tj1 . In this way, when ignoring the constant items, the log-likelihood function can be formulated as

( lnLi ðHÞ ¼  lnb0 þ þ

i X






l0  a0 2 =b0 þ

i 2

X

   ln j2 t j  t j1 þ lj  lj1 = j2 t j  t j1 j¼1

2

Z 
 4ln fl t  t  x  l þ x j1 j j1 j1 j j1

tj

!2

uðs; hÞds

t j1

j¼1

39, 
 = = flj1 tj  tj1 5 2 ;

ð29Þ

Note that, because the drift parameter li is treated as a hidden variable which is given by Eq. (24), so estimating the unknown parameter vector H ¼ ½a0 ; b0 ; f; j; h by maximizing Eq. (29) directly is impossible. To solve the problem, we consider the EM algorithm which provides a possible framework for estimating the unknown parameters involving hidden variables. A basic assumption of the EM algorithm is that the hidden variables can be estimated by the monitoring data. Our
 problem meets the assumption since p li X0!i can be obtained by Eq. (27). A detail of the EM algorithm for our problem is given as follows. EM algorithm: ^ ðkÞ , we have E-step: Taking the expectation operator on the side of Eq. (29) with respect to li X0!i ; H i

  ^ ðk Þ ¼ E lnLi HjH i l

i

ðkÞ

jX 0!i ;H^ i

flnLi ðHÞg;

ð30Þ

h i ^ ðkÞ ¼ aðkÞ ; bðkÞ ; fðkÞ ; jðkÞ ; hðkÞ denotes the estimated parameters in the k-th step conwhere k is the number of iterations and H i 0;i 0;i i i i ditional on X0!i .

    To calculate the conditional expectation, i.e., El X ;H^ ðkÞ flnLi ðHÞg, we need to obtain El X ;H^ ðkÞ lj , El X ;H^ ðkÞ l2j , j 0!i i j 0!i i j 0!i i i      i  i El X ;H^ ðkÞ lj lj1 , El X ;H^ ðkÞ l1 and E . In this paper, we estimate these conditional expectations based ln l ð k Þ j j ^ li jX0!i ;H i j 0!i i j 0!i i i i on the properties of variance–covariance and Rauch-Tung-Striebel (RTS) smoothing algorithm [28].   ^ ðkþ1Þ by maximizing lnLi HjH ^ ðkÞ with H, i.e., M-step: Calculate the unknown parameters H i i ðkþ1Þ

^ H i

 ¼ argmax El H

^ ðkÞ i X0!i ;Hi

j

 flnLi ðHÞg :

ð31Þ

  ^ ðkÞ =@ H ¼ 0. Eq. (31) can be solved by directly taking @lnLi HjH i Iterate the E-step and M-step until a criterion of convergence is satisfied, i.e.,

^ ðkþ1Þ ^ ðkÞ Hi  H i 6 q; ðkÞ ^ Hi

ð32Þ

where q is a given precision requirement. 3.1.3. Updating of the RUL distribution Finally, the PDF of the updated RUL at the current monitoring time point t i can be obtained by using the law of total probability, i.e.,

Z f Ri ðr i jX0!i ; f; hÞ ¼

þ1

1



 f Ri ri jX0!i ; li ; f; h p li X0!i dli ;

ð33Þ

where the model parameters have been derived above. Comparing Eq. (33) with Eq. (23), we can observe that the entire history of the monitoring data to date are utilized to update the RUL distribution. In this way, the assumption that RUL is related to the way how degradation accumulated has been satisfied. If the expectation of RUL is needed, some numerical integration algorithms need to be further considered. 3.2. Abnormal monitoring data eliminating According to the above-mentioned steps, the model parameters and RUL distribution of the unit at the current monitoring time point ti has been obtained. When the next degradation monitoring data is available, we can repeat those steps to update the model parameters and RUL distribution until the monitoring is stopped. However, as mentioned in Section 1, there exist

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Fig. 2. The prediction region of the new degradation monitoring data.

abnormal monitoring data because of highly variability, so we believe that we should judge whether a new monitoring data is normal or not before using it to update the model parameters and RUL distribution. In the following, we provide a method to predict the degradation path of the unit at the next monitoring time point by using the state-space model proposed in Section 2. A prediction region will also be provided so that we can achieve the purpose of abnormal monitoring data eliminating. Fig. 2 shows a vivid description of the prediction region. According to Eq. (25), we know that the degradation increment follows a normal distribution, i.e.,

xiþ1  xi  N

Z

li

t iþ1

ti

!

uðs; hÞds; fli ðtiþ1  ti Þ ;

ð34Þ

where the model parameters have been derived based on X0!i ¼ fx0 ; x1 ;    ; xi g. Thus, the expected value of the degradation path at the next monitoring time point tiþ1 can be predicted by

xiþ1 ¼ xi þ li

Z

t iþ1

uðs; hÞds:

ð35Þ

ti

In the meanwhile, the prediction region of the degradation path at t iþ1 is established based on the 3r-interval criterion in this paper. The upper limit of the prediction region is given as

xiþ1;up ¼ xi þ li

Z

t iþ1

ti

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

uðs; hÞds þ 3 fli ðtiþ1  ti Þ;

ð36Þ

and the lower limit is given as

xiþ1;low ¼ xi þ li

Z ti

t iþ1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

uðs; hÞds  3 fli ðtiþ1  ti Þ:

ð37Þ

When the observed value of the next monitoring data is available, it is compared with the prediction region
 xiþ1;low ; xiþ1;up . We believe that the monitoring data is normal when it is within the prediction region, and contrarily, it is abnormal when without the prediction region and should be eliminated. The abnormal monitoring data eliminating algorithm established by using the 3r-interval criterion is not a strict screening method. It only takes effects for the monitoring dataset with obvious abnormal points deviating from the overall degradation trend and will not cause the information loss. In addition, it could be easily applied for all types of parametric models with periodic or non-periodic monitoring data. 3.3. Model selection As mentioned in the introduction, lots of RUL prediction models can be considered when dealing with the degradation monitoring data. In some applications, it will be quite difficult for us to select a confirmed RUL prediction model based on physical properties of the degradation process or statistical features of the degradation data. To measure the efficiency of different RUL prediction models, a reasonable model selection criterion is needed. Traditional model selection methods

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379

are mainly committed to analyze the goodness-of-fit between the degradation monitoring data and the candidate models, however, a good fitting may not be necessarily able to yield an accurate prediction. In this view, we define a new model selection criterion which measures the prediction accuracy directly in this paper and compare the proposed RUL prediction model with other existing models. The new model selection criterion, named as PAC, is defined as a prognostic measure that evaluates the probability mass of the predicted RUL of the unit at time ti within the e-bounds e ¼ ð1  eÞt F  t i and eþ ¼ ð1 þ eÞtF  ti , where t F represents the true failure time, i.e.,

PAC ¼ Pfe 6 r i 6 eþ g:

ð38Þ

For the proposed RUL prediction model:

Z PAC ¼

eþ

e

Z f Ri jX0!i ;f;h ðr i jX0!i ; f; hÞdr i ¼

eþ

e

Z

þ1

1



 f Ri r i jX0!i ; li ; f; h p li X0!i dli dr i :

ð39Þ

The RUL prediction model having the maximal value of PAC is the best one. The PAC helps to assess the prognostic ability of candidate RUL prediction models when degradation data and true failure time are known. Note that, the choice of e depends on the operation requirements of the considered unit [38]. Fig. 3 gives a descriptive framework of the approach proposed in this paper.

Fig. 3. A descriptive framework of the proposed approach.

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4. Case study In the following, we illustrate the above framework for online RUL prediction by using the degradation monitoring data of the identical thrust ball bearings. 4.1. Test description Bearing is a machine element which constrains relative motion to only the desired motion and reduces friction between moving parts. Identical thrust ball bearing, one of the rolling-element bearings, is commonly used in which rolling elements are placed between the turning and stationary races to prevent sliding friction. Lots of factors such as load and temperature have a significant effect on the life of identical thrust ball bearing. In the working state, the rolling elements of the bearing rotates at high speed, which is inevitable to cause the rotation wear. With the accumulation of wear, the bearing becomes deformed and an increase of the vibration occurs. In this view, vibration-based degradation signals can reflect the rotation wear indirectly. Thus, it is reasonable to measure vibration signals to evaluate the health state of the bearing. An experimental setup has been designed to perform accelerated testing on a set of identical thrust ball bearings [39]. In their work, vibration signals are collected based on the evolution of the vibration level with respect to time. Wiener-processbased degradation models have been used to analyze this set of data and the model parameters were estimated [40]. Fig. 4 describes the regenerated vibration signals of three of their test bearings. It should be noted that the original signals are consisted of two parts, i.e., stable vibration state and rapid degradation state. We only regenerated the vibration signals associated with the second state. When the vibration signal reaches the threshold D = 0.02546 Vrms, the bearing is defined to failure. The threshold here was calculated by using their threshold minus the initial value of the vibration signal as we have adjusted the initial value to zero. In addition, vibration signals were measured every two minutes. According to Fig. 4, it is easy to find that the second bearing, i.e., sample 2, has run to failure within the accelerated testing. The vibration signal of sample 2 reaches the failure threshold at the 129-th monitoring time point and thus we can obtain its real lifetime under the accelerated conditions, i.e., tF ¼ 129  2 ¼ 258 minutes. It should be noted that its signal approaches the failure threshold at the 113-th monitoring point but it does not cross the threshold and therefore does not meet the definition of failure. In addition, we will illustrate that it is actually an abnormal monitoring point in the following. The exact real lifetimes of the other two samples are unknown in this case study. 4.2. Model verification Considering the evolving trend of the vibration signals, as shown in Fig. 4, we appoint uðs; hÞ as a power function form, i.e., uðs; hÞ ¼ absb1 where h ¼ ða; bÞ. The rationality of the choice can be verified on the basis of degradation mechanism or the test of goodness-of-fit. To verify the rationality of our proposed RUL prediction model, we firstly analyzed the vibration signals of three bearings based on Model 2 which assumes that the drift parameter and the diffusion parameter are independent with each other. By using the maximum likelihood estimation (MLE) method, the corresponding model parameters of each bearing are estimated and the results are shown in Table 1. Based on the parameter estimation results shown in Table 1, it is not difficult to find that sample 1 is the fastest degraded bearing, followed by sample 2, and sample 3 is the slowest degraded bearing. In addition, sample 1 has the biggest diffusion parameter, followed by sample 2, and sample 3 has the smallest diffusion parameter. This is consistent with our description in Section 1.2, i.e., when the degradation of a unit is faster, the variation of the degradation over time is also higher. To further find out the specific relationship between degradation rate and degradation variation, a parameter correlation test

Fig. 4. The modified vibration signals of three test bearings.

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H. Wang et al. / Mechanical Systems and Signal Processing 127 (2019) 370–387 Table 1 The MLEs of the drift and diffusion parameters under Model 2. Sample 1 Drift parameter

l

Diffusion parameter

Sample 2 5

r

Sample 3

6:565  10

5

5:859  10

4:967  105

5:036  107

4:601  107

3:488  107

Fig. 5. Result of the parameter correlation test.

between the drift parameter and the square of the diffusion parameter derived under different samples has been carried out. As shown in Fig. 5, the correlation coefficient equals 0.9676 which means that there really exists the quantitative relationship between the drift parameter and the square of diffusion parameter, as described in Theorem 1. The mechanism equivalence parameter f ¼ 0:007613 in this case study.

4.3. RUL prediction In the following, we use our proposed framework to conduct RUL prediction of the three bearings. From the first monitoring point, both Kalman filter and EM algorithm are utilized to update the model parameters and RUL distribution in an adaptive way. To make a comparison, the parameter updating and RUL prediction are carried out based on both Model 2 and Model 3. The predicted degradation paths of three bearings are shown in Figs. 6–8, respectively. For both sample 1 and sample 3, we can find that the application of the two models obtained acceptable prediction results, and our proposed Model 3 performs slightly better than Model 2 because the dispersion of its predicted degradation path is smaller compared with the observed degradation path. However, for sample 2, we can find that the application of the two models only obtained acceptable prediction results before about 150 min and the prediction results become worse and worse after that. Especially, from the monitoring time Q , an unacceptable dispersion occurs for Model 2 and the dispersion is about 0.01 Vrms for Model 3, compared with the actual observed degradation path. It is obvious that the dispersion is caused by a sudden rising of the monitoring process. Actually, the sudden rising point Q is an abnormal monitoring data as the degradation path drops to the normal level after that. Fig. 7 demonstrates that the abnormal monitoring data has a great impact on the prediction results. Therefore, as mentioned in Section 3, we should test whether the monitoring data is abnormal or not when a new monitoring point is available. Based on the proposed algorithm, abnormal monitoring data eliminating has been carried out for the three bearings. Results show that all of the monitoring data are normal for sample 1 and sample 3, however totally 10 abnormal points are picked out from the monitoring process of sample 2, as presented in Fig. 9. They are distributed around 4 sudden changes which have been circled with red. We think that these sudden changes may be caused by various kinds of uncertainties, i.e., an external disturbance or measurement errors. According to the screened monitoring data of sample 2, we carried out the parameter updating and RUL prediction again. Fig. 10 presents the predicted degradation paths of sample 2 according to the total and screened monitoring data. Without the impact of abnormal points, the prediction results become better. For Model 2, the dispersion has an obvious decrease, and for Model 3, we can find that the prediction results match the actual observed degradation path well, which means that our proposed model has a better goodness-of-fit with the screened monitoring data of the identical thrust ball bearings.

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Fig. 6. The predicted degradation paths of sample 1.

Fig. 7. The predicted degradation paths of sample 2.

Fig. 8. The predicted degradation paths of sample 3.

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Fig. 9. Abnormal monitoring data eliminating of sample 2.

Fig. 10. The predicted degradation paths of sample 2 based on the screened monitoring data.

Table 2 Values of PAC under different RUL prediction models and datasets. Monitoring time ½e ; e  

þ

t0

t50

t 100

Q

[245.1, 270.9]

[145.1, 170.9]

[45.1, 70.9]

[19.1, 44.9]

Total dataset

Model 2 Model 3

44.8% 49.7%

65.1% 71.2%

78.9% 82.2%

17.7% 21.4%

Screened dataset

Model 2 Model 3

44.8% 49.7%

65.1% 71.2%

84.1% 89.9%

92.6% 98.2%

4.4. Model selection As mentioned earlier, the real lifetime of sample 2 is 258 min under the given accelerated testing conditions and thus the actual RUL values at each monitoring time point are known. In order to give a more direct comparison of the prognostic ability of the two RUL prediction models, we calculated the PAC based on Eq. (39) at several monitoring time points of sample 2, including t 0 , t50 , t 100 as well as Q , when choosing e ¼ 0:05. Results are presented in Table 2. It is easy to find that all of the values of PAC derived under Model 3 are larger than the values derived under Model 2. For example, based on the screened monitoring data, the PAC at the monitoring time Q is 92.6% under Model 2 and 98.2% under Model 3. Note that, at the monitoring time point t 0 and t50 , the values of PAC derived under both the total dataset and screened dataset are same because no abnormal monitoring data are eliminated before t 50 . At the monitoring time point

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Fig. 11. The estimated RUL and the actual RUL of sample 2 at different monitoring points.

Q , the predicted degradation paths under the total dataset have exceeded the threshold level due to the sudden rising of the monitoring process, thus the values of PAC are quite small. Further, we estimated the mean of the RUL of sample 2 at each monitoring point by using the updated PDF, i.e., Eq. (33), based on the screened monitoring data. Results are shown in Fig. 11. We can find that the mean of the estimated RUL and the actual RUL match each other well, especially for Model 3. This also verifies that our proposed model performs better when used for the identical thrust ball bearings. 5. Conclusions In this paper, a complete framework was constructed for online RUL prediction, which includes Wiener- process-based RUL prediction models, parameter updating procedure, abnormal monitoring data eliminating, as well as model selection criterion. We think the major contributions of this work are: (i) an improved Wiener process model with adaptive drift and diffusion was proposed for online RUL prediction by considering the proportion relationship between degradation rate and degradation variation, which makes it more appropriate for describing the real degradation process compared with other existing RUL prediction models; (ii) both Kalman filter and EM algorithm are used to update the model parameters and RUL distribution when a new monitoring data is available so that the estimated RUL is related to the entire degradation monitoring data to date; (iii) an algorithm for abnormal monitoring data eliminating was established based on the 3rinterval criterion to improve the prediction accuracy; and (iv) an model selection method named as PAC was proposed to measure the prediction accuracy of different prediction models. The advantages of the proposed framework are demonstrated via an application to identical thrust ball bearings. We believe that the results presented in Section 4.3 and Section 4.4 indicate clearly the value of the proposed model and the importance of abnormal monitoring data eliminating, as the prediction accuracy has been improved significantly. One promising and useful future research direction is to investigate the relationship between degradation rate and degradation variation under other stochastic degradation models, i.e., gamma process model [41] or inverse Gaussian process model [42], and develop their corresponding RUL prediction models. We believe that the consideration of the relationship is reasonable and necessary in degradation analysis as it reflects the degradation mechanism equivalence, which is the basis of prediction. Also, it will be useful to consider our proposed algorithm for abnormal monitoring data eliminating under other prediction models when highly variability exists in the monitoring process. Acknowledgments This work was supported by the National Natural Science Foundation of China [No. 61473014]. Appendix A. Proof of Theorem 1 Assume that the degradation process of the unit is performed under two different environmental stresses, i.e., the p-th

 stress level and the q-th stress level. Then denote the CDF of the FHT by F p t p under the p-th stress level and F q tq under

 the q-th stress level respectively. If there exists F p tp ¼ F q t q , the acceleration factor is usually defined as

K p;q ¼ tq =t p :

ðA1Þ

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According to the definition, the acceleration factor is actually related to the value of reliability. However, a reliabilityrelated acceleration factor will increase with the working time and thus its practical value is limited. To meet engineering requirements, we often assume that the acceleration factor is invariant between two constant stress levels. Under this assumption, the acceleration factor is determined only by the values of stress levels and independent of other parameters when the degradation mechanism remains unchanged. To achieve this, for any time tp , the following relationship should be satisfied, i.e.,



 F p t p ¼ F q K p;q  t p :

ðA2Þ

This is known as the acceleration factor invariant principle, which can also be denoted as




 dF p tp
 dF q K p;q  tp  ¼ K p;q  f q K p;q  tp ; f p tp ¼ ¼ K p;q 
dt p d K p;q  t p

ðA3Þ



 where f p tp and f q tq represent the PDFs of the FHT under the p-th and the q-th stress levels, respectively. For the basic Wiener process model, the FHT follows IG distribution. The PDFs of FHT under the two environmental stresses are given as

0  2 1 D  l t p
 p D B C f p t p ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp@ A; 2r2p t p 2pr2p t 3p

ðA4Þ

and

0  2 1 D  l t q
 q D B C f q t q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp@ A; 2t 2 r 3 2 q q 2prq t q

ðA5Þ

respectively. Substitute Eq. (A4) and Eq. (A5) into the acceleration factor invariant principle, and we obtain

K p;q


 rq K 3=2 f p tp p;q
¼ ¼ exp D rp f q K p;q  t p

!

!

lq lp t p l2q K p;q l2p D2 1 1  2 þ  2 þ  2 2 rq rp 2 rq rp 2t p r2q K p;q r2p

!! :

ðA6Þ

To ensure that K p;q is a deterministic factor, items including tp in Eq. (A6), i.e., the second and third items in the exponential term, should be eliminated or have zero coefficients, thus we have

l2q K p;q l2p  2 ¼ 0; r2q rp 1

r2q K p;q



1

r2p

¼ 0:

ðA7Þ

ðA8Þ

Based on Eq. (A7) and Eq. (A8), we can obtain

K p;q ¼

r2q l2p r2p  ¼ : r2p l2q r2q

ðA9Þ

which is equivalent to

lp =r2p ¼ lq =r2q ;

ðA10Þ

K p;q ¼ lp =lq :

ðA11Þ

Substituting Eq. (A10) and Eq. (A11) into Eq. (A6), we find that the value of the exponential term equals to one and the right side of Eq. (A6) is reduced to K p;q ; i.e., the left side of Eq. (A6). This proves that Eq. (A10) and Eq. (A11) are the correct solutions of Eq. (A6). Eq. (A10) means that the ratio of drift parameter and the square of diffusion parameter remains constant under different stress levels if the degradation mechanism remains unchanged. For a specific unit operating under a stable condition, the cumulative structural damage of the unit is constantly changing during the whole degradation process, and therefore the output characteristic of its vibration signals is constantly different when giving the same external excitation. However, when the inner degradation mechanism of the product remains unchanged within the whole degradation process, the relationship between the drift and diffusion parameters are still satisfied. Assume that the ratio in Eq. (A10) equals 1=f, then Theorem 1 is proved.

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Han Wang received his B.S. degree in 2014 from Beihang University. He is currently a Ph.D. candidate at School of Reliability and Systems Engineering, Beihang University. His research interests include stochastic processes, design of experiment and data analysis.

Xiaobing Ma received his Ph.D. degree in 2006 from Beihang University. He is currently a professor at School of Reliability and Systems Engineering, Beihang University. He is also a research fellow in the Key Laboratory on Reliability and Environmental Engineering Technology, Beihang University. His research interests include reliability data analysis, durability design, and system life modelling.

Yu Zhao received his Ph.D. degree in 2005 from Beihang University. He is currently a professor at School of Reliability and Systems Engineering, Beihang University. He is also the associate director of the Key Laboratory on Reliability and Environmental Engineering Technology, Beihang University. His research interests include reliability engineering, reliability evaluation and verification, quality management, and application of statistics techniques.