An Improved Exponential Model for Predicting Remaining Useful Life ...

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An Improved Exponential Model for Predicting Remaining Useful Life of Rolling Element Bearings Naipeng Li, Yaguo Lei, Member, IEEE, Jing Lin , and Steven X. Ding

Abstract—The remaining useful life (RUL) prediction of rolling element bearings has attracted substantial attention recently due to its importance for the bearing health management. The exponential model is one of the most widely used methods for RUL prediction of rolling element bearings. However, two shortcomings exist in the exponential model: 1) the first predicting time (FPT) is selected subjectively; and 2) random errors of the stochastic process decrease the prediction accuracy. To deal with these two shortcomings, an improved exponential model is proposed in this paper. In the improved model, an adaptive FPT selection approach is established based on the 3σ interval, and particle filtering is utilized to reduce random errors of the stochastic process. In order to demonstrate the effectiveness of the improved model, a simulation and four tests of bearing degradation processes are utilized for the RUL prediction. The results show that the improved model is able to select an appropriate FPT and reduce random errors of the stochastic process. Consequently, it performs better in the RUL prediction of rolling element bearings than the original exponential model. Index Terms—Exponential model, first predicting time (FPT), particle filtering (PF), remaining useful life (RUL) prediction, rolling element bearings.

I. I NTRODUCTION

R

OLLING element bearings are widely used in rotating machinery. They generally work in tough environment; hence, different kinds of faults may occur frequently. Any fault of a bearing probably causes breakdown of the entire machine, which could lead to catastrophic consequences. If

Manuscript received January 8, 2015; revised April 24, 2015; accepted May 30, 2015. Date of publication July 10, 2015; date of current version November 6, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 51222503 and Grant 51475355, in part by the Provincial Natural Science Foundation Research Project of Shaanxi under Grant 2013JQ7011, and in part by the Fundamental Research Funds for the Central Universities under Grant 2012jdgz01 and Grant CXTD2014001. N. Li, Y. Lei, and J. Lin are with the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China (e-mail: [email protected]; [email protected]. edu.cn; [email protected]). S. X. Ding is with the Institute of Automatic Control and Complex Systems, University of Duisburg-Essen, 47057 Duisburg, Germany (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2015.2455055

the remaining useful life (RUL) of the bearing is predicted in advance, the catastrophe could be avoided by predictive maintenance. Furthermore, the predictive maintenance is able to make the machine have a maximum uptime with minimum maintenance costs. Therefore, the RUL prediction of rolling element bearings has attracted increasingly more attention in recent years [1]–[4]. The RUL prediction methods can be categorized into datadriven and model-based methods [5], [6]. Data-driven methods attempt to derive the degradation process of a machine from measured data using machine learning techniques. Gebraeel et al. developed an artificial neural network-based method to predict the RUL of bearings [7]. Maio et al. proposed a method based on relevance vector machine for estimating the bearing RUL [8]. In addition, several studies have been published on the neuro-fuzzy system-based prediction methods [9]–[11]. Datadriven methods could be beneficial, when mechanical principles are not straightforward or mechanical systems are so complex that developing an accurate model is prohibitively expensive. The prediction accuracy of data-driven methods, however, depends on not only the quantity but also the quality of the measured data, which is full of challenges for real applications. Model-based methods are to set up mathematical or physical models to describe the degradation process of a machine, and estimate model parameters using measured data. Liao et al. proposed a prediction method based on proportional hazard and logistic regression models to predict the RUL of rolling element bearings [12]. Tian et al. proposed a proportional hazard modelbased method for the RUL prediction of the systems consisting of bearings [13]. Liao employed the Paris model combined with a genetic programming method to predict the RUL of bearings [14]. Model-based methods could incorporate both expert knowledge and measured information. Consequently, they may work well in the RUL prediction of rolling element bearings. Among all model-based studies, the exponential model is one of the most popular methods. It was first established by Gebraeel et al. in [15], where model parameters were updated using a Bayesian approach to incorporate measured information. After that, many variants and applications of the exponential model have been reported in the RUL prediction and health management [16]–[19]. Gebraeel further improved the model by updating the model parameters using multiple historical degradation signals acquired through condition

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LI et al.: IMPROVED EXPONENTIAL MODEL FOR PREDICTING RUL OF ROLLING ELEMENT BEARINGS

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Fig. 1. Bearing degradation process.

monitoring [16]. Si et al. combined Bayesian updating with the expectation maximization (EM) algorithm for parameter estimation and offered a closed-form RUL distribution [19]. It can be seen from these studies that the exponential model works well for the RUL prediction of systems with exponential-like degradation processes. However, there are still two shortcomings in this model. One shortcoming is that the first predicting time (FPT) is selected subjectively. To explain the concept of FPT, a simple example for describing the degradation process of a rolling element bearing is presented in Fig. 1. The root mean square (RMS) of vibration signals is used as the degradation index in this example. It is seen that the degradation process of the bearing generally consists of two stages, i.e., I: the normal operation stage and II: the failure stage. The degradation index in Stage I is stable, which means that the bearing is healthy. Once a fault occurs, the degradation process changes from Stage I to Stage II, where the degradation index generally increases exponentially as the fault gets severer. The major task in Stage I is to monitor the health state and detect incipient faults. When a fault is detected, the predicting process is triggered, and the bearing RUL is predicted during this process. The time to start predicting is defined as FPT, as shown in Fig. 1. It is important to select an appropriate FPT value for the exponential model. If an inappropriate FPT value is selected, either interference noises could be included in the predicting process or critical information may be excluded from the predicting process. Both cases could decrease the RUL prediction accuracy. The appropriate FPT should be the time of fault occurrence, which is difficult to be detected because incipient faults of a bearing often occur randomly and the fault characteristics are quite weak. Therefore, the FPT in the exponential model is generally selected subjectively, which restricts the applications of the exponential model. Several approaches for selecting FPT have been reported in literature, such as the engineering norm ISO 10816 and the approach based on the longest time constant of a machine and statistical properties of a candidate baseline [20], [21]. In these approaches, the FPT is selected based on an alarm, which is set according to the statistical properties of large numbers of systems. Therefore, the approaches fail to adjust the alarm adaptively according to the variation of an individual system. The other shortcoming of the exponential model is that random errors of the stochastic process decrease the RUL predic-

Fig. 2. (a) Simulated degradation process of a bearing and (b) RUL prediction result.

tion accuracy. This can be explained using a simulation shown in Fig. 2, where a stochastic process of a bearing degradation is simulated and the RUL of the bearing is predicted using the exponential model [19]. The error-free process describes the actual degradation process of the bearing, without the interference of random errors. It is noticed that there is an obvious negative correlation between random errors of the stochastic process and deviations of the prediction result. For example, the positive random error at time t1 leads to the negative deviation between the predicted RUL B1 and the actual RUL B1 . Vice versa, the negative random error at time t2 causes the positive deviation between B2 and B2 . This negative correlation can be explained as follows. Taking time t1 for example, the value of the degradation process increases from A1 to A1 because of the positive random error. As a result, the fault severity is overestimated compared with the actual severity. A severer fault generally indicates a shorter RUL. Therefore, the overestimated severity leads to a shorter predicted RUL than the actual one. To overcome these two shortcomings in the exponential model, this paper proposes an improved exponential model for predicting the RUL of rolling element bearings. The major contributions of this work are: 1) an adaptive FPT selection approach is established based on the 3σ interval; and 2) random errors of the stochastic process are reduced utilizing particle filtering (PF), and therefore, the RUL prediction accuracy is raised. The remainder of the paper is organized as follows: In Section II, we briefly describe the basic theory of the exponential model and PF, respectively. Section III shows the improved exponential model for predicting the RUL of rolling element bearings. In Section IV, the model is evaluated using a simulation of a degradation process of bearings. In Section V, the model is demonstrated using accelerated degradation tests of bearings. Conclusions are drawn in Section VI.

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II. B ASIC T HEORY

conditional joint probability density function (PDF) of S1:k is calculated as follows:

A. Exponential Models As mentioned earlier, many variants of the exponential model have been reported in literature [16]–[19]. The model considered in this work furthers the idea in [19]. In this exponential model, the parameters are estimated using the Bayesian updating and EM algorithm, and the RUL is offered as a closed-form distribution. 1) Development of Degradation Models: A degradation process is stochastic in nature due to inherent randomness in manufacturing and operating. Therefore, it is natural to model a degradation process as a stochastic one {xk = x(tk ), tk ≥ 0}, where xk is the system state at time tk . Generally, a degradation model consists of deterministic and stochastic parts. The deterministic part represents a constant physical phenomenon, which is common to all systems of a given population, whereas the stochastic part captures the variation of the degradation process of an individual system [16], [22]. In the exponential model, it is assumed that the system state can be represented by measurements, and the state xk at time tk is given by   σ2   tk xk = ϕ + θ exp β tk + σB(tk ) − (1) 2 where ϕ is a known constant; θ and β  are random variables characterizing the stochastic part; σ is a constant representing the deterministic part; and σB(tk ) is a Brownian motion (BM) following a normal distribution of N (0, σ 2 tk ), which represents random errors in the stochastic process. For convenience, the exponential model is transformed into the logged format and is defined as   σ2 sk = ln[xi − ϕ] = ln(θ ) + β  − tk + σB(tk ) 2 (2)

= θ + βtk + σB(tk )

where θ = ln(θ ) follows N (μ0 , σ02 ) and β = β  − σ 2 /2 follows N (μ1 , σ12 ). It is assumed that ϕ = 0, to simplify the analysis. 2) Parameter Estimation: Suppose that we have observed a sequence of measurements S1:k = {s1 , s2 , . . . , sk }. Since random errors σB(tk ) are independent identically distributed (i.i.d.) normal random variables, if θ and β are known, the

2(i+1) σ ˆk

1 = k



= μθ,k , σ ˆ0,k

(i+1)

= μβ,k , σ ˆ1,k

μ ˆ1,k

√

1

k

2πσ 2 Δt

⎤ k 2 2  (s − θ − βt ) (s − s − βΔt) 1 1 j j−1 ⎦ (3) × exp ⎣− − 2 Δt 2σ 2 t1 2σ j=2 ⎡

where Δt = tj − tj−1 is the constant time interval. Then, the joint posterior PDF of θ and β conditioned on S1:k is still normal resulted from the normal distribution of θ and β, i.e., 2 2 , μβ,k , σβ,k , ρk ), with [15] θ, β|S1:k ∼ N (μθ,k , σθ,k μθ,k





s1 σ02 +μ0 σ 2 t1 σ 2 +σ12 tk −σ02 t1 sk σ12 +μ1 σ 2 −0.5σ 4 = (σ02 +σ 2 t1 ) (σ12 tk +σ 2 )−σ02 σ12 t1 2 σθ,k



σ02 σ 2 t1 σ 2 +σ12 tk = 2 (σ0 +σ 2 t1 ) (σ12 tk +σ 2 )−σ02 σ12 t1

μβ,k





sk σ12 +μ1 σ 2 −0.5σ 4 σ02 +σ 2 t1 −σ12 s1 σ02 +μ0 σ 2 t1 = (σ02 +σ 2 t1 ) (σ12 tk +σ 2 )−σ02 σ12 t1 2 σβ,k



σ12 σ 2 t1 σ02 +σ 2 t1 = 2 (σ0 +σ 2 t1 ) (σ12 tk +σ 2 )−σ02 σ12 t1

ρk

√ −σ0 σ1 t1 =  2 (σ0 + σ 2 t1 ) (σ12 tk + σ 2 )

(4)

where we know that the posterior estimations of θ and β can be easily updated once new measurements are available. However, there are unknown parameters needed to be estimated before updating, which can be expressed as Θ = [σ 2 , μ0 , σ02 , μ1 , σ12 ]. The unknown parameters will be estimated using the EM 2(i) (i) 2(i) (i) 2(i) ˆ (i) = [ˆ σk , μ ˆ 0,k , σ ˆ0,k , μ ˆ1,k , σ ˆ1,k ] as the algorithm. Given Θ k ˆ (i) )]/∂Θk = 0, then the estimation in the ith step, let ∂[ (Θk |Θ k ˆ (i+1) is obtained as (5) parameter estimation in the next step Θ k [19], shown at the bottom of the page.

2 2 s21 − 2s1 (μθ,k + μβ,k t1 ) + σθ,k + σβ,k

2 + 2(ρk σθ,k σβ,k + μθ,k μβ,k ) + t1 μ2β,k + σβ,k t1

⎞ 2 2 2 2 k (s − s ) − (s − s )Δtμ + (Δt) + σ μ  j j−1 j j−1 β,k β,k β,k ⎠ + Δt j=2

(i+1)

μ ˆ0,k

 p(S1:k |θ, β) =

2(i+1)

2 = σθ,k

2(i+1)

2 = σβ,k

(5)

LI et al.: IMPROVED EXPONENTIAL MODEL FOR PREDICTING RUL OF ROLLING ELEMENT BEARINGS

3) RUL Prediction: The RUL Lk at tk is defined as follows: Lk = inf {lk : s(lk + tk ) ≥ γ|S1:k }

(6)

where γ is the prespecified failure threshold, and s(lk + tk ) = sk + βlk + σW (lk ), for lk ≥ 0

(7)

W (lk ) = B(lk + tk ) − B(tk )

(8)

with

2) SIS Algorithm: The recursive propagation of the a posteriori PDF is only a conceptual solution, which cannot be calculated analytically, because it requires the evaluation of complicated high-dimensional integrals. Therefore, the SIS algorithm is employed to approximate the a posteriori PDF, numerically. SIS approximates the state PDF using a set of particles {αi0:k }i=1:Ns sampled from an importance PDF q(αk |z1:k ), where Ns is the number of particles. Using SIS, the state of the system can be approximated using the following discrete density:

which is a standard BM. Given the estimated parameters from (5), the PDF of the RUL can be expressed by the following [19]: γ − sk fLk |S1:k (lk |S1:k ) = 

 2 l +σ 2πlk3 σβ,k ˆk2 k ⎡ ⎤ 2 (γ − sk − μβ,k lk ) ⎦

 , lk ≥ 0. (9) × exp ⎣− 2 l +σ 2lk σβ,k ˆk2 k

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p(αk |z1:k ) ≈

Ns 



wki δ α0:k − αi0:k

(14)

i=1

where wki is the weight of particle αi0:k . Once a new measurement is available, the particle weights can be updated and normalized as follows:



p zk |αik p αik |αik−1 wi i i i i

, wki = N k . (15) wk = wk−1 s i q αk |αk−1 , z1:k i=1 wk In general, the importance PDF q(αk |z1:k ) is chosen as p(αk |αk−1 ), and then, (15) is simplified as follows:

B. PF Algorithm PF is derived from traditional filtering algorithms, such as Kalman filtering and extended Kalman filtering [23]. Bayesian theory is used in PF, which attempts to evaluate the state of a system based on measurements. A sequential importance sampling (SIS) algorithm is developed and also applied in PF [24]. Based on the Bayesian theory and SIS, PF is particularly effective in estimating states of nonlinear and/or non-Gaussian systems, and it has been widely used in the fields of diagnosis and prognosis [25]–[31]. 1) Bayesian Theory: It is assumed that the state of a system αk at tk can be described in the following state-space model: αk = fk (αk−1 , ωk )

(11)

where hk : Rnα × Rnν → Rnz is the measurement function, and νk is the i.i.d. measurement noise. Based on the knowledge of the transition function and the previous state estimation, the a priori PDF of the present state is calculated as follows:  p(αk |z1:k−1 ) = p(αk |αk−1 )p(αk−1 |z1:k−1 )d αk−1 . (12) Once a new measurement is available, the a priori PDF is updated according to (13) to generate the a posteriori PDF. That is, p(αk |z1:k ) =

p(zk |αk )p(αk |z1:k−1 ) . p(zk |z1:k−1 )

i=1

(13)

wki

.

(16)

3) Resampling Step: One problem in the SIS algorithm is the particle degeneracy, i.e., after a few iterations, all but one particle have negligible weights. To deal with this problem, a resampling step is utilized. The basic idea of resampling is to eliminate particles with small weights and to concentrate upon particles with large weights. A suitable measure of the degeneracy of particles is the effective sample size introduced in [32] and defined as follows:

(10)

where fk : Rnα × Rnω → Rnα is the state transition function, and ωk is the i.i.d. state noise. Generally, we cannot obtain the system state but uncertain measurements. Therefore, the system state should be evaluated based on the measurements. The relationship between them is denoted as follows: zk = hk (αk , νk )



wi i wki = wk−1 p zk |αik , wki = Nsk

Neff =

Ns ∗

1 + var wki

(17)

∗

where wki = p(αik |z1:k )/q(αik |αik−1 , z1:k ) is referred to the true weight. The effective sample size cannot be calculated exactly, but we can estimate it by ˆeff =  1 . N Ns i 2 i=1 wk

(18)

ˆeff falls below a threshold NT , a new particle set When N i∗ {αk }i=1:Ns will be generated by resampling Ns times, follow∗ ing the principle of P (αik = αik ) = wki , and the weights are i reset to wk = 1/Ns . III. I MPROVED E XPONENTIAL M ODEL The flowchart of the improved exponential model for predicting RUL of rolling element bearings is shown in Fig. 3. Vibration signals are acquired from rolling element bearings, and then, kurtosis and RMS are extracted from the vibration signals, one of which is used for health monitoring and the other is used for RUL prediction. It is well known that kurtosis is sensitive to incipient faults but is not informative for the

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new trigger mechanism is applied to restrict the interference of random noises, i.e., the predicting process is triggered when l + 1 consecutive kurtosis values exceed the 3σ interval. The process of the trigger mechanism is described as follows: First, let l = 0, find the first time when the kurtosis value exceeds the 3σ interval, and select the time as FPT0 . Second, let l = l+1, find the time tf when the consecutive l+ 1 kurtosis values {mf +k }k=0:l satisfy {|mf +k −μ| > 3σ}k=0:l , and select the time tf as FPTl . Third, let l increases from 1 until l satisfies FPTl = FPTl−1 , and output FPTl as the final FPT value. The theory behind this trigger mechanism is that abnormal states caused by random noises are almost impossible to appear l + 1 times consecutively during the normal operation stage. When the selected FPT starts to keep stable with the increase of l, it may imply that a fault has occurred. After the FPT is decided, RMS is input into the degradation model for the RUL prediction of the bearing. In order to reduce random errors of the stochastic process, PF is employed in this study to estimate the health state. The RUL prediction process of the improved exponential model is explained as follows. ˆ02 , μ0 = First, the model parameters are initialized as σ 2 = σ 2 2 , μ1 = μβ,0 and σ12 = σβ,0 using the approach μθ,0 , σ02 = σθ,0 proposed in [15]. Then, initial particles {si0 }i=1:Ns are sampled ˆ 0 ) ∼ N (s0 , σ ˆ02 Δt). Once the measurement sk from p(˜ s0 |s0 , Θ at tk is available, model parameters are updated using (4) and further estimated with (5). With the estimated parameters and the following variant of the degradation model: sk = sk−1 + βΔt + σW (Δt)

(19)

the PDF of the state is described as follows:

Fig. 3. Flowchart of the improved exponential model.

development of faults, while RMS is able to reflect the increase of the vibration energy with the development of faults [33]. That is why in this study, kurtosis and RMS are adopted as the monitoring and predicting indexes, respectively. In the normal operation stage, kurtosis is used for health monitoring and FPT selection. Once an FPT is decided, the predicting process is triggered, and RMS is utilized as the predicting index for the RUL prediction of the bearing. In order to select the appropriate FPT, an adaptive approach is established based on the 3σ interval. At first, the history data of the bearing in the normal operation stage are utilized to decide the 3σ interval [μ − 3σ, μ + 3σ], by calculating the mean μ and standard deviation σ of kurtosis. Then, the 3σ interval is utilized to identify normal and abnormal states of the bearing. When new kurtosis mf at tf is available, it is compared with the 3σ interval. If mf is beyond the 3σ interval, the health state is considered to be abnormal. The abnormal state may be caused by either a fault or random noises. In this approach, a

1 ˆ k) =  p( sk |ˆ sk−1 , Θ 

2 Δt + σ 2πΔt σβ,k ˆk2 ⎡ ⎤ 2 ( sk − sˆk−1 − μβ,k Δt) ⎦ 

× exp ⎣− (20) 2 Δt + σ 2Δt σβ,k ˆk2 2 ˆ k = [ˆ σk2 , μθ,k , σθ,k , μβ,k , where s˜k is the actual state at tk ; Θ 2 σβ,k ] are the estimated parameters; sˆk−1 is the estimated state at tk−1 , which is different from the measurement sk−1 . It should be noticed that, if sˆk−1 is replaced by sk−1 , (20) will change to ˆ k ), which is the PDF of the state acquired using p(˜ sk |sk−1 , Θ the original exponential model. Now, we need to focus on how to estimate the actual state s˜k at tk based on the measurement sk . Here, we take ˆ k ) as the importance PDF, which contains the sk−1 , Θ p(˜ sk |ˆ information of the state estimation at tk−1 . Then, particle sets {sik }i=1,2,...,Ns are sampled from (20). These particles can only approximate the a priori PDF of the state. In order to approximate the a posteriori PDF, particle weights should be updated according to the measurement sk as follows: 

2  sk − sik 1 wki i i i wk = wk−1  exp − = . , w  k Ns i 2ˆ σk2 tk 2πˆ σk2 tk i=1 wk (21)

LI et al.: IMPROVED EXPONENTIAL MODEL FOR PREDICTING RUL OF ROLLING ELEMENT BEARINGS

With the particles and their corresponding weights, the a posteriori PDF of the system state can be approximated using the following discrete density: ˆ k) ≈ p( sk |sk , Θ

Ns 



wki δ s˜k − sik .

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TABLE I RUL P REDICTION P ROCESS U SING THE I MPROVED E XPONENTIAL M ODEL

(22)

i=1

Through consecutive iteration and updating, the particle whose motion curve is closer to the actual degradation process acquires a higher weight. The health state of the bearing is estimated by sˆk =

Ns 

wki sik .

(23)

i=1

With the weighted average of updated particles, the estimated state sˆk is closer to the actual state than the measurement sk . ˆeff in (18) is After the particle weights have been updated, N ˆeff calculated to measure the degeneracy of particles. When N i∗ falls below a threshold NT , a new set of particles {sk }i=1:Ns are resampled, and the particle weights are reset to wki = 1/Ns . Finally, the PDF of the RUL is calculated using γ − sˆk fLk |S1:k (lk |S1:k ) = 

 2 l +σ 2πlk3 σβ,k ˆk2 k ⎡ × exp ⎣−

⎤ (γ − sˆk − μβ,k lk ) ⎦

 , lk ≥ 0. (24) 2 l +σ 2lk σβ,k ˆk2 k 2

The RUL prediction process using the improved exponential model is summarized in Table I. IV. S IMULATION Here, a simulation of a degradation process of rolling element bearings is utilized to evaluate the effectiveness of the improved exponential model in the bearing RUL prediction. A. Simulations of the Bearing Degradation Process We simulate the vibration signals that represent the degradation process of rolling element bearings [34]. The parameters of the bearing in the simulation are as follows: out race diameter of 29.1 mm, inner race diameter of 22.1 mm, 13 rollers, and contact angle of 0◦ . The bearing has an outer race fault, and its rotating speed is 1800 r/min. The sampling frequency is 25.6 kHz, and each sample contains 2560 data points, i.e., 0.1 s, and the sampling is repeated every 10 s. A sample in the normal operation stage and a sample in the failure stage are shown in Fig. 4(a) and (b), respectively. There exist obvious impacts with a period of τ = 0.0059 s in the sample of the failure stage, which is corresponding to the fault characteristic frequency of 169 Hz. To simulate the whole degradation process of the bearing, the signals are repeated 200 times with the increasing fault severity, and Fig. 4(c) shows the vibration signals of the whole lifetime.

Fig. 4. Simulation of the bearing: (a) a sample in the normal operation stage, (b) a simple in the failure stage, and (c) vibration signals of the whole lifetime.

B. RUL Prediction Based on the Simulation The improved exponential model is used to predict the bearing RUL in this simulation. As described in Section III, kurtosis

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ˆ k ) and p(˜ ˆ k ) at Fig. 7. Comparison between p(˜ sk |sk−1 , Θ sk |sk , Θ tk = 830 s.

Fig. 5. FPT selection in the simulation: (a) Kurtosis and (b) RMS.

Fig. 8. (a) Bearing state estimation and (b) RUL prediction in the simulation.

Fig. 6. Parameter estimation in the simulation.

and RMS are extracted from the vibration signals, which are used for FPT selection and RUL prediction, respectively. The FPT selection result is shown in Fig. 5. It is seen that kurtosis is sensitive to the incipient fault of the bearing but not informative for the development of the fault. RMS has an obvious exponential-like growth during the degradation process but reflects nothing characterizing the fault occurrence. Therefore, it is reasonable to adopt kurtosis as the monitoring index and use RMS as the predicting index. In Fig. 5(b), the RMS is divided into two stages by the FPT. Before the FPT, the values of RMS are stable, while they increase exponentially after the FPT. Therefore, tf = 330 s indicates the initial time of the exponential-like degradation process and is suitable to be the FPT in this simulation. The values of RMS after tf = 330 s are input into the improved exponential model for RUL prediction, and the particle

number is set to 1000. The parameter estimation process is shown in Fig. 6. It is observed that all parameters are not estimated accurately at the beginning of the predicting process. When enough RMS values are used in the parameter estimation, all of these parameters converge to the actual values. To visualize the state estimation process of the improved expoˆ k ) of the original nential model, the state PDF p(˜ sk |sk−1 , Θ ˆ k ) of the exponential model [19] and the state PDF p(˜ sk |sk , Θ improved model in the middle of the process, i.e., tk = 830 s, are presented in Fig. 7. It is seen that, with the help of the state estimation process of PF, the improved exponential model actually changes the result of the original model. For further identifying the effectiveness of the improved exponential model in state estimation and RUL prediction, the results of the improved exponential model are compared with those of the original exponential model and a commonly used PF-based prediction model, i.e., the Paris model [27]. In this process, the same FPT and particle number are used for them. The comparison results are shown in Fig. 8. The actual RUL is calculated with RULk = tEoL − tk , where RULk is the actual RUL at tk , and tEoL is the time index at the end of lifetime [35].

LI et al.: IMPROVED EXPONENTIAL MODEL FOR PREDICTING RUL OF ROLLING ELEMENT BEARINGS

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TABLE II S CORES OF T WO P ERFORMANCE M ETRICS IN THE S IMULATION

Fig. 10. Rolling element bearings before and after a test.

Fig. 9. Overview of the experimental system.

In Fig. 8(a), the state estimation result of the original model almost coincides with the measurement; thus, the estimation result still includes random errors of the stochastic process. The Paris model performs slightly better than the original exponential model, but it still has random fluctuation in the estimation result. The result of the improved model, however, reflects a smoother degradation process and succeeds in restraining random errors of the stochastic process. In Fig. 8(b), the RUL prediction results of all three models produce large errors at the beginning. The prediction results of the original model and the Paris model have larger fluctuation than that of the improved model. As time goes on, all of them converge to the actual RUL. In addition, the improved model converges fastest and provides the most accurate result among them. This corresponds to our requirements for RUL prediction that more accurate prediction results are expected when the machine is closer to the final failure. The results could be explained as follows. At the beginning, RMS values are not enough to accurately estimate the model parameters. Once the model parameters are estimated appropriately, random errors of the stochastic process become the major causes of the prediction errors. In fact, the improved exponential model is able to reduce random errors by using PF. Therefore, it obtains a best prediction result in this simulation. In order to compare the performance of the three models quantitatively, two commonly used performance metrics, i.e., cumulative relative accuracy (CRA) and convergence proposed in [35], are calculated. CRA quantifies how accurately a prediction method performs at specific time indexes. The range of scores for CRA is [0, 1], where the perfect score is 1. Convergence quantifies how fast a method approaches the actual RUL. A lower score means a faster convergence. CRA and convergence are calculated at the time indexes from the half to the end of lifetime, since these time indexes are closer to the final failure and, therefore, more meaningful than the early stages for RUL prediction. The scores of the two metrics are

Fig. 11. Vibration signals of four bearings: (a) bearing 1, (b) bearing 2, (c) bearing 3, and (d) bearing 4.

summarized in Table II. It is noticed that the CRA score of the improved model is the highest among them, which means that the improved model has the most accurate prediction result. The convergence score of the improved model is the lowest among them, which implies that the improved model approaches the actual RUL fastest. V. E XPERIMENTAL D EMONSTRATIONS Here, vibration signals of the whole lifetime acquired from accelerated degradation tests of rolling element bearings are used to verify the effectiveness of the improved model.

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Fig. 12. FPT selection results: (a) bearing 1, (b) bearing 2, (c) bearing 3, and (d) bearing 4.

A. Introduction to the Tests and Vibration Data An experimental system named PRONOSTIA [36] is shown in Fig. 9. This system is designed to test methods for bearing fault detection, diagnosis, and RUL prediction. In order to conduct accelerated degradation tests of bearings in a few hours, a radial force, which is equal to the bearing’s maximum dynamic load of 4 kN, is applied on the tested bearings. The force is generated by a cylinder pressure, and the pressure is delivered through a pressure regulator. During the tests, the rotating speed of the bearing keeps 1800 r/min. Accelerometers are fixed on the outer race of the bearing, and vibration signals are captured. The sampling frequency is 25.6 kHz. Each sample contains 2560 data points, i.e., 0.1 s, and the sampling is repeated every 10 s. The vibration signals are transmitted into a PC for data visualization and storage through a National Instruments (NI) data acquisition (DAQ) card. The bearing useful-life ends at the time when the amplitude of the vibration signal exceeds 20 g [37]. Each bearing is naturally degraded during the tests without seeding a fault in advance. Depending on the diversity of different bearings and the randomness of degradation processes, the fault modes may be slightly different for distinct bearings. As a result, the vibration signal pattern is different for each tested bearing. Fig. 10 shows pictures of a tested bearing before and after a test. It is seen that different kinds of faults occur on the balls and inner race of the bearing. The vibration signals during the whole lifetime of four tested bearings are shown in Fig. 11. The signal amplitudes of bearings 1 and 2 have gradually increasing trends, which indicates that the faults get severe gradually, whereas the signal amplitudes of bearings 3 and 4 show rapid increases at the end of lifetime, thus representing abrupt degradation processes. Taking bearing 4 as an example, more detailed information about the vibration signals in the normal operation stage and the failure stage is presented in Fig. 11. It is shown that there exist obvious impacts in the vibration signals of the failure stage than those of the normal operation stage.

B. FPT Selection As described in the proposed method, kurtosis and RMS are first extracted from the vibration signals. Based on kurtosis,

TABLE III S ELECTED FPT OF F OUR B EARINGS

the FPT selection approach proposed in this paper is used to decide the FPT of the four bearings. To make a comparison, another approach proposed by Ginart et al. [20] is also used to select the FPT. The results are shown in Fig. 12, and all FPT values are given in Table III, where “N/A” means that no FPT is selected during the whole lifetime. It is shown in Fig. 12 that the alarm set by Ginart’s approach is much higher than the 3σ interval and fails to distinguish the normal and the abnormal states. On the contrary, the 3σ interval is able to detect the abnormal states. However, during the normal operation stage, there are some abnormal states caused by random noises instead of faults, which are marked by “∗” in Fig. 12. Due to the antiinterference ability of our approach, the time when faults occur is appropriately selected as the FPT. Fig. 13 plots all RMS values of the four bearings and the corresponding FPT decided by the two approaches. It is seen that RMS has a more obvious degradation trend than kurtosis in Fig. 12, with the development of faults. Since the faults of bearings 1 and 2 get severe gradually, it is slightly difficult to exactly give the time of the fault occurrence. Consequently, the FPT values selected by our approach have a slight time delay, as shown in Fig. 13. The delayed time, however, is much smaller than that of Ginart’s approach, particularly in the case of bearing 1. In addition, the small delay is acceptable for the RUL prediction of bearings. For bearings 3 and 4, they behave abrupt degradation processes. Therefore, it is not as difficult as bearings 1 and 2 to select the FPT. Our approach still performs better than Ginart’s approach in selecting the FPT of bearings 3 and 4. In order to further identify the effectiveness of the 3σ interval, 1σ−4σ intervals are utilized in our approach to select the FPT of four tested bearings. The selection results are shown in Fig. 14. It is seen that the 2σ and 4σ intervals select much too late FPT for bearing 1. For bearing 2, the 1σ and 2σ intervals offer much too early FPT. For bearing 3, all of them have the same FPT. And for bearing 4, the 1σ interval presents much too early FPT. In conclusion, the 3σ interval has the best performance among these intervals.

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Fig. 13. RMS division results: (a) bearing 1, (b) bearing 2, (c) bearing 3, and (d) bearing 4.

Fig. 14. FPT selection results with different σ intervals: (a) bearing 1, (b) bearing 2, (c) bearing 3, and (d) bearing 4.

Fig. 15. RMS estimation results: (a) bearing 1, (b) bearing 2, (c) bearing 3, and (d) bearing 4.

Fig. 16. RUL prediction results: (a) bearing 1, (b) bearing 2, (c) bearing 3, and (d) bearing 4.

C. RUL Prediction The RUL of four bearings is predicted using the improved exponential model with a particle number of 1000. The RMS estimation results are shown in Fig. 15, and the RUL prediction

results are presented in Fig. 16. For comparison, the original exponential model [19] and the Paris model [27] are used to predict the RUL of the four bearings as well. The same FPT and particle number are used for them, and the estimation

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TABLE IV S CORES OF T WO P ERFORMANCE M ETRICS FOR T ESTED B EARINGS

and prediction results are also presented in Figs. 15 and 16, respectively. Fig. 15 indicates that the estimation results of the original model almost coincide with the actual RMS, while two PF-based methods, i.e., the Pairs model and the improved exponential model, may reduce random errors of the stochastic processes and therefore reflect the global degradation processes of the bearings. In Fig. 16, all of them generate inaccurate results at the beginning because of the lack of RMS values. When enough RMS values are available and used, the original exponential model and the Paris model still produce large prediction errors, particularly in the case of bearing 2. The improved exponential model, however, converges to the actual RUL fastest and presents the most accurate prediction results among them. This should be explained as follows. The original exponential model is unable to accurately estimate the health state of the bearings; thus, the prediction results are influenced by random errors of the stochastic process. The Paris model performs better than the original exponential model in the state estimation with the help of PF, but it may not estimate the model parameters effectively. Therefore, the results of the Paris model still have large random errors. The improved exponential model incorporates the superiority of PF in the state estimation and the advantage of the Bayesian updating and EM algorithm in the parameter estimation. Consequently, it presents the best prediction results among the three models. To quantify the prediction performance of the three models, CRA and convergence [35] are calculated at the time indexes from the half to the end of the lifetime. The scores of the two metrics are summarized in Table IV. It is noticed that the exponential model has the highest CRA scores and the lowest convergence scores among the three models for all of the tested bearings. Therefore, the improved exponential model has the most accurate prediction results and the fastest convergence speed among them. In conclusion, the improved exponential model performs best in the RUL prediction of the bearings. VI. C ONCLUSION To enhance the performance of the original exponential model in RUL prediction of rolling element bearings, an improved exponential model has been proposed in this paper. There are two major contributions in this work. First, an adaptive FPT selection approach is established based on the 3σ interval, in which a new trigger mechanism is applied to restrict the interference of random noises. Second, PF is utilized to estimate the health state of the bearings and therefore

reduce random errors of the stochastic process. A simulation and four tests of bearing degradation processes are used to demonstrate the effectiveness of the improved model. In order to identify the benefits of our adaptive FPT selection approach, it is compared with Ginart’s approach. The results show that our approach performs better in restricting the interference caused by random noises and selects a more appropriate FPT. To show the superiority of state estimation using PF, the performance of the improved exponential model is compared with those of the original exponential model and the Paris model. The results clearly demonstrate the effectiveness of the improved model in reducing random errors of the stochastic process and increasing the accuracy of RUL prediction for bearings. Although this study has improved the prediction accuracy of the exponential model by selecting an appropriate FPT and reducing random errors, the failure threshold still influences the prediction accuracy of the exponential model. The fact is that the failure threshold is generally set subjectively and limited research has been carried out on adaptively setting failure thresholds in RUL prediction. Therefore, we will investigate adaptive threshold setting approaches in our future research. R EFERENCES [1] Z. G. Tian, “An artificial neural network method for remaining useful life prediction of equipment subject to condition monitoring,” J. Intell. Manuf., vol. 23, no. 2, pp. 227–237, Apr. 2012. [2] A. Malhi, R. Q. Yan, and R. X. Gao, “Prognosis of defect propagation based on recurrent neural networks,” IEEE Trans. Instrum. Meas., vol. 60, no. 3, pp. 703–711, Mar. 2011. [3] Y. N. Qian, R. Q. Yan, and S. J. Hu, “Bearing degradation evaluation using recurrence quantification analysis and Kalman filter,” IEEE Trans. Instrum. Meas., vol. 63, no. 11, pp. 2599–2610, Nov. 2014. [4] Z. G. Tian, L. Wong, and N. Safaei, “A neural network approach for remaining useful life prediction utilizing both failure and suspension histories,” Mech. Syst. Signal Process., vol. 24, no. 5, pp. 1542–1555, Jul. 2010. [5] J. Liu, W. Wang, F. Ma, Y. B. Yang, and C. S. Yang, “A data-model-fusion prognostic framework for dynamic system state forecasting,” Eng. Appl. Artif. Intell., vol. 25, no. 4, pp. 814–823, Jun. 2012. [6] D. T. Liu et al., “Lithium-ion battery remaining useful life estimation based on fusion nonlinear degradation AR model and RPF algorithm,” Neural Comput. Appl., vol. 25, no. 3/4, pp. 557–572, Dec. 2014. [7] N. Z. Gebraeel, M. A. Lawley, R. Liu, and V. Parmeshwaran, “Residual life predictions from vibration-based degradation signals: A neural network approach,” IEEE Trans. Ind. Electron., vol. 51, no. 3, pp. 694–700, Jun. 2004. [8] F. D. Maio, K. L. Tsui, and E. Zio, “Combining relevance vector machines and exponential regression for bearing residual life estimation,” Mech. Syst. Signal Process., vol. 31, pp. 405–427, Aug. 2012. [9] J. Liu, W. Wang, and F. Golnaraghi, “A multi-step predictor with a variable input pattern for system state forecasting,” Mech. Syst. Signal Process., vol. 23, no. 5, pp. 1586–1599, Jul. 2009. [10] C. C. Chen, B. Zhang, G. Vachtsevanos, and M. E. Orchard, “Machine condition prediction based on adaptive neuro-fuzzy and high-order particle filtering,” IEEE Trans. Ind. Electron., vol. 58, no. 9, pp. 4353–4364, Sep. 2011. [11] A. Soualhi, H. Razik, G. Clerc, and D. D. Dinh, “Prognosis of bearing failures using hidden Markov models and the adaptive neuro-fuzzy inference system,” IEEE Trans. Ind. Electron., vol. 61, no. 6, pp. 2864–2874, Jun. 2014. [12] H. T. Liao, W. B. Zhao, and H. R. Guo, “Predicting remaining useful life of an individual unit using proportional hazards model and logistic regression model,” in Proc. Annu. Rel. Maintain. Symp., Newport Beach, CA, USA, Jan. 23–26, 2006, pp. 127–132. [13] Z. G. Tian and H. T. Liao, “Condition based maintenance optimization for multi-component systems using proportional hazards model,” Rel. Eng. Syst. Safety, vol. 96, no. 5, pp. 581–589, May 2011.

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Naipeng Li received the B.S. degree in mechanical engineering from Shandong Agricultural University, Shandong, China, in 2012. He is currently working toward the Ph.D. degree in mechanical engineering in the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, China. His research interests include machinery condition monitoring, intelligent fault diagnostics, and remaining useful life prediction of rotating machinery.

Yaguo Lei (M’15) received the B.S. and Ph.D. degrees in mechanical engineering from Xi’an Jiaotong University, Xi’an, China, in 2002 and 2007, respectively. He is currently a Full Professor of mechanical engineering at Xi’an Jiaotong University. Prior to joining Xi’an Jiaotong University in 2010, he was a Postdoctoral Research Fellow with the University of Alberta, Edmonton, AB, Canada. He was also an Alexander von Humboldt Fellow with the University of Duisburg-Essen, Duisburg, Germany. His research interests focus on machinery condition monitoring and fault diagnosis, mechanical signal processing, intelligent fault diagnostics, and remaining useful life prediction. Dr. Lei is a member of the American Society of Mechanical Engineers; a Senior Member of the Chinese Mechanical Engineering Society; and an Editorial Board Member of Neural Computing and Applications, Advances in Mechanical Engineering, The Scientific World Journal, International Journal of Mechanical Systems Engineering, Chinese Journal of Engineering, International Journal of Applied Science and Engineering Research, etc.

Jing Lin received the B.S., M.S., and Ph.D. degrees from Xi’an Jiaotong University, Xi’an, China, in 1993, 1996 and 1999, respectively, all in mechanical engineering. He is currently a Professor with the State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University. From July 2001 to August 2003, he was a Postdoctoral Fellow with the University of Alberta, Edmonton, AB, Canada, and a Research Associate with the University of Wisconsin–Milwaukee, Milwaukee, WI, USA. From September 2003 to December 2008, he was a Research Scientist with the Institute of Acoustics, Chinese Academy of Sciences, Beijing, China, under the sponsorship of the Hundred Talents Program. His current research directions are in mechanical system reliability, fault diagnosis, and wavelet analysis. Dr. Lin was a recipient of the National Science Fund for Distinguished Young Scholars in 2011.

Steven X. Ding received the Ph.D. degree in electrical engineering from the Gerhard Mercator University of Duisburg, Duisburg, Germany, in 1992. From 1992 to 1994, he was a Research and Development Engineer with Rheinmetall GmbH, Dusseldorf, Germany. From 1995 to 2001, he was a Professor of control engineering with the University of Applied Science Lausitz, Senftenberg, Germany, where he served as Vice President during 1998–2000. He is currently a Full Professor of control engineering and the Head of the Institute for Automatic Control and Complex Systems (AKS) at the University of Duisburg-Essen, Duisburg. His research interests include model-based and data-driven fault diagnosis, fault-tolerant systems, real-time control, and their applications in industry, with a focus on automotive systems and chemical processes.