A particle filtering-based approach for remaining useful life predication of rolling element bearings Naipeng Li, Yaguo Lei*, Zongyao Liu, Jing Lin State Key Laboratory for Manufacturing Systems Engineering Xi’an Jiaotong University No. 28 Xianning West Road 710049, Xi’an, China *Corresponding author:
[email protected] (Yaguo Lei) Abstract—Rolling element bearings are one of the most widely used components in rotating machinery. However, they are also the components which frequently suffer from damage. Remaining useful life (RUL) prediction of rolling element bearings has received considerable attention, since it can avoid failure risks, and ensure availability, reliability and security. Model-based methods are commonly used in RUL prediction because of their high accuracy in long-time prediction. In modelbased methods, a degradation indicator which describes the whole degradation process of bearings, however, is very critical but difficult to be extracted. A model function, used to predict the evolution trend and the RUL of bearings, is difficult to develop as well. In this paper, a particle filtering (PF)-based approach is developed to predict the RUL of rolling element bearings. In this approach, two modules are included, i.e. indicator calculation module and PF-based prediction module. In the first module, a new degradation indicator is calculated based on correlation matrix clustering and weight algorithm. This indicator fuses different characteristics of multiple features, includes more fault information and therefore has a better prediction tendency. In the second module, a PF-based approach is proposed to predict the RUL of bearings. Different from the traditional PF-based approach, a new algorithm of parameter initialization is introduced to calculate the initial parameters of the state space model. Experimental data of rolling element bearings are used to demonstrate the effectiveness of this approach. For comparison, another RUL prediction approach based on adaptive neuro-fuzzy inference system (ANFIS) is also utilized to process the experimental data. The result shows that the proposed approach can effectively calculate the appropriate degradation indicator, initialize the model parameters and perform better in RUL prediction than the ANFIS-based approach for rolling element bearings. Keywords- Remaining useful life prediction; Degradation indicator; Particle filtering; Parameter intialization; Rolling element bearing
I.
INTRODUCTION
Rolling element bearings are widely used in rotating machinery. They generally work in a tough environment, so different kinds of failures occur frequently. Any failure of a bearing may cause breakdown of the entire machine, which may lead to serious consequence. Predictive maintenance can predict the future state of bearings based on current operating condition and keep the machine having a maximum uptime for This research is supported by National Natural Science Foundation of China (51222503 and 51125022), New Century Excellent Talents in University (NCET-11-0421), Provincial Natural Science Foundation research project of Shaanxi (2013JQ7011) and Fundamental Research Funds for the Central Universities (2012jdgz01). 9781479949434/14/$31.00 ©2014 IEEE
minimum effective maintenance costs. Thus, it has become more and more popular in recent years. To fulfill the goal of predictive maintenance, a lot of research has been carried out on remaining useful life (RUL) prediction, because it can directly serve as a decision variable of predictive maintenance. The RUL prediction methods can be roughly divided into two kinds: data-driven methods and model-based methods. Data-driven methods utilize the monitored operation data collected by sensors to evaluate the degradation of a mechanical system, instead of building models based on comprehensive system physics and human expertise [1]. They could be more beneficial when understanding of first principles of system operation is not straightforward or when the system is so complex that developing an accurate model is prohibitively expensive. The commonly used data-driven methods include artificial neural network (ANN) [2], support vector machine (SVM) [3], relevance vector machine (RVM) [4], neuro-fuzzy system [5] and so on. All of these methods assume that the degradation of the system has an unchanged tendency during its whole lifetime and attempt to predict the future evolution trend of the system based on historical degradation process. In fact, the system has various failure modes in different stages, which may lead to increasing prediction errors as time goes on. On the contrary, model-based methods attempt to set up mathematical models to describe the physical phenomena underlying the failure modes and the process of components degradation, and update parameters of the model with new measured data to predict the state evolution. The commonly used models for rolling element bearings are logistic regression model (LRM) [6], exponential degradation model [7] and Paris’ law model [8], etc. One of the widely used methods for updating model parameters is non-linear stochastic filtering [9]. Using the mathematical model and parameter updating, model-based methods can take advantages of both expert knowledge and real-time condition information. As a result, they work well in long-time prediction. Model-based methods, however, have two problems in predicting RUL of rolling element bearings. One is how to extract an effective degradation indicator from bearing vibration signals; the other is to establish a prediction model to accurately estimate the failure time. Aiming to these two problems, this paper proposes a particle filtering (PF)-based approach for predicting the RUL of rolling element bearings. In the approach, two modules, i.e. indicator calculation module and PF-based prediction module are involved. In the first
module, a degradation indicator is calculated by using of correlation matrix clustering and weight algorithm. The indicator fuses failure information from multiple features and may be representative of the whole failure process. In the second module, the indicator is put into the state space model and the model parameters are initialized and updated using least square fitting and update algorithm respectively. Then the RUL of the bearing is predicted by the prognostic algorithm. The rest of this paper is organized as follows. Section 2 briefly introduces the background of PF. Section 3 is dedicated to a description of the proposed PF-based prediction approach. In Section 4, experimental data of rolling element bearings are used to demonstrate the effectiveness of the approach in the RUL prediction of bearings. A data-driven method based on adaptive neuro-fuzzy inference system (ANFIS) is also applied to compare with the approach. The result shows that the PFbased approach proposed in this paper performs better than the ANFIS-based approach. Some concluding remarks are drawn in Section 5.
II.
BACKGROUND OF PARTICLE FILTERING
PF is derived from traditional nonlinear filtering theory, such as Kalman filtering (KF) and Extended KF [10]. Like traditional nonlinear filtering approaches, Bayesian theory is utilized in PF, which attempts to construct the posterior probability density function (PDF) of the state based on all available information, including the measurements of the state. However, different from other nonlinear filtering approaches, the PDF of the state in PF is described with the Monte-Carlo simulation approach. The underlying principle of PF is the approximation of relevant distributions with particles and their associated weights. In order to reduce the number of particles and improve the computational efficiency, a sampling importance resampling (SIR) algorithm was developed and applied to the PF [11]. Based on the concept of SIR and the use of Bayesian theory, PF is particularly useful in dealing with nonlinear and/or non-Gaussian problems. PF was first introduced into the field of mechanical RUL prediction by Orchard [12]. Cadini et al. combined the Paris-Erdogan model with PF and applied this method to the fatigue crack growth estimation [13]. In PF algorithm, the state of the mechanical system at the discrete time step tk = k ⋅ Δt is described in the following state space model. (1) xk = f k (xk −1, θ k , ωk ) (2) z k = hk (xk ,ν k ) where z xk is the state of the fault dimension ( such as the crack size or the wear area); n n n z fk : R x × R w → R x is the state transition function (possibly nonlinear); n n n z hk : R x × R ν → R z is the measurement function (possibly nonlinear); z θ is a vector of model parameters; k
z z
zk is the degradation indicator that has been calculated from measured data; wk, vk are two independent identically distributed (i.i.d.) noise vector sequences.
Before prediction, the model parameters must be estimated and updated with the Bayesian theory. This task involves two basic steps. In Prediction Step, a priori state estimation is generated from the knowledge of the previous state estimation and the transition function. (3) p( xk | z1:k −1 ) = ∫ p (xk | xk −1 ) p(xk −1 | z1:k −1 ) dxk −1
The Update Step incorporates the new observation data into the priori state estimation p( xk | z1:k −1 ) in order to generate the posteriori state estimation p ( xk | z1:k ) . p ( zk | xk ) p ( xk | z1:k −1 ) (4) p ( xk | z1:k ) = p ( zk | z1:k −1 ) PF approximates the state PDF by using particles having associated discrete probability masses and a set of corresponding normalized importance weights. In practical application, the actual distribution is approximated by a set of samples drawn from an importance distribution q( x0:k | z1:k ) . Then (3) is transformed into the following one [12]. N
( ) (
~ xi ⋅ δ x − xi p( x k | z1:k ) ≈ ∑ w k 0:k 0:k 0:k i =1
)
(5)
where the update for the importance weights is given by the following equation.
wk = wk −1
p ( zk | xk ) p ( xk | xk −1 ) q( xk | x0:k −1, z1:k )
(6)
The importance distribution can be chosen as p( xk | xk −1 ) , and then the update for importance weights is simplified as [13]: (7) wk = wk −1 ⋅ p ( zk | xk ) We use Paris-Erdogan model to describe the dynamic evolution of the rolling element bearing. The grow rate of fault dimension can be expressed using the following equation. dx (8) = C (Δk )n dN where x is the fault dimension, N is the load cycles, C and n are constants related to the material properties, and need to be estimated from experimental data, and Δk is the stress intensity amplitude roughly proportional to the square root of x: (9) Δk = β x In (9), the parameter β is again a constant to be estimated from experimental data. Equation (8) can be transformed into the following one if (9) is substituted. dx (10) = A ⋅ xm dN where A and m are two parameters which need to be estimated from available data. According to the state space model equation, the ParisErdogan model can be rewritten in the following form. (11) xk = xk −1 + A ⋅ xkm−1 ⋅ Δt (12) zk = xk + ν k
where ωk is ignored because it can be handled through the uncertainty in model parameters, zk is assumed to have the linear relationship with xk, and ν k ~ N 0, σ ν2 is the measurement noise caused by operating conditions or equipments. The following distribution is acquired from these two above equations.
(
)
1 ⎛ z - x - A⋅ xkm−1 ⋅ Δt ⎞⎟ − ⎜ k k −1 ⎟ σν 2⎜ ⎠ e ⎝
2
1 (13) 2π σν Once new measured data are available, the particles and the corresponding importance weights can be updated with the help of (13).Then the parameters of the state space model could be adjusted according to the new distribution of the particles. After all of the available data have been used, the final parameters of the model are employed to predict the degradation process and the RUL of the mechanical system [14]. p ( z k | xk ) =
III.
PARTICLE FILTERING-BASED APPROACH
A.
Outline of the algorithm The flow chart of the PF-based approach proposed in this paper is shown in Fig. 1. It is seen that the approach is composed of two modules: indicator calculation module and PF-based prediction module. Feature extraction Bearing vibration signal
Multiple features
Correlation matrix clustering
Typical features
Weight algorithm Indicator calculation module
Parameters initialization
PF-based prediction module
Degradation indicator Least Preprocess square fitting State space model
Update algorithm Parameters updating
Prognostic algorithm RUL prediction
Figure 1. The overall flow chart of the PF-based approach
1. In the indicator calculation module, a degradation indicator is extracted for RUL prediction of rolling element bearings. Prediction approaches assume that the degradation indicator includes the failure information and can reflect the health state of a system. Some features have been used to mechanical RUL prediction such as RMS, kurtosis, crest factor, etc. Previous research work has shown that different features have various sensitivities to varying stages of the degradation
process. For example, spikiness of the vibration signals indicated by crest factor and kurtosis implies incipient defects, whereas the high energy level given by the value of RMS indicates severe defects [15]. A good degradation indicator is expected to include all fault information from different features and keep consistent sensitivity to varying stages of the degradation process. In order to extract a good indicator, a clustering algorithm based on correlation matrix and a weight algorithm are combined in the indicator calculation module. The indicator calculation involves the following three steps. 1) Multiple features in time and time-frequency domain are extracted from the vibration signal. 2) A clustering algorithm based on correlation matrix is utilized to classify these features. 3) Typical features are selected from every cluster, and the degradation indicator is caculated by applying weight algorithm. 2. The PF-based prediction module is designed to develop a state space model to predict the evolution trend and the RUL of bearings. In PF-based approach, the parameters initialization and updating of the state space model are very important, because appropriate parameters can promise a reliable prediction result. But in traditional PF-based approach, initial parameters of the state space model are assumed to known, and the process of parameters initialization is always ignored. In the PF-based prediction module of the approach, an algorithm of parameter initialization is proposed to deal with this problem. After the parameters of the model are initialized, they are fine adjusted with the update algorithm. Then the state space model is utilized to predict the RUL of the bearing by using the prognostic algorithm. The step of this module is as follows. 1) The degradation indicator acquired from the first module is preprocessed in order to reduce the noise of the data and get the desired form. 2) The preprocessed indicator is put into the state space model constructed in advance. The initial parameters of the model are calculated by applying least square fitting methods. 3) Parameters are then adjusted with the help of update algorithm. This process continues until all of the measured data have been used. 4) State space model is used to predict the degradation process and the RUL of the bearing based on the prognostic algorithm. B.
Indicator caculation module In order to fully mine the fault information hidden in vibration signals, two trigonometric function features, nineteen time domain and sixteen time-frequency domain features are extracted from vibration signals. Two new trigonometric function features, Std. of inverse hyperbolic cosine and Std. of inverse hyperbolic sine, are used in this approach, as they have been proven to have a better monotonicity and tendency than traditional features [16]. The formulas of these two new features are given in (14) and (15). More details about the nineteen time domain features are shown in Tab. 1. Wavelet packet decomposition (WPD) has been widely used in feature extraction of bearing condition monitoring [17]. In our approach, a third layer WPD of the signal is utilized, and eight
node energy (NE) and their ratios to ensemble energy (ER) are extracted. Std. of inverse hyperbolic cosine:
(
)
1 ⎤⎞ ⎛ ⎡ σ ⎜⎜ log ⎢ xi + xi2 − 1 2 ⎥ ⎟⎟ ⎦⎠ ⎝ ⎣
(14)
M features
Calculate the correlation matrix, and initialize the clustering number K.
Std. of inverse hyperbolic sine: ⎛
⎡
⎝
⎣
(
)
σ ⎜⎜ log ⎢ xi + xi2 + 1 TABLE I. Dimensional features
P1: Mean P2: Std P3: Variance P4: Skewness P5: Kurtosis P6: Maximun P7: Minmun P8: P-P P9: Squre-mean-root P10: Mean-abslute P11: RMS P12: Peak value
1
⎞ ⎥⎟ ⎦⎠
2 ⎤⎟
(15) 1
Lowest correlation coefficient
P13: Shape factor
Select K cluster centers
P15: Impulse factor P17: Skewness factor P18: Kurtosis value P19: Entropy
Among these extracted features, some of them may have the similar characteristics, which means that the failure information contained by them is redundant. These features need to be clarified into different clusters in terms of their characteristics. So, a new clustering algorithm based on correlation matrix is established to deal with this problem. The principle of the algorithm is that the features in the same clusters have higher correlation while the features in different clusters have lower correlation. Fig. 2 gives the flow chart of the clustering algorithm. It includes the following four procedural steps. 1) Input M featuers and caculate the correlation matrix of these features. Initialize the clustering number K. 2) Select K cluster centers. a) Select the two features which have the lowest correlation coefficient as the first and second center. b) Select ith center whose mean correlation coefficient with selected centers is lowest ( i = 3,..., K ).
Go back to b) until i ≥ K . 3) Classify the remainder M − K features into the K clusters. a) Classify jth feature into the ith cluster which has the highest mean correlation coefficient with this feature ( j = 1,2,..., M − K ; i = 1,2,..., K ). c)
b) Go back to a) until j ≥ M − K .
4) Output the clustering result.
i-1
i
P14: Crest factor
No
i≥K
P16: Clearance factor
…
Lowest mean correlation coefficient
TIME DOMINE FEATURES Non-dimensional features
2
i=i+1
Yes Classify remainder features
1
j
Highest mean correlation coefficient
…
2
j≥M-K
K
No
j=j+1
Yes Output the clustering result Figure 2. Flow chart of the correlation matrix clustering
After the clustering process, all of the features have been classified into K clusters. The next task is to select the typical features from every cluster. In this step, a trendability index is employed to evaluate the fitness of every feature. Trendability can be expressed as a function of a feature and time. When the feature increases or decreases in linearity with time, the trendability value will be ± 1 . Conversely, if the feature is constant or vary randomly with time, the value of this index will be zero. Higher absolute value of this index means that the feature has a better tendency, which is helpful for condition monitoring and RUL prediction. The straight forward formula is given in (16) [16]. N
N
N
t =1
t =1
t =1
N ∑ (t ⋅ Ft ) − ∑ t ⋅ ∑ Ft T =
(16)
2 2 ⎡ N ⎛N ⎞ ⎤ ⎛ N ⎞ ⎤⎡ N ⎢ N ∑ t 2 − ⎜ ∑ t ⎟ ⎥ ⎢ N ∑ Ft 2 − ⎜ ∑ Ft ⎟ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢ t =1 ⎝ t =1 ⎠ ⎥⎦ ⎝ t =1 ⎠ ⎥⎦ ⎢⎣ t =1 ⎣
where T is the trendability index value, N is the length of original data, and Ft is the feature of tth sample. Correlation can be positive, negative, perfect or no correlation. Thus the value of T can be from - 1 ≤ T ≤ +1 .
One feature whose absolute value of trendability is the biggest is selected from every cluster. The K features constitute the final typical set. We assume that the rolling element bearing is normal at the beginning of the process, and the first p points of every feature in the typical set are considered as the normal space. In order to assess the severity of degradation, the Manhattan distance between the current space and the normal space is calculated and weighted. A bigger weight value is applied to the feature which has a higher trendability, while a smaller weight value is applied to the feature which has a lower trendability. The purpose of conducting the weight algorithm is to fuse the typical set into a final degradation indicator and keep the indicator having a higher trendability. Based on the process mentioned above, the indicator algorithm can be given as follows.
to update the weights of particles according to the new measured data. After that, the parameters of the model can be adjusted to describe the evolution of the indicator more precisely. This process continues until all of the measured data have been used. Then the prediction algorithm is employed to predict the RUL of the rolling element bearing. Degradation indicator
Smooth the indicator by applying “LOESS” filter [15]
Resample the indicator according to expected time interval
1) Caculate the trendability of every features. 2) Select one feature from every cluster, which has the biggest trendability in its own cluster. 3) Form the typical set with the selected K features. 4) Normalize the typical set with extremums of learning data. 5) Initialize the normal space dimension p, and caculate the mean vector of the space V with the following equation. p
Vi =
1 ∑ Fi,t (i = 1,..., K ) p t =1
Recombine the measured data xk-1= measure data (1: end-1) xk = measure data (2: end)
xk = xk−1 + A⋅ xkm−1
(17) Calculate the initial parameters A and m by utilizing least square fitting
6) Normalize the trendability of features in the typical set. T (18) Wi = K i (i = 1,..., K ) ∑ Ti
Output the initialized parameters
i =1
7) Caculate the Manhattan distance between F and V of every feature,and weight the distance with the normalized trendability. K
(
Dt = ∑ Wi ⋅ Fi, t − Vi i =1
)
(t = p + 1,..., N )
Figure 3. Flow chart of the paremeters initializetion
IV.
(19)
C.
Particle filtering-based prediction module In this PF-based approach, Paris-Erdogan model is adopted to describe the degradation of the rolling element bearing, and the state space model can be described as (11) and (12). Then, in order to acquire the parameters which can describe the degradation process of the indicator and predict the RUL of the bearing, two steps are necessary in the approach: parameter initialization and parameter updating.
1) Parameter initialization: The details of this step are shown in Fig.3. In this step, the degradation indicator is smoothed by applying “LOESS” filter to reduce the noise. The smoothed indicator is then resampled according to expected time interval. Then the model function of (11) is fitted with the help of least square fitting methods, and the initial parameters A and m can be calculated. During this step, the parameters of the model are roughly selected. 2) Parameter updating: If we want to get more accurate parameters, they must be fine adjusted according to the realtime observed data. In this step, Bayesian approach is utilized
EXPERIMENTAL DEMONSTRATION
A.
Experiment introduction Experimental data which comes from IEEE PHM 2012 prognostic challenge is employed to test the proposed approach [18]. The experimentation platform called PRONOSTIA is shown in Fig. 4. This platform is designed to test and validate the bearing fault detection, diagnostic and prognostic approaches. PRONOSTIA includes three parts: a rotating part, a degradation generation part and a measurement part. The fault of the bearing in this platform is not artificially processed but normally occurred. Meanwhile, this platform can conduct the bearing degradation in a few hours, so it is suitable to operate the bearing accelerated life test [19].
Three different operation conditions are involved in the challenge data, and the first condition is: 1800 rpm and 4000 N. There are two learning bearings and five test bearings in this condition. In our experiment, the first learning bearing and the first test bearing is considered. The sampling frequency is 25.6 kHz, the length of the data is 2560 (i.e. 1/10 s), and recording is repeated every ten seconds. Two vibration signals (horizontal and vertical) are recorded at the same time. RUL is defined as time when accelerometer exceeds 20 m/s2 [20].
Pressure regulator
Cylinder Pressure Force sensor
Test bearing
Accelerometers
AC Motor
Speed sensor
Speed reducer
Torquemeter
Coupling
Thermocouple
Figure 4. Overview of PRONOSTIA
(a)
50
-50
0
7175680
0
7175680
(b)
50 0 -50
(c)
50
-50
0
6080000
(d)
50 0 -50
0
6080000
In Fig. 7 (b), the degradation indicator of the test bearing involves three stages obviously, normal stage, failure development and severe failure. In normal stage, it fluctuates within a narrow range which means that the bearing is still normal during this time. In failure development, the curve increases gradually, and this stage continues for a long time during the whole process. This is because the failure has a very long development experience until the final failure. In severe failure stage, an abrupt fluctuation occurs, and the bearing is destroyed in a short time. Comparing Fig. 7(b) with Fig. 6, we can see that the degradation indicator can reflect more information than any one of the original features, and the degradation progress of the bearing can be reflected obviously by the fused indicator. 30
30
P-P of H
NI DAQ card
insensitive to the occurring and development of failure, but sensitive to the severe failure at the end of the lifetime. The differences among them are that, P-P of H has some fluctuation noise in the early-stage and 1st ER of V has fluctuation noise at the late-stage. 1st ER of H increases gradually in the whole lifetime, which means that the energy ratio of 1st frequency band in H becomes higher and higher. But it becomes smooth when the failure develops into severe stage. Mean of V keeps its small fluctuation until the last time when the amplitude of the fluctuation increases abruptly. While, Mean of H keeps the small fluctuation until the end. 8th ER of V has an obvious decrease trend at the late-stage, and 3rd ER of H keeps the decrease trend from the beginning, but stops at the latter part of the time. From the analysis we can see that, some features are sensitive to the severe stage of the failure, some can reflect the information of the incipient failure, and some even reflect nothing about the state. None of the features can reflect all information about the degradation process.
Skewness of H
Fig. 5 shows the vibration signals of these two bearings, where (a) and (b) display the horizontal (H) direction and vertical (V) direction of the learning bearing, respectively, and (c) and (d) give H and V of the test bearing, respectively.
20 10 0
0
1000
From Fig. 6 we can see that, skewness of H, P-P of H and 1 ER of V have similar performance. All of them are st
1st ER of H
1st ER of V
1000
2000
0
1000
2000
0 0 0.2
Mean of H
2
1000
2000
0
1000
2000
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2000
0
-0.2
-2
0 0.04
1000
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100
0.02
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0.5
3rd ER of H
37 original features are extracted from H and V, respectively. The clustering number is set to be 8, and 74 features are divided into 8 clusters. One feature having the biggest trendability in its own cluster is selected. The selected eight features form the typical set, which is shown in Fig.6. Set the normal space dimension p as 100 and the final degradation indicators of these two bearings are shown in Fig. 7.
0.5
0
Mean of V
Degradation indicator caculation The algorithm approach mentioned in Section 3 B is used to process the experimental data. From Fig. 5 we can see that the maximum amplitude of the vibration signals has exceeded 50 m/s2. To be consistent with [20], the bearings data is truncated at the time when the vibration amplitude exceeds 20 m/s2. So, the lifetimes of these two bearings are 27600s and 17790s, respectively.
0
1
8th ER of V
B.
10 0
2000
1
Figure 5. Vibration signals of bearings
20
0 0
1000
Time (10s)
2000
0
Time (10s)
Figure 6. Typical set of the test bearing
0.6 0.4 0.2 0
0
1000 2000 Sever 3000 Time (10s) failure
0.8
Failure development 0.6
Normal stage
0.4 0.2 0
0
500
1000 1500 Time (10s)
2000
Figure 7. Degradation indicator of the bearings
C.
RUL prediction By applying the initialization algorithm in Section 3 C, the parameters of the state space model are initialized as A = 0.0100 and m = 0.9282 . Then the parameters are updated according to the degradation indicator from 0 to 170min. The parameters updating process of the test bearing is shown in Fig. 8. The blue lines are the medians of the parameters and the red lines are the 95% confidence interval (CI) of the parameters. After the updating process, the medians of the parameter are changed to be A = 0.0187 and m = 1.2710 . Median
95% CI
0.04
A
0.02 0 -0.02
0
50
100
150
m
2.4 1.2 0
result we can see that, both methods can predict the degradation trend and acquire the failure time based on a threshold value. However, the curve predicted by PF-based approach is smoother than that of ANFIS-based approach. This is because the curve of PF-based approach is determined by the model function, while, the curve of ANFIS-based approach is determined by the inner structure of the inference system. The PF-based approach can reflect the overall trend of the indicator, while the ANFIS-based approach can reflect the local fluctuation of the curve. The RUL of the bearing is determined by the overall trend not the local fluctuation. So, for RUL prediction of the bearing, PF-based approach is more useful than ANFIS-base approach. Moreover, the PF-based approach can provide a probability distribution of RUL, which is more practical than a certain number for real applications. For the convenience of comparison, the predict median of PF is considered as prediction result of RUL. The lifetime of the test bearing is 17790s. After resampling according to the time interval Δt = 1 min , the real lifetime becomes 296min. At t = 170 min the real RUL of the bearing is 126min. The prediction result of PF-based approach is 116min, and the relative error is -8.62%. The prediction result of ANFIS-based approach is 96min, and the relative error is -23.81%. The PFbased approach has a less prediction error than the ANFISbased approach. Through the comparison, it is believed that the PF-based approach presented in this paper is more effective in RUL prediction than ANFIS-based approach for rolling element bearings. Training data Test data Fitting median of PF Fitting 95% CI of PF Predict median of PF Predict 95% CI of PF Predict result of ANFIS 0.8
Degradation indicator
(a) Degradation indicator of the learning bearing (b) Degradation indicator of the test bearing
0.8
0.6 0.4 0.2 0
0
50
100
0
80
160
240
320
Time (min)
150
Time (min) Figure 9. Prediction result of the test bearing Figure 8. Parameters updating process of the test bearing
A threshold value of the degradation indicator is essential for RUL prediction of the bearing. In this approach, the threshold value of the test bearing is set in terms of failure amplitude of the learning bearing. As shown in Fig. 7(a), the failure amplitude of the learning bearing is 0.6, so the threshold is selected to be 0.6 based on the observation of indicator. For comparison, a commonly used data-driven method based on ANFIS [21] is also used to predict the RUL of the bearing. The prediction result of the test bearing is shown in Fig. 9. From the
V.
CONCLUSIONS
This paper develops a PF-based approach for the RUL prediction of rolling element bearings. This approach is composed of two modules: indicator calculation module and PF-based prediction module. In the degradation indicator calculation, multiple features are extracted from original data, and a clustering algorithm based on correlation matrix is utilized to cluster the features. Then a trendability index is used
to evaluate and select the typical features. After that, the degradation indicator is calculated from the typical set using a weight algorithm. In PF-based prediction module, an adaptive algorithm of parameter initialization is proposed to calculate the initial parameters of the state space model. After parameter initialization step, the parameters are updated with the help of Bayesian theory. Then, the state space model is employed to predict the degradation process and RUL of bearings. Experimental data is employed to testify the approach. Through comparison with ANFIS-based approach, the result reveals that the approach proposed in this paper is more effective in RUL prediction of rolling element bearings. ACKNOWLEDGMENT The authors wish to thank FEMTO-ST Institute and the organizing committee of IEEE PHM 2012 prognostic challenge for sharing the degradation data of rolling element bearings. REFERENCES [1]
[2]
[3]
[4]
[5]
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