# DCA-Based Real-Time Residual Useful Life Prediction for Critical Faulty Component

Authors: Funa Zhou; Jiayu Wang; Yulin Gao
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

Residual useful life (RUL) prediction is significant for condition-based maintenance. Traditional data-driven RUL prediction method can only predict fault trend of the system rather than RUL of a specific system component. Thus it cannot tell the operator which component should be maintained. The innovation of this paper is as follows: (1) Wavelet filtering based method is developed for early detection of slowly varying fault. (2) Designated component analysis is introduced as a feature extraction tool to define the fault precursor of a specific component. (3) Exponential life prediction model is established by nonlinear fitting of the historical RUL and the fault size characterized by the statistics used. Once online detection statistics is obtained, real-time RUL of the critical component can be predicted online. Simulation shows the effectiveness of this algorithm.


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## Body

## 1. Introduction

With the rapid development of modern industrial technology, reliability, maintainability, and security of large-scale system have widely received attention [1–4]. Abnormal detection, root cause identification, and RUL prediction are the stages for efficient system monitoring.Abnormality is expected to be detected as quickly as possible to prevent major accidents and reduce loss of downtime and maintenance. In the past decades, multivariate statistical analysis techniques such as principal component analysis (PCA) and partial least squares (PLS) [5–7] have been widely used in complex system monitoring [8]. However, abnormal detection and fault diagnosis usually answer the question of “having trouble,” not “how long it will happen.”Detection and maintenance after system failure are usually too late for system with higher security requirement and expensive downtime loss, such as power station, power transmission, petrochemical industry, large-scale iron, steel enterprise, and other industry fields [9].If condition-based maintenance rather than breakdown maintenance is used, severe faults leading to shutdown can be avoided, and a lot of maintenance cost can be saved [10]. Fault prediction technology is a critical step of condition-based maintenance. In recent years, fault prediction has become a hot topic in the field of system monitoring [11]. Artificial neural network (ANN) [12–17], autoregressive model (AR) [18, 19], support vector machine (SVM) [20], vector autoregressive model (VAR), and so forth [21] are commonly used fault prediction methods. But these pure data-driven methods without reference to expert experience can only predict fault trend rather than directly predict RUL online.An important stage of RUL prediction strategy is how to properly assess fault size that reflects the degradation process. If a damage precursor is available, damage precursor based method for RUL prediction is an objective choice [22]. So fault damage precursor as well as the control limits should be determined before online RUL prediction. Existing method for establishing the fault damage precursor can be categorized into 2 classes: data-driven method and model-based method. Accurate fault propagation model to establish fault damage precursor is usually unavailable. However, thanks to the rapid development of sensor technology and condition monitoring technology, large amount of observation data reflecting status of the system can be used to establish damage precursor by using data-driven method [9, 13]. Therefore, data-based method is gradually attracting the attention of researchers. Li et al. use historical normal data to establish PCA model, and 1-norm of the residual vector is used to define the fault damage precursor [23, 24]. RUL can be estimated by multistep recursive prediction using AR model with forgetting factor [25, 26]. Ma et al. [27] use square prediction error based fault reconstruction method to estimate fault size in the first step. Then RUL can be estimated by multilevel recursive prediction. Li et al. [24] propose a fault prediction method for industry process with performance degradation by using multiscale PCA. Fault size computed by fault reconstruction can be used to establish a prediction model based on exponential smoothing technique. These methods share the same deficiency that fault direction is computed by PCA related method [27]. But pattern compounding problem of PCA makes it impossible to establish 1-1 correspondence between the failure component and the reconstructed fault direction [28]. Thus PCA based RUL prediction cannot tell the operator which critical component should be maintained.Designated component analysis (DCA) is a knowledge-guided data-driven feature extraction method which can be used to diagnose root cause of the abnormality [29]. In this paper, DCA is introduced as a feature extraction tool for establishing fault damage precursor and life prediction model. Since small fault is usually buried in noise, pretreatment of filtering is developed for early detection of slowly varying small fault.Remark 1.
It is assumed that failure of critical component defined by expert has disastrous impact on the system. So RUL prediction of critical component is much significant in the sense that unnecessary surplus maintenance can be reduced to save maintenance cost.The remainder of this paper is organized as follows: Section2 describes principal component analysis and designated component analysis. DCA-based real-time prediction for critical component is developed in Section 3. Simulation analysis in Section 4 shows the efficiency of the method proposed. Conclusions and further research are given in the last section of the paper.


## 2. Review of PCA and DCA

### 2.1. PCA Modeling

AssumeY0∈Rn×m is the observation data matrix collected in the normal operation conditions, where m is the number of observation variables and n is the number of samples. Firstly, Y0 is standardized via (1)Y0¯=Y0-11⋯1TMdiag⁡1s1,1s2,…,1sm,where M=Y01Y02⋯Y0m is the mean of Y0,Y0m is the variable of Y0,s=s1s2⋯sm is the standard deviation of Y0.Y
0
¯ can be decomposed into two parts as follows [11]:(2)Y0¯=∑i=1vtipiT+E,where ti∈Rn×1 is the score vector, pi∈Rm×1 is the loading vector, vv<m is the number of significant principal components, and E=∑i=v+1mtipiT is the residual error.After establishing PCA modeling under normal operating conditions, multivariate statistics called squared prediction error (SPE), can be used for fault detection and diagnosis. SPE is described as follows:(3)SPEi=Ei2=YiI-PPTYiT,where Yii=1,2,…,n represents a sample of online observation data, and P is a matrix composed of the first v loading vectors.SPE control limit can be determined via(4)Qα=θ1Cα2θ2h02θ1+1+θ2h0h0-1θ121/h0,(5)θt=∑j=k+1mλjtt=1,2,3,(6)h0=1-2θ1θ33θ22,where λj is the eigenvalues of the covariance matrices and Cα is the threshold of hypothesis testing to normal distribution with confidence level α.The system is considered to be abnormal if(7)SPE≥Qα.


### 2.2. DCA

DCA is a knowledge-guided multivariate statistical feature extraction method [29]. The basic idea is to define a designated pattern via the fault-symptom relation.First, different from the loading vectorpi∈Rm×1 of PCA, the definition of designated pattern Di is defined by a knowledge-guided means via (8)Di=di1di2⋮dim,i=1,2,…,L,where dirr=1,2,…,m can be taken as 0 or 1 according to the fault-symptom relation, and dir=1 means that the symptom r is shown in the ith designated pattern; otherwise dir=0.Table1 lists the typical fault-symptom relation of an air compressor. There are 17 typical faults and 19 observation parameters included in the table. Insufficient supply of lubricating oil or oil blockage, lubricant contamination, motor drive failure, and so on are the typical faults listed in Table 1. First-stage exhaust temperature, intercooler exhaust temperature, two-stage exhaust temperature, oil cooling outlet temperature and motor current, and so forth are the observed parameters [29].Table 1
Typical fault-symptom relation of air compressor.
u
1
u
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u
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u
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u
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u
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u
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u
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u
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u
19
u
20
D
1
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
D
2
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
D
3
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
D
4
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
D
5
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
D
6
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
D
7
0
0
0
1
1
1
0
1
1
1
0
0
0
0
0
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0
0
D
8
0
0
0
0
1
1
0
0
0
1
0
0
1
0
0
0
0
0
0
0
D
9
0
0
0
1
1
1
0
1
1
1
1
1
0
0
0
0
0
0
0
0
D
10
0
0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
0
D
11
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
1
0
0
0
0
D
12
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
D
13
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
D
14
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
D
15
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
D
16
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
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17
0
0
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0
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0
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0
0
0
0
0
0
1From Table1, the fifth row corresponds to the fault of insufficient supply of lubricating oil, the sixth row corresponds to the fault of lubricant contamination, and these fault patterns can be defined as follows [29]: (9)D5=00000010000001000000T,D6=11100010000001000000T.Then, project the observation variabley to designated patterns Di to obtain the designated component wi:(10)wi=DiTy.Similarly to (2), the observation matrix Y0 can be decomposed as (11)Y0=∑i=1LDiWi+E,where L is the number of designated patterns, Wi=wi1,wi2,…,win∈R1×n, and E is the residual matrix.Finally, Shewhart chart of each designated component is used to implement fault diagnosis.


## 2.1. PCA Modeling

AssumeY0∈Rn×m is the observation data matrix collected in the normal operation conditions, where m is the number of observation variables and n is the number of samples. Firstly, Y0 is standardized via (1)Y0¯=Y0-11⋯1TMdiag⁡1s1,1s2,…,1sm,where M=Y01Y02⋯Y0m is the mean of Y0,Y0m is the variable of Y0,s=s1s2⋯sm is the standard deviation of Y0.Y
0
¯ can be decomposed into two parts as follows [11]:(2)Y0¯=∑i=1vtipiT+E,where ti∈Rn×1 is the score vector, pi∈Rm×1 is the loading vector, vv<m is the number of significant principal components, and E=∑i=v+1mtipiT is the residual error.After establishing PCA modeling under normal operating conditions, multivariate statistics called squared prediction error (SPE), can be used for fault detection and diagnosis. SPE is described as follows:(3)SPEi=Ei2=YiI-PPTYiT,where Yii=1,2,…,n represents a sample of online observation data, and P is a matrix composed of the first v loading vectors.SPE control limit can be determined via(4)Qα=θ1Cα2θ2h02θ1+1+θ2h0h0-1θ121/h0,(5)θt=∑j=k+1mλjtt=1,2,3,(6)h0=1-2θ1θ33θ22,where λj is the eigenvalues of the covariance matrices and Cα is the threshold of hypothesis testing to normal distribution with confidence level α.The system is considered to be abnormal if(7)SPE≥Qα.


## 2.2. DCA

DCA is a knowledge-guided multivariate statistical feature extraction method [29]. The basic idea is to define a designated pattern via the fault-symptom relation.First, different from the loading vectorpi∈Rm×1 of PCA, the definition of designated pattern Di is defined by a knowledge-guided means via (8)Di=di1di2⋮dim,i=1,2,…,L,where dirr=1,2,…,m can be taken as 0 or 1 according to the fault-symptom relation, and dir=1 means that the symptom r is shown in the ith designated pattern; otherwise dir=0.Table1 lists the typical fault-symptom relation of an air compressor. There are 17 typical faults and 19 observation parameters included in the table. Insufficient supply of lubricating oil or oil blockage, lubricant contamination, motor drive failure, and so on are the typical faults listed in Table 1. First-stage exhaust temperature, intercooler exhaust temperature, two-stage exhaust temperature, oil cooling outlet temperature and motor current, and so forth are the observed parameters [29].Table 1
Typical fault-symptom relation of air compressor.
u
1
u
2
u
3
u
4
u
5
u
6
u
7
u
8
u
9
u
10
u
11
u
12
u
13
u
14
u
15
u
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u
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u
18
u
19
u
20
D
1
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
D
2
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
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3
0
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1
0
0
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1
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4
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9
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16
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17
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1From Table1, the fifth row corresponds to the fault of insufficient supply of lubricating oil, the sixth row corresponds to the fault of lubricant contamination, and these fault patterns can be defined as follows [29]: (9)D5=00000010000001000000T,D6=11100010000001000000T.Then, project the observation variabley to designated patterns Di to obtain the designated component wi:(10)wi=DiTy.Similarly to (2), the observation matrix Y0 can be decomposed as (11)Y0=∑i=1LDiWi+E,where L is the number of designated patterns, Wi=wi1,wi2,…,win∈R1×n, and E is the residual matrix.Finally, Shewhart chart of each designated component is used to implement fault diagnosis.


## 3. DCA-Based Real-Time Prediction for Critical Component

DCA-based RUL prediction method needs to determine critical component by expert experience. In this paper, the component corresponding to designated patternD1 is assumed to be a critical component.


### 3.1. Fault Damage Precursor Based on Historical Observation

#### 3.1.1. Knowledge-Guided Data-Driven Feature Extraction

Project historical normal observation dataY0 to the designated pattern Di to obtain the designated component vector W0i: (12)W0i=DiTY0,i=1,2,…,L.The designated component W0i is the feature extracted from Y0. In general, statistical distribution of W0i is normal distribution as long as the observation of each observation variable is normal distribution.


#### 3.1.2. Data-Driven Fault Damage Precursor

In the case when there is no fault damage precursor determined by expert or by physical fault propagation model, it is necessary to establish a data-driven fault damage precursor for describing the fault evolution process. For this purpose, the failure control limitq to determine the failure time and the fault trend control limit l1 for early detection of slowly varying fault should first be determined. Specific steps are as follows.Step 1 (failure control limit and fault trend control limit).
As it is analyzed in Section3.1.1, W0i has the normal distribution. By 3σ criteria of hypothesis testing to normal distribution, the critical designated component W01 falls into interval -3σ,3σ with confidence level 99.7% [30, 31]:(13)P-3σ≤W01≤3σ=99.7%.So the failure control limit q can be determined via (14)q=3σ,where σ is the standard deviation of W01.For the sake of determining fault trend control limitl1, wavelet filtering is firstly introduced as a preprocessing tool to historical normal observation data, and then DCA is used to perform feature extraction to the filtered normal observation. The filtering process can be described as follows [32, 33].First, discrete wavelet transform (DWT) for the observation vector of theith observation variable Y0i can be depicted via (15)ajk=∑NhN-2kY0jik,djk=∑NgN-2kY0jik,where k=0,1,2,…,n-1; n is the number of discrete sampling points; aj(k) is the scaling coefficient for signal on the scale j; dj(k) is the wavelet coefficient for signal on the scale j; h is a low-pass filter, g is a high-pass filter, and N is the length of the filter.After DWT, the next step is filtering. How to select the threshold is the key problem because it will directly affect the result of wavelet filter. In this paper, we choose the soft threshold method shown in(16)d^jk=signdjkdjk-T,djk≥T0,djk<T,where T is the filtering threshold.Finally, inverse discrete wavelet transform (IDWT) is implemented to obtain the filtered observation:(17)Y~0i=∑kajkhN-2k+∑kd^jkgN-2k.Based on the filtered normal observation, fault trend control limitl1 can be determined via (19):(18)W0iF=DiTY~0,i=1,2,…,L,(19)l1=3σ1,where σ1 is the standard deviation of W01F.Step 2 (fault trend detection point and the failure point).
Project historical faulty observation dataY1 to the designated pattern D1 to get the designated component W1:(20)W1=D1TY1.The extracted feature W1 can characterize the fault evolution process of the critical component. The failure point tf can be calculated as follows: (21)tf=min⁡k:W1k≥q.
Similarly, the filtered designated component can be computed via (22).(22)W1F=D1TY1F.Once W1F and fault trend control limit l1 have been obtained, the definition of fault trend detection point ts can be expressed as (23)ts=argmink⁡k:W1Fk≥l1.Step 3 (fault damage precursor).
Smoothing technique descripted in (24) is required to get a more smooth fault damage precursor:(24)W1F¯1=W1F1W1F¯2=W1F1+W1F2+W1F33W1F¯3=W1F1+W1F2+W1F3+W1F4+W1F55⋮W1F¯k=W1Fk-c+1+W1Fk-c+2⋯+W1Fkc⋮,where c is the size of the smoothing window; W1Fii=1,2,…,n is the element of vector W1F.
The fault evolution process in the period fromts to tf can be defined as fault damage precursor, denoted by C:(25)Ct-ts+1=W1F¯t,t=ts,ts+1,…,tf.


#### 3.1.3. The RUL Prediction Model

The historical RUL is defined by the deviation betweent and tf:(26)RULt-ts+1=tf-t,t=ts,ts+1,…,tf,where t is the current sampling time.Once the fault precursor defined in (26) is established, the algorithm of establishing RUL prediction model can be divided into three steps.Step 1.
Computerize the deviation between fault damage precursor and the control limit denoted byR:(27)Rt-ts+1=Ct-l1.Step 2.
Once the deviationR and the historical RUL have been obtained, historical data can be determined as follows:(28)J=R,RUL.Step 3.
Exponential life prediction model is established by nonlinear fitting of the historical data. Fitting equation is shown as follows:(29)RUL=c+a×ebR.


### 3.2. Online Fault Diagnosis

Firstly, online observation data after pretreatment of wavelet filtering is denoted byYz. Project Yz to the designated patterns Di to obtain the designated component vector Wiz:(30)Wiz=DiTYz,i=1,2,…,L.The smoothed designated component is denoted byWiz¯. Shewhart charts based method is used in this paper to implement fault diagnosis.


### 3.3. Online Life Prediction

A key step of online RUL prediction for the critical component is to judge whetherRk is positive or negative. If Rk is positive, it means that the system is abnormal. Real-time RUL prediction results can be obtained via exponential life prediction model:(31)Rk=W1z¯k-l1>0,RUL^k=c+a×ebRk,k=ts,ts+1,…,tf.


## 3.1. Fault Damage Precursor Based on Historical Observation

### 3.1.1. Knowledge-Guided Data-Driven Feature Extraction

Project historical normal observation dataY0 to the designated pattern Di to obtain the designated component vector W0i: (12)W0i=DiTY0,i=1,2,…,L.The designated component W0i is the feature extracted from Y0. In general, statistical distribution of W0i is normal distribution as long as the observation of each observation variable is normal distribution.


### 3.1.2. Data-Driven Fault Damage Precursor

In the case when there is no fault damage precursor determined by expert or by physical fault propagation model, it is necessary to establish a data-driven fault damage precursor for describing the fault evolution process. For this purpose, the failure control limitq to determine the failure time and the fault trend control limit l1 for early detection of slowly varying fault should first be determined. Specific steps are as follows.Step 1 (failure control limit and fault trend control limit).
As it is analyzed in Section3.1.1, W0i has the normal distribution. By 3σ criteria of hypothesis testing to normal distribution, the critical designated component W01 falls into interval -3σ,3σ with confidence level 99.7% [30, 31]:(13)P-3σ≤W01≤3σ=99.7%.So the failure control limit q can be determined via (14)q=3σ,where σ is the standard deviation of W01.For the sake of determining fault trend control limitl1, wavelet filtering is firstly introduced as a preprocessing tool to historical normal observation data, and then DCA is used to perform feature extraction to the filtered normal observation. The filtering process can be described as follows [32, 33].First, discrete wavelet transform (DWT) for the observation vector of theith observation variable Y0i can be depicted via (15)ajk=∑NhN-2kY0jik,djk=∑NgN-2kY0jik,where k=0,1,2,…,n-1; n is the number of discrete sampling points; aj(k) is the scaling coefficient for signal on the scale j; dj(k) is the wavelet coefficient for signal on the scale j; h is a low-pass filter, g is a high-pass filter, and N is the length of the filter.After DWT, the next step is filtering. How to select the threshold is the key problem because it will directly affect the result of wavelet filter. In this paper, we choose the soft threshold method shown in(16)d^jk=signdjkdjk-T,djk≥T0,djk<T,where T is the filtering threshold.Finally, inverse discrete wavelet transform (IDWT) is implemented to obtain the filtered observation:(17)Y~0i=∑kajkhN-2k+∑kd^jkgN-2k.Based on the filtered normal observation, fault trend control limitl1 can be determined via (19):(18)W0iF=DiTY~0,i=1,2,…,L,(19)l1=3σ1,where σ1 is the standard deviation of W01F.Step 2 (fault trend detection point and the failure point).
Project historical faulty observation dataY1 to the designated pattern D1 to get the designated component W1:(20)W1=D1TY1.The extracted feature W1 can characterize the fault evolution process of the critical component. The failure point tf can be calculated as follows: (21)tf=min⁡k:W1k≥q.
Similarly, the filtered designated component can be computed via (22).(22)W1F=D1TY1F.Once W1F and fault trend control limit l1 have been obtained, the definition of fault trend detection point ts can be expressed as (23)ts=argmink⁡k:W1Fk≥l1.Step 3 (fault damage precursor).
Smoothing technique descripted in (24) is required to get a more smooth fault damage precursor:(24)W1F¯1=W1F1W1F¯2=W1F1+W1F2+W1F33W1F¯3=W1F1+W1F2+W1F3+W1F4+W1F55⋮W1F¯k=W1Fk-c+1+W1Fk-c+2⋯+W1Fkc⋮,where c is the size of the smoothing window; W1Fii=1,2,…,n is the element of vector W1F.
The fault evolution process in the period fromts to tf can be defined as fault damage precursor, denoted by C:(25)Ct-ts+1=W1F¯t,t=ts,ts+1,…,tf.


### 3.1.3. The RUL Prediction Model

The historical RUL is defined by the deviation betweent and tf:(26)RULt-ts+1=tf-t,t=ts,ts+1,…,tf,where t is the current sampling time.Once the fault precursor defined in (26) is established, the algorithm of establishing RUL prediction model can be divided into three steps.Step 1.
Computerize the deviation between fault damage precursor and the control limit denoted byR:(27)Rt-ts+1=Ct-l1.Step 2.
Once the deviationR and the historical RUL have been obtained, historical data can be determined as follows:(28)J=R,RUL.Step 3.
Exponential life prediction model is established by nonlinear fitting of the historical data. Fitting equation is shown as follows:(29)RUL=c+a×ebR.


## 3.1.1. Knowledge-Guided Data-Driven Feature Extraction

Project historical normal observation dataY0 to the designated pattern Di to obtain the designated component vector W0i: (12)W0i=DiTY0,i=1,2,…,L.The designated component W0i is the feature extracted from Y0. In general, statistical distribution of W0i is normal distribution as long as the observation of each observation variable is normal distribution.


## 3.1.2. Data-Driven Fault Damage Precursor

In the case when there is no fault damage precursor determined by expert or by physical fault propagation model, it is necessary to establish a data-driven fault damage precursor for describing the fault evolution process. For this purpose, the failure control limitq to determine the failure time and the fault trend control limit l1 for early detection of slowly varying fault should first be determined. Specific steps are as follows.Step 1 (failure control limit and fault trend control limit).
As it is analyzed in Section3.1.1, W0i has the normal distribution. By 3σ criteria of hypothesis testing to normal distribution, the critical designated component W01 falls into interval -3σ,3σ with confidence level 99.7% [30, 31]:(13)P-3σ≤W01≤3σ=99.7%.So the failure control limit q can be determined via (14)q=3σ,where σ is the standard deviation of W01.For the sake of determining fault trend control limitl1, wavelet filtering is firstly introduced as a preprocessing tool to historical normal observation data, and then DCA is used to perform feature extraction to the filtered normal observation. The filtering process can be described as follows [32, 33].First, discrete wavelet transform (DWT) for the observation vector of theith observation variable Y0i can be depicted via (15)ajk=∑NhN-2kY0jik,djk=∑NgN-2kY0jik,where k=0,1,2,…,n-1; n is the number of discrete sampling points; aj(k) is the scaling coefficient for signal on the scale j; dj(k) is the wavelet coefficient for signal on the scale j; h is a low-pass filter, g is a high-pass filter, and N is the length of the filter.After DWT, the next step is filtering. How to select the threshold is the key problem because it will directly affect the result of wavelet filter. In this paper, we choose the soft threshold method shown in(16)d^jk=signdjkdjk-T,djk≥T0,djk<T,where T is the filtering threshold.Finally, inverse discrete wavelet transform (IDWT) is implemented to obtain the filtered observation:(17)Y~0i=∑kajkhN-2k+∑kd^jkgN-2k.Based on the filtered normal observation, fault trend control limitl1 can be determined via (19):(18)W0iF=DiTY~0,i=1,2,…,L,(19)l1=3σ1,where σ1 is the standard deviation of W01F.Step 2 (fault trend detection point and the failure point).
Project historical faulty observation dataY1 to the designated pattern D1 to get the designated component W1:(20)W1=D1TY1.The extracted feature W1 can characterize the fault evolution process of the critical component. The failure point tf can be calculated as follows: (21)tf=min⁡k:W1k≥q.
Similarly, the filtered designated component can be computed via (22).(22)W1F=D1TY1F.Once W1F and fault trend control limit l1 have been obtained, the definition of fault trend detection point ts can be expressed as (23)ts=argmink⁡k:W1Fk≥l1.Step 3 (fault damage precursor).
Smoothing technique descripted in (24) is required to get a more smooth fault damage precursor:(24)W1F¯1=W1F1W1F¯2=W1F1+W1F2+W1F33W1F¯3=W1F1+W1F2+W1F3+W1F4+W1F55⋮W1F¯k=W1Fk-c+1+W1Fk-c+2⋯+W1Fkc⋮,where c is the size of the smoothing window; W1Fii=1,2,…,n is the element of vector W1F.
The fault evolution process in the period fromts to tf can be defined as fault damage precursor, denoted by C:(25)Ct-ts+1=W1F¯t,t=ts,ts+1,…,tf.


## 3.1.3. The RUL Prediction Model

The historical RUL is defined by the deviation betweent and tf:(26)RULt-ts+1=tf-t,t=ts,ts+1,…,tf,where t is the current sampling time.Once the fault precursor defined in (26) is established, the algorithm of establishing RUL prediction model can be divided into three steps.Step 1.
Computerize the deviation between fault damage precursor and the control limit denoted byR:(27)Rt-ts+1=Ct-l1.Step 2.
Once the deviationR and the historical RUL have been obtained, historical data can be determined as follows:(28)J=R,RUL.Step 3.
Exponential life prediction model is established by nonlinear fitting of the historical data. Fitting equation is shown as follows:(29)RUL=c+a×ebR.


## 3.2. Online Fault Diagnosis

Firstly, online observation data after pretreatment of wavelet filtering is denoted byYz. Project Yz to the designated patterns Di to obtain the designated component vector Wiz:(30)Wiz=DiTYz,i=1,2,…,L.The smoothed designated component is denoted byWiz¯. Shewhart charts based method is used in this paper to implement fault diagnosis.


## 3.3. Online Life Prediction

A key step of online RUL prediction for the critical component is to judge whetherRk is positive or negative. If Rk is positive, it means that the system is abnormal. Real-time RUL prediction results can be obtained via exponential life prediction model:(31)Rk=W1z¯k-l1>0,RUL^k=c+a×ebRk,k=ts,ts+1,…,tf.


## 4. Simulation

### 4.1. Simulation Data

In this section,m=20, n=1000 are used for simulation. Normal observation data is the compound of ten coexisting change patterns:(32)Y0=∑i=1LDiWi¯,where L=10 is the number of designated patterns, D1,D7 represent fault patterns, and the rest are normal random disturbance patterns. Wi¯ is the sample vector of designated component for simulation. In MATLAB, Wi¯ can be defined by the function of “randn” and some linear operations. Assuming Wi¯ is normally distributed,(33)W1¯=randn⁡1,n,W2¯=0.5W1¯+0.8randn⁡1,n,W3¯=0.5W1¯+0.5W2¯,W4¯=0.5W2¯+0.1randn⁡1,n,W5¯=W3¯+0.2W4¯,W6¯=0.2W1¯+0.3W4¯,W7¯=0.3randn⁡1,n,W8¯=0.2W7¯+0.2randn⁡1,n,W9¯=0.1W8¯+0.2randn⁡1,n,W10¯=0.5randn⁡1,n.From 130th to 1000th sampling point, the influence of fault patternsD1 becomes greater(34)f1=g11-e-t1/τ1,where g1=6,t1∈0,13,τ1=8.From 300th to 1000th sampling point, the influence of fault patternsD7 becomes greater (35)f7=g71-e-t7/τ7,where g7=4,t7∈1,8,τ7=6.


### 4.2. Data Feature Extraction Based on DCA

Figure1 shows SPE chart of PCA. The control limit with confidence level α=0.003, which can be calculated by (4). From Figure 1, the abnormally can be detected from the 411th sample time. It can be concluded from Figure 1 that the detection time using PCA is not consistent with the failure point of any component.Figure 1
PCA based fault detection.Therefore, DCA is introduced to overcome the pattern compounding deficiency of PCA. DCA-based faults diagnosis results are depicted in Figure2. The dotted red line is the control limit. As shown in Figure 2, the Shewhart charts correspond to D1,D7 beyond the upper control limit from 451 and 501, respectively. The results indicate that DCA is an effective multiple faults diagnosis method for multiple faults diagnosis.Figure 2
DCA-based faults diagnosis.


### 4.3. Early Fault Feature Extraction Based on DCA

In this part, wavelet based denoising technique is used to extract early fault feature. The filtered SPE chart of PCA is shown in Figure3. The blue line denotes the filtered SPE (abbreviated as F-SPE in the following); the black point line denotes the filtered SPE after smoothing (abbreviated as F-S-SPE in the following). The fault trend can be detected from the 154th sample point.Figure 3
WF-PCA based early fault detection.Figure4 shows the faults results of DCA preprocessed by wavelet filter (WF-DCA). The blue line denotes the filtered designated components (the F-dcs); the black line denotes the filtered dcs after smoothing (the F-S-dcs). It can be seen from Figure 4 that the fault trend of the 1st dc can be detected at 161. The 7th dc is abnormal from 326.Figure 4
WF-DCA-based early faults diagnosis.These simulation results imply that the early fault detection can be achieved by wavelet filtering which can increase the signal-to-noise ratio (SNR) of the fault signal. This paper determines the fault trend detection point using wavelet filtering technique.


### 4.4. RUL Prediction Model

In this paper, the system component corresponding toD1 is assumed to be a critical component. In order to get more precise RUL prediction model, smoothing technique is used to postprocess the fault precursor. The smoothed evolution precursor between fault trend detection point and failure point can be defined as the damage precursor to establish RUL prediction model.For establishing RUL prediction model based on WF-PCA, the failure timetf=411 can be determined from Figure 1. The fault trend detection point ts=154 can be obtained from Figure 3. The F-S-SPE between 154 and 411 can be defined as the fault damage precursor. Use RUL=a×ebR as the fitting function to establish RUL prediction model, where R is the deviation between the value of F-S-SPE and the control limit. Figure 5 shows the fitted RUL prediction model based on WF-PCA.Figure 5
The fitted RUL prediction model based on PCA.Figure6 shows the fitted RUL prediction model of the critical component based on WF-DCA. The first F-S-dc between 161 and 451 can be defined as fault damage precursor, where tf and ts can be obtained from Figures 2 and 4, respectively.Figure 6
The fitted RUL prediction model of the critical component based on DCA.


### 4.5. Online Life Prediction and Fault Diagnosis Based on DCA

Once online data is obtained, we can predict the online RUL based on the aforementioned prediction model. Figure7 depicts the online fault detection results by online F-S-SPE. It can be seen from Figure 7 that the fault trend can be detected from the 152nd sample point.Figure 7
Online early fault detection based on WF-PCA.To further demonstrate the superiority of the proposed method, two commonly used prediction algorithms are employed for comparison, namely, AR and BPNN. For BPNN and AR, fault trend prediction and recursive RUL prediction are implemented in two separate stages which makes them unavailable to predicting the RUL online. The parameters of BPNN are shown in Table2. In this paper, the order of AR model is 1, and the regression parameter is φ. The recursive RUL prediction process can be formulated as follows:(36)W1z¯k+s=φW1z¯k+s-1≥q,k=ts,ts+1,…,tf,where s is the number of recursive steps and q is the failure control limit.Table 2
Parameters of BPNN in pretraining.
Parameters
BPNN
Layers
3
Neurons in each layer
[
1,10,1
]
Learning rate
0.0001
Epochs
1000Figure8 shows the online RUL prediction result of PCA. The blue line is the real RUL. The dotted red line is the predicted RUL by the method proposed in this paper. The dotted black line is the predicted RUL by AR. The dotted blue line is the predicted RUL by BPNN. As we can see from Figure 8, the prediction accuracy of the dotted red line is higher than that of the dotted black line and the dotted blue line. Extra recursive steps are required for both AR and BPNN based RUL prediction model. The method proposed in this paper can avoid this problem and reduce the prediction error.Figure 8
Online RUL prediction based on PCA.Figure9 depicts the online faults diagnosis results based on WF-DCA. From Figure 9, the fault trends corresponding to D1 and D7 can be detected from 165 and 324, respectively. Figure 10 shows the online prediction result of the critical component. The blue line is the real RUL. The dotted red line is the predicted RUL by nonlinear fitting method. The dotted black line is the predicted RUL by AR. The dotted blue line is the predicted RUL by BPNN. It can be seen from Figure 10 that fitted RUL prediction model is more efficient than other models. The mean of RUL prediction error is listed in Table 3. It can be concluded from Table 3 that the proposed method is a good choice for the RUL prediction of the critical component.Table 3
Mean of RUL prediction error.
Statistics
Fitting
AR
BPNN
SPE
10.536
15.969
14.450
dc1
9.973
24.156
20.035Figure 9
Online early fault diagnosis based on DCA.Figure 10
The online prediction result of the critical component.


## 4.1. Simulation Data

In this section,m=20, n=1000 are used for simulation. Normal observation data is the compound of ten coexisting change patterns:(32)Y0=∑i=1LDiWi¯,where L=10 is the number of designated patterns, D1,D7 represent fault patterns, and the rest are normal random disturbance patterns. Wi¯ is the sample vector of designated component for simulation. In MATLAB, Wi¯ can be defined by the function of “randn” and some linear operations. Assuming Wi¯ is normally distributed,(33)W1¯=randn⁡1,n,W2¯=0.5W1¯+0.8randn⁡1,n,W3¯=0.5W1¯+0.5W2¯,W4¯=0.5W2¯+0.1randn⁡1,n,W5¯=W3¯+0.2W4¯,W6¯=0.2W1¯+0.3W4¯,W7¯=0.3randn⁡1,n,W8¯=0.2W7¯+0.2randn⁡1,n,W9¯=0.1W8¯+0.2randn⁡1,n,W10¯=0.5randn⁡1,n.From 130th to 1000th sampling point, the influence of fault patternsD1 becomes greater(34)f1=g11-e-t1/τ1,where g1=6,t1∈0,13,τ1=8.From 300th to 1000th sampling point, the influence of fault patternsD7 becomes greater (35)f7=g71-e-t7/τ7,where g7=4,t7∈1,8,τ7=6.


## 4.2. Data Feature Extraction Based on DCA

Figure1 shows SPE chart of PCA. The control limit with confidence level α=0.003, which can be calculated by (4). From Figure 1, the abnormally can be detected from the 411th sample time. It can be concluded from Figure 1 that the detection time using PCA is not consistent with the failure point of any component.Figure 1
PCA based fault detection.Therefore, DCA is introduced to overcome the pattern compounding deficiency of PCA. DCA-based faults diagnosis results are depicted in Figure2. The dotted red line is the control limit. As shown in Figure 2, the Shewhart charts correspond to D1,D7 beyond the upper control limit from 451 and 501, respectively. The results indicate that DCA is an effective multiple faults diagnosis method for multiple faults diagnosis.Figure 2
DCA-based faults diagnosis.


## 4.3. Early Fault Feature Extraction Based on DCA

In this part, wavelet based denoising technique is used to extract early fault feature. The filtered SPE chart of PCA is shown in Figure3. The blue line denotes the filtered SPE (abbreviated as F-SPE in the following); the black point line denotes the filtered SPE after smoothing (abbreviated as F-S-SPE in the following). The fault trend can be detected from the 154th sample point.Figure 3
WF-PCA based early fault detection.Figure4 shows the faults results of DCA preprocessed by wavelet filter (WF-DCA). The blue line denotes the filtered designated components (the F-dcs); the black line denotes the filtered dcs after smoothing (the F-S-dcs). It can be seen from Figure 4 that the fault trend of the 1st dc can be detected at 161. The 7th dc is abnormal from 326.Figure 4
WF-DCA-based early faults diagnosis.These simulation results imply that the early fault detection can be achieved by wavelet filtering which can increase the signal-to-noise ratio (SNR) of the fault signal. This paper determines the fault trend detection point using wavelet filtering technique.


## 4.4. RUL Prediction Model

In this paper, the system component corresponding toD1 is assumed to be a critical component. In order to get more precise RUL prediction model, smoothing technique is used to postprocess the fault precursor. The smoothed evolution precursor between fault trend detection point and failure point can be defined as the damage precursor to establish RUL prediction model.For establishing RUL prediction model based on WF-PCA, the failure timetf=411 can be determined from Figure 1. The fault trend detection point ts=154 can be obtained from Figure 3. The F-S-SPE between 154 and 411 can be defined as the fault damage precursor. Use RUL=a×ebR as the fitting function to establish RUL prediction model, where R is the deviation between the value of F-S-SPE and the control limit. Figure 5 shows the fitted RUL prediction model based on WF-PCA.Figure 5
The fitted RUL prediction model based on PCA.Figure6 shows the fitted RUL prediction model of the critical component based on WF-DCA. The first F-S-dc between 161 and 451 can be defined as fault damage precursor, where tf and ts can be obtained from Figures 2 and 4, respectively.Figure 6
The fitted RUL prediction model of the critical component based on DCA.


## 4.5. Online Life Prediction and Fault Diagnosis Based on DCA

Once online data is obtained, we can predict the online RUL based on the aforementioned prediction model. Figure7 depicts the online fault detection results by online F-S-SPE. It can be seen from Figure 7 that the fault trend can be detected from the 152nd sample point.Figure 7
Online early fault detection based on WF-PCA.To further demonstrate the superiority of the proposed method, two commonly used prediction algorithms are employed for comparison, namely, AR and BPNN. For BPNN and AR, fault trend prediction and recursive RUL prediction are implemented in two separate stages which makes them unavailable to predicting the RUL online. The parameters of BPNN are shown in Table2. In this paper, the order of AR model is 1, and the regression parameter is φ. The recursive RUL prediction process can be formulated as follows:(36)W1z¯k+s=φW1z¯k+s-1≥q,k=ts,ts+1,…,tf,where s is the number of recursive steps and q is the failure control limit.Table 2
Parameters of BPNN in pretraining.
Parameters
BPNN
Layers
3
Neurons in each layer
[
1,10,1
]
Learning rate
0.0001
Epochs
1000Figure8 shows the online RUL prediction result of PCA. The blue line is the real RUL. The dotted red line is the predicted RUL by the method proposed in this paper. The dotted black line is the predicted RUL by AR. The dotted blue line is the predicted RUL by BPNN. As we can see from Figure 8, the prediction accuracy of the dotted red line is higher than that of the dotted black line and the dotted blue line. Extra recursive steps are required for both AR and BPNN based RUL prediction model. The method proposed in this paper can avoid this problem and reduce the prediction error.Figure 8
Online RUL prediction based on PCA.Figure9 depicts the online faults diagnosis results based on WF-DCA. From Figure 9, the fault trends corresponding to D1 and D7 can be detected from 165 and 324, respectively. Figure 10 shows the online prediction result of the critical component. The blue line is the real RUL. The dotted red line is the predicted RUL by nonlinear fitting method. The dotted black line is the predicted RUL by AR. The dotted blue line is the predicted RUL by BPNN. It can be seen from Figure 10 that fitted RUL prediction model is more efficient than other models. The mean of RUL prediction error is listed in Table 3. It can be concluded from Table 3 that the proposed method is a good choice for the RUL prediction of the critical component.Table 3
Mean of RUL prediction error.
Statistics
Fitting
AR
BPNN
SPE
10.536
15.969
14.450
dc1
9.973
24.156
20.035Figure 9
Online early fault diagnosis based on DCA.Figure 10
The online prediction result of the critical component.


## 5. Conclusions

Traditional RUL prediction is based on autoregression which cannot ensure a real-time RUL prediction since necessary computation time is required for online recursive prediction. In addition in order to implement RUL prediction for critical component that has disastrous impact on the system, DCA is introduced as a fault feature extraction tool for a certain system component to overcome the pattern compounding problem of PCA. The fault feature extracted by DCA can be used to define the fault precursor. For the sake of establishing RUL prediction model, wavelet filter technology based preprocessing is used for early detection of those slowly varying faults. Finally, exponential fitting based online RUL prediction model is developed to get the real-time online prediction of RUL.



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*Source: 8492139-2017-05-21.xml*