RUL prediction of electronic controller based on multiscale characteristic analysis

Mechanical Systems and Signal Processing xxx (2017) xxx–xxx

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

RUL prediction of electronic controller based on multiscale characteristic analysis Dan Xu, Shao-Bo Sui, Wei Zhang ⇑, Mengli Xing, Ying Chen, Rui Kang School of Reliability and Systems Engineering, Science & Technology on Reliability & Environmental Engineering Laboratory, Beihang University, Beijing 100191, China

a r t i c l e

i n f o

Article history: Received 15 July 2016 Received in revised form 22 October 2017 Accepted 20 November 2017 Available online xxxx Keywords: Electronic controller Performance degradation Remaining useful life Ensemble empirical mode decomposition Gaussian process regression

a b s t r a c t The reliability of an electronic controller, which is usually determined by analysing its performance degradation under the working conditions and environmental stresses, has a significant impact on aircraft engine safety. In this paper, a hybrid degradation model, which combines multiscale characteristic analysis (MCA) with modified Gaussian process regression (GPR), is proposed to predict the remaining useful life (RUL) of a controller under various working conditions. Ensemble empirical mode decomposition (EEMD) is utilized to decompose the original data into a number of independent intrinsic mode functions (IMFs) consisting of both degradation and fluctuation information. Characteristic analysis, including series extraction and importance measurement, is conducted to identify the main characteristics hidden in the target IMF. To describe the degradation path under various working conditions, an equivalent function describing the relationship between the degradation rate and stress levels is constructed. It is applied to modify the mean function of the GPR model to describe the relationship between the input of the time series and the degradation trend extracted via EEMD. Meanwhile, in the modified GPR model, combinations of three kinds of covariance function are used to capture the scale characteristics of periodicity and mutability. Then, these individual GPRs are aggregated into the final degradation model. Based on that, the prognostic probability distribution of a controller’s RUL can be calculated numerically via a Monte Carlo (MC) simulation. Finally, the effectiveness and accuracy of the proposed method are verified. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The electronic controller, which controls the running state of an aircraft engine to satisfy the thrust demands under different working conditions, is the core part of an aircraft engine. Its performance directly affects the flying quality and reliability of an aircraft [1]. The working conditions, such as the high temperature, vibration, humidity and external magnetic field, will cause device degradation, resulting in a shifting of the output performance characteristic. Thus, to ensure the normal operation of an engine and the stable flight of an airplane, degradation analysis and accurate estimation of the RUL have become more and more important. The raw performance data of an electronic controller has nonstationary, nonlinear and multiscale characteristics. These characteristics contain not only the tendency information, which is influenced by the stress condition and time, but also the ⇑ Corresponding author. E-mail address: [email protected] (W. Zhang). https://doi.org/10.1016/j.ymssp.2017.11.036 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: D. Xu et al., RUL prediction of electronic controller based on multiscale characteristic analysis, Mech. Syst. Signal Process. (2017), https://doi.org/10.1016/j.ymssp.2017.11.036

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cycle information (caused by the control signal) and mutability information (resulting from an emergency event and the failure components). It is very difficult to capture the exact performance characteristics buried in multiscale characteristics. Regarding electronic signal analysis, the methods, including wavelet decomposition (WD), wavelet packet decomposition (WPD) and empirical mode decomposition (EMD), are widely used in signal denoising, feature extraction, signal decomposition and fault identification [2–5]. As the automatic decomposition methods, WD and WPD have the shortcoming of threshold sensitivity. EMD suffers from mode mixing in the decomposition results [6]. EEMD exhibits good adaptability and is adopted to decompose data series coupled with multiscale characteristics to overcome the deficiencies of EMD and WD [5–7]. For further accurate extraction of fault characteristics, it is not sufficient to use only the above decomposition methods to analyse a signal with multiscale characteristics; hence, hybrid decomposition methods have recently been applied. A data-characteristic-driven reconstruction method consisting of EEMD and artificial intelligence (AI) models was proposed to model the periodicity and mutability of oil prices [5]. Another proposed hybrid learning paradigm composed of EEMD and LSSVR was proven to be helpful for time series prediction with high volatility [7]. In [3], WD and Gaussian process regression (GPR) were applied to model the local regeneration and degradation trend based on physical models and chemical reactions. These methods can be used to accurately extract multiscale signals with time-frequency characteristic in some respects but have limitations when used to process the signals with stress impact. Changes in the environmental stresses and operating conditions, such as the temperature, vibration and input control signal, will influence the degradation path. This complicates the degradation modelling. Although Petri net, the piecewise deterministic Markov process (PDMP) and particle filtering (PF) have been introduced to model the evolution of controller performance characteristics under various working conditions [8–10], they are not suitable for solving the problem of the performance degradation rate being highly dependent on the working conditions. The influence of multiscale characteristics and uncertainties in physical systems makes it difficult to predict the decomposed modes accurately. Although some physical degradation models of the critical components in an electronic controller have been established [11,12], a physical degradation model of an electronic controller is difficult to establish because of its complex structures and the uncertainties. These uncertainties include variable configuration parameters, flexible working patterns, and physical model uncertainties [13]. Here, the data-driven-based approach is adopted to model the controller degradation. Of the data-driven models, the regression model and AI technique are two main types. Regression models, such as the random walk model [14,15], autoregressive integrated moving average model [16] and vector autoregression model [17], have been widely used in performance degradation assessment and fault detection. Regarding the AI techniques, the artificial network algorithm [18], fuzzy logic algorithm [19], self-organizing map [2,20] and support vector machine [21,22] have been widely adopted in the degradation analysis of gears, bearings and batteries. However, these methods ignore the uncertainties caused by environmental stresses and prediction. Recently, PF and GPR were adopted to predict the degradation of products considering these uncertainties [3,23–26]. The PF method combines the system measurements with established physical and empirical models [3,23]. Due to the lack of a physical model of the system, the PF method is not applicable for an electronic controller. Compared with the PF method, GPR is a pure data-driven prediction method with less computation cost [3]. Because of its good mathematical properties, in this paper, the mean function and kernel function of GPR are modified to model the influence of multiscale characteristics in decomposed modes. In this paper, a novel hybrid method combined with multiscale analysis and the modified GPR model is proposed to tackle an accurate characteristic extraction and RUL estimation of an electronic controller under various conditions. In the characteristic extraction, the EEMD-based FFT, F-test and correlation coefficient are utilized together to accurately extract the multiscale characteristics hidden in the raw data. In the modified GPR model, an equivalent function based on the physical acceleration model is used to modify the mean function. The proper combinations of kernel functions and the modified mean function are used to model the multiscale characteristics of decomposed modes. Finally, all predicted modes are integrated into the final prediction to estimate the RUL using the modified GPR model. The remaining part of this paper is organized as follows: In Section 2, the degradation mechanism of the electronic controller is briefly introduced. The formulation process of the proposed method and the RUL estimation are discussed in Section 3. A case study and further discussion are presented in Section 4. Finally, in Section 5, the conclusion is provided.

2. Degradation mechanism of electronic controller An electronic controller has four main performance characteristics: oil needle position, guide blade position, water gate position and thermocouple sensor signal. The degradation of an electronic controller arises from the degradation of the internal components, which results in the shifting of the output performance characteristics. The working ability of an electronic controller is determined by these performance characteristics. According to the information offered by the manufacturers, if one of these performance characteristics exceeds the threshold, the electronic controller cannot fulfil its requirements. The increment thresholds of the four output performance characteristics are the same. Thus, the RUL of an electronic controller is determined by the performance characteristic parameter with the fastest degradation rate. Via historical test data analysis, it is found that the oil needle position parameter has the fastest degradation rate, as shown in Fig. 1. Thus, failure analysis is required for the oil needle position parameter. A schematic diagram of the oil needle position parameter is shown in Fig. 2. Analysis of the circuit diagram reveals that the drifting of the oil needle position parameter is mainly caused by the degradation of the optocoupler. Please cite this article in press as: D. Xu et al., RUL prediction of electronic controller based on multiscale characteristic analysis, Mech. Syst. Signal Process. (2017), https://doi.org/10.1016/j.ymssp.2017.11.036

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Fig. 1. Increments of performance characteristics of an electronic controller.

Fig. 2. Schematic diagram of the oil needle position parameter.

The optocoupler encapsulates the photosensor and the luminescent device inside one shell, as shown in Fig. 3. The input voltage load on the optocoupler causes the luminescent device to shine light onto the photosensor. Then, the electrical characteristic of the photosensor changes with the light, resulting in a corresponding photogenerated current in the output circuit. Finally, the trans-impedance amplifier (TIA) transforms this current into an output voltage. This process realizes the conversion of an electric signal to a light signal to an electric signal without a physical relationship between the input and output [27]. The current transmission ratio (CTR) is the main characteristic of the optocoupler. It is the output current divided by the LED forward current, as shown in [28].

CTR ¼

Icollector ILED

ð1Þ

where Icollector is the output current and ILED is the LED forward current. The degradation of the CTR over time reflects the optocoupler performance degradation [28,29]. The reduction in efficiency of the LED is the main cause of CTR degradation [29]. The ambient conditions, such as high temperature, will accelerate this degradation as the LED efficiency degrades with temperature [30]. In the literature [30], an accelerated degradation test of an optocoupler has been conducted under 125 °C, 150 °C and 175 °C. The degradation results show that the CTR degrades over time and with increasing temperature. This explains the degradation of the oil needle position in the mechanism.

3. RUL estimation based on EEMD and GPR 3.1. Problem statement The objective of this study is to estimate the RUL of an electronic controller at tpresent. Considering the existing variable factors influencing its degradation, it is reasonable to treat the performance degradation evolution as a stochastic process. These factors include microstructural characteristics, manufacturing characteristics, load stress and ambient conditions. The degradation state and RUL at ttest are random variables. Therefore, the RUL is not a predicted value RULðtpresent Þ but a probability distribution that takes the uncertainties into account at tpresent. However, the time series of the performance characteristics are nonstationary, nonlinear and have multiscale characteristics. In this paper, the multiscale characteristics are divided into two categories: time-frequency impact and stress impact caused by environmental stresses and various operating conditions. Thus, a method for accurate multiscale characteristics extraction and RUL prediction is introduced, which contains the signal decomposition, characteristic analysis and individual prediction based on the stochastic process. First, the original time series are divided into a number of IMFs via EEMD coupled with multiscale characteristics. Characteristic analysis is used to determine the main characteristic hidden in IMFs. Then, the GPR is modified to model the influence of multiscale characteristics on IMFs. The prediction of IMFs is carried out using the modified GPR separately. Finally, all forecasting values are included in the final prognosis result to estimate the RUL. Based Please cite this article in press as: D. Xu et al., RUL prediction of electronic controller based on multiscale characteristic analysis, Mech. Syst. Signal Process. (2017), https://doi.org/10.1016/j.ymssp.2017.11.036

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Fig. 3. Simplified structure diagram of the optocoupler.

on the proposed method, a hybrid approach integrating EEMD, data analysis and modified GPR, as shown in Fig. 4, is carried out. The raw degradation DyðtÞ is calculated using Eq. (2):

DyðtÞ ¼ yðtÞ  yð0Þ

ð2Þ

where yðtÞ represents the measurement value at time t, yð0Þ is the initial value, and DyðtÞ denotes the degradation variable of the performance characteristic. 3.2. Time-frequency characteristic extraction and analysis of multiscale signal In this section, EEMD and characteristic analysis are adopted to analyse the multiscale characteristics of signals. Here, the performance characteristic of time series is considered as a hybrid signal consisting of multiscale components [3]. The cycle information, mutable information and degradation tendency can be divided into different frequency scales using the decomposition method. Characteristic analysis is performed to determine the main characteristics hidden in the divided subsignals. 3.2.1. Data decomposition based on EEMD In contrast to traditional frequency domain decomposition approaches such as the fast Fourier transform (FFT) and WD, EEMD is an empirical, direct and self-adaptive signal decomposition method that decomposes a time series based on its traits. Although EEMD is based on EMD, it can effectively address the mode mixing that appears during EMD [31].

Fig. 4. Framework of the proposed hybrid EEMD-GPR approach.

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EMD is a self-adaptive data process and shows high efficiency in dealing with nonstationary and nonlinear time series. It decomposes complex data into a finite number of IMFs according to their characteristics. Meanwhile, the feature information buried in raw data series is also divided into different scales. The EMD process and its standards are presented in the literature [14,16,20]. However, whether or not the true IMF is obtained depends on the distribution of extreme values in EMD. If the extrema of a signal appear to follow a maldistribution, then mode mixing will appear in the decomposition results. EEMD introduces white noise to settle this problem. Before decomposition, the original signal and the white noise are aggregated into a new signal. Then, the new signal can be separated automatically into different frequency scales based on the fact that the white noise is distributed equally throughout the time-frequency space [5]. The white noise is offset via ensemble averaging according to the following statistical rule [31]:

e e ¼ pffiffiffiffiffiffi

ð3Þ

NE

where e is the standard deviation of the difference between the input and the decomposition results, the added white noise, and NE is the number of ensemble members. Given the input original degradation DyðtÞ, the EEMD process is as follows [6,24]: Step Step Step Step

1 2 3 4

e is the amplitude of

Add white noise to the original degradation time series DyðtÞ. Decompose the data sequence with the added white noise into a series of IMFs via EMD. Repeat steps 1–2 n times with different white noise. After n iterations, the decomposed IMFs are the final EEMD of DyðtÞ.

3.2.2. Multiscale characteristic analysis The aim of this section is to analyse the multiscale characteristics of the original data and discover the main factors hidden in the decomposed IMFs. As seen in Table 1, the characteristics hidden in the oil needle position mainly include the periodicity, mutability and central trend. To discover the main characteristics hidden in the target IMF, three substeps are involved in the data characteristic analysis: (1) data characteristic testing, (2) characteristic series extraction and (3) characteristic importance measurement [7]. Data characteristic testing is conducted to check whether the IMF has the related characteristic. Characteristic series extraction and characteristic importance measurement are designed to identify the main characteristic hidden in the data series. In data characteristic testing, the time-frequency impact of multiscale characteristics is tested, including the periodicity and mutability. There are several effective tools for confirming the periodicity characteristic hidden in IMFs, such as R/S analysis, autocorrelation analysis, and power spectral analysis. In this paper, we use the fast Fourier transform to analyse the data, with the cycle information being determined by the maximal spectrum power obtained via time scale estimation. Mutability is caused by breakpoints in the data. The chow test is chosen to detect the potential breakpoint, and the given potential breakpoint is tested via the F-test [32].

F¼

SSESSE1 SSE2 m1 SSE1 þSSE2 N 1 þN 2 2m2

 F ðm þ 1; N1 þ N2  2m  2Þ

ð4Þ

where SSE is the sum of the residual square for the whole data series in data modelling, with N 1 þ N 2  m  1 degrees of freedom, and SSE1 ; SSE2 are the sums of the residual square of data subsets before and after the breakpoint, respectively, with N 1  m  1 and N 2  m  1 degrees of freedom. Characteristic variables are introduced to form the characteristics sequence in characteristic series extraction. For cyclical characteristic series, average cyclical dummies P c ðtÞ with period s are utilized [7].

8 b1 > > > . . . > > : bs

t ¼ 1; 1 þ s; 1 þ 2s; 3 þ 3s; . . . t ¼ 2; 2 þ s; 2 þ 2s; 2 þ 3s; . . . 

ð5Þ

t ¼ s; 2s; 3s; 4s; . . .

where b1 ; b2 ;    ; bs are the average cyclical dummies. Table 1 Data characteristics hidden in degradation data series. Data characteristic

Implication

Periodicity

This characteristic corresponds to the repetition of data in the same order with peaks and troughs from the beginning to the end [5] This characteristic corresponds to the effect of some irregular events on the controller This characteristic corresponds to the long-term tendency of the original data

Mutability Tendency

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In the mutability series construction, dummy variables P m1 ðtÞ and P m2 ðtÞ representing the structural changes in the level and slope at breakpoint tk are introduced as [5,33,34]



Pm1 ðtÞ ¼  Pm2 ðtÞ ¼

0

t < tk

ð6Þ

1 t P tk 0

t  tk

t < tk

t P tk

ð7Þ

The tendency of the characteristic series is represented by P T ðtÞ [5]

PT ðtÞ ¼ t

ð8Þ

Finally, the main characteristic is distinguished from the comparison of correlation coefficients between the characteristic series and the corresponding IMF [7]. The correlation coefficients are calculated as

qPY ¼

covðP; YÞ

rP rY

¼

h i E ðP  lP ÞðY  ly Þ

ð9Þ

rP rY

where P refers to the characteristic series; Y represents the corresponding IMF; covðP; YÞ is used to calculate the covariance of P and Y; rP and rY indicate the standard deviations of sequence P and sequence Y, respectively; and lP and lY are the mean values of the characteristic series P and IMF Y, respectively. Accordingly, the GPR model is built based on the characteristic of each IMF in the next individual prediction. 3.3. Modified GPR model considering various conditions and RUL estimation In this section, the GPR is modified to model the influence of multiscale characteristics on IMFs individually. Then, the final prediction is calculated from these individual IMF predictions using the ensemble principle. Finally, the RUL is estimated using the MC method. In GPR modification, the equivalent function based on the acceleration function is applied to modify the mean function. This is done to describe the influence of the stress impact of the multiscale characteristic. The covariance function is chosen based on the time-frequency impact of the multiscale characteristic of IMFs. GPR is chosen to analyse the uncertainties caused by environmental stresses and prediction. 3.3.1. Modified GPR model As mentioned in Section 3.1, the degradation evolution is considered as a stochastic process. Due to the good adaptability in dealing with complicated nonlinear, high-dimensional and small sample data series [35], GPR is adopted to model the degradation evolution. It yields predicted values with inferred probability. Assume that the degradation process DyðtÞ is a Gaussian process (GP), which is defined by its mean function mðtÞ and covariance function kðt; t0 Þ. The GP is a collection of random variables with a joint Gaussian distribution [36]:

DyðtÞ  GPfmðtÞ; Kðt; tÞg

ð10Þ

mðtÞ ¼ E½DyðtÞ

ð11Þ

kðt; t0 Þ ¼ E½ðDyðtÞ  mðtÞÞðDyðt 0 Þ  mðt0 ÞÞ

ð12Þ 0

0

where t ¼ ft1 ; t2 ; . . . ; tN g is the input argument matrix; t and t are different elements in t; kðt; t Þ is the covariance calculated using the covariance function; and Kðt; tÞ is the covariance matrix consisting of values of kðt; t0 Þ calculated from all possible pairs in t. In terms of the function space, all statistical characteristics of the Gaussian process are defined by its mean function mðtÞ and covariance function kðt; t0 Þ. In Section 3.2.1, the degradation data were decomposed via EEMD into a series of IMFs. The IMF evolution can also be considered as a GP. Here, we adopt GPR to model each IMF according to its main characteristic. Proper forms of the mean function and covariate function will improve the accuracy of the multiscale characteristic model and prediction. Thus, it is fair to conduct individual modelling of each IMF by modifying the mean function and covariate function. In the GPR model, the mean function is utilized to describe the degradation trend [19]. Considering the influence of the nonlinearity and various working conditions, a simple linear mean function is not sufficient. In this study, an equivalent function built based on an acceleration function is utilized to modify the mean function describing the relationship between the degradation and stresses. The mean function under the constant working condition is as follows:

mTendency ðtjSi Þ ¼ bðSi Þta þ Dy0

06i6k

ð13Þ

where Si is the environmental stress; Dy0 represents the initial value; bðSi Þ indicates the degradation rate under stress Si ; k is the order of the working condition; and a is a parameter independent of the environmental stress Si . When operating the controller, the changeable environmental stress is the temperature T i . Please cite this article in press as: D. Xu et al., RUL prediction of electronic controller based on multiscale characteristic analysis, Mech. Syst. Signal Process. (2017), https://doi.org/10.1016/j.ymssp.2017.11.036

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According to the Arrhenius acceleration equation, bðT i Þ is represented as [37]:

 bðT i Þ ¼ exp a 

b¼

b 273 þ T i



ð14Þ

E K

ð15Þ

where a; b are the unknown parameters; E is the activation energy; and K is Boltzmann’s constant. Under the following assumptions: (1) the failure mechanisms of the electronic controller under different working conditions remain invariable and (2) the degradation paths of the controller under different working conditions have the same form with different parameters; the mean function, modified to capture the degradation tendency under different working conditions, is represented by

8 bðT 1 Þta þ y0 > > > > > bðT 2 Þðt  t1 þ s1 Þa > > < a mTendency ðtjT i Þ ¼ bðT 3 Þðt  t2 þ s2 Þ > > > ... > > > > : bðT k Þðt  tk1 þ sk1 Þa

0 6 t < t1 t1 6 t < t2 t2 6 t < t3

ð16Þ

t k1 6 t < t k

mTendency ðsi1 jT i Þ ¼ mTendency ðt i1 jT i1 Þ

ð17Þ 0

where si1 is the time compensation corresponding to the temperature T i .The covariance function kðt; t Þ is used to determine the variation around the mean function [35]. Here, it is used to model the influence of the time frequency. Three common types of covariance functions are used in this study, i.e., periodic covariance function kp ðt; t0 Þ, squared exponential covariance function kSE ðt; t0 Þ and rational quadratic covariance function kRQ ðt; t 0 Þ, to model the cycle characteristic, mutability characteristic and fluctuation of the IMF. These three covariance functions are as follows. Periodic covariance function (P)

  0   2 2 wðt  t Þ l kp ðt; t 0 Þ ¼ d2f exp 2 sin 2p

ð18Þ

Squared exponential covariance function (SE) 0

kSE ðt; t Þ ¼

d2f

exp 

ðt  t0 Þ 2l

2

!

2

ð19Þ

Rational quadratic covariance function (RQ) 0

kRQ ðt; t Þ ¼

d2f

1þ

ðt  t 0 Þ 2al

2

2

!a ð20Þ

where d2f is the variance of the time series; l is the characteristic length scale; and w denotes the angular frequency. It is found that the combination of multiple covariance functions is better than a single covariance function. For the IMF coupled with the stress impact on multiscale characteristics, we take mTendency as its mean function to model the influence of stress on the degradation trend. Regarding the other IMFs, they change around zero. A constant mean function is thus sufficient. Based on the analysis above, the modified GPR model, consisting of mean function mTendency and a covariance function combining SE and RQ, is utilized to model the IMF with tendency information, where mTendency is used to model the influence of stress. For IMFs with a time-frequency impact of multiscale characteristics, a covariance function combining P and RQ is utilized to model the periodicity characteristic, while a covariance function combining SE and RQ is adopted to present the mutability. 3.3.2. Comprehensive prediction based on modified GPR model In the IMF individual regression, a set of measurements D ¼ fðti ; Dyij Þji ¼ 1; 2; . . . N; j ¼ 1; 2 . . . Mg is taken as the training data for the Gaussian process, where i ¼ 1; 2; . . . ; N; j ¼ 1; 2; . . . ; M denote the measurement times and number of sequences of the IMFs, respectively; ti is the input of the model; and Dyij is the value at time t i in IMF j. Considering noise, the Gaussian process Dyj is then represented as

Dyj ¼ f ðtÞ þ e

ð21Þ

where e indicates the independent noise following a Gaussian distribution e  Nð0; r2n Þ; r2n is the variance of the noise signal; t ¼ ft 1 ; t 2 ; . . . ; tN g is the input matrix; and Dyj is the measurement matrix. Please cite this article in press as: D. Xu et al., RUL prediction of electronic controller based on multiscale characteristic analysis, Mech. Syst. Signal Process. (2017), https://doi.org/10.1016/j.ymssp.2017.11.036

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The prior distribution of y, its mean function and its covariance function are derived from Eq. (10) as

  Dyj  GP mj ðtÞ; Kðt; tÞ þ r2n I

ð22Þ

m ¼ mðt; hÞ

ð23Þ

k ¼ kðt; t 0 ; cÞ

ð24Þ

where h; c are the unknown parameters, usually called hyperparameters, of the mean and covariance functions. I is a unit matrix with the same dimensions as for Kðt; tÞ, which is the covariance matrix consisting of all values of kðt; t0 Þ. The accuracy of the prediction is dependent on the hyperparameters H ¼ fh; cg. The hyperparameters are optimized by maximizing the marginal likelihood of the training data D via the conjugate gradient method in our study. The marginal likelihood is as shown in [19].

L ¼ logðpðDyj jt; HÞÞ ¼ 

1 1 N log jKj  ðDyj  mj ðtÞÞT K 1 ðDyj  mj ðtÞÞ  logð2pÞ 2 2 2

ð25Þ

Given a new test input t test , the prior joint distribution of the training sample and targets is described as

"

# 



Dyj ðttrain Þ mj ðttrain Þ Kðttrain ; ttrain Þ Kðt test ; ttrain Þ  ; T Dyj ðttest Þ mj ðttest Þ K ðt test ; ttrain Þ Kðt test ; t test Þ

ð26Þ

Then, the posterior distribution of predicted target value yðt test Þ at time ttest is predicted using Eq. (26)

j ðt test Þ; r2j ðt test ÞÞ PðDyj ðt test Þjttrain ; Dyj ðttrain Þ; t test Þ  NðDy j ðt test Þ; r where Dy

2 Dyj ðt test Þ

ð27Þ

are the mean and variance of the Gaussian distribution, respectively:

Dyj ðt test Þ ¼ mj ðt test Þ þ KT ðt test ; ttrain ÞK1 ðttrain ; ttrain ÞðDyj ðttrain Þ  mj ðttrain ÞÞ

ð28Þ

r2j ðttest Þ ¼ Kðttest ; ttest Þ  KT ðttest ; ttrain ÞKðttest ; ttrain Þ þ r2n

ð29Þ

Since the IMFs are a linear expansion of raw degradation data in EEMD, the final degradation prediction of the electronic controller is obtained from all individual predictions of the IMFs via ADD [5]. The final prediction follows a Gaussian distribution

Dyprediction ðt prediction Þ  N

X

mi ðtprediction Þ;

X

r2i ðtprediction Þ

ð30Þ

where mi ðt prediction Þ and r2i ðt prediction Þ are the predicted mean value and variance of IMF i, respectively. To measure the accuracy of the prognosis, three common criteria are used: relative accuracy (RA), root mean square error (RMSE), and mean absolute percentage error (MAPE). The definitions of the RA, RMSE, and MAPE are as follows

RAðtÞ ¼ 1 

jDya ðtÞ  Dyp ðtÞj Dya ðtÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PNðtÞ ^ 2 i¼1 ðDyi  Dya Þ RMSEðtÞ ¼ N MAPEðtÞ ¼

 NðtÞ  ^a  100 XDyi  Dy   N i¼1 Dyi 

ð31Þ

ð32Þ

ð33Þ

where Dya denotes the actual degradation measured at time t and Dyi ; Dyp are the prognosis values. 3.3.3. RUL estimation based on modified GPR model According to the information offered by the manufacturers, if one of these performance characteristic increases over the threshold, the electronic controller cannot fulfil its requirements. This means that when DyðtÞ exceeds the given threshold wy , the electronic controller fails. The failure time sf is also called the first passage time, which is defined as

sf ¼ inffYðtÞ P wy g

ð34Þ

Thus, the remaining useful life RULðtÞ at time t is defined as

RULðtÞ ¼ inffs P t; YðsÞ P wy g  t

ð35Þ

where s denotes a future time, which can be written as

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Fig. 5. Operation condition for the controller.

Test sample of electronic controller (inside chamber)

Comprehensive test chamber

Input control signal setup interface of electronic controller system

Data acquisition unit

Vibration table

Input control unit of electronic controller

Fig. 6. Degradation test setup and acquisition system.

Fig. 7. The increment of oil needle position.

RULðtÞ ¼ sf  t

ð36Þ

From Eqs. (35) and (36), we can see that the first passage time is the key for RUL estimation. In this study, an MC simulation is conducted to calculate the first passage time and its probability density. Assume that the step size Dt > 0, present time is tpresent and test time t j ¼ jDt þ tpresent ; j ¼ 1; 2; . . . :n. The probability distribution PðDyðt j ÞjY train ; ttrain ; t j Þ of cumulative degradation Dyðtj Þ at test time tj was derived in Section 3.3.2.

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Fig. 8. Decomposition results of the oil needle position parameter.

PðDyðt j ÞjDytrain ; ttrain ; t j Þ  N

X

mi ðtj Þ;

X

r2i ðtj Þ

ð37Þ

where mi ðt j Þ and r2i ðt j Þ are the predicted mean value and variance of IMF i, respectively. The simulation is conducted as follows: Please cite this article in press as: D. Xu et al., RUL prediction of electronic controller based on multiscale characteristic analysis, Mech. Syst. Signal Process. (2017), https://doi.org/10.1016/j.ymssp.2017.11.036

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Step 1 set i = 0 and the number of trajectories N (i.e., the maximum number of replications) Step 2 if i 6 N, i ¼ i þ 1; and j ¼ 0; produce the present degradation state Dyðtpresent Þ from the initial distribution PðDyðt present ÞjDytrain ; t train ; t present Þ Step 3 if Dyðt j Þ < wy , calculate the probability distribution PðDyðtj ÞjDytrain ; t train ; tj Þ, produce Dyðtj Þ from the distribution, and set j ¼ j þ 1; Step 4 if Dyðt j Þ < wy , stop and return to step 2, and record the first passage time t j1 Step 5 if i > N, terminate the procedure; the simulation PðRULðt j ÞÞ ¼ passage times at time tj

nrecord ðt j Þ , N

where nrecord is the number of recorded first

Consequently, we obtain the statistic of the first passage time. The probability density distribution of the first passage time can be estimated using methods such as kernel density estimation [38]. Finally, we can obtain the probability distribution of the RUL based on step 5 and Eq. (36). The estimation error is controlled by shortening the time interval.

4. Case study and discussion In this section, the proposed prognosis method will be validated via a comparison with the experimental data obtained from the degradation test of the electronic controller under different working conditions. Table 2 Periodicity analysis results for IMFs. Mode

Cycle (hour)

Mode

Cycle (hour)

IMF IMF IMF IMF IMF

0.8352 2.0717 5.3569 39.8877 128.9368

IMF IMF IMF IMF IMF

340.3788 578.7577 >Sample size >Sample size >Sample size

1 2 3 4 5

6 7 8 9 10

Table 3 Mutability analysis results for IMFs. Mode

Breakpoints Time (hour)

IMF IMF IMF IMF IMF IMF

1 2 3 4 5 6

IMF 7 IMF 8 IMF 9 IMF 10

– – 36.15 1770.4 28.333 1072.33 1152.83 2022.8 904.933 1631.083 312.356 1418.4 1481.9 –

Chow-test Serial number

S-value

88 1429 69 1092 1125 1567 1023 1350 749 1244 1278 –

– – 4.3938 3.0548 878.361 8.3766 6.1107 17.7565 3.2039 3.2502 3.0495 3.1085 3.0772 –

Mutability P-value

0.0125 0.0474 0.0000 0.0002 0.0023 0.0000 0.0408 0.0408 0.0479 0.0449 0.0463 –

No No Yes Yes Yes Yes

Yes Yes Yes No

Table 4 Main characteristic analysis results for IMFs. Mode

IMF IMF IMF IMF IMF IMF IMF IMF IMF IMF

1 2 3 4 5 6 7 8 9 10

Characteristic importance analysis

Main characteristic

Tendency

Cyclicity

Mutability

0.0008 0.0091 0.0091 0.1074 0.0145 0.0118 0.0046 0.1516 0.2494 0.9234

0.4518 0.4109 0.2188 0.1732 0.3320 0.4182 0.6525 – – –

– – 0.0053 0.0325 0.0128 0.0238 0.0117 0.2996 0.4838 –

Cyclicity Cyclicity Cyclicity Cyclicity Cyclicity Cyclicity Cyclicity Mutability Mutability Tendency

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D. Xu et al. / Mechanical Systems and Signal Processing xxx (2017) xxx–xxx Table 5 Mean and covariance functions of GPR for each IMF. Mode

IMF IMF IMF IMF IMF IMF IMF IMF IMF IMF

1 2 3 4 5 6 7 8 9 10

Main characteristic

Cyclicity Cyclicity Cyclicity Cyclicity Cyclicity Cyclicity Cyclicity Mutability Mutability Tendency

GPR Mean function

Covariance function

mconstant mconstant mconstant mconstant mconstant mconstant mconstant mconstant mconstant mTendency

kRQ kRQ kRQ kRQ kRQ kRQ kRQ kRQ kRQ kRQ

þ kp þ kp þ kp þ kp þ kp þ kp þ kp þ kSE þ kSE þ kSE

Fig. 9. Results of comparison between proposed hybrid method and actual degradation: (a) modelling results of proposed hybrid method for both history data and prediction compared with actual degradation; (b) detailed comparison between prediction results of proposed hybrid method and actual degradation.

4.1. Experiment description The electronic controller degradation test is performed under different working conditions. Considering that the degradation thresholds of the output performance characteristics are the same, the oil needle position parameter, which has the fastest degradation rate, is chosen to estimate the RUL of the electronic controller and validate the proposed method. The operation condition for the controller is shown in Fig. 5. During the test, one sample, with an ambient temperature varying from 110 °C to 90 °C, was tested. The test setup and acquisition system are shown in Fig. 6. The electronic controller was put into three comprehensive test chambers, which were used to simulate the ambient temperature stress and vibration stress. The inputs of the electronic controller were set to simulate the operation mode of a controller on an airplane. The four output performance characteristics were sampled for 1 min at 5 Hz every 25 min. Then, the measured values were obtained by averaging the values sampled over 1 min. Finally, Eq. (2) was used to calculate the degradation of the output analogue voltage signal of the electronic controller. 4.2. Data decomposition and multiscale characteristic analysis As analysed above, the fastest degradation signal determines the RUL of an electronic controller. Thus, the oil needle position parameter is chosen to verify our proposed hybrid method and estimate the RUL of an electronic controller. Fig. 7 shows the cumulative degradation of the oil needle position parameter. We observe that periodicity and mutability exist in the degradation process and that the degradation rates change with the temperature stresses. The EEMD results are listed in Fig. 8. The original data series is decomposed into ten IMFs. It is obvious that IMF 10 has a monotonous degradation tendency, eliminating high frequency fluctuation, and cycle and mutability, which occur at a low frequency. Other IMFs, which vary around zero, exhibit cycle and mutability characteristics that reflect the control process. Please cite this article in press as: D. Xu et al., RUL prediction of electronic controller based on multiscale characteristic analysis, Mech. Syst. Signal Process. (2017), https://doi.org/10.1016/j.ymssp.2017.11.036

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Then, we conduct multiscale characteristic analysis for each IMF. The periodicity analysis results are listed in Table 2. In the periodicity analysis, not all history data are used. The data measured from 400 h to 1000 h are ignored because of the absence of measurement values and changeable measurement cycles. It can be seen that the cycles of IMFs 1–3 are relatively less than 6 h, especially that of IMF 1, which is 0.835 h, while there is no obvious cycle for IMFs 8, 9, and 10, which means that the main time scales of the corresponding cycles are greater than the sample size. Regarding mutability, the chow test is introduced to investigate the potential breakpoint in the original data corresponding to a sudden change in the system. The time at which breakpoints occur and the corresponding statistical values and p-values are listed in Table 3. Regarding the breakpoint, its corresponding p-value in the chow test is determined to be smaller than the significance level, i.e., 0.05. No breakpoint is found for IMFs 1–2 and IMF 10, which means that IMFs 1–2 and IMF 10 do not have mutability. The main characteristics of each IMF are estimated in terms of the correlation coefficient between the target IMF and its corresponding characteristic sequence. The results in Table 4 show that the periodicity is the main characteristic of IMFs 1–7, whereas IMF 8 and IMF 9 possess mutability, and the main characteristic of IMF 10 is tendency.

Fig. 10. Results of comparison between hybrid method with the linear mean function and actual degradation: (a) modelling results of hybrid method with linear mean function GPR for both history data and prediction compared with actual degradation; (b) detailed comparison between prediction results of hybrid method with the linear mean function and actual degradation.

Fig. 11. Results of comparison between GPR with the modified mean function and actual degradation (a) modelling results of GPR with modified mean function for both history data and prediction compared with actual degradation; (b) detailed in comparison between prediction results of GPR with the modified mean function and actual degradation.

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4.3. Prediction and comparison Each IMF is modelled using the modified GPR according to its main characteristics, as listed in Table 5. Considering the periodicity of IMFs 1–7, a constant mean function and a covariance function combining P (kp ) and RQ (kRQ ) are selected. For IMF 8 and IMF 9, a constant mean function and a covariance function combining RQ (kRQ ) and SE (kSE ) are used to describe their mutability. For IMF 10, the tendency of which is influenced by stress, the modified mean function mTendency and a covariance function combining RQ (kRQ ) and SE (kSE ) are used to describe its tendency characteristic. Finally, all predicted IMF values are aggregated into the final prediction of the degradation data via ADD. The accurate and rapid estimation of degradation is important for RUL prediction. To validate the efficiency and accuracy of the proposed hybrid approach, the 3 common criteria mentioned in Section 3.3 are adopted, namely, relative accuracy (RA), root mean square error (RMSE), and mean absolute percentage error (MAPE). The comparison of the 3 criteria is conducted among the proposed hybrid method, GPR model with the modified mean function, and hybrid method with the linear mean function to prove the accuracy of the proposed hybrid approach. Compared with the proposed hybrid method, the GPR model has a modified mean function but lacks EEMD and multiscale characteristic analysis, while the hybrid method lacks the modified mean function. The GPR model and hybrid method with the linear mean function are taken as the comparison groups. We use the initial 1902 h of history degradation data as the training data to predict the degradation value from 1902 h to 2353 h under 90 °C. We take the mean value of the predicted degradation distribution as the prognostic degradation value.

(b)

(a)

(c) Fig. 12. Comparison of prediction error results of the proposed hybrid method, GPR model with the modified mean function, and hybrid method with the linear mean function. (a) RMSE (b) MAPE (c) RA.

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Fig. 13. Uncertainty analysis results for hybrid EEMD-GPR prediction.

Fig. 14. Survival probability of RUL.

3

x 10

-3

Probability density distribuiton caculated by MC Fitted probability density distribuiton

2.5

Density

2

1.5

1

0.5

0

200

400

600

800

1000

1200

Data Fig. 15. Probability density distribution of RUL.

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Table 6 Hyperparameter estimation results for IMF with periodicity. Mode

mconstant

IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7

kp

4.5198e05 1.8051e05 1.8050e05 1.8050e05 1.1943e05 2.3463e04 2.3463e04

kRQ

l

w

df

l

df

a

1.4948 1.0445 1.0445 1.0445 0.2297 1.1854 1.1854

12.2642 12.2627 12.2619 12.2657 13.3861 12.2649 12.2649

1.6987 1.7946 1.7946 1.7948 2.1090 1.7605 1.7605

0.4211 0.0962 0.0962 0.0962 0.0103 0.4531 0.4531

6.7107 8.4134 8.4134 8.4134 8.8844 8.5799 8.5799

2.5726 3.0240 3.0239 3.0243 3.0703 2.9712 2.9712

Table 7 Hyperparameter estimation results for IMF with mutability. Mode

mconstant

kRQ

7.7724e04 6.8547e04

IMF8 IMF9

kSE

l

df

a

l

df

5.8095 5.9519

5.7168 6.1385

4.9087 5.0568

8.4600 8.4914

6.4047 6.5042

Table 8 Hyperparameter estimation results for IMF with tendency. Mode

IMF10

mtendency

kRQ

kSE

a

b

a

y0

l

df

a

l

df

12.0017

2.1959e+3

0.2077

7.9598e4

0.4662

8.0085

3.0709

9.1723

5.9996

Then, the RMSE, MAPE and RA of the predicted value are calculated to compare the capacities of the proposed hybrid method, GPR model with the modified mean function, and hybrid method with the linear mean function. The prediction results are shown in Figs. 9-11. We can observe that the data before 1902 h for all three methods fit well in the modelling, but in the prediction process, there is disparity among these three methods, as shown in Figs. 9(b)–11(b). Fig. 12 shows the comparison results in terms of the RMSE, MAPE and RA, where the blue line denotes the proposed hybrid method; the red line denotes the hybrid method with the linear mean function; and the yellow line denotes the GPR model with the modified mean function. Fig. 12(a) displays the comparison results in terms of the RMSE, which reveal that the proposed hybrid method has the lowest RMSE for long-term prediction. In Fig. 12(b), the comparison of the MAPE values of all three methods shows that the proposed hybrid method has the best performance most of the time, while the GPR model has the poorest accuracy after 2050 h. Regarding the RA comparison results, all three methods have values larger than 0.86, but it can be seen that the hybrid EEMD-GPR achieves the largest RA value among the three methods as time progresses, as shown in Fig. 12(c). All these comparisons indicate that the proposed hybrid method well predicts the degradation of the electronic controller and has the highest accuracy. Moreover, these phenomena become increasingly apparent. Fig. 13 shows the analysis of uncertainty in the EEMD-GPR prediction in terms of the probability distribution. Fig. 13 presents the 95% confidence interval of the prediction against the actual degradation, where the red line denotes the actual degradation and the blue line denotes the forecast values. The time points 1912, 2045, 2196 and 2338 are chosen to calculate the 95% confidence intervals, the results of which are [0.0105, 0.0136], [0.0105, 0.0138], [0.0107, 0.0139] and [0.0109, 0.0142], respectively. The corresponding actual values are 0.012 V, 0.0125 V, 0.0125 V and 0.0130 V. It can be seen that all actual degradation values fall within the 95% confidence intervals. All the 2353 h history data are used here to predict the RUL under 90 °C. For the parameter estimation, the conjugate gradient method is adopted. The estimations of hyperparameters in the mean functions and the covariance functions are shown in Tables 6–8. With the threshold of 0.018 supplied by the manufactures, the survival probability and probability density distribution of the RUL can be calculated via an MC simulation, as shown in Figs. 14 and 15. According to the curves, the reliable lifetime is 473 h, with a survival probability of 0.9.

5. Conclusions The main contribution of this study is that it proposes a novel hybrid method based on multiscale characteristic analysis to estimate the RUL of an electronic controller under different stress levels. In the proposed approach, the time-frequency characteristics hidden in multiscale signals are accurately extracted using the FFT, F-test and correlation coefficient methods Please cite this article in press as: D. Xu et al., RUL prediction of electronic controller based on multiscale characteristic analysis, Mech. Syst. Signal Process. (2017), https://doi.org/10.1016/j.ymssp.2017.11.036

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on the basis of the IMFs decomposed via EEMD. In the modified GPR model, an equivalent function based on the acceleration model is used to modify the mean function to present the influence of stress. Proper combinations of kernel functions (SE, RQ, and P) are utilized to model the time-frequency impact. Then, these individual GPRs are integrated to obtain the final prediction result in terms of the probability density distribution. Finally, the probability distribution of the RUL is calculated using the MC numerical method. The results show that the proposed hybrid method can accurately and effectively extract the multiscale features and estimate the RUL. In feature extraction and multiscale characteristic analysis, it is found that the main characteristic of IMFs 1–7 is periodicity, that of IMF 8 and IMF 9 is mutability, and IMF 10 is influenced by stress. In the modified GPR model, the efficiency and accuracy of the proposed hybrid method in terms of prediction are verified by comparing three indexes, namely, RMSE, MAPE and RA, with those of two other methods: the GPR method and the EEMD-GPR with the linear mean function. Finally, the RUL of an electronic controller at 90 °C is calculated via an MC simulation, and the result is a reliable lifetime of 473 h with a survival probability of 0.9. In this paper, an electronic controller is considered under a single changeable stress, and the RUL is determined by a single performance characteristic parameter. However, in many practical cases, the electronic controller is usually operated under multiple types of changeable stresses. The corresponding evaluation of the RUL becomes much more complicated, as it involves multiple performance characteristics from different sensors. This investigation will be extended to this complex case in future work. 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