remaining useful life prediction for rotating machinery...

Research ArticleRemaining Useful Life Prediction for Rotating MachineryBased on Optimal Degradation Indicator

Aisong Qin Qinghua Zhang Qin Hu Guoxi Sun Jun He and Shuiquan Lin

Guangdong Provincial Key Laboratory of Petrochemical Equipment Fault Diagnosis Guangdong University ofPetrochemical Technology Maoming 525000 China

Correspondence should be addressed to Qin Hu huqinbinzhou163com

Received 6 September 2016 Revised 1 March 2017 Accepted 8 March 2017 Published 22 March 2017

Academic Editor Mickael Lallart

Copyright copy 2017 Aisong Qin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Remaining useful life (RUL) prediction can provide early warnings of failure and has become a key component in the prognosticsand health management of systems Among the existing methods for RUL prediction the Wiener-process-based method hasattracted great attention owing to its favorable properties and flexibility in degradation modeling However shortcomings existin methods of this type for example the degradation indicator and the first predicting time (FPT) are selected subjectively whichreduces the prediction accuracy Toward this end this paper proposes a new approach for predicting the RUL of rotatingmachinerybased on an optimal degradation indictor First a genetic programming algorithm is proposed to construct an optimal degradationindicator using the concept of FPT Then a Wiener model based on the obtained optimal degradation indicator is proposed inwhich the sensitivities of the dimensionless parameters are utilized to determine the FPT Finally the expectation of the predictedRUL is calculated based on the proposed model and the estimated mean degradation path is explicitly derived To demonstratethe validity of this model several experiments on RUL prediction are conducted on rotating machinery The experimental resultsindicate that the method can effectively improve the accuracy of RUL prediction

1 Introduction

Rotating machinery are the most widely used mechanicalequipment in industry and generally consist of motor shaftgear box bearing and load [1 2] Any fault in rotatingmachinery can cause the breakdown of the entire machinewhich can lead to catastrophic consequences According tostatistics one-third to one-half of the expenditure is wastedthrough ineffective maintenance [3] If the remaining usefullife (RUL) can be predicted in advance catastrophes can beavoided and predictive maintenance can be implemented tomaximize machine uptime and minimize maintenance costsTherefore there is a pressing need to continuously developand improve the current RUL prediction technologies [4]

At present the main method used for RUL estimationis a physics-based failure model and a data-driven methodThe data-driven method has become the mainstream in thefield of RUL prediction and it relies only on the availablepast observed data and statistical models [5 6] The data-driven method can be beneficial when mechanical principles

are not straightforward or when mechanical systems are verycomplex Considering that rotation machinery has complexdegradation processes due to nonlinearity stochasticity andnonstationarity the data-driven method is suitable for esti-mating the RUL of rotating machinery The main existingdata-driven methods are based on regression-based modelsGamma processes Wiener processes or Markovian-basedmodels As the Wiener degradation model is a frequentlyused model its prognostics have been introduced in recentresearch Guan et al modeled the constant-stress accelerateddegradation test as a Wiener process in [7] and simula-tions were conducted to demonstrate the effectiveness ofdegradation modeling Huang et al proposed a nonlinearheterogeneous Wiener process model to characterize degra-dation trajectories in [8] Wang et al presented an adaptivemethod of RUL estimation based on a generalized Wienerdegradation process in [9] The nonlinearity and temporaluncertainty were jointly taken into account in the proposeddegradation model

HindawiShock and VibrationVolume 2017 Article ID 6754968 12 pageshttpsdoiorg10115520176754968

2 Shock and Vibration

However there is little research concerning the construc-tion of an optimal degradation indicator and the selectionof first predicting time (FPT) based on the Wiener processdegradation model It is well known that the extraction andconstruction of a degradation indicator plays an importantrole in RUL prediction The purpose of a degradation indi-cator construction for prognostics is to identify a set ofdegradation indicators that can clearly present the degra-dation process However in most of the current literaturethe selection of a degradation indicator usually dependson the authorsrsquo subjective judgment and experience Thedegradation indicatorrsquos ability to capture the trend of theprogression process has not been discussed Usually thedegradation curve of an excellent degradation indicator hasgood monotonicity which is explained in [10] where anactual gyrorsquos drift data are illustrated In reality howeverthe performance variables at any time take on stochasticproperties because of the differences in the products thisis often not the case for most of the individual productunits where the degradation curve is monotonous Thatis the performance degradation curve is nonmonotonic inpractice In [11] an actual measurement series for a bearingfailure history was collected from a pump in the fieldAlthough the measurement showed a generally increasingtrend there were large fluctuations at multiple places In [12]the practical degradation path of a ball bearing was shownthe degradation curve was quite steady until ball-bearingfailure occurred

To our knowledge the studies on an optimal degradationindicator model are very limited A few exceptions are theworks in [13ndash15] Gomes et al presented a method forcombining the measured parameters into a single indicatorfor monitoring the condition of systems in [13] Liu etal proposed a health-index extraction and optimizationframework requiring only the operating parameters forbattery degradationmodeling and RUL estimation in [14] Toderive optimal prognostic features Liao employed the Parismodel combined with a genetic programming (GP) methodto predict the RUL of bearings in [15] In this literaturethe FPT was not considered while using GP to generate theoptimal prognostic features The time to start predicting wasdefined as the FPT in [16] as shown in Figure 1 If the FPTis not adopted a large amount of data will be required tobuild an optimal degradation indicator which will affect theconvergence and complexity of GP and make the optimaldegradation indicator nonideal Through determining anappropriate FPT the data from the FPT to the failure timecan be utilized to construct an optimal degradation indicatorwhichwill greatly reduce the amount of data and be beneficialin constructing an optimal degradation indicator In additionLiao assumed in [15] that the fault growth of the bearingfollowed the Paris law However the fault growth of thebearing came into being as the result of the accumulationof many small cracks and not the principal crack extensionwhich implied that the Paris law did not apply to bearingsIt is common knowledge that establishing physical modelsfor complicated rotating machinery is difficult Thus only

Degradation processFirst predicting time

005

115

225

335

4

RMS

20 40 60 80 100 1200Time

Figure 1 Degradation process and first predicting time

a data-driven approach can be adopted to estimate theRUL However little reported literature can be found onconstructing an optimal degradation indicator based onthe FPT and addressing the Wiener degradation model forrotating machinery

Based on the above discussions an integrated RUL pre-diction method was proposed based on GP and the conceptsof FPT and Wiener process degradation modeling whichaimed to solve the problem that the degradation progressionmight be nonmonotonic

This research work has the following theoretical andpractical contributions

(i) Our method can easily construct an optimal degra-dation indicator using GP owing to the involvementof the FPT which can greatly reduce the data in theterminator set during GP

(ii) The optimal degradation indicator extracted throughGP has a better monotonic trend than the conven-tional degradation indicator Moreover the optimaldegradation indicator contains more degradationinformation than a conventional degradation indica-tor which characterizes the degradation process well

(iii) Based on the optimal degradation indicator aWiener-process-based degradation model is estab-lished to estimate the RUL To demonstrate theeffectiveness of our method we apply it to a real testbedThe results show that our developed method canmake accurate RUL predictions

The remaining parts of this paper are organized as followsIn Section 2 the related works are surveyed and presentedIn Section 3 we describe the RUL prediction frameworkSection 4 briefly reviews the GP algorithm and the time-domain parameters and Section 5 describes the Wiener pro-cess degradation modeling for RUL prediction Introductionto the experimental equipment and the construction of anoptimal degradation indicator are given in Section 6 TheRUL prediction and the comparison of the proposed methodwith other predictionmethods are presented in Section 7 andSection 8 concludes this paper

Shock and Vibration 3

2 Related Works

Researchers had conducted considerable efforts on RULestimation and developed a variety of prediction meth-ods Owing to their favorable mathematical propertiesand physical interpretations Wiener processes have beenwidely adopted to characterize degrading systems [10 17ndash19] Si et al presented a more general nonlinear Wiener-process-based model for RUL estimation in [20] Unlike theusual monotonic assumption they formulated an analyticalapproximation of the RUL distribution Wang and Carrin [21] considered a Wiener process with random effectsbut without measurement errors for the lifetime modelingof bridge beams Tseng and Peng proposed an integratedWiener process to model the cumulative degradation path ofa productrsquos quality characteristics in [22] Balka et al reviewedsome methods of cure rate models based on the first passagetimes using Wiener processes in [23] Mishra and Vanliin [24] used principal component regression and Wienerprocess degradation modeling for predicting the RUL of astructure from the Lamb wave sensor data The principalcomponent regression was used for extracting damage-sensitive features and the Wiener process was developedto model the random growth Tang et al addressed theeffects of model misspecification of the linearWiener processfor RUL estimation in [25] Tang et al in [26] proposedreal-time RUL prediction based on nonlinear Wiener-baseddegradation processes with measurement errors The simu-lation results showed that considering measurement errorscould significantly improve the accuracy of real-time RULpredictionWang et al proposed an additiveWiener-process-based prognostic model for hybrid deteriorating systemsin [27] by employing an additive Wiener process modelthat consisted of a linear degradation part and a nonlinearpart The above survey of the related works and discussionsposes an interesting challenge for prognostic studies throughWiener-process-based degradation modeling

In terms of feature extraction Kotani et al used GPto search for the terms of polynomial expressions whichwere the features extracted to improve the performance ofpattern recognition in [28] Muni et al in [29] presentedan online feature selection algorithm using GP The pro-posed methodology simultaneously selected a good subsetof features and constructed a classifier Guo et al in [30]proposed an approach-based GP for feature extraction fromraw vibration data The created features were used as theinputs to a neural classifier for the identification of six bearingconditions Bechhoefer et al fused condition indicators in[31] to obtain a degradation indicator achieving fault detec-tion and threshold setting for gears state-space models wereused to estimate the RUL Zhou et al proposed an indirectdegradation indicator construction method for estimatingthe rotating machineryrsquos health conditions in [32] and useda Monte Carlo approach to predict the RUL Sun et al in[33] presented the application of a state-space model for theprognostics of an engineering system subject to degradationA health index was inferred from a set of sensor signals tocharacterize the hidden health state of the system Zeng etal proposed a new equipment degradation state recognition

Wiener process degradation modeling

Vibration data collection

Prediction result

Original degradation

indicator

Optimal degradation

indicator

Multiple degradationindicators

constructedby GP

Selection mutation andcrossover operations

Opt

imal

degr

adat

ion

indi

cato

r sele

ctio

nRU

L pr

edic

tion

Utilizing Bayesian method and EM algorithm to estimate model

parameters

Figure 2 Remaining useful life prediction framework

and fault prognostics method based on the kernel principalcomponent analysis and hidden semi-Markov model in [34]The kernel principal component analysis method was usedto construct an optimal set of prognostic parameters whilepreserving the most important characteristics of the inputdata

From the above survey of related works we can observethat there is a continuing trend to develop an integratedmethod for RUL prediction which can fully utilize theadvantages of multiple methods Owing in part to the lowprediction accuracy of the rotating machinery RUL resultsthis naturally leads to our primary objective of this paperwhich is to improve the RUL prediction accuracy for rotatingmachinery

3 Remaining Useful LifePrediction Framework

To estimate the RUL of the rotating machinery an integratedRUL prediction framework is proposed based on the GP andWiener process degradation modeling as shown in Figure 2The proposed framework can be implemented as per thefollowing steps

Step 1 (vibration data collection) For monitoring rotatingmachinery a number of sensors of various types (veloc-ity transducer acceleration transducer and displacementtransducer) are mounted on the bearings of the rotatingmachinery to measure the initial vibration signals Run-to-failure vibration data are collected providing an overallindication of the mechanical systemrsquos health By calculatingthe probability density functions of the vibration signals themost commonly used vibration parameters can be obtained


Step 2 (optimal degradation indicator extraction) Basedon the acquired condition monitoring data the dimensionparameters (eg root mean square mean value and peakvalue) and dimensionless parameters (eg impulse indexmargin index and kurtosis index) in the time domain arechosen as the original degradation indicators Then multipledegradation indicators are constructed through GP withthe introduction of FPT According to this approach theinitial features are combined and optimized and optimaldegradation indicators are formed after the run-to-failuredata processing under the FPT Then the fitness functionis adopted to measure the performances of the generatedfeatures The optimal degradation indicator can well charac-terize the system health condition

Step 3 (building Wiener-process-based degradation model)To achieve degradation modeling and RUL estimation forrotating machinery the optimal degradation indicator ismodeled as a Wiener process The Wiener-process-baseddegradation is utilized to describe the equipment degradationprocess

Step 4 (utilizing Bayesian method and EM algorithm to esti-mate model parameters) Based on the monitored degrada-tion data a parameter estimation approach for a degradationmodel obtained through the collaboration between Bayesianupdating and the expectation maximization (EM) algorithmis presented The Bayesian method is used to update the driftcoefficient and the EM algorithm is utilized to update allother parameters The obtained estimation in each iterationis unique and optimal

Step 5 (prediction result) Finally a practical case study isprovided to show that the presented approach models thedegradation process estimates the model parameters andgenerates a prediction result

4 Brief Review of Genetic ProgrammingAlgorithm and Time-Domain Parameters

41 Genetic Programming Algorithm GP has been proposedas a machine learning method in different fields and it hasthe advantage of selecting and constructing features Thebasic idea is described below First GP randomly creates aninitial population (generation 0) which consists of a numberof individuals in a tree structure Then a fitness functionis assigned to calculate the fitness value of each individualAccording to the principle of selecting the superior andeliminating the inferior the proximate optimum solutionor the optimal solution for one generation can be foundby selecting genetic operators (selected operator crossoveroperator mutation operator etc) to optimize the populationadaptively

The tree structure is a common representation of a GPindividual Each individual is amath expression using the treestructure As shown in Figure 3 the individual is representedasradic119911 times (119910 + 2) + ln119909

The nodes of the tree are classified into two types A nodelocated inside the tree is the operator and the nodes at the

yz

sqrt

2

times

+

+

x

ln

Figure 3 Form of genetic programming individual

leaves of the tree are the terminators making mathematicalexpressions easy to evolve and evaluate For example theoperators in Figure 3 are the basic operators such as ldquo+rdquo ldquotimesrdquoldquolnrdquo and ldquosqrtrdquo The terminators contain variables ldquo119909rdquo ldquo119910rdquoand ldquo119911rdquo and the constant ldquo2rdquoWe can use these operators andterminators to combine more complicated expressions

In practical application a few details must be determinedbefore running the GP algorithm

(1) Selection of the operator set this paper uses ldquo+rdquoldquominusrdquo ldquolowastrdquo ldquordquo ldquosqrtrdquo ldquoexprdquo ldquologrdquo and ldquofabsrdquo as theoperator set

(2) Selection of the terminals set five conventionaldimensionless parameters are used as the terminalsset in this paper

(3) Design of the fitness function the fitness functionis a measure for evaluating the fitness levels of theindividuals

(4) Setting the population size evolutionary generationprobability settings of genetic operators and termina-tion criterion

42 Time-Domain Parameters Most of the existing prognos-tic techniques use condition-monitoring indexes to representhealth Using the vibration signals in the time domainto extract the degradation indicators is the most com-mon method In this paper the dimension parameters anddimensionless parameters are utilized to characterize thedynamics and nonlinearity of the degradation progressionThe dimension parameters are the square root of amplituderoot mean square mean value and peak value which aredefined as follows [35]

119909119889 = [int+infinminusinfin

|119909|119897 119901 (119909) 119889119909]1119897 =

119883119903 119897 = 12 119883 119897 = 1119883rms 119897 = 2119883119901 119897 997888rarr infin

(1)


where 119909 denotes the vibration amplitude 119901(119909) denotes theprobability density function of vibration amplitude119883119903 is thesquare root of amplitude119883 is themean value119883rms is the rootmean square and119883119901 is the peak value


radic|119909|119901 (119909) 119889119909]2 = ( 1119873119873sum119894=1

radic10038161003816100381610038161199091198941003816100381610038161003816)2

119883 = intinfinminusinfin

|119909| 119901 (119909) 119889119909 = 1119873119873sum119894=1

10038161003816100381610038161199091198941003816100381610038161003816 119883rms = radicintinfin

minusinfin1199092119901 (119909) 119889119909 = radic 1119873

119873sum119894=1

1199092119894 119883119901 = lim

119897rarrinfin[int+infinminusinfin

|119909|119897 119901 (119909) 119889119909]1119897 = 119909max

(2)

The dimensionless parameter is a better diagnostic char-acteristic The dimensionless parameters are defined as fol-lows


|119909|119897 119901 (119909) 119889119909]1119897[int+infinminusinfin

|119909|119898 119901 (119909) 119889119909]1119898 (3)

Specifically

(i) if 119897 = 2119898 = 1 have waveform index 119878119891(ii) if 119897 rarr infin119898 = 1 have impulse index 119868119891(iii) if 119897 rarr infin119898 = 12 have margin index 119862119871119891(iv) if 119897 rarr infin119898 = 2 have peak index 119862119891(v) kurtosis index 119870V = 1205731198834rms In the formula 120573

is a dimension parameter named kurtosis which isdefined as 120573 = intinfin

minusinfin1199094119901(119909)119889119909 119883rms is a dimension

parameter named root mean square

In practice the above-mentioned dimensionless param-eters are frequently used The dimension parameters varyin different work conditions and may be influenced bydisturbances (eg speed load and sensitivity of the instru-ment) that cause data deviation However the dimensionlessparameters are sensitive to faults instead of work conditionsMoreover they can classify some types of faults correctlyand efficiently In consideration of the advantages of dimen-sion and dimensionless parameters an approach to applyGP combining dimension parameters with dimensionlessparameters is proposed to construct the optimal degradationindicator which can characterize the degradation processmore comprehensively

5 Wiener Process Degradation Modeling withRandom Effects

Asmentioned abovemany variants of theWienermodel havebeen reported in literature In the model considered in thiswork the parameters are estimated using Bayesian updating

and the EM algorithm The Wiener process is typically usedfor modeling degradation processes where the degradationincreases linearly in time with random noise The rate ofdegradation is characterized by the drift coefficient

51 Degradation Model In general the Wiener processdegradationmodeling with random effect can be representedas

119883 (119905) = 120573119905 + 120590119861 (119905) (4)

where 120573 is the drift coefficient 120590 is the diffusion coefficientand 119861(119905) is the standard Brownian motion representing thestochastic dynamics of the degradation process 120573 is consid-ered as a random variable following a normal distribution of119873(120583120573 1205902120573) which represents individual differences betweenthe pieces of equipment Thus the estimated parameters inthe model have 120583120573 1205902120573 and 1205902 For convenience we denoteΘ = (120583120573 1205902120573 1205902) as a parameter vector to represent the modelparameters

Lifetime is usually defined as119879 = inf119905 | 119883(119905) ge 120578 whichmeans that the random degradation process 119883(119905) 119905 ge 0first reaches a prespecified failure threshold 120578The probabilitydensity function and the expectation of lifetime 119879 can bedirectly obtained as

119891119879 (119905) = 120578radic21205871199053 (1205902

120573119905 + 1205902)

times exp(minus (120578 minus 120583120573119905)22119905 (1205902120573119905 + 1205902))

119864 (119879) = 119864 (119864 (119879 | 120573)) = 119864( 120578120573)= 1205781205902120573

exp(minus 120583212057321205902120573

)int1205831205730

exp( 119909221205902120573

)119889119909= radic2120578120590120573 119863( 120583120573radic2120590120573)

(5)

where 119863(119911) = exp(minus1199112) int1199110exp(1199092)119889119909 is the Dawson integral

for a real 119911 which is known to exist

52Model Parameter Estimation Based on EMAlgorithm [36]To estimate the parameters we assume that there are 119899 testeditems and that the degradation data 119883119894 = (1199091198941 119909119894119898)1015840of the 119894th item are available where 1199091198941 119909119894119898 denotes thedegradation observations at time 1199051 119905119898 For simplicity let119883 = (1198831 119883119899)1015840 andΩ = (1205731 120573119899) where119883 denotes thedegradation dataset and 120573119894 denotes the drift coefficient of the119894th item


For a given 119883119894 and Θ(119896) the sampling of the 119894th item isdistributed by the following expression

119891 (119883119894 | 120573119894 Θ) = 1prod119898119895=1radic21205871205902Δ119905119895times exp(minus 119898sum

119895=1

(Δ119909119894119895 minus 120573119894Δ119905119895)221205902Δ119905119895 )(6)

with Δ119909119894119895 = 119909119894119895minus119909119894119895minus1 and Δ119905119895 = 119905119895 minus119905119895minus1 where Δ119909119894119895 denotesthe degradation incremental of the 119894th item from time 119905119895minus1 to119905119895 and Δ119905119895 denotes the time interval

The log-likelihood function can be written as

ℓ (Θ | 119883Ω) = 119899sum119894=1

[ln119901 (119883119894 | 120573119894 Θ) + ln119901 (120573119894 | Θ)]= minus12

119899sum119894=1

[[(119898 + 1) ln 2120587 + 119898sum119895=1

lnΔ119905119895 + 119898 ln1205902

+ 119898sum119895=1

(Δ119909119894119895 minus 120573119894Δ119905119895)21205902Δ119905119895 + ln1205902120573 + (120573119894 minus 120583120573)21205902120573

]] (7)

Let us assume thatΘ(119896) = 120583(119896)120573 1205902(119896)120573

1205902(119896) is the estimatein the 119894th step based on 119883 With 119883119894 and Θ(119896) known theposterior distribution of 120573119894 will still be normal that is 120573119894 sim119873(120583(119896)119894 1205902(119896)119894 )

In the Bayesian framework the posterior distribution of120573119894 can be updated via the Bayesian rule as follows

119901 (120573119894 | 119883119894 Θ119896) prop 119901 (119883119894 | 120573119894 Θ119896) 119901 (120573119894 | Θ119896)prop exp[[minus

119898sum119895=1

(Δ119909119894119895 minus 120573119894Δ119905119895)221205902(119896)Δ119905119895 ]] exp[[[minus(120573119894 minus 120583(119896)120573 )

2

21205902(119896)120573

]]]prop exp

minus12 [[[

119898sum119895=1

1205732119894 Δ1199051198951205902(119896)

minus 119898sum119895=1

2120573119894Δ1199091198941198951205902(119896) + 1205732119894 minus 2120573119894120583(119896)120573 + (120583(119896)120573)2

1205902(119896)120573

]]]

prop expminus

12 [[(1199051198981205902(119896) minus 11205902(119896)

120573

)1205732119894 minus 2( 1199091198941198981205902(119896) + 120583(119896)1205731205902(119896)120573

)120573119894]]

prop expminus[120573119894 minus (1199091198941198981205902(119896)120573 + 120583(119896)

1205731205902(119896)) (1199051198981205902(119896)120573 + 1205902(119896))]2

21205902(119896)1205902(119896)120573

(1199051198981205902(119896)120573 + 1205902(119896))

(8)

Owing to the property of the normal distribution of 120573119894 |119883119894 Θ119896 we obtain119901 (120573119894 | 119883119894 Θ119896) = 1

radic21205871205902(119896)119894 exp[[minus(120573119894 minus 120583(119896)119894 )221205902(119896)119894 ]] (9)

with

120583(119896)119894 = 1199091198941198981205902(119896)120573 + 120583(119896)1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) 1205902(119896)119894 = 1205902(119896)

1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) (10)

where we can learn that the posterior estimation of 120573119894 can beeasily updated once a new observation is available Now letus focus on calculating the maximum-likelihood estimationΘ = (120583120573 2120573 2) using the EM algorithmΘ can be estimatedthrough two steps the E-step and the M-step

In the E-step the expectation 119864[ℓ(Θ | 119883Θ(119896))]can becomputed as follows

119864 [ℓ (Θ | 119883Θ(119896))] = minus12119899sum119894=1

[[[(119898 + 1) ln 2120587 + 119898sum

119895=1

lnΔ119905119895 + 119898sdot ln1205902+ 119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 ((120583(119896)119894 )2 + 1205902(119896)119894 )1205902Δ119905119895

+ ln1205902120573 + (120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583120573 + 12058321205731205902120573

]]]

(11)

Then in the M-step letting 120597119864[ℓ(Θ | 119883Θ(119896))]120597120579 = 0we obtain Θ(119896+1) as follows120583(119896+1)120573 = 1119899

119899sum119894=1

120583(119896)119894 1205902(119896+1)120573 = 1119899

119899sum119894=1

[(120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583(119896+1)120573 + (120583(119896+1)120573 )2] 1205902(119896+1) = 1119899119898

sdot 119899sum119894=1

119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 (120583(119896)119894 )2 + 1205902(119896)119894Δ119905119895

(12)

The above steps are iterated multiple times to produce asequence Θ(0) Θ(1) Θ(2) of increasingly good approxi-mations Θ = (120583120573 2120573 2) For each iteration the analyticalsolution for updating the model parameters is derived Theiterations are usually terminated when the EM algorithmconverges

6 Experimental Demonstrations

In this section we provide a practical case study to illustratethe application of our model and compare the performanceof our model with that of other models


Figure 4 Test bed

Figure 5 Rolling element bearing

61 Introduction to the Experimental Equipment and DataAcquisition The bearing is a key device in rotating machin-ery and its operating state has a direct influence on therotating machinery condition The rotating machinery oper-ation at very high speeds can lead to bearing wear As thewear accumulates the bearing will become deformed andsuch deformation may lead to incipient faults in the rotatingmachinery The increasing faults result in the failure of thebearing and of the rotating machinery Past data show thatalmost 80 of the failures in rotating machinery result fromthe wear of rolling element bearings which were extensivelyinvestigated in the literature

A picture of a test rig is provided in Figure 4 and it showsthe major components of the test rig such as the industrialmotor shaft and test bearingThe bearings are instrumentedwith accelerometers in both the axial and radial directions onthe bearing housing An illustration of a bearing is providedin Figure 5

We use the vibration accelerometer to gather the vibra-tion signals The sampling frequency is 1 kHz and eachsample contains 4096 data points The failure threshold isdetermined by the vibration level ISO 2372 and ISO 10816The rotation speed is maintained at 1800 RPM A radialload of 25MPa pressure is applied onto the bearing andtwelve failure data of bearings are collected Among themeight bearings resulted in outer race failures which are

Vibr

atio

n am

plitu

de

minus04minus03minus02minus01

001020304

500 1000 1500 2000 2500 3000 3500 40000Sampling number

Figure 6 Vibration wave under the normal condition

Vibr

atio

n am

plitu

deminus08minus06minus04minus02

002040608

500 1000 1500 2000 2500 3000 3500 40000Sampling number

Figure 7 Vibration wave in the failure stage

Table 1 Failure data of bearings

Bearing number 1 2 3 4 5 6 7 8Failure time (h) 900 918 954 876 922 894 988 1002

summarized in Table 1 The failure times are recorded as thetimes at which the observed values cross the threshold Thetotal life expectancy of the eight bearings is 932 h

The monitoring data of the first sample are illustratedin Figures 6 and 7 and show the time-domain vibrationwaves under the normal condition and in the failure stagerespectively For our monitored rotating machinery with aterminated life of 900 h 450 monitoring data were collectedat regular condition monitoring intervals of 2 h

By calculating the vibration signal four dimensionparameters and five dimensionless parameters were mea-sured For convenience the square root of amplitude meanvalue root mean square peak value waveform indeximpulse index margin index peak index and kurtosis indexare represented by 119878119872 119877 119875119882 119868 119879 119871 and119870 respectivelyBy visually inspecting the nine features no obvious trendwas found in an individual feature Among these features thesquare root of amplitude showed the best increasing trend asshown in Figure 8 It showed a smooth trend at the beginningwhich however fluctuated toward the end of life

The degradation parameter possessing the worst mono-tonicity was the mean value as shown in Figure 9 A suddenincrease in the mean value appeared only just before thebearing failed

62 Optimal Degradation Indicator Extraction The FPTselection results are shown in Figures 10 and 11 It is observed


0

01

02

03

04

05

06

07

Squa

re ro

ot o

f am

plitu

de

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 8 Square root of amplitude

times10minus3

minus4

minus3

minus2

minus1

0

1

2

3

4

Mea

n va

lue

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 9 Mean value

that the waveform index and kurtosis index are sensitive tothe incipient faults in the bearing The waveform index andkurtosis indexwere divided into two stages by the FPT BeforeFPT the values of the waveform index and kurtosis indexwere stable however they increased suddenly after the FPTTherefore FPT = 630 h indicates the initial time of the degra-dation process The degradation data after FPT are used toconstruct new degradation parameters

The specific steps to achieve an optimal degradationindicator for application in RUL prediction are as follows

Step 1 Experimental dataset acquisition is as follows Run-to-failure data of bearings are obtained

Step 2 Basic parameters of GP are determined Table 2 liststhe basic parameters of GP

Step 3 119872 initial individuals are generated randomly Theinitial individual is expressed in 119873 = 119891(119878119872 119877 119875119882 119868119872119871119870) represented new degradation parameter which means

Wav

efor

m in

dex

Degradation processFPT

12

125

13

135

14

145

15

155

16

165

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 10 Waveform index

Kurt

osis

inde

x


0

2

4

6

8

10

12

14

16

18

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 11 Kurtosis index

that the new degradation parameter is a function thatdepends on the square root of amplitude mean value rootmean square peak value waveform index impulse indexmargin index peak index and kurtosis index

Step 4 The data of the degradation index after FPT = 630 hare computed as the inputs to119873Themonotonicity is used forevaluating the fitness levels of the individuals In this paperthe fitness function [15] is defined by

fitness = 10038161003816100381610038161003816100381610038161003816of 119889119889119865 gt 0119899 minus 1 minus of 119889119889119865 lt 0119899 minus 110038161003816100381610038161003816100381610038161003816 (13)

where 119899 is the number of observations in a period 119865represents a feature and 119889119889119865 is the derivative The larger


Table 2 Basic parameters of GP

Parameters SettingsObject Building the new degradation indicator

Set of terminals119878 (square root of amplitude)119872 (mean value) 119877 (root mean square)119875 (peak value)119882 (waveform index) 119868 (impulse index) 119879 (marginindex) 119871 (peak index)119870 (kurtosis index)

Set of functions + minus lowast exp log sqrtParameters Population size119872 = 1000

Evolutionary generation 119866 = 50Probability settings of genetic operator Crossover probability 085

Mutation probability 015Method of selection Tournament selection method the size of tournament is fiveTermination criterion Required maximum of evolutionary generationMax depth of tree Six

Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)

Figure 12 Optimal degradation indicator

the fitness value the better the monotonicity of the newdegradation parameter

Step 5 According to the genetic parameters new individualsare generated using the following series of actions reproduc-tion crossover and mutation

Step 6 Steps 3 4 and 5 are executed repeatedly until thealgorithm running termination criterion is met

Analyzing the operation results of GP the best indi-vidual is

119873 = ln (119879 minus 119890119878) + (119882119875 + 119877 lowast radic119879)119878 (14)

where S R P W and 119879 represent the square root ofamplitude root mean square peak value waveform indexand margin index respectively

Figure 12 shows the plot of the optimal degradation indi-cator obtained from the FPT-to-failure test A clear increasingtrend is exhibited by the optimal degradation parameter from

Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)

Figure 13 Calculated optimal degradation indicator data of eightbearings

the FPT concept RUL prediction can be implemented bymodeling the optimal degradation indicator

7 RUL Prediction

For illustration the optimal degradation parameter is usedas the prediction index to collect the monitoring data forthe remaining seven bearings The collected monitoring dataof eight bearings are illustrated in Figure 13 and are used todemonstrate the method developed in this work The valuesof the optimal degradation indicator after 119905 = 630 h are inputinto the Wiener model for RUL prediction Now we modelthe degradation process as a Wiener process based on theoptimal degradation indicator The failure threshold is 825When the degradation value increases to 825 the bearingwillbe considered to have failed

Based on the data shown in Figure 13 and the parameterestimationmethod presented in the above section (parameterestimation is illustrated after each iteration until convergencein Figure 14) we can observe that the parameters are stable


120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029

1 2 3 4 5 6 70Iteration number(a)

times10minus3

25

3

35

4

45

5

120590120573

1 2 3 4 5 6 70Iteration number

(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590

1 2 3 4 5 6 70Iteration number(c)

Figure 14 Parameter estimation (a) 120583120573 (b) 120590120573 and (c) 120590

over four iterations 120583120573 = 00288 120573 = 00027 and =01270Once the estimated parameters are obtained the expec-

tation of RUL can be calculated

119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)

The entire life is calculated with 119879EoL = FPT + RUL Wecan predict that the entire life of the bearing is 919 h whichagrees well with the actual mean experimental results Thisdemonstrates that our method can accurately estimate theRUL

For further verification of the effectiveness of our modelfor RUL prediction the Wiener-model-based conventionaldegradation indicator using the FPT is used to predict theRUL of the eight bearings as well The model is referred to asModel 1 in this paper The RMS is used to construct Model1 We have 120583120573 = 00427 120573 = 00210 and = 00152

through the parameter estimations and the prediction resultis calculated as 705 h for Model 1 which has a relatively largedifference with the experimental result We can see that theobtained result of our model is better than the one givenby the Wiener model based on the conventional prognosticfeatures This demonstrates that our model can improve theaccuracy of RUL estimation In addition the estimated meandegradation path based on the established model can beexpressed as 119864[119883(119905)] = 120583120573119905 The estimated mean degradationpaths using our model and Model 1 are shown in Figure 15

We can see from Figure 15 that the predicted results ofour model gradually match with the actual sample meanwhich illustrates that our model has a good fitting degreeHowever Model 1 does not consider the monotonicity ofthe degradation index which makes the RUL demonstrate agreater variation than the actual mean experimental resultThe experiment demonstrates that our proposed integratedRUL prediction method can work well and efficiently

Shock and Vibration 11D

egra

datio

n da

ta

Sample meanOur modelModel 1

0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)

Figure 15 Comparison of the estimated mean degradation paths

8 Conclusions

In order to improve the prediction accuracy of rotatingmachinery an integrated RUL prediction method based onGP andWiener process degradation modeling was proposedin this paper The GP algorithm was used to find a betterdegradation indicator using the concept of FPT By selectingan appropriate FPT value small amounts of data couldbe used to construct the optimal degradation indicatorand a better degradation indicator could be obtained AWienermodel was proposed for RUL prediction based on theobtained optimal degradation indicator As the input to thedegradation model the optimal degradation indicator fusingmultiple vibration features could contain more vibrationsignatures and provide a more noticeable trend than theconventional degradation indicators Using the measureddata the parameter estimations for the stochastic parametersin themodel were updated recursively by using the conditionmonitoring observations based on the Bayesian method andEM The expectation of the predicted RUL was calculatedbased on the proposed model and the estimated meandegradation path was explicitly derived Experimental resultsindicated that the method could effectively improve theaccuracy of RUL prediction

Although we demonstrated the usefulness of our pro-posed model there are still many open problems that mustbe studied For example in this case study eight bearingsthat were tested resulted in outer race failures This impliedthat the health and prediction algorithm was evaluated withrespect to only one failure mode of the bearing In realitythe defect can be either an inner race defect or a rollerelement defect Ourmethod has not been fully evaluatedwithrespect to multiple failure modes In addition the presentedmodel needs multiple history datasets of similar equipmentwhichmay take a long time to obtainThese problems requireconsiderable further research

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research was partially supported by the NSFC underGrants no 61473094 and no 61673127 the Internationaland Hong Kong Macao amp Taiwan Collaborative InnovationPlatform and Major International Cooperation Projects ofColleges in the Guangdong Province (no 2015KGJHZ026)Thework described in this paper was also partially supportedby the open foundation of the Guangdong Provincial KeyLaboratory of Petrochemical Equipment Fault Diagnosisunder Grants no GDUPTKLAB201604 and no GDUPTK-LAB201603 the Technical Project of Maoming City (no201618) and the Guangdong Province Natural Science FundProject (no 2016A030313823)

References

[1] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics state of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[2] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[3] A S Y Heng Intelligent prognostics of machinery health utilisingsuspended condition monitoring data [PhD thesis] QueenslandUniversity of Technology Brisbane Australia 2009

[4] Z-Q Wang C-H Hu W Wang and X-S Si ldquoAn additivewiener process-based prognostic model for hybrid deteriorat-ing systemsrdquo IEEE Transactions on Reliability vol 63 no 1 pp208ndash222 2014

[5] J Z Sikorska M Hodkiewicz and L Ma ldquoPrognostic mod-elling options for remaining useful life estimation by industryrdquoMechanical Systems and Signal Processing vol 25 no 5 pp1803ndash1836 2011

[6] X-S Si W Wang C-H Hu and D-H Zhou ldquoRemaininguseful life estimationmdasha review on the statistical data drivenapproachesrdquo European Journal of Operational Research vol 213no 1 pp 1ndash14 2011

[7] Q Guan Y Tang andA Xu ldquoObjective Bayesian analysis accel-erated degradation test based on Wiener process modelsrdquoApplied Mathematical Modelling vol 40 no 4 pp 2743ndash27552016

[8] Z Huang Z Xu W Wang and Y Sun ldquoRemaining useful lifeprediction for a nonlinear heterogeneous wiener process modelwith an adaptive driftrdquo IEEE Transactions on Reliability vol 64no 2 pp 687ndash700 2015

[9] XWang N Balakrishnan and B Guo ldquoResidual life estimationbased on a generalized Wiener degradation processrdquo ReliabilityEngineering and System Safety vol 124 pp 13ndash23 2014

[10] X-S Si W Wang C-H Hu M-Y Chen and D-H Zhou ldquoAWiener-process-based degradationmodel with a recursive filteralgorithm for remaining useful life estimationrdquo MechanicalSystems and Signal Processing vol 35 no 1-2 pp 219ndash237 2013


[11] Z Tian ldquoAn artificial neural network method for remaininguseful life prediction of equipment subject to condition mon-itoringrdquo Journal of Intelligent Manufacturing vol 23 no 2 pp227ndash237 2012

[12] Z-X Zhang X-S Si and C-H Hu ldquoAn age- and state-dependent nonlinear prognostic model for degrading systemsrdquoIEEE Transactions on Reliability vol 64 no 4 pp 1214ndash12282015

[13] J P P Gomes R K H Galvao T Yoneyama and B P LeaoldquoA new degradation indicator based on a statistical anomalyapproachrdquo IEEE Transactions on Reliability vol 65 no 1 pp326ndash335 2016

[14] D Liu J ZhouH Liao Y Peng andX Peng ldquoAhealth indicatorextraction and optimization framework for lithium-ion batterydegradation modeling and prognosticsrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 45 no 6 pp 915ndash928 2015

[15] L Liao ldquoDiscovering prognostic features using genetic pro-gramming in remaining useful life predictionrdquo IEEE Transac-tions on Industrial Electronics vol 61 no 5 pp 2464ndash2472 2014

[16] N Li Y Lei J Lin and S X Ding ldquoAn improved exponentialmodel for predicting remaining useful life of rolling elementbearingsrdquo IEEE Transactions on Industrial Electronics vol 62no 12 pp 7762ndash7773 2015

[17] X Wang ldquoWiener processes with random effects for degrada-tion datardquo Journal of Multivariate Analysis vol 101 no 2 pp340ndash351 2010

[18] Z-S Ye Y Wang K-L Tsui and M Pecht ldquoDegradation dataanalysis usingwiener processeswithmeasurement errorsrdquo IEEETransactions on Reliability vol 62 no 4 pp 772ndash780 2013

[19] X-S Si W Wang M-Y Chen C-H Hu and D-H Zhou ldquoAdegradation path-dependent approach for remaining useful lifeestimation with an exact and closed-form solutionrdquo EuropeanJournal of Operational Research vol 226 no 1 pp 53ndash66 2013

[20] X-S Si W Wang C-H Hu D-H Zhou and M G PechtldquoRemaining useful life estimation based on a nonlinear diffu-sion degradation processrdquo IEEE Transactions on Reliability vol61 no 1 pp 50ndash67 2012

[21] WWang andM Carr ldquoAn adapted Brownionmotionmodel forplant residual life predictionrdquo in Proceedings of the Prognosticsand System Health Management Conference (PHM rsquo10) pp 1ndash7Macao January 2010

[22] S-T Tseng and C-Y Peng ldquoOptimal burn-in policy by using anintegrated Wiener processrdquo IIE Transactions vol 36 no 12 pp1161ndash1170 2004

[23] J Balka A F Desmond and P D McNicholas ldquoReview andimplementation of cure models based on first hitting times forWiener processesrdquo Lifetime Data Analysis vol 15 no 2 pp 147ndash176 2009

[24] S Mishra and O A Vanli ldquoRemaining useful life estimationwith lamb-wave sensors based on wiener process and principalcomponents regressionrdquo Journal of Nondestructive Evaluationvol 35 no 1 article 11 2016

[25] S Tang X Guo and Z Zhou ldquoMis-specification analysis of lin-ear Wiener process-based degradation models for the remain-ing useful life estimationrdquo Proceedings of the Institution ofMechanical Engineers Part O Journal of Risk and Reliability vol228 no 5 pp 478ndash487 2014

[26] S-J Tang X-S Guo C-Q Yu Z-J Zhou Z-F Zhou and B-C Zhang ldquoReal time remaining useful life prediction based on

nonlinear Wiener based degradation processes with measure-ment errorsrdquo Journal of Central South University vol 21 no 12pp 4509ndash4517 2014


[28] M Kotani S Ozawa M Nakai and K Akazawa ldquoEmergenceof feature extraction function using genetic programmingrdquo inProceedings of the 3rd International Conference on Knowledge-Based Intelligent Information Engineering Systems (KES rsquo99) pp149ndash152 September 1999

[29] D P Muni N R Pal and J Das ldquoGenetic programmingfor simultaneous feature selection and classifier designrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 36 no 1 pp 106ndash117 2006

[30] H Guo L B Jack and A K Nandi ldquoFeature generation usinggenetic programming with application to fault classificationrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 35 no 1 pp 89ndash99 2005

[31] S Kadry ldquoDiagnostics and prognostic of engineering systemsmethods and techniquesrdquo in Data Driven Prognostics for Rotat-ing Machinery E Bechhoefer Ed chapter 6 pp 1ndash13 IGIGlobal Hershey Pa USA 2012

[32] Y Zhou Y Sun J Mathew R Wolff and L Ma ldquoLatentdegradation indicators estimation and prediction a MonteCarlo approachrdquoMechanical Systems and Signal Processing vol25 no 1 pp 222ndash236 2011

[33] J Sun H Zuo W Wang and M G Pecht ldquoApplication of astate space modeling technique to system prognostics based ona health index for condition-based maintenancerdquo MechanicalSystems and Signal Processing vol 28 no 3 pp 585ndash596 2012

[34] Q Zeng J Qiu G Liu and X Tan ldquoResearch on equipmentdegradation state recognition and fault prognostics methodbased on KPCA-hidden semi-Markov modelrdquo Chinese Journalof Scientific Instrument vol 30 no 7 pp 1341ndash1346 2009

[35] Q H Zhang Q Hu G Sun et al ldquoConcurrent fault diagnosisfor rotating machinery based on vibration sensorsrdquo Interna-tional Journal of Distributed Sensor Networks vol 9 no 1 pp59ndash72 2013

[36] A P Dempster N M Laird and D B Rubin ldquoMaximumlikelihood from incomplete data via the EM algorithmrdquo Journalof the Royal Statistical Society Series B vol 39 no 1 pp 1ndash381977

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



However there is little research concerning the construc-tion of an optimal degradation indicator and the selectionof first predicting time (FPT) based on the Wiener processdegradation model It is well known that the extraction andconstruction of a degradation indicator plays an importantrole in RUL prediction The purpose of a degradation indi-cator construction for prognostics is to identify a set ofdegradation indicators that can clearly present the degra-dation process However in most of the current literaturethe selection of a degradation indicator usually dependson the authorsrsquo subjective judgment and experience Thedegradation indicatorrsquos ability to capture the trend of theprogression process has not been discussed Usually thedegradation curve of an excellent degradation indicator hasgood monotonicity which is explained in [10] where anactual gyrorsquos drift data are illustrated In reality howeverthe performance variables at any time take on stochasticproperties because of the differences in the products thisis often not the case for most of the individual productunits where the degradation curve is monotonous Thatis the performance degradation curve is nonmonotonic inpractice In [11] an actual measurement series for a bearingfailure history was collected from a pump in the fieldAlthough the measurement showed a generally increasingtrend there were large fluctuations at multiple places In [12]the practical degradation path of a ball bearing was shownthe degradation curve was quite steady until ball-bearingfailure occurred

To our knowledge the studies on an optimal degradationindicator model are very limited A few exceptions are theworks in [13ndash15] Gomes et al presented a method forcombining the measured parameters into a single indicatorfor monitoring the condition of systems in [13] Liu etal proposed a health-index extraction and optimizationframework requiring only the operating parameters forbattery degradationmodeling and RUL estimation in [14] Toderive optimal prognostic features Liao employed the Parismodel combined with a genetic programming (GP) methodto predict the RUL of bearings in [15] In this literaturethe FPT was not considered while using GP to generate theoptimal prognostic features The time to start predicting wasdefined as the FPT in [16] as shown in Figure 1 If the FPTis not adopted a large amount of data will be required tobuild an optimal degradation indicator which will affect theconvergence and complexity of GP and make the optimaldegradation indicator nonideal Through determining anappropriate FPT the data from the FPT to the failure timecan be utilized to construct an optimal degradation indicatorwhichwill greatly reduce the amount of data and be beneficialin constructing an optimal degradation indicator In additionLiao assumed in [15] that the fault growth of the bearingfollowed the Paris law However the fault growth of thebearing came into being as the result of the accumulationof many small cracks and not the principal crack extensionwhich implied that the Paris law did not apply to bearingsIt is common knowledge that establishing physical modelsfor complicated rotating machinery is difficult Thus only

Degradation processFirst predicting time

005

115

225

335

4

RMS

20 40 60 80 100 1200Time

Figure 1 Degradation process and first predicting time

a data-driven approach can be adopted to estimate theRUL However little reported literature can be found onconstructing an optimal degradation indicator based onthe FPT and addressing the Wiener degradation model forrotating machinery

Based on the above discussions an integrated RUL pre-diction method was proposed based on GP and the conceptsof FPT and Wiener process degradation modeling whichaimed to solve the problem that the degradation progressionmight be nonmonotonic

This research work has the following theoretical andpractical contributions

(i) Our method can easily construct an optimal degra-dation indicator using GP owing to the involvementof the FPT which can greatly reduce the data in theterminator set during GP

(ii) The optimal degradation indicator extracted throughGP has a better monotonic trend than the conven-tional degradation indicator Moreover the optimaldegradation indicator contains more degradationinformation than a conventional degradation indica-tor which characterizes the degradation process well

(iii) Based on the optimal degradation indicator aWiener-process-based degradation model is estab-lished to estimate the RUL To demonstrate theeffectiveness of our method we apply it to a real testbedThe results show that our developed method canmake accurate RUL predictions

The remaining parts of this paper are organized as followsIn Section 2 the related works are surveyed and presentedIn Section 3 we describe the RUL prediction frameworkSection 4 briefly reviews the GP algorithm and the time-domain parameters and Section 5 describes the Wiener pro-cess degradation modeling for RUL prediction Introductionto the experimental equipment and the construction of anoptimal degradation indicator are given in Section 6 TheRUL prediction and the comparison of the proposed methodwith other predictionmethods are presented in Section 7 andSection 8 concludes this paper


2 Related Works





Prediction result


indicator

Optimal degradation

indicator


constructedby GP


Opt

imal

degr

adat

ion

indi

cato

r sele

ctio

nRU

L pr

edic

tion


parameters
















yz

sqrt

2

times

+

+

x

ln










|119909|119897 119901 (119909) 119889119909]1119897 =

119883119903 119897 = 12 119883 119897 = 1119883rms 119897 = 2119883119901 119897 997888rarr infin

(1)




radic|119909|119901 (119909) 119889119909]2 = ( 1119873119873sum119894=1

radic10038161003816100381610038161199091198941003816100381610038161003816)2


|119909| 119901 (119909) 119889119909 = 1119873119873sum119894=1



119873sum119894=1

1199092119894 119883119901 = lim


|119909|119897 119901 (119909) 119889119909]1119897 = 119909max

(2)




|119909|119898 119901 (119909) 119889119909]1119898 (3)

Specifically










119883 (119905) = 120573119905 + 120590119861 (119905) (4)



119891119879 (119905) = 120578radic21205871199053 (1205902

120573119905 + 1205902)


119864 (119879) = 119864 (119864 (119879 | 120573)) = 119864( 120578120573)= 1205781205902120573

exp(minus 120583212057321205902120573

)int1205831205730

exp( 119909221205902120573

)119889119909= radic2120578120590120573 119863( 120583120573radic2120590120573)

(5)







119895=1

(Δ119909119894119895 minus 120573119894Δ119905119895)221205902Δ119905119895 )(6)



ℓ (Θ | 119883Ω) = 119899sum119894=1

[ln119901 (119883119894 | 120573119894 Θ) + ln119901 (120573119894 | Θ)]= minus12

119899sum119894=1

[[(119898 + 1) ln 2120587 + 119898sum119895=1

lnΔ119905119895 + 119898 ln1205902

+ 119898sum119895=1


]] (7)





119898sum119895=1


2

21205902(119896)120573

]]]prop exp

minus12 [[[

119898sum119895=1

1205732119894 Δ1199051198951205902(119896)

minus 119898sum119895=1

2120573119894Δ1199091198941198951205902(119896) + 1205732119894 minus 2120573119894120583(119896)120573 + (120583(119896)120573)2

1205902(119896)120573

]]]

prop expminus

12 [[(1199051198981205902(119896) minus 11205902(119896)

120573

)1205732119894 minus 2( 1199091198941198981205902(119896) + 120583(119896)1205731205902(119896)120573

)120573119894]]


1205731205902(119896)) (1199051198981205902(119896)120573 + 1205902(119896))]2

21205902(119896)1205902(119896)120573

(1199051198981205902(119896)120573 + 1205902(119896))

(8)



with

120583(119896)119894 = 1199091198941198981205902(119896)120573 + 120583(119896)1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) 1205902(119896)119894 = 1205902(119896)

1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) (10)



119864 [ℓ (Θ | 119883Θ(119896))] = minus12119899sum119894=1

[[[(119898 + 1) ln 2120587 + 119898sum

119895=1

lnΔ119905119895 + 119898sdot ln1205902+ 119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 ((120583(119896)119894 )2 + 1205902(119896)119894 )1205902Δ119905119895

+ ln1205902120573 + (120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583120573 + 12058321205731205902120573

]]]

(11)


119899sum119894=1

120583(119896)119894 1205902(119896+1)120573 = 1119899

119899sum119894=1

[(120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583(119896+1)120573 + (120583(119896+1)120573 )2] 1205902(119896+1) = 1119899119898

sdot 119899sum119894=1

119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 (120583(119896)119894 )2 + 1205902(119896)119894Δ119905119895

(12)





Figure 4 Test bed





Vibr

atio

n am

plitu

de


001020304

500 1000 1500 2000 2500 3000 3500 40000Sampling number


Vibr

atio

n am

plitu


002040608

500 1000 1500 2000 2500 3000 3500 40000Sampling number










0

01

02

03

04

05

06

07

Squa

re ro

ot o

f am

plitu

de

100 200 300 400 500 600 700 800 9000Time (hours)


times10minus3

minus4

minus3

minus2

minus1

0

1

2

3

4

Mea

n va

lue

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 9 Mean value






Wav

efor

m in

dex


12

125

13

135

14

145

15

155

16

165

100 200 300 400 500 600 700 800 9000Time (hours)


Kurt

osis

inde

x


0

2

4

6

8

10

12

14

16

18

100 200 300 400 500 600 700 800 9000Time (hours)













Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)









Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)



7 RUL Prediction




120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 Related Works





Prediction result


indicator

Optimal degradation

indicator


constructedby GP


Opt

imal

degr

adat

ion

indi

cato

r sele

ctio

nRU

L pr

edic

tion


parameters
















yz

sqrt

2

times

+

+

x

ln










|119909|119897 119901 (119909) 119889119909]1119897 =

119883119903 119897 = 12 119883 119897 = 1119883rms 119897 = 2119883119901 119897 997888rarr infin

(1)




radic|119909|119901 (119909) 119889119909]2 = ( 1119873119873sum119894=1

radic10038161003816100381610038161199091198941003816100381610038161003816)2


|119909| 119901 (119909) 119889119909 = 1119873119873sum119894=1



119873sum119894=1

1199092119894 119883119901 = lim


|119909|119897 119901 (119909) 119889119909]1119897 = 119909max

(2)




|119909|119898 119901 (119909) 119889119909]1119898 (3)

Specifically










119883 (119905) = 120573119905 + 120590119861 (119905) (4)



119891119879 (119905) = 120578radic21205871199053 (1205902

120573119905 + 1205902)


119864 (119879) = 119864 (119864 (119879 | 120573)) = 119864( 120578120573)= 1205781205902120573

exp(minus 120583212057321205902120573

)int1205831205730

exp( 119909221205902120573

)119889119909= radic2120578120590120573 119863( 120583120573radic2120590120573)

(5)







119895=1

(Δ119909119894119895 minus 120573119894Δ119905119895)221205902Δ119905119895 )(6)



ℓ (Θ | 119883Ω) = 119899sum119894=1

[ln119901 (119883119894 | 120573119894 Θ) + ln119901 (120573119894 | Θ)]= minus12

119899sum119894=1

[[(119898 + 1) ln 2120587 + 119898sum119895=1

lnΔ119905119895 + 119898 ln1205902

+ 119898sum119895=1


]] (7)





119898sum119895=1


2

21205902(119896)120573

]]]prop exp

minus12 [[[

119898sum119895=1

1205732119894 Δ1199051198951205902(119896)

minus 119898sum119895=1

2120573119894Δ1199091198941198951205902(119896) + 1205732119894 minus 2120573119894120583(119896)120573 + (120583(119896)120573)2

1205902(119896)120573

]]]

prop expminus

12 [[(1199051198981205902(119896) minus 11205902(119896)

120573

)1205732119894 minus 2( 1199091198941198981205902(119896) + 120583(119896)1205731205902(119896)120573

)120573119894]]


1205731205902(119896)) (1199051198981205902(119896)120573 + 1205902(119896))]2

21205902(119896)1205902(119896)120573

(1199051198981205902(119896)120573 + 1205902(119896))

(8)



with

120583(119896)119894 = 1199091198941198981205902(119896)120573 + 120583(119896)1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) 1205902(119896)119894 = 1205902(119896)

1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) (10)



119864 [ℓ (Θ | 119883Θ(119896))] = minus12119899sum119894=1

[[[(119898 + 1) ln 2120587 + 119898sum

119895=1

lnΔ119905119895 + 119898sdot ln1205902+ 119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 ((120583(119896)119894 )2 + 1205902(119896)119894 )1205902Δ119905119895

+ ln1205902120573 + (120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583120573 + 12058321205731205902120573

]]]

(11)


119899sum119894=1

120583(119896)119894 1205902(119896+1)120573 = 1119899

119899sum119894=1

[(120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583(119896+1)120573 + (120583(119896+1)120573 )2] 1205902(119896+1) = 1119899119898

sdot 119899sum119894=1

119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 (120583(119896)119894 )2 + 1205902(119896)119894Δ119905119895

(12)





Figure 4 Test bed





Vibr

atio

n am

plitu

de


001020304

500 1000 1500 2000 2500 3000 3500 40000Sampling number


Vibr

atio

n am

plitu


002040608

500 1000 1500 2000 2500 3000 3500 40000Sampling number










0

01

02

03

04

05

06

07

Squa

re ro

ot o

f am

plitu

de

100 200 300 400 500 600 700 800 9000Time (hours)


times10minus3

minus4

minus3

minus2

minus1

0

1

2

3

4

Mea

n va

lue

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 9 Mean value






Wav

efor

m in

dex


12

125

13

135

14

145

15

155

16

165

100 200 300 400 500 600 700 800 9000Time (hours)


Kurt

osis

inde

x


0

2

4

6

8

10

12

14

16

18

100 200 300 400 500 600 700 800 9000Time (hours)













Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)









Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)



7 RUL Prediction




120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















yz

sqrt

2

times

+

+

x

ln










|119909|119897 119901 (119909) 119889119909]1119897 =

119883119903 119897 = 12 119883 119897 = 1119883rms 119897 = 2119883119901 119897 997888rarr infin

(1)




radic|119909|119901 (119909) 119889119909]2 = ( 1119873119873sum119894=1

radic10038161003816100381610038161199091198941003816100381610038161003816)2


|119909| 119901 (119909) 119889119909 = 1119873119873sum119894=1



119873sum119894=1

1199092119894 119883119901 = lim


|119909|119897 119901 (119909) 119889119909]1119897 = 119909max

(2)




|119909|119898 119901 (119909) 119889119909]1119898 (3)

Specifically










119883 (119905) = 120573119905 + 120590119861 (119905) (4)



119891119879 (119905) = 120578radic21205871199053 (1205902

120573119905 + 1205902)


119864 (119879) = 119864 (119864 (119879 | 120573)) = 119864( 120578120573)= 1205781205902120573

exp(minus 120583212057321205902120573

)int1205831205730

exp( 119909221205902120573

)119889119909= radic2120578120590120573 119863( 120583120573radic2120590120573)

(5)







119895=1

(Δ119909119894119895 minus 120573119894Δ119905119895)221205902Δ119905119895 )(6)



ℓ (Θ | 119883Ω) = 119899sum119894=1

[ln119901 (119883119894 | 120573119894 Θ) + ln119901 (120573119894 | Θ)]= minus12

119899sum119894=1

[[(119898 + 1) ln 2120587 + 119898sum119895=1

lnΔ119905119895 + 119898 ln1205902

+ 119898sum119895=1


]] (7)





119898sum119895=1


2

21205902(119896)120573

]]]prop exp

minus12 [[[

119898sum119895=1

1205732119894 Δ1199051198951205902(119896)

minus 119898sum119895=1

2120573119894Δ1199091198941198951205902(119896) + 1205732119894 minus 2120573119894120583(119896)120573 + (120583(119896)120573)2

1205902(119896)120573

]]]

prop expminus

12 [[(1199051198981205902(119896) minus 11205902(119896)

120573

)1205732119894 minus 2( 1199091198941198981205902(119896) + 120583(119896)1205731205902(119896)120573

)120573119894]]


1205731205902(119896)) (1199051198981205902(119896)120573 + 1205902(119896))]2

21205902(119896)1205902(119896)120573

(1199051198981205902(119896)120573 + 1205902(119896))

(8)



with

120583(119896)119894 = 1199091198941198981205902(119896)120573 + 120583(119896)1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) 1205902(119896)119894 = 1205902(119896)

1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) (10)



119864 [ℓ (Θ | 119883Θ(119896))] = minus12119899sum119894=1

[[[(119898 + 1) ln 2120587 + 119898sum

119895=1

lnΔ119905119895 + 119898sdot ln1205902+ 119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 ((120583(119896)119894 )2 + 1205902(119896)119894 )1205902Δ119905119895

+ ln1205902120573 + (120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583120573 + 12058321205731205902120573

]]]

(11)


119899sum119894=1

120583(119896)119894 1205902(119896+1)120573 = 1119899

119899sum119894=1

[(120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583(119896+1)120573 + (120583(119896+1)120573 )2] 1205902(119896+1) = 1119899119898

sdot 119899sum119894=1

119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 (120583(119896)119894 )2 + 1205902(119896)119894Δ119905119895

(12)





Figure 4 Test bed





Vibr

atio

n am

plitu

de


001020304

500 1000 1500 2000 2500 3000 3500 40000Sampling number


Vibr

atio

n am

plitu


002040608

500 1000 1500 2000 2500 3000 3500 40000Sampling number










0

01

02

03

04

05

06

07

Squa

re ro

ot o

f am

plitu

de

100 200 300 400 500 600 700 800 9000Time (hours)


times10minus3

minus4

minus3

minus2

minus1

0

1

2

3

4

Mea

n va

lue

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 9 Mean value






Wav

efor

m in

dex


12

125

13

135

14

145

15

155

16

165

100 200 300 400 500 600 700 800 9000Time (hours)


Kurt

osis

inde

x


0

2

4

6

8

10

12

14

16

18

100 200 300 400 500 600 700 800 9000Time (hours)













Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)









Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)



7 RUL Prediction




120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












radic|119909|119901 (119909) 119889119909]2 = ( 1119873119873sum119894=1

radic10038161003816100381610038161199091198941003816100381610038161003816)2


|119909| 119901 (119909) 119889119909 = 1119873119873sum119894=1



119873sum119894=1

1199092119894 119883119901 = lim


|119909|119897 119901 (119909) 119889119909]1119897 = 119909max

(2)




|119909|119898 119901 (119909) 119889119909]1119898 (3)

Specifically










119883 (119905) = 120573119905 + 120590119861 (119905) (4)



119891119879 (119905) = 120578radic21205871199053 (1205902

120573119905 + 1205902)


119864 (119879) = 119864 (119864 (119879 | 120573)) = 119864( 120578120573)= 1205781205902120573

exp(minus 120583212057321205902120573

)int1205831205730

exp( 119909221205902120573

)119889119909= radic2120578120590120573 119863( 120583120573radic2120590120573)

(5)







119895=1

(Δ119909119894119895 minus 120573119894Δ119905119895)221205902Δ119905119895 )(6)



ℓ (Θ | 119883Ω) = 119899sum119894=1

[ln119901 (119883119894 | 120573119894 Θ) + ln119901 (120573119894 | Θ)]= minus12

119899sum119894=1

[[(119898 + 1) ln 2120587 + 119898sum119895=1

lnΔ119905119895 + 119898 ln1205902

+ 119898sum119895=1


]] (7)





119898sum119895=1


2

21205902(119896)120573

]]]prop exp

minus12 [[[

119898sum119895=1

1205732119894 Δ1199051198951205902(119896)

minus 119898sum119895=1

2120573119894Δ1199091198941198951205902(119896) + 1205732119894 minus 2120573119894120583(119896)120573 + (120583(119896)120573)2

1205902(119896)120573

]]]

prop expminus

12 [[(1199051198981205902(119896) minus 11205902(119896)

120573

)1205732119894 minus 2( 1199091198941198981205902(119896) + 120583(119896)1205731205902(119896)120573

)120573119894]]


1205731205902(119896)) (1199051198981205902(119896)120573 + 1205902(119896))]2

21205902(119896)1205902(119896)120573

(1199051198981205902(119896)120573 + 1205902(119896))

(8)



with

120583(119896)119894 = 1199091198941198981205902(119896)120573 + 120583(119896)1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) 1205902(119896)119894 = 1205902(119896)

1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) (10)



119864 [ℓ (Θ | 119883Θ(119896))] = minus12119899sum119894=1

[[[(119898 + 1) ln 2120587 + 119898sum

119895=1

lnΔ119905119895 + 119898sdot ln1205902+ 119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 ((120583(119896)119894 )2 + 1205902(119896)119894 )1205902Δ119905119895

+ ln1205902120573 + (120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583120573 + 12058321205731205902120573

]]]

(11)


119899sum119894=1

120583(119896)119894 1205902(119896+1)120573 = 1119899

119899sum119894=1

[(120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583(119896+1)120573 + (120583(119896+1)120573 )2] 1205902(119896+1) = 1119899119898

sdot 119899sum119894=1

119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 (120583(119896)119894 )2 + 1205902(119896)119894Δ119905119895

(12)





Figure 4 Test bed





Vibr

atio

n am

plitu

de


001020304

500 1000 1500 2000 2500 3000 3500 40000Sampling number


Vibr

atio

n am

plitu


002040608

500 1000 1500 2000 2500 3000 3500 40000Sampling number










0

01

02

03

04

05

06

07

Squa

re ro

ot o

f am

plitu

de

100 200 300 400 500 600 700 800 9000Time (hours)


times10minus3

minus4

minus3

minus2

minus1

0

1

2

3

4

Mea

n va

lue

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 9 Mean value






Wav

efor

m in

dex


12

125

13

135

14

145

15

155

16

165

100 200 300 400 500 600 700 800 9000Time (hours)


Kurt

osis

inde

x


0

2

4

6

8

10

12

14

16

18

100 200 300 400 500 600 700 800 9000Time (hours)













Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)









Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)



7 RUL Prediction




120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119895=1

(Δ119909119894119895 minus 120573119894Δ119905119895)221205902Δ119905119895 )(6)



ℓ (Θ | 119883Ω) = 119899sum119894=1

[ln119901 (119883119894 | 120573119894 Θ) + ln119901 (120573119894 | Θ)]= minus12

119899sum119894=1

[[(119898 + 1) ln 2120587 + 119898sum119895=1

lnΔ119905119895 + 119898 ln1205902

+ 119898sum119895=1


]] (7)





119898sum119895=1


2

21205902(119896)120573

]]]prop exp

minus12 [[[

119898sum119895=1

1205732119894 Δ1199051198951205902(119896)

minus 119898sum119895=1

2120573119894Δ1199091198941198951205902(119896) + 1205732119894 minus 2120573119894120583(119896)120573 + (120583(119896)120573)2

1205902(119896)120573

]]]

prop expminus

12 [[(1199051198981205902(119896) minus 11205902(119896)

120573

)1205732119894 minus 2( 1199091198941198981205902(119896) + 120583(119896)1205731205902(119896)120573

)120573119894]]


1205731205902(119896)) (1199051198981205902(119896)120573 + 1205902(119896))]2

21205902(119896)1205902(119896)120573

(1199051198981205902(119896)120573 + 1205902(119896))

(8)



with

120583(119896)119894 = 1199091198941198981205902(119896)120573 + 120583(119896)1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) 1205902(119896)119894 = 1205902(119896)

1205731205902(119896)

1199051198981205902(119896)120573 + 1205902(119896) (10)



119864 [ℓ (Θ | 119883Θ(119896))] = minus12119899sum119894=1

[[[(119898 + 1) ln 2120587 + 119898sum

119895=1

lnΔ119905119895 + 119898sdot ln1205902+ 119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 ((120583(119896)119894 )2 + 1205902(119896)119894 )1205902Δ119905119895

+ ln1205902120573 + (120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583120573 + 12058321205731205902120573

]]]

(11)


119899sum119894=1

120583(119896)119894 1205902(119896+1)120573 = 1119899

119899sum119894=1

[(120583(119896)119894 )2 + 1205902(119896)119894 minus 2120583(119896)119894 120583(119896+1)120573 + (120583(119896+1)120573 )2] 1205902(119896+1) = 1119899119898

sdot 119899sum119894=1

119898sum119895=1

(Δ119909119894119895)2 minus 2120583(119896)119894 Δ119909119894119895Δ119905119895 + (Δ119905119895)2 (120583(119896)119894 )2 + 1205902(119896)119894Δ119905119895

(12)





Figure 4 Test bed





Vibr

atio

n am

plitu

de


001020304

500 1000 1500 2000 2500 3000 3500 40000Sampling number


Vibr

atio

n am

plitu


002040608

500 1000 1500 2000 2500 3000 3500 40000Sampling number










0

01

02

03

04

05

06

07

Squa

re ro

ot o

f am

plitu

de

100 200 300 400 500 600 700 800 9000Time (hours)


times10minus3

minus4

minus3

minus2

minus1

0

1

2

3

4

Mea

n va

lue

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 9 Mean value






Wav

efor

m in

dex


12

125

13

135

14

145

15

155

16

165

100 200 300 400 500 600 700 800 9000Time (hours)


Kurt

osis

inde

x


0

2

4

6

8

10

12

14

16

18

100 200 300 400 500 600 700 800 9000Time (hours)













Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)









Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)



7 RUL Prediction




120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Figure 4 Test bed





Vibr

atio

n am

plitu

de


001020304

500 1000 1500 2000 2500 3000 3500 40000Sampling number


Vibr

atio

n am

plitu


002040608

500 1000 1500 2000 2500 3000 3500 40000Sampling number










0

01

02

03

04

05

06

07

Squa

re ro

ot o

f am

plitu

de

100 200 300 400 500 600 700 800 9000Time (hours)


times10minus3

minus4

minus3

minus2

minus1

0

1

2

3

4

Mea

n va

lue

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 9 Mean value






Wav

efor

m in

dex


12

125

13

135

14

145

15

155

16

165

100 200 300 400 500 600 700 800 9000Time (hours)


Kurt

osis

inde

x


0

2

4

6

8

10

12

14

16

18

100 200 300 400 500 600 700 800 9000Time (hours)













Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)









Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)



7 RUL Prediction




120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

01

02

03

04

05

06

07

Squa

re ro

ot o

f am

plitu

de

100 200 300 400 500 600 700 800 9000Time (hours)


times10minus3

minus4

minus3

minus2

minus1

0

1

2

3

4

Mea

n va

lue

100 200 300 400 500 600 700 800 9000Time (hours)

Figure 9 Mean value






Wav

efor

m in

dex


12

125

13

135

14

145

15

155

16

165

100 200 300 400 500 600 700 800 9000Time (hours)


Kurt

osis

inde

x


0

2

4

6

8

10

12

14

16

18

100 200 300 400 500 600 700 800 9000Time (hours)













Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)









Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)



7 RUL Prediction




120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Opt

imal

deg

rada

tion

indi

cato

r

1

2

3

4

5

6

7

8

9

50 100 150 200 2500Time (hours)









Threshold

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

deg

rada

tion

indi

cato

r

50 100 150 200 2500Time (hours)



7 RUL Prediction




120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120583120573

002

0021

0022

0023

0024

0025

0026

0027

0028

0029


times10minus3

25

3

35

4

45

5

120590120573


(b)

0121

0122

0123

0124

0125

0126

0127

0128

120590





119864 (119879) = radic2120578120590120573 119863( 120583120573radic2120590120573) = 289 (15)






egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










egra

datio

n da

ta


0

2

4

6

8

10

12

50 100 150 200 2500Time (hours)


8 Conclusions





Acknowledgments


References









































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









remaining useful life prediction for rotating machinery...

Documents