Bearing Remaining Useful Life Prediction Based on an Improved Back Propagation Neural Network

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1 nternatonal Journal of Performablty Engneerng, Vol. 10, No. 6, September 2014, pp RAMS Consultants Prnted n nda Bearng Remanng Useful Lfe Predcton Based on an mproved Back Propagaton Neural Network XNGHU ZHANG 1*, LE XAO 2, and JANSHE KANG 1 1 Mechancal Engneerng College, Shjazhuang, CHNA 2 The State Key Lab of Mechancal Transmsson, Chongqng Unversty, Chongqng, CHNA (Receved on March 29, 2014; revsed on June 7, 2014) Abstract: Bearngs are the key components n most of rotatng machneres. Ther falures can lead to catastrophc dsasters. The accuracy of remanng useful lfe (RUL) predcton has a great nfluence on the preventve mantenance actvty. RUL predcton based on standard back propagaton neural network (BPNN) already exsts. However, tranng standard BPNN needs more tme and sometmes t may converge to local optma whch can have contrary nfluence on the accuracy. Exstng BPNN mprovng works used dynamc learnng rate, momentum tem and utlzed genetc algorthms or other random researchng algorthm to optmze the adjustment of connect weghts n the network. n ths paper, an mproved BPNN based on Levenberg-Marquardt algorthm and momentum tem s proposed. t can predct the bearng s RUL wth a good performance. Fnally, the bearng smulaton lfe data sets are used to valdate the proposed method. The results show that the predcton accuracy of the proposed method s superor to other exstng BPNNs. Keywords: Bearng, remanng useful lfe, back propagaton neural network, Levenberg- Marquardt, smulaton 1. ntroducton As an mportant content of condton based mantenance (CBM) [1], remanng useful lfe (RUL) predcton attracted many researchers attenton. Among exstng RUL predcton methods, neural networks based RUL predcton methods are very popular [2-4]. They all used Levenberg-Marquardt (LM) algorthm to tran the back propagaton neural network (BPNN). Through revewng the exstng artcles, t s found that few people pad attenton to the algorthm s mprovement to enhance the predcton performance. So, n order to mprove the BPNN s performance on RUL predcton, ths paper proposes a hybrd optmzaton method both usng dynamc momentum and dampng LM algorthm to get more accurate RUL results. From reference [5], we can know predcton error deeply mpacts the mantenance acton whch s based on the falure rate threshold. 2. mproved BPNN For a standard BPNN, the learnng rate s a fxed value. Adjustment of weghts s based on the gradent descend method whch s related to last teraton and fxed adjustment step, namely, learnng rate. Meanwhle, ths method just consders the frst-order dervatves of error functon, and fxed learnng rate s the man contrbuton of oscllaton. As a good combnaton of Gaussan-Newton and steepest descend, LM algorthm consders the secondorder dervatves of error functon and makes the convergence more fast. Momentum tem connects the adjustment of weghts n ths teraton and the prevous. t s a good way to weaken oscllaton. The form of the jonted LM and momentum tem s as follows w(t+1)w(t)+ w(t)+λ(t) α(t) w(t-1) (1) t means the teraton steps. λ s the dampng factor n LM algorthm whch could be regarded as the learnng rate n the standard momentum method. α s the momentum coeffcent n the * Correspondng author s emal: dynamcbnt@gmal.com 653

2 654 X. Zhang, L. Xao, and J. Kang momentum tem. α should be updated accordng to the performance of the teraton. f the error reduces n ths teraton, t means the prevous weght update s benefcal to convergence, the researchng drecton s correct. Therefore α should be bgger for encouragng towards the correct drecton. Otherwse, the momentum rate should declne. The weghts update can be formulated as follows α( t) 2α( t 1) E( t) < E( t 1) α( t) α( t) E( t) E( t 1) (2) α( t) α( t 1) 2 E( t) > E( t 1) f α s bgger than 1, t s 1. That lmts α n the nterval (0, 1). The prncple of λ update s dfferent from α update. For a gven λ, f the error reduces, λ should declne to perform LM as the analogous Gauss-Newton algorthm whch has the advantage of fast convergence for local search. Otherwse, λ ncreases to make LM analogous Steepest-descend algorthm for better searchng global optma. By ths way, λ s dynamc and the convergence s more effcent. The rule of the adjustment of λ s as follows. λ( t) 2λ( t 1) E( t) > E( t 1) λ( t) λ( t) E( t) E( t 1) (3) λ( t) λ( t 1) 2 E( t) < E( t 1) Through equaton (1)-(3), the weght update s performed by the hybrd optmzaton of LM algorthm and momentum method. The convergence speed s faster. The overall learnng and predcton process s same to the standard BPNN. Only dfference s the tranng algorthm. Meanwhle, actvaton functon s another aspect to speed up tranng. The hyperbolc tangent actvaton functon s used for the neurons n the hdden layer. Because we fnd that the hyperbolc tangent actvaton functon could learn faster than the sgmod actvaton functon. The lnear actvaton functon s used for the neurons n the output layer. Many works have proved that the number of the neurons n the hdden layer s mportant for convergence. Too many or too few neurons make the network more tme for tranng. Tradtonally, t s set by expertse or random. n ths paper, we decde the number of the neurons n the hdden layer accordng to the emprcal formula whch s shown as follows hn n + on + σ (4) n and on are the number of the neurons n the nput layer and output layer respectvely. σ s a postve nteger n nterval (1, 10). 3. RUL Predcton Based on mproved BPNN The features extracted from the vbraton sgnals can be as the nput vector of the neural network. The outputs of the network are the RUL predcton results whch usually are the tme to falure. However, n ths paper, the outputs are the RUL percent. Ths s based on the assumpton that smlar feature vector should have smlar lfe percentage. Ths s very dfferent from other RUL predcton methods whch all have strong relatons wth the degradaton trajectores. Smlar to Tan [5], we can take an example to explan ths dea. Suppose a bearng s falure tme s 1,000 hours. At an nspecton pont, the tme s 300 hours. n ths method s vew, we say the component completed ts 30% lfe and also has 70% RUL untl falure. The feature vector at ths nspecton pont s supposed as [1.1, 1.3, 2.4, 3.6, 4.5, 5.4]. Then, when a same type s new bearng operates some tme, the feature vector extracted from the vbraton sgnals s [1.2, 1.1, 2.6, 3.3, 4.3, 5.7], we can judge ths component has 70% RUL. Because the feature vector s very smlar to the tranng feature vector at 30% lfe. Because

3 Bearng Remanng Useful Lfe Predcton Based on An mproved Back Propagaton Neural Network 655 the stochastc character of bearng s lves, even the smlar feature vectors may have dfferent lfe percentage due to the dfferent component. So, for better predcton, ths BPNN must traned by a lot of lfe data sets. 4. Verfcaton Usng Smulaton Data Based on the amount of experment data sets, Gebraeel et al. [6] shows the bearng degradaton sgnal grows exponentally. So, Zhang et al. [7] developed a bearng lfe data smulaton method. Accordng to ths method, we smulated sxteen bearng lfe data sets whch can be used to valdate the proposed method. Except of ths, some statstcal ndexes were developed by Yuan and Uday [8]. Feature vector can be extracted from the vbraton sgnals usng wavelet packet analyss. The db4 s selected as the wavelet base, and the decomposton level s set to three. The energy of wavelet packet coeffcents s used as the feature vector. n engneerng applcaton, f we use some features to track the bearng degradaton, t s preferably to use one good feature to defne the falure but the peak value. n order to gve an nstance, the frequency band energy of 0~750 Hz s used as the ndcator to defne the bearng falure. The degradaton path of these sxteen bearngs defned by the frequency band can be depcted n Fgure 1. The lves of these bearngs are gven n Table 1. Frequency band Energy Threshold Tme (hours) Fgure 1: Energy Values of Smulated Vbraton Data n order to evaluate the RUL predcton performance, we develop three new evaluaton ndexes to not only avod unmeanng values but also make the results comparable from dfferent tranng methods. The proposed ndexes are as follows ndex1 ndex2 ndex3 d y ( d ) 2 y 2 ( d y) 1 s the total number of testng sample, d s the desred output of the network and y s the real output. ndex1 s the devatons from verdcalty. means the absolute value. ndex1 doesn t need to take the nf nto consderaton, because, s a postve nteger, the denomnator won t be zero. f the ndex1 s bgger, t means the predcted results are much (5) (6) (7)

4 656 X. Zhang, L. Xao, and J. Kang devated from real value. ndex2 s the error functon of BPNN n ths paper, t s another way to evaluate the mean error power. ndex3 s the error center. t s the mean of the absolute error. f the results from equaton (5) and equaton (7) are small smultaneously, t means, the predcted results are closed to the real value. f the results from equaton (5) s bgger, whle the results from equaton (7) s small, t means the error fluctuates heavly. By equatons (5) and (7), t s an effectve way to reflect the devaton of the results from the real value. Table 1: Lves of Tranng and Testng Data Sets Bearng No. Bearng 1 Bearng 2 Bearng 3 Bearng 4 Bearng 5 Tranng (hour) Bearng No. Bearng 6 Bearng 7 Bearng 8 Bearng 9 Bearng 10 Tranng (hour) Bearng No. Bearng 11 Bearng 12 Bearng 13 Bearng 14 Bearng 15 Testng (hour) Bearng No. Bearng 16 Testng (hour) 3330 As an llustraton, the predcton results for Bearng 16 are shown n Fgure Predcted RUL value Real RUL value RUL (%) Tme (hour) Fgure 2: Result of Predcton for Bearng 16 n order to demonstrate the effectveness of the proposed method, t s compared wth two networks proposed by El-Alfy [9] and Nørgaard [10]. We denote them as ELM and NLM respectvely. The comparson results of sx bearngs can be shown n Table 2. Table 2: Average ndexes Value for the Sx Testng Data Sets Evaluaton ndcator No ndex ndex2ndex3 ELM BPNN NLM BPNN proposed LM BPNN Conclusons n ths paper, an mproved BPNN was developed to predct the bearng s RUL. The better performance of the proposed method s valdated by comparng t wth two other tradtonal mproved algorthms. t s very crtcal for makng approprate mantenance decson and savng the cost. n the future, the nfluence of mantenance cost by RUL predcton accuracy wll be researched.

5 Bearng Remanng Useful Lfe Predcton Based on An mproved Back Propagaton Neural Network 657 Acknowledgment: The research s partally supported by the Chna NSFC under grant number References [1] Jardne, A. K. S., D. Ln, and. Banjevc, D. A Revew on Machnery Dagnostcs and Prognostcs mplementng Condton-based Mantenance. Mechancal Systems and Sgnal Processng 2006; 20(7): [2] Mazhar, M.., S. Kara, and H. Haebernck. Remanng Lfe Estmaton of Used Components n Consumer Products: Lfe Cycle Data Analyss by Webull and Artfcal Neural Networks. Journal of Operatons Management 2007; 25(6): [3] Mahamad, A.K., S. Saon, and T. Hyama. Predctng Remanng Useful Lfe of Rotatng Machnery based Artfcal Neural Network. Computers & Mathematcs wth Applcatons 2010; 60(4): [4] Tan, Z. G. An Artfcal Neural Network Method for Remanng Useful Lfe Predcton of Equpment subject to Condton Montorng. Journal of ntellgent Manufacturng 2012; 23(2): [5] Tan, Z. G., T. Jn, B. R. Wu, and F.F. Dng. Condton-based Mantenance Optmzaton for Wnd Power Generaton Systems under Contnuous Montorng. Renewable Energy 2011; 36: [6] Gebraeel, N., A. Elwany, and J. Pan. Resdual Lfe Predctons n the Absence of Pror Degradaton Knowledge. EEE Transactons on Relablty 2009; 58(1): [7] Zhang, X.H., Kang, J.S. and Jn, T. Degradaton modelng and mantenance decson based on Bayesan belef network. EEE Transactons on Relablty 2014; 63(2): [8] Yuan, F. and Uday Kumar. Statstcal ndex Development from Tme Doman for Rollng Element Bearngs. nternatonal Journal of Performablty Engneerng 2014; 10(3): [9] El-Alfy, E.S.M. Detectng Pxel-value Dfferencng Steganography usng Levenberg-Marquardt Neural Network. EEE Symposum on Computatonal ntellgence and Data Mnng 2013; [10] Nørgaard, M. Neural Network based System dentfcaton Toolbox. Tech. Report. 00-E-891, Department of Automaton, Techncal Unversty of Denmark, Xnghu Zhang s a Ph.D. student of Mechancal Engneerng College, Shjazhuang, Chna. Hs research s focused on fault detecton, dagnostcs and prognostcs, and performance-based contractng. Le Xao s a Ph.D. student of Chongqng Unversty, Chongqng, Chna. Her man research drecton s the remanng useful lfe predcton based on data-drven models. Janshe Kang receved the Ph.D. degree n Mechatroncal Engneerng from Bejng nsttute Technology, Bejng, Chna. He s a professor n Mechancal Engneerng College. Hs research nterests nclude relablty analyss and optmzaton, and condton montorng.