RBF Neural Network Model Training by Unscented Kalman Filter and Its Application in Mechanical Fault Diagnosis

Transcription

1 Appled Mechancs and Materals Submtted: ISSN: , Vols , pp Accepted: do:.428/ Onlne: rans ech Publcatons, Swtzerland RBF Neural Networ Model ranng by Unscented Kalman Flter and Its Applcaton n Mechancal Fault Dagnoss Xu-Sheng Gan,a, Hua-Png L 2,b, Ha-Long Gao 3,c XJng College, X an, Shaanx, 723, Chna 2 PLA, Ar Force X an Flght Academy, X an, Shaanx, 736, Chna 3 Unt 9332 of PLA, Qqhar, Helongang, 672, Chna a ganxusheng23@63.com, b hpl@mal.xdan.edu.cn, c ghllyx@sna.com Keywords: Mechancal Equpment; Fault Dagnoss; Neural Networ; Unscented Kalman Flter Abstract. o mprove the ablty of fault dagnoss for mechancal equpment, a Radal Bass Functon Neural Networ (RBFNN) dagnoss method based on Unscented Kalman Flter (UKF) algorthm s proposed. In the algorthm, at frst, UKF algorthm s used to estmate the parameters of RBFNN, and then the proposed method s ntroduced nto the fault dagnoss of mechancal equpment. he smulaton ndcates that the establshed model has a good dagnoss performance for mechancal fault dagnoss. Introducton In modern ndustry producton, the fault dagnoss technology of mechancal equpment always receved much attenton, f the fault of a equpment occurs and s not detected and removed, t not only cause the serous damages of the equpment tself, even may cause the serous consequences leadng to machnery losts and casualtes. herefore, the researches on mechancal fault dagnoss has very mportant sgnfcance for the producton system. In recent years, the artfcal neural networ s popularly appled n mechancal fault dagnoss. It has made a great breathrough n theory, caused the attenton of domestc academa and ndustry. Radal Bass Functon neural networ (RBFNN) s proposed by Moody et al [], and s also called theocalzed receptve feld networ. It s a smulaton of operaton of neural networ of human bran. At present, RBFNN has be wdely used n mechancal fault dagnoss [2][3]. Based on above analyss, an UKF learnng algorthm [4][5] s ntroduced n tranng of RBFNN to mprove the dagnoss ablty for mechancal fault. he smulaton valdates ts feasblty and effectveness. Radal bass functon neural networ RBFNN can approxmate any nonlnear functon, can process the regularty n the system that s not able to be analyzed, and has good generalzaton ablty wth fast convergence speed. It has been successfully appled n the feld such as nonlnear functon approxmaton, tme seres analyss, data classfcaton, pattern recognton, nformaton processng, mage processng, system modelng, control and fault dagnoss and so on. RBFNN s the topology structure wth the feedbac feature. Assume that the networ has three layers: the nput layer s compose of the sgnal source nodes; the second layer s the hdden layer where the transform functon s RBF functon whch s the nonlnear transfer functon that radally attenuate to center pont, RBF functon s the local dstrbuton, and adopts Gauss ernel. he correspondng output of hdden layer s n 2 ( ) = ( ( ) ) = exp{ [ ( ) ] / 2 } z t x t s x t s a () where z ( t) s the output of -th hdden unt; x( t) s the vector of -th nput pattern; s s the transform center vector of -th unt n hdden layer; a s the control parameters correspondng to -th center vector. All rghts reserved. No part of contents of ths paper may be reproduced or transmtted n any form or by any means wthout the wrtten permsson of rans ech Publcatons, (ID: , Pennsylvana State Unversty, Unversty Par, USA-/5/6,9:3:3)

2 2384 Advanced Manufacturng and Informaton Engneerng, Intellgent Instrumentaton and Industry Development he thrd layer s the output layer, the output node s: l y ( t) = w ( ) z ( t) + θ ( t) = w ( ) z ( t),( m) = = l (2) where m s the number of output nodes; l s the number of hdden nodes w ( ) = θ ( ), z ( t) = (3) he nodes of output layer n the networ s the lnear processng unt that has the very good classfcaton propertes on -th nput mode. Suppose that d ( t ) s the expected output of -th node of t-th mode, then the obectve functon of the error s m 2 f ( ) = λ [ d ( t) y ( t)] 2 (4) t= t= Unscented Kalman flter algorthm UKF s a new Kalman flter for solvng the nonlnear system on the bass of the Unscented ransformaton (U), whch has attracted great attenton. UKF approxmates the random varable dstrbuton by generatng a dscrete dstrbuton comprsng the mnmum number of pont that preserves the same frst and second order moments, and has the faster convergence rate and hgher precson wthout computng Jacoban matrx [6]. Suppose that a nonlnear dscrete system s defned as follows: x = f ( x ) + w (5) y = h( x, u ) + v where x s the unnown state of system. y s the measurement output of system. w s the process nose whch s the whte nose wth zeros mean and covarance Q. v s the measurement nose whch s the whte nose wth zeros mean and covarance R. For the above-defned system, the steps of UKF s as follows:. Intalzaton x = E x [ ] P = E ( x x)( x x) 2. Calculate the sgma-pont x usng the method as ntroduced n the reference [7] x = x + P X sgma 3. me update x = f ( x ) + w x = n + ω x, = = ω,, = P x x x y = h( x, u ) + v y = ω y, = 4. Measurement update yy = ω,, = P y y y xy = ω,, = P x x y (6) (7) (8) (9) () () x + Q (2) (3) (4) y + R (5) y (6)

3 Appled Mechancs and Materals Vols K = P ( P ) xy yy x x = + K ( y y ) P = P K P K yy (7) (8) (9) Realzaton of RBFNN tranng based on UKF algorthm he process for tranng RBFNN by UKF algorthm (UKF-RBFNN) s to use UKF algorthm to perform an optmzaton process of networ parameters as follows: center c, wdth σ and connecton weght w for the optmal soluton. Before optmzaton, these parameters need to form a vector θ = { c, σ, w } easy to calculaton n advance. he flow of UKF-RBFNN s shown n Fg.. Smulaton valdaton Fg. Realzaton flow of UKF-RBFNN In the valdaton experment of UKF-RBFNN, the gear and bearng n a certan type of tractor gear box are selected as the obect of fault dagnoss vbraton test, and the part vbraton sgnals are measured under the condton of gear and bearng falure. Accordng to the actual applcaton stuaton, the smulaton experments manly study 3 faults for second shaft [8] as follows: ) ndentaton of outer rng raceway of front bearng; 2) desquamaton of nner rng raceway of front bearng; 3) broen teeth of IV gear drven gear. he curve n tme doman and frequency doman are extracted under dfferent rotatonal speed, n the same tme, we also can obtan the ampltude value of the curves n frequency doman and frequency under dfferent frequences, as shown n able. able Ampltude and ts maxmum and of dagnoss samples under dfferent frequences

4 2386 Advanced Manufacturng and Informaton Engneerng, Intellgent Instrumentaton and Industry Development UKF-RBFNN s used to establsh the mechancal fault dagnoss model for udgng the fault dagnoss samples of gear and bearng n tractor gear box transmsson, able 2 shows the dagnoss results for the samples. Because the error convergence factor s less than., so we consder that the results can meet the requrements of fault dagnoss. able 2 Dagnoss error and results MSE n Fg. 2 Convergence curve of UKF-RBFNN Concluson A mechancal fault dagnoss model based on RBF neural networ traned by UKF algorthm s proposed for gear box. In the model, UKF algorthm s used to optmze the center, wdth and connecton weght of RBF neural networ. he smulaton result ndcates that proposed UKF-RBFNN model has a good dagnoss performance for mechancal fault of gear box, and has also a wde actual applcaton perspectve for fault dagnoss n other felds. References [] J. Moody, C. Daren. Fast learnng n networs of locally-tuned processng unts. Neural Computaton, (2), (989), [2] S. J. Hanson, D. J. Burr. Mnows-r bac-propagaton: learnng n connectonst models wth non-eucldean error sgnals. Neural Informaton Processng Systems, (987), [3] S. Chert, P. M. Crant, C. F. N. Cown. Orthogonal least square algorthm for radal bass functon networs. IEEE ransacton on Neural Networs, 2(2), (99), [4] E. A. Wan, R. van der Merwe. he unscented Kalman flter for nonlnear estmaton. Proceedngs of IEEE Symposum 2, Lae Louse Alberta, Canada, Oct. (2) [5] S. J. Juler, J. K. Uhlmann, H. F. Durrant-whyte. A new approach for the nonlnear transformaton of means and covarances n flters and estmators. IEEE ransactons on Automatc Control, 45(3), (2), [6] S. J. Juler, J. K. Uhlmann. Unscented flterng and nonlnear estmaton. Proceedngs of the IEEE, 92(3), (24), [7] S. J. Juler. he sphercal smplex unscented transformaton. Proceedngs of the Amercan Control Conference, Denver, (23) [8] X. H. Zhang, Y. Le. Applcaton of BP neural networ n mechancal fault dagnoss. Nose and Vbraton Control, 28(5), (28), 95-97

5 Advanced Manufacturng and Informaton Engneerng, Intellgent Instrumentaton and Industry Development.428/ RBF Neural Networ Model ranng by Unscented Kalman Flter and ts Applcaton n Mechancal Fault Dagnoss.428/