Principal Component Analysis Based Fault Detection and Diagnosis of Active Magnetic Bearing System

Transcription

1 MIT International Journal of Mechanical Engineering, Vol. 5, No. 1, January 2015, pp Principal Coponent Analysis Based Fault Detection and Diagnosis of Active Magnetic Bearing Syste Mohaad Wasee M.Tech. (CAD/CAM), MED, MNNIT, Allahabad, Allahabad , India. Eail ID: wasee2k14@gail.co Praveen Kuar Agarwal Assistant Professor, MED, MNNIT, Allahabad, Allahabad , India. Eail ID:p.k.agarwal@outlook.co ABSTRACT Active Magnetic Bearings (AMBs) support the rotor without any physical contact. Therefore, copared to the frictional contact and hydrodynaic bearings, AMBs provide advantages of reduced frictional lossesand can support high speed rotors. Stable operation of AMBs is priarily depends on the working condition of its position sensors and actuators. Fault or failure in sensors or actuators of AMB syste can result in undesired rotor dynaics. Hence, to ensure the safe operation and reliable perforance of AMB syste, fault detection and diagnosis (FDD) of AMB syste is very essential. Principal coponent analysis (PCA) is a odel-free and robust statistical ethod which can detect and diagnose the faults in engineering systes with high accuracy. Therefore, in the present work, PCA based ethodology is used for the fault detection and diagnosis in sensors and actuators of AMB syste. Siulations have been carried out to diagnose these faults of AMB syste. Q-statistic or square prediction error is used for diagnosing bias and ultiplicative faults. Keywords: Active Magnetic Bearing (AMB), Fault detection and diagnosis (FDD), Principal coponent analysis (PCA), Bias faults, Multiplicativefaults INTRODUCTION Active agnetic bearing (AMB) is a echatronic device which levitates the rotor by anipulating the attractive electroagnetic forces. Due to its ability in operating without echanical friction between the rotor and stator and high-precision operation, they are ost suitable for high speed application such as turbo achinery and achine tools Schweitzer et.al. [1]. The desired operation of AMB ainly depends on perforance of displaceent sensors and actuators of the syste. Fault or failure of any one of the sensor and actuator can result in destructive rotor dynaics behaviour. Therefore, in order to avoid such failure of entire syste, fault detection and diagnosis (FDD) of AMBs under its operating condition becoes very essential. In the present work, an attept is ade to detect and diagnose the sensor and actuator faults in AMBs. Many FDD ethods for AMBs have been discussed in the literature. Ki and Lee [2] proposed state estiation FDD ethod to detect sensor faults in AMBs. An abrupt change in transient state and dual faults can t be detected by this ethodology. Losch [3] used position estiation technique to detect the sensor faults in AMBs. Hu et al. [4] used ulti value logic algebra ethod to identifying the sensor faults in AMBs, but it was incapable to diagnosed dual faults at a tie. Cade et al. [5] ipleented wavelet analysis technique (digital signal processing approach) to identify the rotor displaceent in AMBs. Garcia et al. [6] used redundancy based FDD ethod to forulate and predict sensor alfunction and abnoral operating conditions in AMBs, however ultiple sensor faults can t be predict. By this ethodology. Tsai et al. [7] applied Luenberger state estiation technique to diagnose the ultiple sensor faults in AMBs by using the atheatical odel of AMB. Beckerle et al. [8] proposed balancing filter approach based on parity equation to identify unknown faulty states in AMBs. However, all the above FDD ethods are odel and redundancy based and require the precise atheatical odelling as well as result in increased coplexity and cost of AMB syste. There-

2 MIT International Journal of Mechanical Engineering, Vol. 5, No. 1, January 2015, pp fore, in this paper, odel-free principal coponent analysis (PCA) based FDD ethod has been proposed for AMB sensors and actuators faults diagnosis. PCA is a siple ultivariable statistical ethod with very high diensional accuracy Edward [9]. Both stationary and dynaic faults in the syste can be detected and isolated by correlating data into lower diensional subspace (Wang and Xiao, 2003). FAULTS IN AMBs Faults are defined as the unperitted deviations of a signal fro its noral state Iserann [10]. It is a state that ay lead to undesired operation or failure of the entire syste. Faults in AMB syste can be broadly classified as external and internal faults. A fault is considered to be external when either it anifests itself as or its effect can be replicated by external disturbance acting on the syste. Internal faults cannot be represented by external disturbances as they affect the actuation, easureent or control processes and thereby the syste dynaics. Sensors and actuators are the iportant coponents of AMB syste. Their perforance directly affects the entire operation of the syste. Sensor faults occur due to various factors, such as anufacturing defects, wear and tear with long-ter usage and incorrect calibration or ishandling. Basically, there are three types of faults in sensors and actuators of AMB such as; bias fault, ultiplicative fault and noise addition Ki and Lee [3]. EIGHT-POLE MAGNETIC BEARING MODELLING Eight-pole heteropolar configuration of AMB is considered in the present work. Geoetry of eight-pole agnetic bearing is shown in Fig.1. All the poles of the AMB are considered identical. Forces generated along the positive X and Y axes Agarwal and Chand, [11] are given by Eqs. (1) and (2): 8 2 φ F x = cos θ (1) 2 µ A = φ F y = sin θ (2) = 1 2µ 0 A where, µ o is the agnetic pereability of free space, φ, A and θ are the agnetic flux, pole face area and pole orientation angle of the th pole respectively. In the present work, fuzzy logic controller (FLC) is used to obtain the stable rotor position of AMBs. Bias current together with control current is supplied separately to each pole of eightpole AMBs. PRINCIPAL COMPONENT ANALYSIS (PCA) PCA is a ultivariate statistical analysis ethod Edward [9]. A principal coponent is defined as a linear transforation of the original variables into new set of variables. Original variables are norally correlated while new variables are uncorrelated or orthogonal to each other. Variables in odern engineering syste are ulti-diensional and correlated. Due to the redundancy of the variables, the original variables can be represented by saller nuber of principal coponents. PCA uses latent variables instead of every easured variable in the process so they can better explain the behaviour of the process. Instead of analyzing all the variables, the PCA ethod focuses on analyzing these principal coponents when onitoring the condition of a syste. Fig. 1: Eight-pole agnetic bearing geoetry (a) PCA odel developent Step I: Noralize the data. First data is centered into zero ean and then into unit variance. Centering is done by subtracting the ean of each colun of the atrix D fro corresponding eleent of that colun given by Eq. (3). D = [{ d ean( x )}{ d ean( d )}.. { d ean( x )}] (3) n centered Then each colun of the ean centered atrix is divided by the corresponding colun s standard deviation given by Eq. (4) [ { / ( )} { / ( )}.. { / ( )} ] D = d std d d std d d std d (4) std n Step II: Calculate the covariance atrix C. The covariance atrix is given by Eq. (5). ' std D Dstd C = n 1 (5) Where, covariance atrix C is real syetrical atrix. Step III: Finding the loading vectors (P) using Singular Value Decoposition (SVD) of covariance atrix C Qinghua, [12] given by Eq. (6). [ ] ( ) P, S, P = SVD C = P S P ', with PP = P P = I (6)

3 MIT International Journal of Mechanical Engineering, Vol. 5, No. 1, January 2015, pp Step IV: Optial nubers of principal coponents are selected based on the scree plot ethod as shown in Fig. 3. It is a graphical ethod in which the principal coponents are arranged in decreasing order of their eigenvalues. Step V: To calculate the threshold of Q statistic or square prediction error (SPE) in residual subspace harrow et al. [13]. Upper liit of Q statistic is calculated using Jackson & Mudholkar forula according Eq. (7): h 0 ( ) = h C 2se se h 1 h + + se se i = λ 1 = a+ 0 α Q se 1 with se α 1 2 i (7) (b) Data generation Siulation odel of AMB with fuzzy logic controller developed in MATLAB prograing environent is used for generating the training data. Various displaced positions of the rotor are considered and the corresponding currents in all the coils of AMB are recorded. The traectories of rotor fro the displaced position to the stable position in all four quadrants of XY- plane are shown in Fig. 2. The siulation is carried out for the 32 displaced rotor positions. For each displaced rotor position, data of 10 variables (8 actuators and 2 position sensors) are noted in 100 equal tie intervals of size s, starting fro displaced position to stable position of the rotor. The training data is arranged in the for of a atrix given by Eq. (8) D = [ I1 I2 I3 I4 I5 I6 I7 I8 X Y ] (8) where, I 1, I 2, I 3, I 4, I 5, I 6, I 7, I 8 are currents to all the eight-pole fro actuators and X,Y are displace position of the rotor along X and Y axis read fro the position sensors respectively. Fig. 2: Rotor traectory in XY-plane Fig. 3: SCREE plot of PCA odel (c) Training of PCA odel 3200 saples under noral operating condition were used to construct training atrix. Now training data is used to find out the optial nuber of principal coponents using PCA odel. To deterine the optial nuber of principal coponents of PCA odel, Scree test Edward [9] is used. Four PCs corresponding to higher eigenvalues are selected in PCA odel. The upper liit (threshold) of Q-statistic is deterined as by taking the confidence level 95%. RESULTS AND DISCUSSION Various siulation tests were conducted to verify the PCA strategy for detecting and diagnosing actuators and displaceent sensors faults of AMB syste. In Test-1, bias (10%) fault was added at tie sec in one of the sensor or actuator of AMB syste. Due to this fault the Q-statistic value is increased significantly and exceeded the threshold liit. It found that fault is detected in first actuator soon after the fault occurred as shown in the Fig. 4 (a). To diagnose the type of the fault (i.e. bias or ultiplicative) in the faulty actuator, Q-statistic value is plotted with respect to tie as shown in Fig. 4 (b). It can be observed that once the fault has occurred, Q-statistic value is increased and exceeded the threshold liit and then it becoes constant with respect to tie. It indicates that there is a bias fault in the first actuator.

4 MIT International Journal of Mechanical Engineering, Vol. 5, No. 1, January 2015, pp Fig. 4: (a) Q-statistic plot of siulation Test-1 for fault detection 4 Fig. 5: (b) Q-statistic plot of siulation Test-2 for fault diagnosis CONCLUSION Fig. 4: (b) Q-statistic plot of siulation Test-1 for fault diagnosis In Test-2, ultiplicative fault was introduced at tie sec in one of the sensor or actuator of AMB syste. Fig. 5 (a) shows that fault is in the position sensor along Y-axis. Fro Fig. 5 (b) it can be observed that the Q-statistic value of the faulty sensor varies linearly with respect to tie. It indicates that there is ultiplicative type of fault in position sensor along Y-axis. In the present work, principal coponent analysis based fault detection and diagnosis ethodology has been proposed to detect and diagnose actuator and sensor faults of AMB syste. Modelling of agnetic bearing has been represented to deterine forces generated by eight-pole agnetic bearing along X and Y axis respectively. Q-statistic or square prediction error evaluated fro the principal coponents is utilized to detect the actuator and sensor fault of AMB. Once the fault is detected, Q-statistic plot with respect to tie have been eployed to diagnose the type of fault (i.e. bias and ultiplicative). The siulation results show that the proposed PCA based fault detection and diagnosis technique can effectively detect and diagnose different types of faults in AMB syste coponents. REFERENCES 1. Schweitzer, G., Bleuler, H. and Traxler, A., Active Magnetic Bearing, Zurich, Switzerland vdf Hochschulverlag A G (1994). 2. Ki, S.J. and Lee, C.W., Diagnosis of Sensor Faults in Active Magnetic Bearing Syste Equipped With Built-In Force Transducers, IEEE/ASME transactions on echatronics, vol. 4, no. 2 (1999). 3. Losch, F., Detection and correction of sensor and actuator faults in active agnetic bearing syste, 8th international syposiu on agnetic bearing, Japan (2002). 4. Hu Y.F., Ku S.P., Zhou Z.D., Su Y.X., Multi-valued Logic and its Application in the Fault Diagnosis of the Sensors of Magnetic Bearings, Proceedings of the Third International Conference on Machine Learning and Cybernetics, Shanghai (2004). Fig. 5: (a) Q-statistic plot of siulation Test-2 for fault detection 5. Cade, I.S., Keogh, P.S., Sahinkaya, M.N., Fault Identification in Rotor/Magnetic Bearing Systes Using Discrete Tie Wavelet Coefficients, IEEE/ASME transactions on echatronics, vol. 10, no. 6 (2005).

5 MIT International Journal of Mechanical Engineering, Vol. 5, No. 1, January 2015, pp García, F., Castelo, P., Pazos, P., Rolle, C., On AMBs Diagnosis by Analytical Redundancy, IEEE Conference on International Syposiu on Industrial Electronics, (2007). 7. Tsai, N.C., King, Y.H., Lee, R.M., Fault diagnosis for agnetic bearing systes, Mechanical Systes and Signal Processing, 23, (2009). 8. Beckerle, P., Schaede, H., Butzek, N., Rinderknecht S., Balancing filters: An approach to iprove odel-based fault diagnosis based on parity equations, Mechanical Systes and Signal Processing, 29, (2012) 9. Edward, J., User s guide to principal coponents, Wiley, Iserann, R., Model-based fault-detection and diagnosis status and applications, Annual Control Review, 29,71 85 (2005). 11. Agarwal, P.K. and Chand S., Fault-tolerant control of three-pole active agnetic bearing, Expert Systes with Applications, 36, (2009) 12. Qinghua, H.E., Xiangyu, H.E. and Jianxin Z., Fault detection of excavator s hydraulic syste based on dynaic principal coponent analysis, J. Cent. 13. Harrou, F., Nounou, M.N. and Nounou, H.N., Statistical Detection of Abnoral Ozone Levels Using Principal Coponent Analysis,International Journal of Engineering & Technology IJET-IJENS, No:06, Vol:12 (2012). 14. South Univ. Technol., 15: (2008). Alkaya A. and Eker I., Variance sensitive adaptive threshold-based PCA ethod for fault detection with experiental application, ISA Transactions, 50, (2011). 15. Wang, S. and Xiao, F., AHU sensor fault diagnosis using principal coponent analysis ethod, Energy and Buildings, 36, (2004).