Tuning of Extended Kalman Filter for Speed Estimation of PMSM Drive using Particle Swarm Optimization

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1 Tuning of Extene Kalman Filter for Spee Etimation of PMSM Drive uing Particle Swarm Optimization Mahewara Rao Chintala 1, Ramana Pilla 2, Ayya Rao SLV Tummala 3 1 M.Tech Scholar, Dept. of EEE, GMR Intitute of Technology, Rajam, Srikakulam, AP , Inia 2 Aociate Profeor, Dept. of EEE, GMR Intitute of Technology, Rajam, Srikakulam, AP , Inia 3 Aitant Profeor, Dept. of EEE, GMR Intitute of Technology, Rajam, Srikakulam, AP , Inia ABSTRACT Thi paper preent a pee etimation techniue for Permanent Magnet Synchronou Motor (PMSM) bae on Extene Kalman Filter (EKF). Particle Swarm Optimization (PSO) metho i ue to optimize the noie covariance matrice of EKF, thereby enuring filter tability an accuracy in pee etimation. The propoe metho will be performe in two tep; in the firt the covariance matrice are optimize in off-line manner an then in the econ tep thee error covariance matrice value are injecte into EKF to etimate rotor pee. Simulation reult how that the covariance matrice improve the convergence of etimation proce an uality of the ytem performance. Keywor: Permanent Magnet Synchronou Motor, Extene Kalman Filter, Particle warm optimization, Objective function. 1. INTRODUCTION Current inutry tren avocate the PMSM a the firt preference for motor control application eigner. It unparallel feature uch a high power enity, fat ynamic repone an high efficiency in comparion with other motor in it category, together with ecreae manufacturing cot an improve magnetic propertie, make the PMSM a goo recommenation for large-cale prouct implementation [1-2]. However, conventional motor control nee a pee enor or an optical encoer to meaure the rotor pee with better accuracy. Senor preent ome iavantage uch a rive cot, machine ize, reliability an noie immunity; therefore, a enorle control without poition an pee enor for PMSM rive become a popular reearch topic in literature [3-4]. Variou control algorithm like Sliing Moe Oberver (SMO) [ 5], reuce orer oberver [6], Full orer oberver [7], Extene Kalman Filter(EKF) [8-9], Moel Reference Aaptive Sytem (MRAS) [10], Fuzzy logic [11] an Artificial Neural Network [12] are propoe in the literature for pee an poition etimation of PMSM. Among the propoe algorithm EKF i one of the promiing oberver, if the noie covariance matrice are known; offer bet poible filtering of the noie in meaurement an of the ytem. If the rotor pee coniere a an extene tate an i incorporate in the ynamic moel of a PMSM then the EKF can be ue to re-linearize the non-linear tate moel for each new value of etimate. A a reult, EKF i the bet olution for the pee etimation of a PMSM. But EKF etimation mainly epen on noie covariance matrice Q & R. They can be obtaine by coniering the tochatic propertie of the correponing noie. Since thee are uually unknown, in mot cae the EKF matrice are eigne an tune by trial-an-error proceure. But it i a time conuming proce, to overcome thi problem the covariance matrice are tune uing Genetic algorithm [13]. In thi paper, a new alternative metho combination of EKF-PSO i ue to tune the covariance matrice Q & R. In the firt tep, fining the optimal value of covariance matrice in off-line metho an finally thee value are place in the correponing matrice an run in on-line to etimate rotor pee. 2.MATHEMATICAL MODELING The voltage euation for a PMSM in the rotor reference frame can be expree a [14]. V R i L p i L i --- (1) e V R i L p i L i --- (2) e e f Volume 5, Iue 7, July 2016 Page 100

2 The electromagnetic torue of PMSM i ecribe a 3 T P i L L i 2 e n f --- (3) The motion euation i expree a follow a r J Br Tl Te --- (4) t 3.PROPOSED SCHEME OF EKF BASED SENSOR LESS SPEED CONTROL OF PMSM DRIVE * r v v i a i b i c abc e r i i v v Figure 1 Block iagram of propoe pee etimation ytem of PMSM 3.1 Extene Kalman Filter (EKF) Kalman Filter i a mathematical moel that run in parallel to the actual ytem an provie the etimation of the tate of linear ytem. But the rawback of Kalman filter i that it i a time conuming proce when applie for nonlinear ytem. For the implementation of nonlinear ytem, thee function of the tate variable change with every time tep, a a reult the iteration cannot be pre-compute. Short coming of thi moel can be overcome by uing Extene Kalman filter (EKF). In the moel ecribe by euation (1), (2) an (3), the current of -axi an the pee are electe a tate variable, an the input variable u an output variable y are efine a follow: i x i V i, u V, y i r The icrete time ytem euation are x f ( x, u, w ) --- (5) k k 1 k1 k1 k1 y h ( x, v ) --- (6) k k k k Where x k i tate vector, u k i input vector, w k i ranom tate noie, y k i the noiy obervation or meaure variable vector an v k i the meaurement noie. 3.2 EKF Algorithm Step 1: State vector an covariance matrice are initialize i.e. x (0), P(0), Q, R Step 2: Fin Jacobian matrice fk 1 an hk uing fk 1 Fk (7) xk hk H k --- (8) x k Volume 5, Iue 7, July 2016 Page 101

3 Step 3: Preiction of tate matrix an error covariance matrice x f ( x, u ) x(0) --- (9) k1 k 1 k 1 k 1 P F P F Q --- (10) T k1 k 1 k 1 k 1 k 1 Step 4: Correction tate Calculation of Kalman gain matrix T T K P H / H P H R --- (11) Upate tate preiction Etimation of error covariance matrix k k 1 k k k 1 k k k k 1 k k k k 1 x x K y H x --- (12) 1 Pk I KkHk Pk --- (13) 3.3 EKF etimation for PMSM rive The ynamic tate euation of PMSM are i 1 V R i Pn r L i t L L i V t L L 1 Ri PnW r L i Pn r f r 1 3 B Tl Pn i f L L i r t J 2 J J The above euation written in the form of tate pace reprentation i R Pn rl t L L L i i Pn r L R P n f 1 V i 0 t L L V L L r r 3P n L L i Pn f B t 2J 2J J Dicrete time repreentation of above euation i --- (14) --- (15) --- (16) --- (17) R Pn rl T T T 0 L L L i Pn rl R P n f 1 V fk 1 xk 1, uk 1, wk T 1 T T i 0 T L L V L L r 3P n L L i Pn f B T T 1 T 2J 2J J The Graient matrix i given a R Pn rl 1 T T 0 L L f K 1 Pn rl R P n f F k 1 T 1 T T --- (19) x k L L L 3Pn L L i 3Pn f B T T 1 T 2J 2J J --- (18) Volume 5, Iue 7, July 2016 Page 102

4 h K H k xk (20) 4. PARTICLE SWARM OPTIMIZATION (PSO) Particle warm optimization, a warm intelligence-bae global ranom earch algorithm, i propoe by Kenney an Eberhart inpire from artificial life reearch reult [15]. It regar all iniviual in the population a particle without ma an volume in the D-imenional earch pace an each particle move at a certain pee to the bet poition of it own hitory P bet an the bet poition of it neighborhoo hitory g bet in the olution pace, in orer to achieve the evolution of caniate olution. Figure 2 Flow chart of particle warm optimization 5. TUNING OF EKF USING PARTICLE SWARM OPTIMIZATION The critical tep in a Kalman filter eign i to obtain a numerical evaluation of the filter parameter pecifie by the initial tate x (0), an the covariance matrice P(0), Q an R. Thi proce i calle tuning an it involve an iterative earch for the coefficient value that yiel the bet etimation performance poible. Changing the covariance matrice Q an R affect both the tranient an the teay tate operation of the filter. Increaing Q woul inicate increae in either noie riving the ytem or uncertainty in the moel. Thi will increae the value of the tate covariance element. The filter gain will alo increae thereby weighting the meaurement more heavily, an the filter tranient performance i fater. Similarly, increaing the covariance R inicate that the meaurement are ubjecte to a tronger corruptive noie an houl be weighte le by the filter. Coneuently the value of the gain matrix K will ecreae, an the tranient performance i lower. For the initial tate covariance matrix Po, the iagonal term repreent variance or mean uare error in knowlege of the initial conition. Varying P (0) yiel a ifferent magnitue tranient characteritic. The tranient uration will be the ame an the teay tate conition are unaffecte. The covariance matrice Q, R an P (0) are aume to be iagonal ue to lack of ufficient tatitical information to evaluate their off-iagonal term. The main objective function of thi paper i election of optimum value of Q an R. Thee value are electe manually by uing trial & error metho. But thi i very time conuming proce. To urmount thi problem, covariance matrice are tune by uing Particle Swarm Optimization (PSO). The objective function F= w 1 e 1 +w 2 e 2 +w 3 e 3 Where Volume 5, Iue 7, July 2016 Page 103

5 2 2 e i i 1 act et e i i 2 act et 2 e 3 r act r et Proper election of weight i eential in tuning ele thee weight may lea to large error. Thee weight are electe a follow w 1 =0.125, w 2 =0.05, w 3 = e F w1e 1 w2e2 w3e3 Figure 3 EKF-PSO block iagram 6. RESULTS AND DISCUSSIONS In the imulation i, i, v, v are input variable of EKF algorithm ani, i,ω r are the etimate tate variable. In orer to mimic the conition of real ytem Gauian white noie are ae to feeback value of i, i are et to an ample time of the white noie block i et to ec. It houl be note that the convergence of the PSO metho to the optimal olution epen on the parameter c 1, c 2, w min, an w max value. During the imulation, thee value are et to c 1 =2, c 2 =2, w min, =0.5 an w max =0.9 repectively Number of Generation Table 1: PSO-EKF etimation Number of iteration = 50(PSO) Diagonal matrix Q Diagonal matrix R Objective function value 5 [ ] [ ] [ ] [ ] [ ] [ ] Figure 4 Evolution of fitne function relative to PSO-EKF Volume 5, Iue 7, July 2016 Page 104

6 Optimize parameter of matrice Q & R of EKF with their correponing ISE obtaine by propoe PSO-EKF metho an it performance i given in Figure 4. Table 1 how the convergence of PSO-EKF proce an ISE i ecreae with increaing of generation count. Here bet objective function value i obtaine for generation count 20, correponing value are injecte into EKF an run in online manner. Finally the tate of the EKF i etimate a hown in below figure. Figure 5 Meaure an etimate current of i an i Figure 6 Actual an etimate rotor pee Figure 7 Actual an etimate rotor pee uner variable pee The meaure an etimate waveform of i an i are hown in figure 5. Due to convergence problem of tate covariance matrix P, the etimate -axe current having large ripple upto0.03 ec. After 0.03 ec matrix P i converge, then tate variable i an i are track the actual value. From Figure 6 it i evient that the etimate pee matche with the actual pee near ec. The effectivene of PSO-EKF metho i evaluate uner two cae, in one cae the pee i varie from 200 rpm to 500 rpm at 0.5 ec an in another cae motor i run at 500 rpm at 0.5 ec a hown in figure 7 an in both cae the etimate pee i converge accoringly ue to precie value of matrice Q & R. Volume 5, Iue 7, July 2016 Page 105

7 7. CONCLUSION In thi paper, EKF bae enorle pee control of PMSM rive ha been preente to how the reult of etimate value of pee an -axe tator current. The performance of EKF i mainly epen on error covariance matrice Q & R, which are uitably electe. Thee matrice are improve the ytem convergence an uality of etimation. The imulation reult how the uperior performance in term of ettling time, reuction of noie an overall ytem tability. Appenix A: Simulation Parameter value of PMSM rive: Parameter Symbol Numerical value Reitance of tator R ohm Direct axi inuctance of tator L H Quarature axi inuctance of tator L H Flux linkage ψ f 0.12 Wb Inertia of rotor J Kg/m 2 friction coefficient B Nm/ 2 Pair of pole P n 3 Rate pee ω 1000 rpm Reference [1]. R. Krihnan an A. J. Beutler, "Performance an eign of an axial fiel permanent magnet ynchronou motor ervo rive", Proc. IEEE IAS Annu. Meeting, pp , [2]. P. Ramana, K. Alice Mary, M. Surya Kalavathi an A.Swathi Parameter Etimation of Permanent Magnet Synchronou Motor-A Review, i-manager Journal on Electrical Engineering, Vol.9, No.4, (2015), [3]. Houe M, A Rahman M. A. Spee an poition enorle permanent magnet ynchronou motor rive, Canaian Conference on Electrical an Computer Engineering, 1994,vol.2, pp [4]. J. Kim an S. Sul, New approach for high performance PMSM rive without rotational poition enor, IEEE Tran. Power Electron., Vol. 12, No. 5, pp , Sep [5]. B. K. Kim, W. K. Chung, an K. Ohba, Deign an performance tuning of liing-moe controller for high-pee an high-accuracy poitioning ytem in iturbance oberver framework, IEEE Tran. In. Electron., vol. 56, No. 10, pp , Oct [6]. P.Ramana, K. Alice Mary, M. Surya Kalavathi an J. Hareeh Kumar, Deign of a Non-linear Reuce an Full orer Oberver for an Inverter Fe Permanent Magnet Synchronou Motor Drive, Inian Journal of Science an Technology, Vol.9, No.10,(2016), 1-8. [7]. S. Po-ngam an S. Sangwongwanich, "Stability an Dynamic Performance Improvement of Aaptive Full-Orer Oberver for Senorle PMSM Drive," in IEEE Tranaction on Power Electronic, vol. 27, no. 2, pp , Feb [8]. S. Bolognani, L. Tubiana an M. Zigliotto, "Extene Kalman filter tuning in enorle PMSM rive," in IEEE Tranaction on Inutry Application, vol. 39, no. 6, pp , Nov.-Dec [9]. R. Dhaouai, N. Mohan an L. Norum, "Deign an implementation of an extene Kalman filter for the tate etimation of a permanent magnet ynchronou motor," in IEEE Tranaction on Power Electronic, vol. 6, no. 3, pp , Jul [10]. Zhang Bingyi, Chen Xiangjun, Sun Guanggui an Feng Guihong, "A Poition Senorle Vector-control Sytem Bae on MRAS for Low Spee an High Torue PMSM Drive," 2005 International Conference on Electrical Machine an Sytem, Nanjing, 2005, pp [11]. Faa-Jeng Lin, Chih-Hong Lin an Po-Hung Shen, "Self-contructing fuzzy neural network pee controller for permanent-magnet ynchronou motor rive," in IEEE Tranaction on Fuzzy Sytem, vol. 9, no. 5, pp , Oct [12]. M. A. Rahman an M. A. Houe, "On-line aaptive artificial neural network bae vector control of permanent magnet ynchronou motor," in IEEE Tranaction on Energy Converion, vol. 13, no. 4, pp , Dec [13]. Shi KL, Wong YK, Ho SL, Spee etimation of an inuction motor rive uing an optimize extene kalman filter., IEEE Tranaction on Inutrial Electronic 49: [14]. D. Xu, S. Zhang, an J. Liu, Very-low pee control of PMSM bae on EKF etimation with cloe loop optimize parameter, Elevier, ISA Tranaction,Vol. 52, (2013), Volume 5, Iue 7, July 2016 Page 106

8 [15]. Yahia Laamari, Kheireine Chafaa an Belkacem Athamena, Particle warm optimization of an extene Kalman filter for pee an rotor flux etimation of an inuction motor rive, Electrical Engineering, Springer, Vol.97, (2015), Author, Mahewara Rao Chintala receive B.Tech egree, firt cla with itinction from JNTUK, Kakinaa in He i currently puruing M.Tech in Power an Inutrial Drive at GMR Intitute of Technology, Rajam, AP, Inia Ramana Pilla receive B.Tech egree, firt cla with itinction from JNTU, Hyeraba in He receive M.Tech egree, firt cla with itinction from JNTU, Hyeraba in He i in the teaching profeion for lat 15 year. He i puruing hi part-time Ph.D at JNTUH, Hyeraba, AP, Inia. He ha publihe 30 Reearch Paper. Hi reearch interet inclue control ytem an electrical machine rive. At preent, he i working a Aociate Profeor in GMR Intitute of Technology, Rajam, AP, an Inia Ayyarao S L V Tummala receive B.Tech egree, firt cla with itinction from JNTU, Hyeraba in He receive M.Tech egree, firt cla with itinction from JNTU, Hyeraba in He i in the teaching profeion for lat 10 year. He i puruing hi part-time Ph.D at ANU, Guntur, A.P, an Inia. Hi reearch interet inclue application of optimization techniue an control & etimation of power ytem. At preent, he i working a Aitant Profeor in GMR Intitute of Technology, Rajam, AP, Inia. Volume 5, Iue 7, July 2016 Page 107