SIMULATION OF PERMANENT MAGNET SYNCHRONOUS MOTOR CONTROL SYSTEM WITH EXTENTED KALMAN FILTER

Transcription

1 JAI & LLS. All ight eeved. SIMULAION OF PERMANEN MAGNE SYNCHRONOUS MOOR CONROL SYSEM WIH EXENED KALMAN FILER 1 AO XU, 2 ZHENGBIN ZHANG 1 Aoc. Pof., Depatment of Automation, Shenyang Aeopace Univeity, P. R. China 2 Potgaduate Student, Depatment of Automation, Shenyang Aeopace Univeity, P. R. China 1 xutao@au.edu.cn, 2 binbincool.o@163.com ABSRAC With the development of Pemanent Magnet Synchonou Moto (PMSM), it i applied to moe and moe aea. he contol ytem of PMSM become the mot impotant iue in application. Becaue the enole contol ytem poee ome good pefomance, it attact many eeache inteet. hi pape peent the application of Extended Kalman Filte (EKF) in PMSM peed contol ytem. A PMSM vecto contol ytem model i etablihed with Matlab/Simulin. he mathematical model of two-phae tationay coodinate ytem ( α - β coodinate ytem) i adopted fo lineaization of the nonlinea mathematical model of PMSM. EKF algoithm etimate the otation angle θ and otation peed ω of the oto of PMSM in the poce of ytem contol. he imulation eult how that the contol ytem wo moothly in the cae of the upe-peed, low-peed and with load. EKF etimation of the otation angle θ and peed ω tac actual angle and peed fom eno accuately. Keywod: Pemanent Magnet Synchonou Moto (PMSM), Modeling and Simulation, Extended Kalman Filte (EKF), Vecto Contol 1. INRODUCION With the development of the powe electonic, mico-electonic technology, new moto theoy and the pemanent magnet mateial, PMSM ha been pomoted apidly and application aea ha been extended in ecent yea. Becaue of the pemanent magnet excitation of PMSM, the moto tuctue i elatively imple and the poceing and aembly cot ae educed damatically. Meanwhile, eliminate the need fo eo-pone collecto ing and buhe and impove the eliability of the opeation of the PMSM. heefoe, PMSM poee many good pefomance uch a imple tuctue, mall ize, high efficiency, high powe facto and low moment of inetia compaed with taditional electically ynchonou moto. With the impovement of pemanent magnet mateial, the vecto contol ytem of PMSM achieve a wide ange of peed and poition contol with high peciion and high dynamic pefomance in cae of mall o medium powe load, high eliability and wide peed ange of the evo ytem. hi pape focue on EKF algoithm and etablihe imulation model of vecto contol ytem fo PMSM with Matlab/Simulin. Afte pogamming the EKF algoithm by S function, enole vecto contol ytem of PMSM i tudied baed on the imulation model. hi pape i oganized a follow. Section II peent the elated wo of PMSM in ecent yea. Section III peent the tuctue of PMSM vecto contol ytem. In Section IV, we intoduce the mathematical model of PMSM. In Section V, we decibe the tate etimation with EKF. Section VI how the imulation analyi of PMSM contol ytem. Finally, we peent meaningful concluion. 2. RELAED WORK Many eeache ae inteeted in PMSM and the application of it ae pomoted gadually [1]. Geneally, PMSM contol ytem i divided into peed eno contol and peed enole contol. he fome acquie otation angle and otation peed though mechanical eno to modulate the moto peed in eal time in ode to achieve the deied peed value. he latte adopt the mathematical model of PMSM to etimate the angle and peed of the oto in eal time with ome method, including model efeence adaption, high- 144

2 JAI & LLS. All ight eeved. fequency ignal injection, tate obeve, Kalman optimal etimation, intelligent contol and et cetea [2]-[6]. Among them, model efeence adaption i imple. But it depend on the accuacy of paamete of the PMSM model. When the moto wo in cae of low otation peed, the highfequency ignal injection method ha cetain advantage. But it will poduce toque ipple, additional loe and equiement of aliency. Intelligent contol algoithm poee the chaacteitic of elf-leaning and adaption. But it i moe complex than the poceeding method and i not eay to modulate tuctue and deign paamete [7]. Compaed with othe method, the EKF algoithm can effectively uppe the affect of ytem eo and meauement eo on tate etimation. Meanwhile, it poee uch good pefomance a fat convegence peed and high accuacy fo tate etimation. EKF i a bette peed obeve in thoe enole contol method [8]. 3. SRUCURE OF PMSM VECOR CONROL SYSEM Vecto contol ytem tuctue of PMSM i hown in figue 1, including Popotional Integal (PI) modulato, Space Vecto Pule Wih Module (SVPWM), EKF etimation module, Cla-Pa tanfom module, thee-phae invete and PMSM module. PMSM adopt Field Oiented Contol (FOC) contolling method with dual cloed-loop contol cheme of peed and cuent egulated by PI modulato. Sytem etimate otation angle θ and otation peed ω fom EKF in eal time. Hee, ω i the feedbac vaiable of peed contol loop and θ i the paamete of the Pa tanfom and invee tanfom to implement elevant calculation. * ω + PI ω * i d * iq + i d + i d i v q q PI dq, v PI d α,β i q θ EKF Etimation dq, α,β v α SV v β PWM i α α,β i β abc,, v α v β i α i a i b U dc 3-Phae Invete PMSM i β Fig. 1 Senole Vecto Contol Sytem Stuctue of PMSM 4. MAHEMAICAL MODEL OF PMSM he mathematical model of PMSMS can be divided into thee ind of epeentation by the axi ytem, including the thee-phae tationay coodinate ytem (A-B-C hafting coodinate ytem), the tato phae tationay coodinate ytem ( α - β coodinate ytem) and the oto two-phae otating coodinate ytem (d-q coodinate ytem). he mathematical model utilized fo EKF etimation i the α - β coodinate ytem in thi pape. heefoe, thi coodinate ytem i epeented a follow. he voltage equation of PMSM in the α - β coodinate ytem ae equation (1) and (2). i diα U = R + L ω N λ in θ α α p (1) i diβ U = R + L + ω N λ co θ β β p (2) Fom equation (1) and (2), we have di R N pλ α Uα = iα + ω inθ + (3) L L L diβ R N pλ U i β = β ω coθ + (4) L L L In digital ytem, the ampling cycle i vey hot. And each ampling peiod can be conideed a contant. So, we have the next equation (5) and (6). dω = 0 (5) dθ = ω (6) Hee, U α and U β peent the axi voltage of α and β epectively, i α and i β peent the axi cuent of α and β epectively. R i the equivalent phae eitance of the PMSM tato and L i the equivalent phae inductance of the PMSM tato. N i the numbe of moto pole p and λ i the oto flux. ω i the mechanical angula velocity of the oto and θ i the angula poition of the oto. Fom the equation (3)-(6), we have the tate equation and output equation of PMSM in equation (7) and (8). dx Ax Bu = + (7) y = Hx (8) 145

3 JAI & LLS. All ight eeved. Hee, iα iβ U α iα x =, u = w, = U β iβ θ R N λ p 0 inθ 0 L L R N λ p, A = 0 coθ 0 L L N 0 p y, 1 0 L B = 0, H =. L Suppoe ampling peiod i, we have the dicete epeentation fom equation (7) and (8). heefoe, dicete tate equation of PMSM ae equation (9) and (10) a follow. x f I A x B u (9) = (x-1,u -1) = ( + ) y = h x ) = H x ( (10) 5. SAE ESIMAOIN WIH EKF Kalman filte i the tate etimation fo linea ytem. EKF extend the application of Kalman filte into nonlinea ytem. Becaue contol ytem of PMSM i a non-linea ytem, thi pape adopt EKF to etimate tate fo PMSM. he geneal expeion of EKF ae equation (11) and (12) [6]. dxˆ = A( xˆ) xˆ + Bu + K( y yˆ) (11) yˆ = Hxˆ (12) Stuctue cheme of EKF equation i hown in figue 2 a follow. u Sytem Kalman filte K x x H H y - y Fig. 2 Stuctue Scheme of EKF y - y + Afte the ytem noie and obevation noie ae involved into the contol ytem, non-linea dicete expeion of PMSM ae equation (12) and (13). x = f( x, u ) + W = ( I+ A ) x + B u + W (12) Fig. 3 Sytem Simulation Diagam Hee, y = hx ( ) + V = H x+ V (13) W and V ae zeo-mean Gauian white noie equence. Covaiance matixe of W and 146

4 JAI & LLS. All ight eeved. V ae cov (W) E { WW } Q = = and { } R = cov (V) = E VV epectively. When lineation poce fo the nonlinea dicete tate equation (12) of PMSM, uppoe the optimal etimation xˆ of tate vaiable x i nown ahead of time at a moment. Fom equation (12), extend f () into aylo eie at xˆ and omit thoe item moe than econd-ode [9]. he nonlinea function acquied i equation (14). f x f( xˆ ) + (x - xˆ ) + W (14) xˆ Now, define that f ( f xˆ, 1) -1 Φ = = xˆ xˆ x ˆ 1 x-1 R λ N λ N p p 1 0 inθ ω coθ L L L R λ N λ N p p = 0 1 coθ ω inθ L L L N 1 p Etimate the tate vaiable x at diffeent time with ecuive EKF algoithm. ˆ ˆ (15) x = f( x,u ) = ( I+ A ) x + B u P =Φ Pˆ Φ + Q (16) -1 1 = ( + ) (17) K PH H PH R xˆ = x + K ( y H x ) (18) Pˆ = P KHP (19) In the equation above, x~ peent the tate vaiable at time and it i pedicted by optimal x tate etimation ˆ -1 pediction value at time.. Alo, it i called tate K i the gain matix of Kalman filte and P ~ i the covaiance matix of tate pediction value x~. Pˆ i the covaiance matix of optimal tate etimation xˆ. 6. SIMULAION ANALYSIS OF PMSM CONROL SYSEM Speed enole contol ytem imulation model of PMSM i etablihed with Matlab/Simulin. he imulation model diagam i hown in figue 3. In the imulation model, PMSM model i deigned with SimpoweSytem oolbox and it paamete ae configued be the efeence liteatue [7]. R = 2.875Ω, L = 8.5mH, λ = 0.175Wb, N = 4 =4 p 2 J = g m, In the S function fo EKF etimation, configue the initial tate x = [ 0 ;0; 0; π 2], the initial covaiance matix of tate eo P=diag(0.5,0.5,100,10), the initial covaiance matix of ytem noie Q=diag(5,5,200,1) and the initial covaiance matix of obevation noie R=diag(0.5,0.5). In the poce of EKF etimation, the calculation of the Kalman filte gain K i elated with Q, R and P. If the covaiance matix Q change, R will influence the dynamic chaacte and tability of the EKF. Meanwhile, that the Q and R ae appopiate o not geatly influence the accuacy of the EKF etimation and convegence of the algoithm. Geneally, Q and R ae unnown ahead of time. he detemine pinciple i jut conideed baed on the tatitical chaacteitic of the ytem and meauement noie. In the actual ytem, they ae geneally detemined by expeience and imulation eult. hi pape chooe appopiate value by the peviou expeience fom liteatue [10]. Afte the epetition of imulation and adjutment of the tial-and-eo, thoe initial value ae configued a befoe. In the poce of ytem imulation, configue the efeence peed a 400ad/. Afte 0.2, peed become 200ad/. he imulation duation i 0.5. Expeiment imulation eult of peed i hown in figue 4. In the figue 4(a), the ed cuve peent the actual meauement fom peed eno. A can be een fom the figue above, ytem ditubance emege in the poce of ytem tating and tate tanition. But, the tanient to table tate i vey quic. In the figue 4(b), blue cuve peent the etimation of peed with EKF in enole condition. Compaed with actual peed meauement, etimation fom EKF lightly lag in the poce of ytem tating and peed tanition. 147

5 JAI & LLS. All ight eeved. But, they till meet the pefomance equiement of the contol ytem and capable of tacing the actual meauement peed value. Afte the ytem tanition to teady-tate, enole contol i conitent with contol by eno accuately. (a) Actual peed with eno CONCLUSION Diffeent contol model of PMSM ae tudied by imulation in thi pape. We compae the enole contol ytem with the eno contol ytem by Matlat/Simulin. Simulation eult how that EKF algoithm accuately tac otation poition and otation peed of PMSM when the accuate PMSM model i available. heefoe, EKF tate etimation meet the pefomance equiement of contol ytem and poee the imila pefomance of eno contol ytem. Yet, the tacing eo of the etimation model i elatively lage in the beginning tage of the peed etimation and the peed change tage, epecially in the beginning tage. o deceae the tacing eo, moe complicated model will be tudied, uch a the intelligent contol ytem that may play the impot ole in the poce of ytem contolling. REFRENCES: (b) Etimation of peed with EFK Fig. 4 Simulation Reult Of Speed Angle poition etimation when enole and angle poition meauement with eno ae hown in figue 5, in which ed cuve peent the angle meauement and blue cuve peent the angle etimation fom EKF. It clealy how that the etimation ae nealy identical with the actual meauement. Simulation eult how the feaibility of EKF etimation in the PMSM contol ytem in cae of enole condition. Fig. 5 Simulation Reult of Angula Poition [1] Wang Jian, Oveview Of he Moden AC Sevo Sytem echnology And he Maet Development, Sevo Contol, Vol. 12, No. 9, June 2008, pp [2] Wang Chengyuan, Xia JiaKuan, Yang Junyou and et al, Moden Mote Contol echnology, Beijing: Machiney Induty Pe, Augut 2006, pp [3] S Bolognani, L ubiana and M Zigliotto. Extended Kalman Filte uning in Senole PMSM Dive, IEEE anaction on Induty Application, 2003, Vol. 6, No. 39, Novembe 2003, pp [4] Jiang Jun, Shen Yanxia and Ji Zhicheng, Speed and Rote Poitoin Etimation fo PMSM Baed on EKF, Acta Simulata Sytematic Sinica, Vol. 17, No. 7, July 2005, pp [5] Jeong Y, Loenz R D, Jahn M and et al, Initial Roto Poition Etimation of an Inteio Pemanent Magnet Synchonou Machine uing Caie Fequency Injection Method, IEEE anation on Induty Application, Vol. 41, No. 1, May 2005, pp [6] AlbetQiu and Bin Wu. Senole Contol of Pemanent Magnet Synchonou Moto Uing Extended Kalman Filte, IEEE Enegy Conveion Conge and Expo (CCECE), Electical and Compute Engineeing (Canada), May , pp

6 JAI & LLS. All ight eeved. [7] Xue Shugong, Wei Liheng and Ling Youzhu, Senole Vecto Contol of Pemanent Magnet Synchonou Mote Baed on Extend Kalman Filte, Electic Machine & Contol Application, Vol. 38, No. 8, Augut 2011, pp [8] Wang Chenchen and Li Yongdong, A Speed Senole Vecto Contol Method of IM Uing Extended Kalman Filte, Jounal of inghua Univeity (Science and echnology), Vol. 48, No. 10, Octobe 2008, pp [9] Fu Mengyin, Deng Zhihong and Yan Liping, Kalman Filte Pinciple and it Application in Navigation Sytem, Beijing: Science Pe, Apil 2010, Second Edition, pp [10] An Quntao, Sun Li and Li Bo, Vaiable Paamte EKF fo Speed Etimation of PMSM, Electic Machine and Contol, Vol. 11, No. 6, June 2007, pp