EKF- BASED SENSORLESS DIRECT TORQUE CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTOR: COMPARISON BETWEEN TWO DIFFERENT SELECTION TABLES

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1 ansaction in Machine, Powe electonics an Dives ISSN: Online Publication, June 21 MD-C17/GJO EKF- BASED SENSORLESS DIREC ORQUE CONROL OF PERMANEN MAGNE SYNCHRONOUS MOOR: COMPARISON BEWEEN WO DIFFEREN SELECION ABLES Habib Kaiem, Mustapha Messaoui, Lassaâ Sbita an Mohame Naceu Abelkim National Engineeing School of Gabes (ENIG), unisia Receive Apil 29, Revise June 29, Accepte Novembe 29 Abstact In this pape a sensoless iect toque contol (DC) scheme of pemanent magnet synchonous moto (PMSM) is popose. o impove the pefomance of the classical DC a moifie contol scheme base on twelve sectos instea of six is pesente an a compaison between the two methos is caie out. he high pefomance of the DC is elate to the accuacy of the flux estimation witch is affecte by paamete vaiation especially the stato esistance ue changes in tempeatue o fequency. heefoe, it is aequate to compensate this paamete vaiation using an online aaptation of the contol scheme by the estimate stato esistance using the Extene Kalman Filte (EKF). he EKF is esigne to estimate the oto spee an stato flux an esistance. Estimate paametes ae use fo the close loop spee sensoless contol opeation of the PMSM. It has been emonstate that the EKF estimation an sensoless DC pefom quite well in spite of the paametes an loa vaiations that hanle by the system. he simulation esults ae pesente to valiate the effectiveness of the oveall contol scheme. Keywos: DC, Synchonous machine, EKF, Senso-less, PM I. Intouction In high-pefomance sevo applications a api an accuate toque contol is esie, pefeably without the use of a motion-state senso [12]. he use of PMSMs combine with the DC scheme offes many oppotunities to achieve this goal. he appellation synchonous moto is eive fom the fact that the oto an the otating fiel of the stato otate togethe at the same spee. he oto tens to align itself with the otating fiel pouce by the stato. he stato has often a thee phase wining. he oto magnetization is cause by the pemanent magnets in the oto. hese motos types ae calle pemanent magnet synchonous motos (PMSMs). PMSMs ae wiely use in high pefomance ives such as inustial obots an machine tools thanks to thei known avantages as: low inetia, high efficiency, high powe ensity, high eliability an fee maintenance. Moe than two ecaes ago, DC was intouce by akahashi fo inuction motos as an altenative to the fiel oiente contol (FOC) technique [l], [2]. his ive stategy can also be applie to PMSM. he iea of combining the avantages of DC an PMSMs into a highly ynamic ive appeae in the liteatue in the late 199 s [4]. he DC stategy iectly contols the invete states base on the eos between the efeence an estimate values of toque an flux. It selects one of eight voltage vectos geneate by a voltage souce invete to keep toque an flux within the limits of two hysteesis compaatos. Compae with Roto Fiel Oiente Contol, DC has many avantages such as less machine paamete epenence (only the stato esistance), simple implementation an quicke ynamic toque esponse. hee is no cuent contolle neee in DC, because it selects the voltage space vectos accoing to the eos of flux linkage an toque. he main awback of the DC is its elatively high toque an flux ipple. o impove the pefomance of the classical DC, the stato flux locus is ivie into twelve sectos instea of six so all six active states will be use in each secto. In the classical DC, the stato flux vecto which is estimate as the integal of the stato voltage vecto as escibe by (5). his estimation suffes fom the well known poblems of the pue integation an on the othe han the stato esistance vaiation effect an this occu especially at low spee opeation ange [14]. heefoe we popose in this wok to use the EKF to obseve simultaneously the oto spee an stato flux an esistance. his obsevation oes not epen on moto paamete except the stato esistance. A constant value of stato esistance is

2 EKF- Base Sensoless Diect oque Contol of Pemanent Magnet Synchonous Moto: Compaison between wo Diffeent Selection ables 47 consiee. Howeve, in pactice, this paamete changes ue to vaiation of stato winings tempeatue. his fact intouces eos in the flux an the electomagnetic toque estimations an the ive may become unstable. So the compensation of the stato esistance vaiation effect becomes necessay [15] [16]. In this pape, the vaiation of stato esistance is online estimate an compensate using the EKF. stato flux an toque eos within the hysteesis ban an obtain the fastest toque esponse [9]. he selection of the voltage vectos in conventional DC is base on eo outputs pouce by the toque an flux hysteesis contolles (τ f an τ ) an is povie in the fom of a switching table as shown in able (1). able (1): he switching table of the classical DC II. Basic Synchonous Moto Moel he ynamic behavio of a synchonous machine is escibe by the following equations witten in space vecto notation an in the -q efeential [5], [6]: Stato Voltage Equations: vs = Ris + & ϕs ϕ v = Ri + ωϕq t ϕq vq = Riq + + ωϕ t Stato Flux Equations: ϕs = Li s s + ϕm ϕ = L i + ϕm ϕ q = Li q q Electomagnetic toque: 3 e = np( is ϕ s) 2 3 e = np( iqϕ iϕq) 2 3 = n ( L L ) i i + ϕ i 2 ( ) e p q q m q Mechanical Equation: ω J + fvω = e l t (4) Whee: vs = v v q is the stato voltage vecto, is = i i q is the stato cuent vecto, ϕs = ϕ ϕ q is the stato flux vecto, ϕ = ϕ is the pemanent magnet flux linkage vecto, m [ ] m R is the stato esistance, L an L q ae the an q axis self stato inuctances, n p the numbe of pole pais, e an l ae the electomagnetic an loa toques, J is the moment of inetia, ω is the oto spee an f v is the fiction facto. III. Diect oque Contol Stategy DC of PMSM is caie out by hysteesis contol of moto stato flux an toque that iectly select one of the six non-zeo an two zeo iscete voltage vectos of the invete [7] Fig. 1. he selection of the voltage vectos is mae so as to estict the moto (1) (2) (3) IV. Flux an oque Estimation In the conventional DC scheme, the stato flux is gotten fom (5), which is eive fom (1) using only the measue stato voltages an cuents [3]. ϕ = v R i t (5) ( ) s s s Fig. 1. Classical DC an its six sectos his equation is the founation fo implementing the flux estimato. It may be implemente iectly, o appoximate by vaious methos to avoi integato ift [1]. At aveage an high spee, the voltage op tem (R.i s ) can be neglecte, so the stato flux expession becomes: ϕ s = Vs (6) t heefoe the stato flux vaiation is null fo a null voltage vecto. ϕ s = (7) t DC equies accuate knowlege of the amplitue an angula position fo the flux contolle: 2 2 ϕ = ϕ + ϕ (8) s α β he angula position of the stato flux vecto must be known so that the DC can chose between an appopiate set of vecto epening on the flux position.

3 48 EKF- Base Sensoless Diect oque Contol of Pemanent Magnet Synchonous Moto: Compaison between wo Diffeent Selection ables ϕ 1 β θs = tan (9) ϕα he electomagnetic toque can be estimate by using the stato cuents measuement an estimate flux as (1): 3 e = np( iqϕ iϕq) (1) 2 V. oque an flux hysteisis compaatos he electomagnetic toque value esulting fom (1) is then compae with the efeence electomagnetic toque, using the thee level hysteesis compaato, epesente in Fig. 2. In this manne, the esult may be incease, ecease o maintain the toque, epening on the compaato output [8], [14]. Fig. 2. hee level hysteesis compaato. τ = 1 ; incease toque, τ = ; maintain toque, τ = 1 ; ecease toque. In a simila way, the flux value will be compae with the efeence flux, but using a two level hysteesis compaato, shown in Fig. 3. he esult will be use to incease o ecease the flux. Fig. 4. hee phase PWM invete switching. he combination of the compaato states, (τ, τ f ) an the position secto of the stato flux ae use by the optimal switching selection table (1) to chose an appopiate voltage vecto to be applie to the invete [9]. VII. welve sectos able In classical DC, as shown by Fig. 1, thee ae two states pe secto that pesent a toque ambiguity. heefoe, they ae not use. It can be seen in the fist secto that the vectos V 1 an V 4, ae not use in the classical DC because they can incease o ecease the toque at the same secto epening on if the position is in its fist 3 egees o in its secon ones. It seems a goo iea that if the stato flux locus is ivie into twelve sectos instea of just six, all six active states will be use pe secto as shown in able (2). Consequently, it is aisen the iea of the twelve sectos DC. his novel stato flux locus is intouce in Fig. 5 Notice how all six voltage vectos can be use in all twelve sectos. Howeve, it has to be intouce the iea of small toque incease instea of toque incease, mainly ue to the fact that the tangential voltage vecto component is vey small an consequently its toque vaiation will be small as well. heefoe, the hysteesis block shoul consist of fou levels instea of two. able (2): he switching table of the 12 sectos DC Fig. 3. wo level hysteesis compaato. τ f =1 ; incease flux, τ f = ; ecease flux. An impotant facto in these opeations is the hysteesis ban of the two compaatos. A too small value may have the effect of losing the contol. he stato flux linkage may excee the values equie by the toleance ban. A naow winow will give bette cuent an flux wavefoms but will also incease the invete switching fequency. VI. Voltage Souce Invete he voltage souce invete can be moelle as shown in Fig. 4, whee S a, S b, S c ae the switching states (1 in case of switch-on, in case of switch-off). Eight output voltage vectos V to V 7 {, 1, 11, 1, 11, 1, 11, 111} ae obtaine fo iffeent switch combinations. Hence, V an V 7 ae zeo voltage vectos. he voltage vecto of the thee- phase voltage invete is pesente as follows: ( ( π ) ( π )) V = (2 / 3) U S + S exp i2 / 3 + S exp i4 / 3 (11) s a b c Fig. 5. he 12 sectos DC an its new sectos

4 EKF- Base Sensoless Diect oque Contol of Pemanent Magnet Synchonous Moto: Compaison between wo Diffeent Selection ables 49 VIII. Extene Kalman Filte he Kalman filte (KF) is a special kin of obseve, which povies optimal filteing of noises in measuement an insie the system if the covaiance matices of these noises ae known. he pocess an the measuement noises ae both assume to be Gaussian with a zeo mean. he elements of thei covaiance matices (Q an R) seve as esign paametes fo the convegence of the algoithm. Fo nonlinea poblems, the KF is not stictly applicable since lineaity plays an impotant ole in its eivation an pefomance as an optimal filte. he EKF attempts to ovecome this ifficulty by using a lineaize appoximation whee the lineaization is pefome about the cuent state estimate [11]. he geneal iagam of the EKF is given by Fig. 6. Fig. 6. he geneal iagam of the Extene Kalman Filte Also the KF has the ability to pouce estimates of states which ae not measuable. his featue is paticulaly impotant fo estimation poblems associate with the PMSM as the oto quantities ae not iectly accessible. If a simultaneous estimate of the machine paamete, let say stato esistance, is neee then it is efine as an auxiliay state vaiable. A new state vecto containing the oiginal states an the paamete is then establishe. In this case, the nonlineaity of the system inceases. heefoe, the EKF is moe convenient than the KF. Using the -q cooinate system an sepaating the machine vaiables state vectos into thei eal an imaginay pats, the well-known PMSM moel expesse in tems of the state vaiables is obtaine fom (1)-(4), an is given by : ( ) &x = A x x+ Bu (12) y = Cx Whee x, u an y ae espectively the state vecto, the input vecto an the output vecto which ae efine as fellow: x i = i q ω R Lq npω L L L R ϕmn p, A= npω Lq Lq Lq 3 np 3 n p fv ( L Lq) iq ϕm 2 J 2 J J 1 L v 1 B = L, u v 1 = q, C i = q 1, y = i q l 1 J he stato flux an the electomagnetic toque ae not estimate iectly using the EKF. hei estimations ae one using espectively equations (2) an (3) an by exploiting the estimate stato cuent i an i q. Let us now see the ecusive fom of the EKF as in [11]. Peiction: xˆ(( k + 1) / k = F( k). xˆ( k/ k) + G( k). u( k) (13) P(( k + 1)/ k) = F( k). P( k/ k). F ( k) + Q (14) Coection: xˆ(( k+ 1) /( k+ 1)) = xˆ(( k+ 1) / k) + K( k+ 1) y( k+ 1) H( k+ 1). xˆ(( k+ 1) / k) (15) 1 ( 1) (( 1)/ ). Kk+ = P k+ k H ( k+ 1). Hk ( ). P(( k+ 1)/ k). H ( k) + R (16) P(( k+ 1) /( k+ 1)) = P(( k+ 1) / k) K( k+ 1). H( k+ 1). P(( k+ 1) / k) (17) Whee the estimation covaiance eo is: Pk ( / k) = E{ ( xk ( ) xk ˆ( ))( xk ( ) xk ˆ( )) } (18) K is the Kalman gain matix. ((k+1)/k) enotes a peiction at time (k+1) base on ata up to an incluing k. (14) an (17) foms the well-known Riccati equation. Equation (12) efines a continuous moel, but as estimation is to be implemente on a igital pocesso, the PMSM continuous moel must be witten in a iscete fom. By applying the Eule fomula a iscete time-vaying non-linea moel is obtaine: A = exp( A ) I + A. (19) B = exp( Aξ). B ξ B. (2) he iscete time vaying nonlinea stochastic moel of the PMSM has the following fom: x( k + 1) = F( k) x( k) + G( k) u( k) (21) yk ( ) = Hk ( ). xk ( ) (22) Whee x(k), u(k) an y(k) ae efine as fellow: x( k) = ( ) ( ) ( ) ( ) i k iq k ω k R k u( k) = ( ) ( ) ( ) v k vq k l k, y( k) = i ( ) ( ) k iq k he pocess an the measuement noise vectos ae anom vaiables an chaacteize by: E w( k) =, E w( k) w( j) = Qδ ; Q (23) { } { } { } { } E v( k) =, E v( k) v( j) = Rδ kj ; R (24) he initial state x() is chaacteize by: E x() = x, E ( x() x )( x() x ) = P (25) { } { } IX. Simulation Results o veify the effectiveness of the popose DC sensoless PMSM ives as illustate in Fig. 7, a igital simulation base on Matlab- Simulink softwae package has been caie out. Moto paametes use in simulation ae given in the appenix. he use sampling peio fo simulation is = 1-5 s. kj

5 5 EKF- Base Sensoless Diect oque Contol of Pemanent Magnet Synchonous Moto: Compaison between wo Diffeent Selection ables vaiation on the flux tajectoy (R = 1.3*R at.7s). We can see that the position an the magnitue of stato flux ae change Fig. 1. Stato cuents esponses Fig. 7. DC bloc iagam of a sensoless PMSM ives. Fig. 8 to 11 show the oto spee, the electomagnetic toque, the stato cuents an flux esponses to a tiangula wave spee efeence taget. he flux efeence is maintaine at its nominal value of.3 Wb an a squae wave loa toque is applie. Fig. 8 shows the actual an estimate esponses of the popose PMSM sensoless scheme. oque contol ynamic of the DC is shown in Fig. 9. It is note that fast tansient toque esponse is obtaine owing to the fast ynamic of the hysteesis contolle. It can be seen fom Fig. 11 that flux is inepenent fom the toque vaiations an the estimate flux magnitue follows the efeence value. Fig. 11. he stato flux magnitue Fig. 12. he actual an estimate stato esistance.4 Fig. 8. Actual an obseve oto spee Fig. 9. Electomagnetic oque Fig. 12 an 13 show espectively the simulation esults of the estimation of the stato esistance an the effect of this paamete Quaatic axis stato flux (Wb) Diect axis stato flux (Wb) Fig. 13. he stato flux tajectoy without compensation of the stato esistance vaiation. he esults in Fig. 14 to 16 illustate the oto spee, the electomagnetic toque, an the stato cuent an flux simulation esponses when a loa toque step of 1 Nm at.5s is applie with compensation of the stato esistance vaiation. he taget spee is change fom a/s to 157 a/s at.1s. he esult clealy shows that the estimate spee follows the actual one with a negligible eo afte loa toque an stato esistance vaiation ( l =1 N.m at.5s, R = 1.3*R at.7s). he tajectoy of the estimate flux

6 EKF- Base Sensoless Diect oque Contol of Pemanent Magnet Synchonous Moto: Compaison between wo Diffeent Selection ables 51 vecto pesente in Fig. 17 shows that the compensation of the stato esistance vaiation is achieve; the magnitue an the position of stato flux ae not affecte when stato esistance is incease. he loa toque vaiation i not affect the stato flux tajectoy. he main awback of the DC is its elatively high toque an flux ipples as shown in Fig. 15 an 17. In all simulation pesente, it can be obseve that EKF gives a goo obsevation of the oto spee, stato cuent, flux an esistance. Fig. 14. Actual an obseve oto spee Fig. 18, 19 an 2 pesents a compaison between the classical DC an the popose twelve sectos DC. It is clealy shown that, the novels DC euces the toque an flux ipples but thee ae not completely eliminate. Electomagnetic toque (Nm) Classical DC,5 12 sectos DC -, ime(s) Fig. 18. Electomagnetic oque ynamics magnitue in classical DC an 12 sectos DC..3 Fig. 15. Electomagnetic oque ynamics Stato flux (Wb) Classical DC.3 12 sectos DC Zoom ime (s) Fig. 19. he stato flux magnitue in classical DC an 12 sectos DC..4 Quaatic axis stato flux (Wb) Fig. 16. Stato cuents esponses Diect axis stato flux (Wb) Fig. 17. he stato flux tajectoy with compensation of the stato esistance vaiation. Quaatic axis stato flux (Wb) Classical DC 12 sectos DC Diect axis stato flux (Wb) Fig. 2. he stato flux tajectoy in classical DC an 12 sectos DC. X. Conclusion In this pape, moeling an simulation of Sensoless DC of PMSM has been pesente. EKF is esigne an evelope fo use in close loop contol to estimate the oto spee, stato flux an esistance. he filteing action of EKF impoves the system pefomance in pesence of stato esistance an loa toque vaiations. he main avantages of this metho ae the high ynamics of toque contol, simple contol algoithm an shot time of calculation. o ovecome the majo awback of the DC, witch is the toque an flux ipples, twelve sectos DC scheme is

7 52 EKF- Base Sensoless Diect oque Contol of Pemanent Magnet Synchonous Moto: Compaison between wo Diffeent Selection ables use. It is clealy shown that the ipples ae euces but not completely eliminate. APPENDIX hee phases PMSM paametes: Rate output powe 15 Watt, Rate phase voltage 22/38V, φ m =.15Wb, R=1.4Ω, L =.66H, L q =.58H, f v =.38N.m.a -1.s, J =.176 kg.m 2, n p = 3. REFERENCES [1] I. akahashi, an. Noguchi, A new quick-esponse an high-efficiency contol stategy fo an inuction moto, IEEE ans. In. Applicat., vol. IA-22, no. 5, pp , [2] M. Ben Hame an L. Sbita, Spee sensoless iniect stato fiel oiente contol of inuction moto base on luenbege obseve, IEEE ISIE, Quebec, Canaa, pp , July 9-12, 26. [3] K.E.B Quinee, E.F Ruppet, an M.E.F Oliveia, Diect toque contol of pemanent magnet synchonous moto ive with a thee-level invete, Powe Electonics Specialists Confeence PESC 6 37th IEEE, pp. 1-6, June 26. [4] L. Zhong, M. F. Rahman, W. Y. Hu, an K. W. Lim, Analysis of iect toque contol in pemanent magnet synchonous moto ives IEEE ans. Powe Electon., vol. 12, no. 3, pp , May [5] D. Sun, J. G. Zhu an Y. H. Kang, Continuous iect toque contol of pemanent magnet synchonous moto base on SVM, Int. Conf. on Electical Machines an Systems ICEMS 23, vol. 2, pp , 9-11 Nov. 23. [6] A. B. Dehkoi, A. M. Gole an. L. Maguie, Pemanent magnet synchonous machine moel fo eal- time simulation, Int. Conf. on Powe Systems ansients IPS 5, No. IPS5 159, Monteal, Canaa, June 19-23, 25. [7]. J. Vyncke, R. K. Boel an J. A Melkebeek, Diect oque Contol of Pemanent Magnet Synchonous Motos, 3RD IEEE Benelux Young Reseaches Symposium in Electical Powe Engineeing, no. 28, Ghent, Belgium, Ap , 26, [ 8] M. Kajouj, S. aibi, N. Golea an M. E. H Benbouzi, Moifie iect toque contol of PMSM ives using ithe signal injection an non-hysteesis contolles, Int. Conf. Sciences an echniques of Automatic contol SA 6, Hammamet, unisia, 26. [9] J. Habibi an S. Vaez-Zaeh, Efficiency Optimizing Diect oque Contol of Pemanent Magnet Synchonous Machines, Poc. IEEE Powe Electonics Specialists Confeence, PESC 25, Recief, Bazil, pp , June 25. [1] Y. Li, H. Hu, J. Chen an W. Jixiong, Peictive contol of toque an flux of inuction moto with animpove stato flux estimato, Powe Electonics Specialists Confeence, PESC 21, vol. 3, pp , 21. [11] C. Vasile, M. Cenat, F. Moloveanu an D. Ioan, Sensoless Spee an Diect oque Contol of Suface Pemanent Magnet Synchonous Machines using an Extene Kalman Filte, 9th IEEE Meiteanean Conf. on Contol an Automation MED'1, no. 5, June 27-29, 21. [12] M. Boussak, Implementation an expeimental investigation of sensoless spee contol of pemanent magnet synchonous moto ive, IEEE ans. In. Electon., vol. 2, no. 6, pp , Nov./Dec. 25. [13] J. Belhaj, Commane iecte en couple 'une machine asynchone application aux systèmes multimachines multiconvetisseus, Ph.D. issetation, Ecole Nationale 'Ingénieus e unis, pp. 185, july. 24. [14] G. S. Buja an M. P. Kazmiekowski, Diect toque contol of PWM invete-fe AC motos a suvey, IEEE ans. In. Electon, vol. 51, no. 4, pp , Aug. 24. [15] M. E. Haque an M. F. Rahman, he effect of stato esistance vaiation on iect toque contolle pemanent magnet synchonous moto ives an its compensation, he Austalasian Univesities Powe Engineeing Confeence Bisbane AUPEC, Sept. 2. [16] M. Messaoui, H. Kaiem, L. Sbita an M. N. Abelkim, A Robust Sensoless Diect oque Contol of Inuction Moto Base on MRAS an Extene Kalman Filte, LJS, AcaemicDiect, Issue 12, p June 28