Sensorless Control of PMSM Based on Extended Kalman Filter

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1 Zong ZHENG,, Yongong LI, Mauric FADEL. Lab. LAPLACE UMR-CNRS, INP-ENSEEIH Ru Charls Camichl, oulous, Franc l.: + / ()... Fax: + / ()... zong.zhng@lapalc.univ-tls.fr mauric.fal@lapalc.univ-tls.fr URL: Dpartmnt of Elctrical Enginring, singhua Univrsity Bijing,, China l.: + / ()... Fax: + / () liy@mail.tsinghua.u.cn URL: Kywors «Prmannt magnt motor», «Synchronous motor», «Vctor control», «Snsorlss control». Abstract his papr proposs a snsorlss control systm bas on xtn Kalman filtr (EKF) for prmannt magnt synchronous motors (PMSM). h EKF uations ar built in rotor flux orint synchronous coorinat, so it can asily b us for ithr non-salint or salint pol motors. Inrtia an othr mchanic paramtrs ar not n in this obsrvr. Rotor sp an position can b stimat xactly an thn a snsorlss control systm is built. h initial rotor position an th mchanic paramtrs ar not n in this systm. By som compnsation in obsrvr uations, th obsrvr can b always stabl an has only on xpct uilibrium point. So th motor can start up from any unknown initial positions. Introuction Prmannt magnt synchronous motors (PMSM) ar mor an mor us bcaus of its high powr nsity, larg toru to inrtia ratio an high fficincy. h rotor flux is gnrat by th prmannt magnt on th rotor. So th rotor flux position is th sam as th rotor lctrical position. An th prcious rotor position is n for th high prformanc control. Sinc mchanic position snsor is usually too xpnsiv, incrass th cost an cras th stability of th systm, mchanic snsorlss control is bcoming a rsarch focus now. Som snsorlss control mthos hav bn propos bfor. Gnrally thr ar th mthos bas on back lctromotiv forc (EMF), mol rfrnc aaptiv systm (MRAS) (xampl in []) an stat obsrvr mtho. h mthos bas on back EMF ar simpl but osn t work wll in low sp rgion bcaus th back EMF is too small compar with th nois. h mtho bas on MRAS also cannot gt a satisfying prformanc in low sp rgion an is gratly pn on th accuracy of th rfrnc mol. h stat obsrvr mtho is not suitabl for th nonlinar mol an is har to know th fback matrix. Also som mthos bas on Kalman filtr hav bn propos, most of which ar foun in th static two-phas coorinat, sinc th stator inuctanc of salint pol motor is a variabl of rotor position in static two-phas coorinat, ths obsrvrs can harly b us for salint pol motors [][][]. EPE - Aalborg ISBN : 99 P.

2 In this papr, a nw stat obsrvr bas on xtn Kalman filtr is us to obsrv th rotor position an sp. h obsrvr mol is st up in th rotor flux orint synchronous coorinat, so it can b us asily in ithr salint or non-salint pol motor bcaus th stator inuctancs in synchronous coorinat ar always constant. Extn Kalman filtr can solv nonlinar uation irctly by numric itration. Kalman filtr also consirs th rrors of th paramtrs an th noiss in th masurmnt, so it is vry robust with th paramtrs rrors an masurmnt noiss. Also th initial rotor position is not ncssary for th start-up. By a propr compnsation in th obsrvr uation, th othr unxpct uilibrium points of th obsrvr ar mov off. h motor can startup succssfully from any unknown initial position[][][]. Obsrvr bas on xtn Kalman filtr In rotor flux orint synchronous coorinat (, axs), PMSM mol is shown in (). I U R I L = + ω I t L L L I U R I L ψ I r = ω ω t L L L L () ω = t θ = ω t Hr I an I ar th currnts in an axs. R is stator rsistanc whil L an L ar th stator phas inuctancs in an axs. For non-salint motors, L is th sam as L. U an U ar stator voltags an R is stator rsistanc. ω is th rotor lctrical angl sp an θ is rotor lctrical angl (rotor flux angl). ψ r is rotor flux amplitu. Rotor sp is consir to chang mor slowly compar with othr variabls. Stat uations for PMSM can b writtn as (). x = gxu (, ) + w () y = C x+ v Hr x= I I ω θ (a) C = (b) I y = I (c) Hr w an v ar ranom isturbancs. In fact w is th procss nois which stans for th rrors of th paramtrs; v is th masurmnt nois which stans for th rrors in th masurmnt an sampl. h nois covarianc matrixs ar fin as follows: Q = cov( w) = E{ ww } (a) R = cov( v) = E{ vv } (b) Extn Kalman filtr can b built by th rivation blow: U R I L I( k) + + ω I L L L U R I L xk ( ) f xk ( ) x ψ + = = + r = I( k) + ω I ω () L L L L ω( k) θ( k) + ω EPE - Aalborg ISBN : 99 P.

3 Dfin matrix F: L L ω I τ L L f L L K F = = ω ( I ) () x L τ L L τ = L / R, τ = L / R ar stator constants. Dfin matrix P as th rror covarianc of obsrvation n P { } {[ ˆ ][ ˆ k = E k k = E xi xi xi xi] } () i= E { } is th computation of xpctation valu. Extn Kalman filtr can b raliz by itration as follows:. Comput th stat aha an th rror covarianc aha. xˆ = xˆ + x (a) kk k k kk k k k k k P = F P F + Q (b). Comput th Kalman gain. K ( ) k = Pk k C C Pk k C + Rk (c). Upat stimation with masurmnt. xˆ ˆ ( ˆ kk = xkk + Kk yk C xkk ) (). Upat th rror covarianc matrix. Pkk = [ I Kk C] Pkk () Bas on xtn Kalman filtr, th snsorlss control systm is shown in Fig.. In this systm, rotor flux orint vctor control is aopt. h -ax currnt is controll to b zro which can gt th largst toru with th smallst phas currnts. Sinc th trminal voltags of motor ar har to masur, th rfrnc voltags ar us in xtn Kalman filtr insta of th ral voltags. * +ω ˆω * I = f (, ˆ ω) * * I = f (, ˆ ω) * * * I * I I, * U, θˆr Start up ability analys Fig.. Block iagram of th snsorlss systm ak non-salint PMSM motor as xampl, in ral rotor flux orint coorinat ( γ δ axs), PMSM mol is: Iγ Uγ R Iγ = + ω Iδ ; (9a) t L L Iδ Uδ R Iδ K = ω I γ ω. (9b) t L L L U γ an U δ ar stator voltags in γ-δ axs. I γ an I δ ar stator currnts in γ-δ axs. EPE - Aalborg ISBN : 99 P.

4 Sinc th ral rotor flux is not known in snsorlss control, in th coorinat orint by stimat rotor position (- axs), thr ar: U = Uγ cosγ Uδ sinγ (a) U = Uδ cosγ + Uγ sin γ I = Iγ cosγ Iδ sinγ (b) I = Iδ cosγ + Iγ sinγ PMSM uations bcom: I U I K = + ω I + ωsinγ t L τ L () I U I K = ω I ωcosγ t L τ L Hr: γ = θ θˆ is rotor position rror. δ γ γ Fig.. Ral an stimat axs In both coorinats, thr always ar: θ = ω () t Elctromagntic toru is: = p ψ I = p ψ ( I cosγ I sin γ) () m r δ r Mchanical movmnt uation of PMSM is: Ω J = m L t ( Ω ) () L ( Ω ) is loa toru an it is a function of rotor sp. If rotor lctrical angl sp is us insta: ω p K = ( Icosγ I sin γ) L( ω) () t J In th obsrvr propos in (), th thir uation for rotor sp just rlis on stat fback. If w just consir th othr thr uations, (suprscript ^ stans for stimat variabls) Iˆ ˆ ˆ U I = + ˆ ω Iˆ (a) t L τ I ˆ ˆ ˆ ˆ ˆ = U I ω I K ˆ ω (b) t L τ L ˆ θ = ˆ ω (c) t Stat obsrvation rrors ar fin as: ˆ ε I I ε = ε ˆ = I I () γ θ ˆ θ Using () () an (), thr ar: EPE - Aalborg ISBN : 99 P.

5 ( I ˆ Iˆ ) K ε = ω ω + ωsinγ ; (a) t L ( ˆ K ε ˆ ) ( cos ˆ = ω I ω I ω γ ω). (b) t L γ = θ ˆ θ = ω ˆ ω (c) t t t h uilibrium points of th systm formr will satisfy: t ε = (9) In th obsrvr bas on th uations formr, bsis th xpct uilibrium point γ =, thr is anothr uilibrium point: ˆ ω = ω = () m = p ψ r ( I cosγ I sin γ) = L ( Ω) Sinc th obsrv sp is zro, iffrnc btwn rfrnc sp an fback sp xists, th output of sp rgulator (rfrnc toru) will arriv at th maximum limitation. But on this point, th lctromagntic toru uals loa toru whil currnt I uals sir toru currnt. So th motor cannot acclrat any mor an will stay in this wrong situation. Particularly, whn loa toru is just friction or block toru which ar most familiar, th uilibrium point is γ =± π / whr th actual lctromagntic toru is zro although currnt I uals th sir valu. On this point, sinc th rotor sp an lctromagntic toru ar all zro, loa toru is also null. Unr th ffcts of th maximum limitation in sp rgulator, th motor will stay in this situation. In practic, whn initial position rror satisfy cosγ >, th ral lctromagntic toru has th sam irction with sir toru, thn PMSM can start up towars th sir sp irction, an th motor will start up succssfully. Othrwis, PMSM rvrss an th obsrvr will gt a wrong sp irction an will convrg to th unxpct uilibrium points. his is vrifi by simulation in nxt sction. In [], anothr uilibrium points ar propos which osn t satisfy (). It s th convrging problm of som obsrvrs an ths points on t xist in our obsrvr. h scon kin of uilibrium points ar mainly trmin by th -ax voltag uation, so w can a som compnsation in th scon uation to brak out this balanc. W chang th uation to: Iˆ Uˆ Iˆ K k RI ˆ ˆ = ω I ˆ ω+ () t L τ L L Hr k is a cofficint positiv. With th compnsation, th unxpct uilibrium points can b avoi. In stay stats, th compnsation will b consir as a littl rror in stator rsistanc paramtr. Its ffcts will b liminat by th robustnss of th systm. Also whn motor is start up succssfully, th cofficint k can b cras artificially an th compnsation can b mov off finally. Simulation rsults Simulations hav bn on in MALAB Simulink to vrify th prformanc of th xtn Kalman filtr. Motor paramtrs ar shown in ABLE I. abl I Motor paramtrs Stator rsistanc R. Ω Stator inuctanc L = L. H Numbr of pol pairs p Rotor magnt flux ψ r. Wb In xtn Kalman filtr, matrixs Q an R in (a) an (b) ar ifficult to b known xactly bcaus th isturbancs w an v ar not known. h only possibl mtho is to ajust th valus of Q an R by practical simulations or xprimnts. In simulation, w us th valus as follows: EPE - Aalborg ISBN : 99 P.

6 .. P = Q. = R =... Rotor sp an position stimation rsults ar shown in Fig. an Fig.. It shows that xtn Kalman filtr can obsrv rotor sp an position xactly. If thr is som initial rotor position rror, whn this rror is too larg, th motor cannot start up an will convrg to th unxpct uilibrium point. As shown in Fig.. With compnsation as shown in (), simulation rsults whn thr is a larg initial position rror ( π /) is shown in Fig.. h motor can start up succssfully unr th ffcts of compnsation. ral sp stimat sp. ral position stimat position rotor sp (ra/s) rotor position (ra) tim(s) Fig.. Estimat an ral rotor sp rotor position (ra) p p ral position stimat position..... tim(s) Fig.. Estimat an ral rotor position rotor sp (ra/s) p ral sp stimat sp rotor position (ra) p ral position stimat position tim(s) -. tim(s).. tim(s) Fig.. Failur start up whn initial position Fig.. Succss start-up with compnsation rror is too larg (simulation) (simulation) Simulations also show that with th sam loa toru an mchanic inrtia, th cofficint of compnsation has no rlations with th initial position rror. With th sam valu of k, rsult of start up without initial position rrors is shown in Fig.. o tst th start up ability with loa toru, simulation rsult with an lctromotiv toru of Nm is shown in Fig.. rotor sp (ra/s) ral sp stimat sp rotor position (ra) ral position stimat position rotor sp (ra/s) ral sp stimat sp rotor position (ra) ral position stimat position. tim(s).. tim(s) Fig.. Start-up with compnsation an no initial position rror -. tim(s).. tim(s) Fig.. Start-up with compnsation an loa toru EPE - Aalborg ISBN : 99 P.

7 Simulations also show that th systm can start up whn thr is a block toru an in stay stat, th compnsation has vry littl ffcts in rotor sp an position stimation. Exprimnt rsults Exprimnts hav bn on on a platform with th DSP C as th controllr. h paramtrs of th motor ar th sam as abl.in fact, th paramtrs can b vari in a larg fil... P =.... Q. = R =... h stimat an ral sps whn th motor rotor mchanical angl sp is acclrat from π ra/s to π ra/s ar shown in Fig.9. If a loa toru impact is us in th rotor, rotor sp uring ynamic stat is shown in Fig.. It can b sn that in both stay an ynamic stats, th stimat sp by EKF obsrvr can also track th ral rotor sp vry wll. ral sp stimat sp ral sp stimat sp rotor sp (ra/s) rotor sp (ra/s) tim(s) Fig. 9. h stimat an ral sps uring acclration tim(s) Fig.. h stimat an ral sps uring loa impact Whn thr ar som rrors in th initial position, th stimat an ral positions uring start up prios ar shown in Fig. an Fig.. h initial valu of th stimat position is always zro whil th ral position is ranom. stimat position ral position stimat position ral position rotor position (ra) rotor position (ra).... tim(s).... tim(s) Fig.. Start-up with littl initial position rror Fig.. Start-up with larg initial position rror (xprimnt) (xprimnt) It can b sn that th motor can start up from any unknown initial position. h xtn Kalman filtr can track th ral rotor position uickly uring start up prios. An thr is fw rvrss or vibrations. It is obvious that th initial position masur or stimation in our systm is not n. In EPE - Aalborg ISBN : 99 P.

8 stay stat, thr ar som stay-stat rrors btwn th ral an stimat rotor positions. hat is bcaus w us th rfrnc voltags insta of th trminal voltags, th voltag rrors caus th position stimation rror. Conclusion his papr propos a snsorlss control systm bas on xtn Kalman filtr for th PMSM. h Kalman filtr can stimat th xact rotor sp an rotor position whil th initial position an mchanic paramtrs ar not n. By propr compnsation in -ax uation, only xpct uilibrium point is kpt. hn th motor can start up at any unknown initial positions an it is not ncssary to stimat th initial position bfor start up. h xtn Kalman filtr is st up at rotor flux orint synchronous axs, so it can b asily us in ithr non-salint or salint motors. h problm is that th covarianc matrixs of noiss can only b trmin by xprimnt sinc th noiss an isturbancs ar not known in practic. Rfrncs [] Yan Liang an Yongong Li, Snsorlss Control of PM Synchronous Motors Bas on MRAS Mtho an Initial Position Estimation ICEMS, Vol., 9-, pp:9 99, Nov. [] S. Bolognani, R. Obo, an M. Zigliotto, Dsp-bas xtn kalman filtr stimation of sp an rotor position of a PM synchronous motor, IECON '9, Vol., pp. 9 -, Spt. 99 [] Silvrio Bolognani, Luca ubiana, an Mauro Zigliotto, Extn Kalman filtr tuning in snsorlss PMSM irvs, IEEE rans. Inustry Applications, Vol. 9, No., pp. -. Nov./Dc.. [] Rach Dhaouai, N Mohan, an Lars Norum, Dsign an Implmntation of an Extn Kalman Filtr for th Stat Estimation of a Prmannt Magnt Synchronous Motor, IEEE rans. Powr Elctronics. Vol.. No.. pp. 9-9, July 99. [] L. Gasc, M.Fal, S. Astir, an L. Calgari, Snsorlss control for PMSM with ruc orr toru obsrvr associat to Kalman filtr, EPE, - Spt.. [] Yoon-Ho Kim, an Yoon-Sang Kook, High prformanc IPMSM Drivs without rotational position snsors Using ruc-orr EKF, IEEE rans. Enrgy Convrsion, Vol., pp., Dc [] B. Nahi Mobarakh, F. Miboy-abar, an F.M. Sargos, A globally convrging obsrvr of mchanical variabls for snsorlss PMSM, PESC, Vol., pp. -9, Jun. EPE - Aalborg ISBN : 99 P.

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