Velocity Observer for PMSM without Rotational Transducer

Transcription

1 Vlocity Obsrvr for PMSM without Rotational Transducr Todd D. Batzl Mbr IEEE and Kwang Y. L Fllow IEEE Abstract In this papr a vlocity stiator is proposd for th purpos of iproving th transint vlocity control prforanc of a prannt agnt synchronous otor (PMSM) oprating without a rotor position or vlocity transducr. An obsrvr is dsignd to stiat rotor vlocity using th lctro-chanical paratrs of th syst asurd stator currnts and th stiatd rotor angl whr th angl stiat is obtaind fro a sparat obsrvr. Spcial attntion is givn to vry low spd opration whr th rotor angl stiat ay b inaccurat du to ill-conditioning of th rotor angl obsrvr undr thos conditions. Exprints ar usd to xain th vlocity control prforanc of th rsulting snsorlss PMSM driv and th robustnss of th syst to xtrnal disturbancs and odl uncrtainty. Indx Trs brushlss achins digital control otor control otor drivs stat stiation vlocity stiation torqu control variabl spd drivs. D I. INTRODUCTION UE to rliability concrns or physical constraints associatd with th us of rotor position and vlocity snsors thr is considrabl intrst in th opration of th prannt agnt synchronous otor (PMSM) without rotor or vlocity transducrs. Many approachs hav bn proposd for stiating both rotor angl and vlocity; but littl attntion has bn givn to vlocity control of rotor spd/position snsorlss PMSM systs for applications whr prcis vlocity tracking prforanc is rquird in transint. High-prforanc PMSM drivs rquir fast and accurat vlocity fdback for optial vlocity control of th achin. Th accuracy and spd of rspons of th vlocity fdback dirctly influncs dynaic and stady-stat vlocity control prforanc. In a position/spd snsorlss PMSM driv howvr th lack of a spd fdback snsor prsnts a considrabl challng to obtain satisfactory spd tracking in transint such as during startup a spd rvrsal or tracking a ti-varying spd coand. Spd stiation tchniqus for th PMSM gnrally fall into on of two catgoris. On thod is to diffrntiat th stiatd rotor position to obtain vlocity [1]. Anothr is to us th ratio of inducd stator f and prannt agnt (PM) flux linkag []. Th forr approach is quit accurat in stady-stat but yilds poor rspons during transint and T. D. Batzl is with th Dpartnt of Elctrical Enginring Pnn Stat Altoona Altoona PA USA (-ail: tdb10@psu.du). K. Y. L is with th Dpartnt of Elctrical Enginring Th Pnnsylvania Stat Univrsity Univrsity Park PA 1680 (-ail: kwangl@psu.du). nar zro spd. Th lattr tchniqu givs a fast rspons but is pron to rrors arising fro asurnt nois and achin paratr uncrtainty. Svral rsarchrs hav proposd iprovnts to vlocity stiation whr both of th spd stiation thods ar usd concurrntly. Th work in [3] uss th drivativ of th stiatd position ovr a priod of ti to liinat stady-stat rrors producd by paratr uncrtainty in th PM flux linkag and stator rsistanc. Th work in [4] suggsts a tchniqu that uss th drivativ of rotor angl for stady-stat angular vlocity dtrination but rlis priarily on th ratio of f to PM flux linkag during transint. Ths works hav donstratd ovrall iprovnt in th vlocity stiat but th bandwidth of th vlocity stiat aftr low-pass filtring is still not sufficint for any snsorlss PMSM driv applications. Th objctiv of th work in this papr is to iprov th vlocity tracking prforanc of th rotor spd/position snsorlss PMSM driv proposd in [5] to addrss th nd for fast and accurat vlocity tracking in transint. Th proposd iprovnts to th vlocity control prforanc of th PMSM driv ar to b carrid out through th application of a vlocity obsrvr that uss asurd stator currnts th odl of th lctrochanical syst and th rotor position stiat drivd fro th obsrvr dscribd in [6]. This rotor position obsrvr was slctd du to its vry robust opration and accurat position stiation. Howvr attntion ust b givn to low-spd syst prforanc whr th rotor angl is not obsrvabl [6]. Exprintal rsults ar usd to donstrat th iprovd vlocity tracking prforanc of th proposd syst copard to th tchniqu usd in [5]. II. ANGULAR VELOCITY ESTIMATION Thr ar two ajor catgoris of vlocity stiators usd in snsorlss PMSM drivs. Th position diffrntiation thod obtains angular vlocity by diffrntiating th stiatd rotor angl. Th back-f thod rlis on asurd trinal voltag and currnt and th PMSM odl paratrs to dtrin vlocity. In this sction both tchniqus ar prsntd and an adaptiv vlocity stiator that is a cobination of ths two tchniqus is discussd. A. Diffrntiation Mthod Th diffrntiation thod for stiating angular vlocity uss th ti drivativ of th stiatd rotor angl ˆ θ ˆ ω = d ˆ θ dt. (1) Authorizd licnsd us liitd to: Baylor Univrsity. Downloadd on January at 1:03 fro IEEE Xplor. Rstrictions apply.

2 Clarly this thod rquirs a stabl rotor position obsrvr. In addition th rotor angl obsrvr should hav pols with a natural frquncy that is high rlativ to th angular vlocity ω [7]. Subjct to ths constraints th stiatd angular vlocity obtaind by (1) approachs th actual valu whn avragd ovr a sufficint priod of ti. Th diffrntiation thod yilds accurat stady-stat spd stiats in th diu to high spd oprating rang; howvr is suscptibl to nois that rsults fro th diffrntiation procss and th incorrct spd stiats that ay rsult at vry low-spd whr th opration of th rotor angl obsrvr is problatic. Thus th diffrntiation thod is of liitd utility during low-spd opration startup and during spd rvrsals. B. Back-f Mthod Th back-f stiator for agnitud of th angular vlocity is [6]: ( v Ri ) + ( v Ri ) ˆ ω = () 3 λ whr v and i rprsnt th trinal voltag in th stationary two-phas rfrnc fra and R and λ ar th phas rsistanc and PM flux linkag constant rspctivly. Fro () it can b sn that only th agnitud of th angular vlocity can b dirctly dtrind. To rsolv th dirction of rotation at sapling intrval k w us: ˆ ω = sgn φ( k) = tan ( φ( k) φ( k 1) ) 1 Ri v v Ri ˆ ω. (3) Th back-f thod producs rlativly accurat stiats of th agnitud of angular vlocity vn at zro and vry low-spd. Howvr dtrining th dirction of rotation at low-spd is problatic and stiation accuracy is dpndnt on th odling crtainty of PMSM paratrs. Furthror any snsing rror associatd with th stator voltag and currnt will rsult in vlocity stiation rror with an ac coponnt at th fundantal frquncy and its scond haronic [5]. To rov th unwantd ac coponnt low-pass filtring of th vlocity stiat is rquird which tnds to rduc vlocity tracking accuracy in transint. C. Adaptiv Vlocity Estiation Th diffrntiation and back-f thod ar sotis cobind to for th adaptiv vlocity stiation thod [3]. Th back-f thod yilds good prforanc at vry low spds but is subjct to rrors arising fro PMSM paratr uncrtainty and rsolution of th dirction of rotation. Th diffrntiation thod howvr is ssntially indpndnt of th paratr stiation rrors so long as th obsrvr pols hav a natural frquncy uch highr than th angular vlocity and th vlocity is sapld ovr a sufficint priod of ti. Thus th strngths of ach thod ay b Fig. 1. Adaptiv vlocity stiation block. cobind to for th vlocity stiation corrction sch shown in Fig. 1. To us (1) to corrct for vlocity stiation rrors obtaind by () a odifir is includd to rprsnt th uncrtainty in th prannt agnt flux linkag: ˆ ω = ( v Ri ) + ( v Ri ) 3 ( λ + ). (4) Th adaptation procss copars th output of (1) with () to gnrat th vlocity stiation rror. As long as th agnitud of th spd stiat of (4) is abov th low-spd thrshold whr th rotor angl obsrvr is assud to b stabl a PI controllr oprats on th stiation rror adjusting th output to corrct for th paratr uncrtainty. Ti constant of th controllr is chosn to b slow to rov nois gnratd by th diffrntiation of th rotor angl. Th adaptiv vlocity stiation tchniqu has donstratd th ability to stiat vlocity accuratly in stady-stat dspit PMSM odling uncrtainty and signal asurnt rrors. Howvr th rquird low-pass filtring of th stiatd vlocity rducs th accuracy during transint. III. MODEL-BASED ANGULAR VELOCITY OBSERVER Hr a odl-basd angular vlocity obsrvr is dvlopd. Th tchniqu was originally dvlopd to iprov vlocity stiation in PMSM srvo drivs using rotary ncodr fdback dvics [8]. In th contxt of this work howvr it shall b usd to provid highly accurat rotor position fdback in a position/vlocity snsorlss PMSM driv. A. Elctro-Mchanical Syst Modl Th lctro-chanical odl for th PMSM is givn by dθ = ω (5) dt dω T Tl Bω = (6) dt J whr θ ω T T B and J rprsnt th chanical l rotor angl chanical angular vlocity otor torqu load torqu viscous friction and inrtia rspctivly. To obtain a third stat quation th load torqu is assud to b a slowly varying quantity such that it is constant ovr th sapling intrval: Authorizd licnsd us liitd to: Baylor Univrsity. Downloadd on January at 1:03 fro IEEE Xplor. Rstrictions apply.

3 dt l dt = 0. (7) Th dvlopd otor torqu for a non-salint PMSM is writtn as [9]: 3 T sin ˆ cos ˆ = λp i θ + i θ (8) whr P is th nubr of pol pairs and ˆ θ is th stiatd chanical rotor angl. B. Vlocity Obsrvr Construction With th stat variabls input vctor and output dfind as x = [ θ ω Tl] (9) u = T (10) y = θ (11) rspctivly th stat spac dfinition of th lctrochanical syst can b writtn x& = Ax+Bu y = Cx (1) whr B 1 A = 0 ; B= 1 ; C = [ J ]. (13) J J Using standard obsrvr dsign tchniqus for a linar syst [10] th for of th proposd angular vlocity obsrvr is: = & ( A-GC) (14) =x-x ˆ (15) whr G and ar th obsrvr gain atrix and stat stiation rrors rspctivly. Th obsrvr is usd to rconstruct th stat variabls yilding th stiatd angular vlocity ω as shown in th block diagra of Fig.. Th stiatd rotor angl θ is usd only to copar with th valu obtaind fro th sparat rotor angl obsrvr. Th obsrvability of th syst is dtrind by th rank of th obsrvability atrix [11]: C CA CA. (16) Th obsrvability atrix is full rank for finit rotor inrtia so that th angular vlocity is obsrvabl for all practical oprating conditions. C. Obsrvr Pol Placnt Th obsrvr ignvalu locations dtrin th spd of convrgnc and th stability of th obsrvr. Lt th dsird ignvalu ( λ ) locations b placd according to: λ 1 = k1 (17) λ = k + jk3 (18) * λ3 = k jk3 = λ. (19) Th constants k 1 and k should b positiv ral nubrs for stability of th obsrvr. Th closd loop ignvalu locations of (17)-(19) ar nforcd by th gain atrix G. Following th procdur outlind in [11] th obsrvr gain atrix to plac th ignvalus at th location spcifid in (17)-(19) is dtrind: k1 + k B J k k3 k B 1k ( B G = k1 k J J ). (0) Jk k + k ( ) 1 3 Th obsrvr gains givn in (0) ar includd in th ovrall vlocity obsrvr block diagra of Fig.. IV. EXPERIMENTAL RESULTS To valuat th prforanc of th vlocity obsrvr in th snsorlss driv siulations wr prford whr th position/vlocity snsorlss controllr shown in Fig. 3 oprats th PMSM. For various oprating conditions coparisons btwn th adaptiv vlocity stiator (sction II) and odl-basd vlocity obsrvr (sction III) Fig.. Vlocity obsrvr block diagra. Authorizd licnsd us liitd to: Baylor Univrsity. Downloadd on January at 1:03 fro IEEE Xplor. Rstrictions apply.

4 Fig. 3. Block diagra of snsorlss PMSM driv syst. a) adaptiv vlocity stiator with filtr. a) adaptiv vlocity stiator with filtr b) odl-basd vlocity obsrvr FIG. 4 STEADY-STATE VELOCITY TRACKING PERFORMANCE COMPARISON. prforanc wr ad. In both cass th rotor position obsrvr dscribd in [6] was usd. In addition voltag and currnt snsing rror was introducd into th siulation so that low-pass filtring of th adaptiv vlocity stiat was ncssary as dscribd in [5]. For all xprints th cutoff frquncy of th low-pass vlocity stiation filtr was st to ¼ of th fundantal lctrical frquncy. For th odlbasd obsrvr th rotor inrtia and friction usd by th odl diffrd fro thir actual valus by a factor of two to intntionally introduc odl uncrtainty to th siulation. b) odl-basd vlocity obsrvr Fig. 5. Startup prforanc coparison. A. Stady-stat Prforanc Th xprint shown in Fig. 4 donstrats stady-stat vlocity track prforanc. Fro th figur th odl-basd vlocity obsrvr shows incrasd accuracy as copard to th prforanc using th adaptiv vlocity stiator. Th Authorizd licnsd us liitd to: Baylor Univrsity. Downloadd on January at 1:03 fro IEEE Xplor. Rstrictions apply.

5 actual and coandd vlocitis diffr by only.01 rp. copard to about 0.6 rp diffrnc for th old vlocity stiator. It should b notd that lowring th cut-off frquncy of th low-pass filtr could rduc th adaptiv vlocity stiation rror at th xpns of accuracy in transint. a) adaptiv vlocity stiator with filtr a) adaptiv vlocity stiator with filtr b) odl-basd vlocity obsrvr Fig. 7. Dynaic vlocity coand tracking coparison. b) odl-basd vlocity obsrvr Fig. 6. Spd rvrsal prforanc coparison. B. Transint Prforanc Startup fro zro spd and spd rvrsals of a snsorlss syst ar challnging sinc rotor angl is not obsrvabl at vry low spd. In this sction prforanc undr transint conditions is xaind. Authorizd licnsd us liitd to: Baylor Univrsity. Downloadd on January at 1:03 fro IEEE Xplor. Rstrictions apply.

6 Fig. 5 shows a startup fro zro spd with an initial rotor angl stiation rror of approxiatly 0.1 radians (typical rror aftr a forcd alignnt). Fro th figur th adaptiv vlocity stiator yilds poor tracking during th startup transint copard with th prforanc using th odlbasd obsrvr which tracks th coand narly prfctly. A spd rvrsal is shown in Fig. 6. Siilar to startup prforanc th odl-basd obsrvr approach producs suprior tracking during th transint through zro spd. Th adaptiv vlocity stiator difficultis around zro spd ar du to svral phnona including a lag introducd by filtring and difficulty in dtrining th dirction of rotation at nar-zro spd. Dynaic prforanc is furthr valuatd in Fig. 7 whr th tracking of a priodic ti-varying sinusoidal spd coand is xaind. Du to th rquird low-pass filtr th adaptiv vlocity stiator is not abl to track th coand. Tracking of th vlocity coand using th odl-basd obsrvr is rasonabl. V. CONCLUSIONS This papr has prsntd th application of a odl-basd vlocity obsrvr intndd to incras th ability of a position/vlocity snsorlss PMSM driv to track vlocity coands in both stady-stat and transint oprating conditions. Siulations hav bn usd to donstrat th ffctivnss of th proposd vlocity obsrvr in trs of stiation accuracy and robustnss. In transint th vlocity obsrvr prfors vry wll and th syst was shown to prfor rliably for both startup and spd rvrsal transints. Coparisons with anothr vlocity stiation tchniqu (adaptiv vlocity stiation) indicat that th odl-basd obsrvr will yild suprior prforanc in transint. In conclusion th odl-basd vlocity obsrvr offrs iprovnt ovr th adaptiv vlocity stiation algorith in cass whr low-pass filtring is rquird to rov unwantd ac coponnts fro th vlocity stiat. Th ac coponnts aris du to haronics in th PMSM back-f wavfor and stator voltag and currnt snsing rrors. [8] X. Dianguo W. Hong and S. Jingzhuo PMSM srvo syst with spd and torqu obsrvr IEEE 34th Annual Powr Elctronics Spcialists Confrnc vol. 1 pp Jun 003. [9] T.D. Batzl and K.Y. L Coutation torqu rippl iniization for prannt agnt synchronous achins with Hall ffct position fdback IEEE Transactions on Enrgy Convrsion v. 13 no. 3 pp [10] W.L. Brogan Modrn Control Thory Englwood Cliffs NJ: Prntic- Hall [11] C. Phillips and R. Harbor Fdback Control Systs 4 th d. Englwood Cliffs NJ: Prntic-Hall VII. BIOGRAPHIES Todd D. Batzl (M 00) rcivd th B.S. and PhD. dgrs in lctrical nginring fro th Pnnsylvania Stat Univrsity Univrsity Park in 1984 and 000 rspctivly and th M.S. dgr in lctrical nginring fro th Univrsity of Pittsburgh Pittsburgh PA in Currntly h is Assistant Profssor of lctrical nginring at Pnn Stat Altoona. His rsarch intrsts includ lctric achins lctric otor controls powr lctronics artificial intllignc applications to control and bddd control systs. Dr. Batzl is a Mbr of th IEEE. Kwang Y. L (F 01) rcivd th B.S. dgr in lctrical nginring fro Soul National Univrsity Kora in 1964 th M.S. dgr in lctrical nginring fro North Dakota Stat Fargo in 1968 and th Ph.D. dgr in systs scinc fro Michigan Stat Univrsity East Lansing in Currntly h is a profssor of lctrical nginring and is Dirctor of Powr Systs Control Laboratory at th Pnnsylvania Stat Univrsity Univrsity Park. H has also bn with Michigan Stat Orgon Stat and Univrsity of Houston. His intrsts includ powr systs control opration and planning and intllignt syst applications to powr systs. Dr. L is and an Associat Editor of IEEE Transactions on Nural Ntworks and an Editor of IEEE Transactions on Enrgy Convrsion. Dr. L is a Fllow of th IEEE. VI. REFERENCES [1] K.W. Li K.S. Low and M.F. Rahan A position obsrvr for prannt agnt synchronous otor driv 0th Intrnational Confrnc on Industrial Elctronics Control and Instruntation (IECON) vol. pp [] L. Sicot S. Siala K. Dbusschr and C. Brgann Brushlss DC otor control without chanical snsors 7th Annual IEEE Powr Elctronics Spcialists Confrnc 1996 vol. 1 pp Jun [3] J. Ki and S. Sul Nw approach for high prforanc PMSM drivs without rotational position snsors IEEE Transactions on Powr Elctronics vol. 1 pp Spt [4] J.X. Shn Z.Q. Zhu and D. How Iprovd spd stiation in snsorlss PM brushlss AC drivs IEEE Trans. on Ind. App. vol. 38 no. 4 pp Aug. 00 [5] T.D. Batzl Iprovd angular vlocity stiation for high prforanc snsorlss PMSM 8 th Annual IEEE Workshop on Powr Elctronics in Transportation pp Oct [6] T. D. Batzl and K. Y. L Slotlss prannt agnt synchronous otor opration without a high rsolution rotor angl snsor IEEE Trans. Enrgy Conv. vol. 15 no. 4 pp [7] T. D. Batzl Elctric propulsion using th prannt agnt synchronous otor without rotor position transducrs Ph.D. dissrtation Dpt. of Elct. Eng. Pnnsylvania Stat Univrsity Univrsity Park 000. Authorizd licnsd us liitd to: Baylor Univrsity. Downloadd on January at 1:03 fro IEEE Xplor. Rstrictions apply.