ROBUST CONTROL OF A SPEED SENSORLESS PERMANENT MAGNET SYNCHRONOUS MOTOR DRIVE

Transcription

1 ROBUST CONTROL OF A SPEED SENSORLESS PERMANENT MAGNET SYNCHRONOUS MOTOR DRIVE A. A. Hassa, a M. Azzam Elctrical Egirig Dpartmt, Faculty of Egirig, El-Miia Uirsity, EL-Miia, Egypt. {aahs, Abstract - This papr prsts th applicatio of th Liar Quartic Gaussia (LQG) cotrollr to th stat stimatio a fback of a sp ssorlss prmat magt sychroous motor (PMSM) ri systm. Th oliar mol of th motor has b liariz o th basis of fil oritatio pricipl. Th staar Kalma filtr tchiu has b us to stimat th sp, positio, a loa toru by masurig oly th stator currt. Th optimal stat fback gais a th Kalma stat spac mol ha b calculat off-li i orr to ruc th computatioal bur. Th propos cotrollr has th aatags of robustss, asy implmtatio a auat prformac i th fac of ucrtaitis. Moror, th loa isturbac ca b rjct without affctig th orall prformac. Computr simulatios ha b carri out i orr to aliat th ffctiss of th propos schm. Th rsults show that accurat trackig prformac of th PMSM has b achi. Ix Trms: prmat magt sychroous motor Liar Quartic Gaussia cotrollr- Kalma filtr. NOMENCLATURE, - stator oltags, α, β α β stator oltags, i, i - stator currts, i α, i β α β stator currts, R s L ω r stator rsistac/phas, L, - stator iuctacs, ω p P φ D T L j Motor agular sp, lctrical agular sp, iffrtial oprator, umbr of pol pairs, prmat magt flux likag, iscous frictio cofficit, Loa toru, momt of irtia 1. INTRODUCTION I rct yars, prmat magt sychroous motor ris ha b wily us i may iustrial applicatios such as robots, rollig mills a machi tools. Th ihrt aatags of ths machis iclu high powr sity, low irtia, a high sp capabilitis. Howr, th cotrol prformac of th PMSM is gratly affct by th ucrtaitis of th plat which usually ar mismatch motor paramtrs, xtral loa isturbac, a umoll a oliar yamics [1]. Aac cotrol tchius such as oliar cotrol [2], aapti cotrol [3], robust cotrol [4], ariabl structur cotrol [5], a itlligt cotrol [6, 7] ha b lop to al with plat ucrtaitis ur arious opratig coitios. I ths cotrol schms, th sp or positio sigal is cssary for stablishig th outr sp loop fback a also i th flux a toru cotrol algorithms. From th iwpoits of rliability, robustss, a cost, sral approachs ha b propos that arss th limiatio of th mchaical ssors. Som approachs ar bas o th motor uatios i orr to xprss rotor positios a sp as fuctios of trmial uatitis [8, 9]. Howr, th ssitiity to motor paramtrs is a major rawback of this mtho. I othr approach, ssorlss PMSM ris ha b lop o th basis of stat obsrrs [2,1,11]. Howr, th orall stability may ot b guarat i ths schms u to crtai assumptios itrouc, complicat cotrollr sig, a fback liarizatio. I a thir approach, th stimatio of th rotor positio a sp ha b propos usig th xt Kalma filtr tchiu [12-15]. Howr, this mtho has som ihrt isaatags such as th ffct of ois charactristic, th computatioal bur, paramtr ssitiity, a th absc of sig a tuig critria. I this papr, th PMSM ri has b cotroll usig th LQG cotrollr. Th structur of th LQG cosists of a Kalma filtr stimator a optimal stat fback gais. Th oliar mol of th motor has b liariz accorig to th fil oritatio pricipl. All th systm stats icluig th sp, positio, a loa toru ha b stimat usig th staar Kalma filtr. Th stator currt is th oly masur sigal. Th computatioal bur has b miimiz to a larg xtt by computig th optimal stat fback gais a th Kalma stat spac mol off-li. Computr simulatios ha b carri out i orr to aluat th ffctiss of th propos schm. Th rsults pro that th propos cotrollr ca gi

2 bttr orall prformac rgarig to high stimatio accuracy, uick rcor from loa isturbac, goo trackig ability a simpl implmtatio. 2. MATHEMATICAL MODEL Th yamic mol of th LBDCM ca b scrib i th - rotor fram as follows [14]: V = R i + pλ ω λ (1) s V = Rsi + pλ + ωλ (2) Whr: λ L i + φ (3) = λ = L i (4) ω = Pω r (5) Th mchaical motio of th PMSM ca b xprss as: T = jpω + Dω + T (6) r r L Whr T is th lctromagtic toru lop by th machi which is gi by: T = 3/ 2) P [ λ i + ( L L ) i i ] (7) ( 3. LINEARISED MODEL Th basic pricipl i cotrollig th PMSM is bas o fil oritatio. This is obtai by lttig th prmat magt flux likag b alig with th - axis, a th stator currt ctor is kpt alog th - axis irctio. This mas that th alu of i is kpt zro i orr to achi th fil oritatio coitio. Sic th prmat magt flux is costat, thrfor th lctromagtic toru is liarly proportioal to th -axis currt which is trmi by clos loop cotrol. As a rsult, maximum toru pr ampr ca b obtai from th machi i aitio to th achimt of high yamic prformac. Applyig th fil oritatio cocpt by lttig i = i uatios (1-7), th liaris mol of th PMSM ca b scrib i a stat spac form as : pi = 1/ L ).( R. i + φ. ω ) (8) ( s 2 pω = (1/ j).(1.5 P φ. i D. ω P. T ) (9) Th rotor positio yamics ca b xprss as: pθ = ω (1) L Assumig that th ukow loa toru has a slow ariatio which ca b mol satisfactorily as [2]: p. T L =. (11) Th stat uatios of th liaris mol of th PMSM ca b writt i a matrix form as : Whr : Rs / L 2 = 1.5P φ / j A px = Ax + Bu (12) y = Cx (13) φ / L D / j 1 [ 1/ ] T P / j =, C [ 1 ] T B L u = a i y =. =, 4. CONTROL STRATEGY I this papr, th LQG cotrollr has b mploy to cotrol a sp ssorlss fil orit PMSM ri. Th LQG is a mor stat spac tchiu for sigig optimal yamic rgulators. It has th followig aatags : 1) It abls to tra off rgulatio prformac a cotrol ffort. 2) It taks ito accout th procss isturbac a masurmt ois. Th LQG cotrollr cosists of a optimal stat fback gai k a a Kalma stat stimator. Th optimal fback gai is calculat such that th fback cotrol law u = kx = k[ i ] T ω θ T L miimizs th prformac ix : T T H = ( x Qx + u Ru)t whr Q a R ar positi fiit or smi fiit Hrmittia or ral symmtric matrics. Th optimal stat fback u = kx is ot implmtabl without full stat masurmt. I our cas, th stats ar chos to b currt, sp, positio a loa toru whil th currt is chos to b th output masur sigal. Th Kalma filtr stimator is us to ri th stat stimatio : T x = i ω θ T L such that u = k x rmais optimal for th output fback problm. Th stat stimatio is grat from [16]:,

3 p x = ( A Bk LC) x + Ly Whr L is th Kalma gai which is trmi by kowig th systm ois a masurmt coariacs Q a R. Howr, th accuracy of th filtr s prformac ps haily upo th accuracy of ths coariacs. O th othr ha th matrics A a B cotaiig th motor paramtrs ar ot ruir to b ry accurat u to th ihrt fback atur of th systm. Th Kalma filtr prforms bst for liar systms. Thrfor, Th oliar mol of th PMSM has b liaris through th us of fil oritatio cocpt. Th optimal stat fback gais a th Kalma stat spac mol ha b calculat offli which rsults i grat saig i computatioal bur. O this basis, th implmtatio of th propos cotrollr bcoms asir a th harwar will b ruc to miimum. 4. SYSTEM CONFIGURATION Th block iagram of th ssorlss fil orit PMSM with th propos LQG cotrollr is show i figur (1). All th comma alus ar suprscript with astrisk i th iagram. Th systm ca b fuctioally ii ito two parts: sp cotrol systm a LQG cotrollr. Th first part cosists of thr loops, o for th sp a th othrs for th - currts. Th sp rror is f to th sp cotrollr i orr to grat th toru currt comma i. Th flux currt comma i is st to zro to satisfy th fil oritatio coitio. Th rfrc currts i a i ar compar with thir rspcti actual currts. Th rsult rrors ar us to grat th oltag commas a which ar cort to thr phas rfrc alus, a, a b c i th stator fram. Ths oltag sigals ar compar with triagular carrir sigal a th output logic is us to cotrol th PWM irtr. Th sco part of th systm cofiguratio is th LQG cotrollr which cosists of Kalma stimator i aitio to optimal stat fback gais. Th Kalma stimator uss th masur -axis currt i orr to stimat all th stats icluig currt, sp, positio a loa toru. Ths stats ar multipli by th corrspoig optimal gais a summ to prouc th cotrol sigal cssary to compsat for th loa isturbac a systm ucrtaitis. Th tir systm has b simulat o th igital computr usig th Matlab / Simulik / Powrlib softwar packag. Th motor us i th simulatio procur has th followig spcificatios : PMSM : 1 kw, 2-pol, 15 rpm Stator rsistac : 1.55 ohm Stator iuctac : 2.5 m.h. Prmat magt flux :.22 N.m./amp. Momt of irtia :.22 kg.m 2 Frictio cofficit :.221 N.m.s/ra ω ω PI i i PI = i PI θ Rot. α β 2 / 3 a b c PWM VSI θ i i α optimal gai i i Rot. i β 2 / 3 ω K T L Kalma stimator PMSM i Fig. (1) Block iagram of th ssorlss propos schm

4 Th gais of th sp a currt cotrollrs ar chos as : Sp loop : kp = 2, ki = 3 - currt loops : kp = 1, ki = 3 Th ois a masurmt coariacs ar st as : Q =.1, R =.1 Also, th alus of Q a R matrics which ar cssary to calculat th optimal fback gais ar st as : Q = [ ], R = RESULTS Computr simulatios ha b carri out i orr to aliat th ffctiss of th propos schm. Th sp, currt, rotor positio, a toru rsposs ar obsr ur arious opratig coitios such as chag i rfrc sp, stp chag i loa, a paramtr ariatio. Figur (2) shows th actual a stimat rsposs of th propos PMSM ssorlss schm. Th machi is start from rst a assum to follow a crtai sp trajctory. Th rfrc sp is assum to b liar urig th first half sco util 1 rpm is rach, a th kpt costat for 1.5 sco. At tim t=2 sc., th rfrc sp is icras liarly agai with th sam iitial slop to 15 rpm, a th kpt costat urig th rmaiig simulatio tim. A loa toru of 4 N.m. is assum to b appli iitially o th machi a stpp to 6 N.m. at t=3.5 sco. Also, th stator rsistac is tu to 12 % of omial alu. It is clar that th stimat sp tracks wll th trajctory of rfrc o with goo accuracy or th whol sp rag xcpt at startig. This is u to th imprfct stimatio of th Kalma filtr urig th trasit stat whr all th sigals ar istort. Moror, th high stat fback gais amplify th istortio of th stimat sigals at startig. I aitio, th assumptio of zro iitial rotor positio is aothr sourc of rror. O th othr ha, a sp ip is otic at th istat of stp icras i loa toru, but it is succssfully rjct withi.15 sc. Also, th followig rmarks ca b coclu from th figur : a) Th ukow loa toru is stimat fastly a accuratly. b) Th -axis currt is wll coupl from th motor sp, a is rgulat uit wll to b zro. c) Th rotor positio agl stimatio is ot affct by th paramtr ucrtaitis, a a stabl machi ri ca b obtai. ) Th siusoial ariatio of th 3-phas stator currts rspos uickly to th chag i loa. Howr, it sms i figur (2) that thr is a iffrc btw th actual a stimat rotor positio which arsly affcts th couplig btw th - a - axs. This is may b attribut to th followig rasos: a) Th Kalma filtr mol, a th optimal stat fback gais ar trmi o th basis of th liaris mol of th motor. b) Zro iitial rotor positio is assum. I orr to ruc th iscrpacy btw th actual a stimat rotor positio, a prcis molig of th systm is ruir. Also, a goo choic of th coariac matrics will impro th filtr prformac. I aitio, th kowig of th iitial rotor positio woul cras th rror to a larg xtt. Fig. (2) Simulatio waforms of th propos schm at high sps with stator rsistac tu to 12% of omial alu (... actual - stimat ) i i

5 Th robustss of th propos ssorlss schm has b tst at low sps a mismatch paramtrs. Figur (3) shows th simulatio waforms wh th sp is ruc liarly from 1 to 5 rpm (about 3.3% of its omial ). Th loa toru is assum to b costat at 4 N.m. urig th simulatio prio. Moror, th stator rsistac, momt of irtia, a frictio cofficit ar all tu to 2% of thir omial alus, whil th stator iuctac is tu to 5% oly. It is clar that goo trackig capability a fast rsposs ha b achi i spit of th mismatch paramtrs. Howr, th iffrc btw th actual a stimat rotor positio, which has b otic i th figur, is for th sam rasos iscuss abo. i i 6. CONCLUSIONS This papr prsts th applicatio of a high yamic optimal rgulator to cotrol th sp a toru of th prmat magt sychroous motor ri systm without a sp ssor. Th cocpt of th fil oritatio has b appli i orr to liaris th oliar mol of th motor. Th staar Kalma filtr tchiu has b mploy to stimat th sp, positio, a loa toru by masurig oly th stator currt. Th computatioal bur has b miimiz to a grat xtt by computig th optimal stat fback gais a th Kalma stat spac mol off-li. Th propos cotrollr has th aatags of robustss, asy implmtatio a goo prformac i th fac of ucrtaitis. Moror, th loa isturbac ca b rjct without affctig th orall prformac. Computr simulatios ha b carri out i orr to aluat th ffctiss of th propos cotrollr. Th rsults pro that accurat trackig prformac of th PMSM has b achi at low sps as wll as high sps. Moror, this schm is robust agaist th paramtrs ariatio a limiats th ifluc of molig a masurmt oiss. REFERENCES [1] F-J Li, Ral tim positio cotrollr sig with toru fforwar cotrol for PM sychroous motor, IEEE Tras. o Iustrial Elctroics, Vol. 44, No. 3, Ju 1997, pp [2] J. Solsoa, M. I. Valla, a C. Murachik, No liar cotrol of a prmat magt sychroous motor with isturbac toru stimatio, IEEE Tras. O Ergy Corsio, Vol. 15, No. 2, Ju 2, pp [3] J. Zhou, a Y. Wag, Aapti backstppig sp cotrollr sig for a prmat magt sychroous motor, IEE Proc., Elctr. Powr Appl. Vol. 149, No. 2, March 22, p [4] S.I. Mistry, a S.S. Nair, Itificatio a cotrol xprimts usig ural sigs, IEEE Cotrol Syst. Magazi, Vol. 14, No. 3, Ju 1994, p [5] J-H L, a M-J You, A w impro cotiuous ariabl structur cotrollr for accuratly prscrib trackig cotrol of BLDD sro motors, Automatica Vol. 4, 24, pp [6] M. A. Rahma, a M. A. Hou, O-li aapti artificial ural twork bas ctor cotrol of prmat magt sychroous motors, IEEE Tras. O Ergy Corsio, Vol. 13, No. 4, Dc. 1998, pp [7] F-J Li, R.-J. Wai, a H-P Ch, A PM sychroous ri with a O-li trai fuzzy ural twork cotrollr, IEEE Tras. O Ergy Corsio, Vol. 13, No. 4, Dc. 1998, pp [8] R. Wu, a G. R. Slmo, A prmat magt motor ri without shaft ssors, IEEE Tras. O Iust. Appl., Vol. 27, No. 5, Spt./Oct. 1991, pp [9] N. Ertugrul, a P.P. Acarly, A w algorithm for ssorlss opratio of prmat magt motors, IEEE Tras. O Iust. Appl., Vol. 3, Ja./Fb. 1994, pp [1] J. Hu, D. M. Dawso, a K. Arso, Positio cotrol of a brushlss DC motor without locity masurmts, IEE Proc., Elctr. Powr Appl. Vol. 142, No. 2, March 1995, pp

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