SAMPLE PAGES TO BE FOLLOWED EXACTLY IN PREPARING SCRIPTS. ADAPTIVE SPEED CONTROL OF PMSMs WITH UNKNOWN LOAD TORQUE

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1 SAMPLE PAGES O BE FOLLOWED EXACLY IN PREPARING SCRIPS ADAPIVE SPEED CONROL OF PMSMs WIH UNKNOWN LOAD ORQUE F. N. Koumbouls * N. D. Kouvaas * G. E. Panagotas * an A. G. Pantelos * Department of Automaton Hals Insttute of echnology Greece Department of Physcs Unversty of Ioannna Greece Abstract: he problem of spee control of a permanent magnet synchronous motor (PMSM wth unnown parameters an unnown external loa torue s formulate an solve as an aaptve control problem that meets the reurements of asymptotc comman followng wth smultaneous asymptotc sturbance attenuaton. he ey pont s to entfy the system wthout nowlege estmaton or measurement of the unnown sturbance. he satsfactory performance of the results s llustrate through applcaton of the propose nonlnear ynamc controller to a PMSM for three cases: frst no external loa torue; secon smoothly startng steay external torue; thr unnown nosy an fast varyng loa torue. Copyrght 5 IFAC Keywors: permanent magnet motors AC machnes aaptve control moel followng control sturbance localzaton ynamc output feebac nonlnear control systems spee control nustral control robust performance NOMENCLAURE ω m Angular velocty L External torue u an u Stator voltages n rect an uarature frame an Stator currents n frame L phase nuctances Φ Magnetc flux R s Phase resstance J Moment of nerta B Vscous loa torue coeffcent p Number of pole pars. INRODUCION PMSMs have attracte sgnfcant attenton ue to ther we range of applcatons (CNC machne tools nustral robots elevators etc as well as ther avantages (hgh effcency hgh torue to nerta rato superor power ensty as compare to other types of electrcal motors (Lessmeer et al In lterature many control algorthms have been evelope to satsfy varous performance reurements for the varables (spee poston torue an current of a PMSM moel. For the problem at han namely spee control of a PMSM there s a long lst of results (Ba et al. 998; Cerruto et al. 995; Chang et al. 994; Ln an Ln 999; Lu an Lu 99; Lu an Cheng 994; Pllsy an Krshnan 99; Rahman an Houe 998; Sepe an Lang 99; Sepe an Lang 99; Xu et al. 998; Zhu et al. an the references theren. Here we focus on aaptve technues (Ba et al. 998; Cherruto et al. 995; Lu an Cheng 994; Rahman an Houe 998; Sepe an Lang 99; Sepe an Lang 99; Xu et al Partcularly n Cherruto et al. (995 a robust controller base on the moel reference aaptve control approach s constructe to compensate the varaton of the system parameters an a sturbance torue observer s employe to balance the reure loa torue. In Sepe an Lang (99a a gtal aaptve controller s constructe through a lnear least suare estmator. hs controller s of ntegral proportonal (IP type whle only vscous an coulombc types of loa torue are consere. In Lu an Cheng (994 three aaptve spee controller

2 wthout shaft sensor s propose. Self-tunng moel followng an moel reference aaptve controllers are apple separately for sensorless guance of the PMSM. In Ba et al. (998 a moel reference aaptve control scheme usng Lyapunov stablty theory s evelope. Furthermore n orer to mprove the robustness an performance a bounary layer ntegral slng moe controller s esgne. In Rahman an Houe (998 spee control s acheve usng an on-lne self-tune artfcal neural networ (ANN that s base on the motor ynamcs an the nonlnear loa characterstcs. he weghts an bases of the ANN scheme are auste both off-lne an on-lne. In Sepe an Lang (99 a spee control scheme base on a sensorless full-state observer s stue theoretcally an expermentally. Fnally n Sepe an Lang (99b a fully gtal aaptve spee controller s evelope an the followng ssues where examne: scretzaton an global lnearzaton of the nonlnear motor system nonlneartes n the nverter nonmnmum phase behavor ue to samplng samplng rate restrctons robust mechancal state estmaton persstent exctaton of the mechancal states mechancal parameter estmaton an valaton of these estmates an unmoele ynamcs. In the present paper a gtal ynamc an nonlnear nrect aaptve controller s evelope to acheve spee control of PMSM. In partcular for the rotor s angular velocty asymptotc comman followng wth smultaneous asymptotc sturbance (loa torue attenuaton wll be acheve whle the rect phase stator current wll be shown to follow exactly the respectve comman. he scretze PMSM moel upon whch the controller s constructe s prouce usng forwar fferences. he parameters of the nonlnear moel come from RLS entfcaton. ang nto account the ffculty of measurng the external loa torue the respectve parameter has not been nclue n the entfcaton algorthm. Conseuently the nfluence of the torue s ncorporate nse the rest parameters of the moel n the sense that the entfe moel parameters converge far from ther real values. he strong pont n our approach that permts entfcaton wthout nowlege of the external loa s that the propose nonlnear ynamc controller s enough robust to satsfy the esgn reurements even n the case where the estmate moel parameters verge from ther orgnal values. he avantages of the above propose scheme les on the fact that no nowlege measurement estmaton or observer of the loa torue s reure to acheve attenuaton of the nfluence of the unnown loa torue to the rotor s spee. Atonally t s mportant to menton that the rect phase stator current s ecouple from the external loa torue.. MAHEMAICAL MODEL OF A PMSM Permanent Magnet Synchronous Motors (PMSMs are moele as follows (see f.e. Leonar 996: ωm pφ B = ωm L t J J J (a s = R p u ωm t L L (b Rs pφ = p ωm ωm u (c t L L L Usng forwar fferences scretzaton metho the screte tme moel of the PMSM taes on the form ωm( L( ωm(( A ( ( = P( B ( u( ωm(( ( u( where a a A ( = a a a b P ( = p B ( = b p b a BJ = a =.5pΦ J a RL = a = L pφ s = RL s p = p p b J = b = b = L a = p ( an where s the samplng pero an the forwar shft operator.. SPEED CONROL WIH KNOWN DAA In the present paper the esgn goal s accurate spee control of the PMSM nepenently from the unnown loa torue. o satsfy ths reurement the esgn scheme of asymptotc comman followng wth smultaneous asymptotc sturbance attenuaton (ACFADA (see Koumbouls 999 Koumbouls an Kouvaas wll be apple to control a PMSM wth unnown loa torue an nown parameters. o satsfy the reurement of ACFADA for the moel ( the followng nonlnear ynamc controller solvng the problem s propose a p u = ωm r (a b b b

3 p a u = ωm ωm b b a F ( G ( ω( r( b b C ( C ( where r ( an r ( are external commans an 7 (b 7 F( = f 7 7 C( = c (4a = 7 7 = 7 G( = g (4b = Clearly the ratonal functons F( / C( an G( / C( are causal. o apply the controller ( to the system ( a Zero Orer Hol D/A converter wll be use to prouce the contnuous tme nput of the system. Applcaton of the controller ( to the system ( results n a nonlnear hybr (contnuous tme screte tme system. If the controller ( s apple to the approxmate scretze system ( the followng close loop system wll be erve ω BG ( ( m = r AC ( ( B ( F( D ( L (5a AC ( ( B ( F( ( = r ( (5b AG ( ( = r AC [ ( ( BF ( (] DF ( ( L (5c C ( [ A( C ( B ( F ( ] he polynomals F( an C( wll be use to place the close loop system poles whle the polynomal G( wll be use to set the external nput to output transfer functon gan to. Note that the controller ( results n a close loop system beng n I/O ecouple form.e. for zero external torue s rven only by r whle s rven only by r. So the angular velocty s controlle only by r whle s controlle only by r. Let the esre characterstc polynomal be of the form A ( = ( ε 9 (6 D where < ε <. o satsfy ths reurement the coeffcents of F ( have to be ( 9 ε 6ε ( 84ε ( 4 6ε f = c a a a f = a c c a f = a c c a 4 5 f = ( ac4 c5 6ε a 6 f4 = ( ac5 c6 84ε a 7 = ( 6ε = ( 9ε f a c c a f = ε a 9 7 whle the coeffcent c has to be f a c a c = 9ε a (7 It can realy be verfe that f one chooses ε > the reurement of sturbance attenuaton s satsfe. Furthermore f the coeffcents of G( are 9 chosen to be g ( 8a ( ε = = 7 unty ampltue of the close loop transfer functon that maps the external comman r to the spee ω m s erve. Hence asymptotc comman followng wth smultaneous sturbance attenuaton s acheve. For the close loop system an the controller to reman stable the polynomal C( wll be chosen to be of the followng specal form 6 C( = ( ρc ma (8 m= where a ρc. he controller polynomal form (8 mples that ts roots are real an eually space at a stance a. Hence the controller s stable f an only f the followng contons hol < ρ c < < ρ c 6a < (9 ang nto account relaton (7 the pole ρ c can be chosen to be of the form ( ε ρc = a a 9 7 ( hus the neualtes n (9 are reuce to ( a ε a ( a ε 9 7 6< < ( It s clear that the neualtes n ( can always be satsfe whle a possble choce for a coul be a = ( a 9ε 7 6 wth ε =.. Observe that snce BJ > then the parameter a s grater than. Also observe that for < ε < t hols that a >. From the close loop system (5 observe that for a step external comman r ( wth ampltue r the part of the spee epenng upon the external

4 comman tens to the ampltue of the external comman. Hence the esgn reurement of asymptotc comman followng has been satsfe. Also for a step loa torue L = L the part of the spee epenng on the loa torue tens to.4497b L. hus the steay state value of the close loop angular velocty s ω = r.4497b. o llustrate the L avantages of the latter result conser the case of the PMSM n Zhu et al. ( where R s =.[ Ω ] J =.6[gr m ] Φ =.8[V sec ra] B =. N m sec ra L =.[ H] an p =. For ths case the poles of the close loop system wll be chosen to be..e. ε =.. For the above ata an for the case of r = 8 ra sec an =.[ sec] t hols that ω = L. Hence asymptotc comman followng wth asymptotc sturbance attenuaton has practcally been acheve va a 7 th orer controller yelng a 9 th orer close loop characterstc polynomal. Wth regar to the behavor of the open loop system t can easly be proven that for gven rect phase stator current let an rotor angular velocty let r an for zero external loa torue the reure rect an uarature stator voltages are eual to u R = ( s Φ (9 u ( ( = BRs p L Φ Φ r p Φ ( Usng the above motor ata an for esre angular velocty 8 ra sec an rect phase stator current.5a t hols that u =.5796V an u = 44.59V. Usng these voltages for fferent external loa torues n the range of ( [ Nm] the steay state of the angular velocty s presente n Fgure (ashe lne. he respectve steay state value for the close loop system s also presente n Fgure (contnuous lne. Accorng to Fgure the steay state of the angular velocty for the close loop system s practcally constant as a functon of the external loa torue whle for the open loop system t changes ramatcally. 4. INDIREC ADAPIVE CONROL SCHEME Conser the screte tme system (. In general the parameters a a a a a p p b b an b are not nown. ang nto account the ffculty to measure the external torue the respectve coeffcent b cannot be entfe. Hence the vectors of the unnown parameters are θ ˆ = a ˆ ˆ a θˆ = aˆ ˆ pˆ b θˆ = aˆ ˆ aˆ pˆ b Ientfcaton ata namely the measurements come from the orgnal contnuous tme moel (. RLS estmaton of θ ˆ ( = at a partcular tme nstant N epens on samples of measurements of spee rotor currents an rotor voltages through the followng recursve relatons (Ästrom an Wttenmar 989. θˆ ( N = θˆ ( N K ( [ ( ( ˆ N y N φ N θ ( N ] K ( N = Q ( N φ ( N φ ( ( ( N Q N φ N Q ( N = Im K ( ( ( N φ N Q N where φ ( N = ωm ( N ( N φ ( N = ( N ωm ( N ( N u ( N φ ( N = ωm ( N ( N ωm ( N ( N u ( N y ( N = ω ( N y ( N = ( N y ( N = ( N m m = m = m = 4 Intalzaton of the RLS entfcaton proceure reures an a pror choce for the ntal values θ ˆ ( an Q (. here s no lmtaton for θ ˆ (. o preserve convergence Q ( must be postve efnte. Accorng to the above algorthm the varatons of the external loa torue nfluence the estmaton of the system parameters an conseuently the choce of the controller through measurement of the spee an the currents. o erve an nrect aaptve scheme the controller parameters wll be compute from the results of the entfcaton algorthm escrbe n ths secton. hs way the controller s as n ( wth the only fference that nstea of the parameters a b an p l. the estmates a ˆ b ˆ an p ˆ. l are use. 5. SIMULAION RESULS o llustrate the avantages of the propose esgn scheme three cases wll be stue for the PMSM ata gven n Secton. In the frst case the

5 performance of the controller wll be teste for a PMSM wth zero external loa torue. In the secon case a smoothly startng torue wth a steay state of.8nm wll be consere.e. t. (.8( (. L t = η t where η( t = e. In the thr case an unnown nosy an fast varyng loa torue (see Fgure wll be consere. Such a torue s usually met n machnng (see f.e. Koumbouls et al. an the references theren. he external commans for all cases are chosen to be of the form r ( 8( ( t = η t an r (.5( ( t = η t. hs type of external commans prove soft startng as well as smooth spee change. In practce the parameters of the PMSM moel n ( are not nown an an entfcaton scheme must be apple. In all three cases presente n the prevous paragraph the RLS entfcaton algorthm presente n Secton 4 has been apple. Also n practce even though the PMSM parameters are not nown there s an a pror estmate comng from smple experments geometrc characterstcs manufacturers manuals guess or arbtrary choce. he ntal estmates of the PMSM parameters let R s J L Φ an B ffer sgnfcantly from ther real values.e. R s =.6R s J =.8J L =.5 L Φ.4 = Φ an B =.B. Hence usng a samplng freuency of Hz the followng ntal vectors of unnown parameters are erve ˆ ( [ ] θ = θ ˆ ( = [ ] ˆ ( [ ] θ = he ntal values of P ( are chosen to be eual to P( = I P( = I an P( = I4 where I m s the m m entty matrx. For comparson reasons the responses of the frst two cases namely the cases wth nonzero external loa torue wll be compare to the respectve responses of the open loop system. he rect an uarature stator voltages of the open loop system wll be chosen to be u ( t = 44.59f ( t an u ( t =.5796f ( t where f ( t = ( η( t. Wth regar to the response of the angular velocty of the close loop system for the case of no external loa torue (see Fgure t can be observe that the angular velocty follows accurately the external comman whle the overshoot s only. ra sec. o llustrate how small s the error between the external comman an the angular velocty conser the so calle relatve error efne to be the h -norm of the error normalze by the h -norm of the external comman. he relatve error compute from to t r (rse tme of the external comman calle rse relatve error s eual to.95% whle the relatve error from t r to calle here steay state relatve error s eual to.6%. Smlarly for the secon case namely the case of smoothly rsng steay external loa torue the angular velocty follows accurately the external comman (see Fgure. he overshoot s less than. ra sec. he rse relatve error s.95% whle the steay state relatve error reuces to.65%. Fnally for the case of unnown an fast varyng external loa torue the angular velocty follows accurately the external comman. he rse relatve error s eual to.97% whle the steay state relatve error s eual to.785% (Fgure. In the presence of external loa torue the open loop system s totally unable to mantan the esre rotor angular velocty. Wth respect to the rect phase stator current for the case of no external loa torue (see Fgure 4 observe that the current follows accurately the external comman. here s no overshoot whle the rse relatve error s.6% whle steay state relatve error s.%. Smlarly for the case of smooth startng steay external loa torue the current follows accurately the external comman. Agan there s no overshoot whle the rse relatve error s compute to be.96% whle the steay state relatve error s.69%. Fnally for the case of unnown external loa torue the current follows accurately the external comman (Fgure 4. he rse relatve error s.54% whle the steay state relatve error reuces to.4%. In the presence of external loa torue the open loop system s unable to mantan the esre rect phase stator current (Fgure 4. Wth respect to the uarature phase stator current (see Fgures 5 an 6 t can be observe that t les wthn acceptable lmts an n all cases wth or wthout the presence of external loa torue oes not sgnfcantly vary. Smlarly the rect an uarature phase stator voltages for all cases are wthn acceptable lmts (see Fgures 7 to 9. For all three cases namely the case of no external torue the case of smoothly startng external loa torue an the case of unnown fast varyng torue the maxmum consume electrcal power s 4.86[ W ] 58.7[ W ] an 44.59[ W ] respectvely. As was expecte the entfe moel parameters for all cases are fferent than the orgnal ones. However ths vergence oes not obstruct the ervaton of a satsfactory close loop performance. hs can be nterprete by the robust sturbance attenuaton characterstcs of the nonlnear ynamc controller propose n Secton. o verfy ths conser Fgures - where the steay responses of the spee an the rect phase stator current versus constant loa torue n [ ][ Nm] are presente

6 respectvely. Accorng to Fgures - the spee an the rect phase stator current of the open loop system are ramatcally affecte by the loa whle n the aaptve close loop cases where the controller epens upon the entfe parameters they reman practcally unaffecte. As was shown above the performance of the close loop system s satsfactory n all cases of external loa torue. he controller s a tme varyng stable ynamc system beng easly mplementable. Inee the poles of the controller namely the roots of C( beng functons of the entfe parameters reman nse the unt sc an slghtly change urng tme. Incatvely see Fgures an where the frst pole ρ c an the stance a are presente for the three loa cases. Before closng ths secton t s mportant to comment on the selecton of the samplng freuency. he goo performance has been acheve usng a rather large samplng pero =.[ sec]. It must be note that for all external loa torue cases extensve computatonal experments have also been execute usng larger samplng peros. It has been observe that n all cases the performance remans satsfactory tll =.[ sec].e. tll 455[ Hz ]. 6. DISCUSSION he problem of spee control of a PMSM wth unnown parameters an unnown external loa torue has been formulate as an aaptve control problem that meets the esgn reurement of asymptotc comman followng wth smultaneous asymptotc sturbance attenuaton. he propose aaptve esgn scheme has been chosen to be nrect an base on RLS entfcaton. he ey pont for sturbance attenuaton was to entfy the system wthout nowlege estmaton or measurement of the unnown sturbance as well as wthout usng loa observer. hs way the torue nfluence moves the moel s entfe parameters far from the moel s parameters real values. he propose esgn scheme has been teste va three computatonal experments to a PMSM non-lnear moel. In the frst the propose controller has been apple to the PMSM wth no external loa torue. In the secon a smoothly startng constant external torue has been apple to the PMSM whle n the thr an unnown nosy an fast varyng loa torue (usually met n machnng has been apple. he responses of the last two cases nclung nonzero external loa torue have been compare to the respectve responses of the open loop system. In the cases of non-zero external loa toruem the open loop system was totally unable to mantan the esre performance. In all cases the controller prouces vsually entcal close loop performances beng almost nepenent from the external loa torue. Before closng t s mportant to pont out that the present esgn scheme s enough smple an computatonally elegant thus offerng tself for mplementaton to most of moern low-level controller archtectures (DSPs µcs PLCs etc. Furthermore the present results appear to contrbute sgnfcantly to many PMSM nustral applcatons that eman hgh precson for example to control strbute an flexble manufacturng systems where precse synchronzaton an nsenstvty to external sturbances appear to be nspensble. ACKNOWLEDGEMEN he present wor has partally been fune by the Hellenc Mnstry of N.E. & R.A. O.P. Eucaton M.. Program Archmees H.I.... REFERENCES Ästrom K.J. an B. Wttenmar (989. Aaptve Control Ason-Wesley New Yor Ba I.C. K.H. Km an M.J. Youn (998. Robust nonlnear spee control of PM synchronous motor usng aaptve an slng moe control technues. IEE Proceengs Electrc Power Applcatons 45 pp Cerruto E. A. Consol A. Ract an A. esta (995. A Robust Aaptve Controller for PM Motor Drves n Robotc Applcatons. IEEE ransactons on Power Electroncs pp.6-7 Chang K...S. Low an.h. Lee (994. An Optmal Spee Controller for Permanent-Magnet Synchronous Motor Drves. IEEE ransactons on Inustral Electroncs 4 pp. 5-5 Koumbouls F.N. G.P. Petropoulos an C.S. Mavrs ( Fault Detecton n Machnng va orue Estmaton. Proceengs of the IFAC Symposum on Manufacturng Moelng Management an Control Ro Greece pp. - 6 Koumbouls FN (999. Inustral Control. New echnologes Etons Athens Koumbouls F.N an N.D. Kouvaas (. Inrect aaptve neural control for precalcnaton n cement plants. Mathematcs an Computers n Smulaton 6 pp. 5-4 Leonar W. (996. Control of Electrcal Drves Sprnger-Verlag Berln Heelberg Lessmeer R. W. Schumacher an W. Leonar (986. Mcroprocessor-controlle AC-servo rves wth synchronous or nucton motors: Whch s preferable?. IEEE ransactons on Inustry Applcatons pp Ln F.J. an Y.S. Ln (999. A Robust PM Synchronous Motor Drve wth Aaptve Uncertanty Observer. IEEE ransactons on Energy Converson 4 pp Lu.H. an C.H. Lu (99. A Multprocessor- Base Fully Dgtal Control Archtecture for

7 PMS Motor Drves. IEEE ransactons on Power Electroncs 5 pp. 4-4 Lu.H. an C.P. Cheng (994. Aaptve Control for a Sensorless Permanent-Magnet Synchronous Motor Drve. IEEE ransactons on Aerospace an Electronc Systems pp Pllay P. an R. Krshnan (99. Control Characterstcs an Spee Controller Desgn for a Hgh Performance Permanent Magnet Synchronous Motor Drve. IEEE ransactons on Power Electroncs 5 pp Rahman M..A. an M.A. Houe (998. On-Lne Aaptve Artfcal Neural Networ Base Vector Control of Permanent Magnet Synchronous Motors. IEEE ransactons on Energy Converson pp. -8 Sepe R.B. an J.H. Lang (99. Real-me Aaptve Control of the Permanent-Magnet Synchronous Motor IEEE ransactons on Inustry Applcatons 7 pp Sepe R.B. an J.H. Lang (99. Real-me Observer-Base (Aaptve Control of a Permanent-Magnet Synchronous Motor wthout Mechancal Sensors. IEEE ransactons on Inustry Applcatons 8 pp Zhu G. L.A. Dessant O. Ahrf an A. Kaour (. Spee racng Control of a Permanent- Magnet Synchronous Motor wth State an Loa orue Observer. IEEE ransactons on Inustral Electroncs 47 pp ω ra sec L Nm Fg.. Steay state of the rotor spee as a functon of the external loa torue (Dashe lne: open loop Sol lne: close loop [ Nm ] L [ ra/sec ] ω ( ( ( (4 (5 Fg.. Angular velocty of the rotor ( - close loop wthout loa - close loop wth smoothly startng constant loa - close loop wth fast varyng loa 4 - open loop wth fast varyng loa 5 - open loop wth smoothly startng constant loa [ A ] (5 (4 ( ( ( Fg. 4. Drect phase stator current ( - close loop wthout loa - close loop wth smoothly startng constant loa - close loop wth fast varyng loa 4 - open loop wth fast varyng loa 5 - open loop wth smoothly startng constant loa [ A ].5.5 ( t [ sec] Fg. 5. Quarature phase stator current ( - close loop wthout loa - close loop wth smoothly startng constant loa - open loop wth smoothly startng constant loa 4 - open loop wth fast varyng loa ( (4 ( Fg.. Unnown an fast-varyng external loa torue

8 .5 [ A ].5 ω ra sec 9 8 ( ( ( 7 ( Fg. 6. Quarature phase stator current of the close loop system wth fast varyng loa L [ Nm] Fg.. Steay state angular velocty as a functon of the external loa torue ( - close loop wthout loa - close loop wth smooth loa - close loop wth fast varyng loa 4 - Open loop 4 ( (4 [ V ] u - ( [ A ] ( ( ( - ( Fg. 7. Drect phase stator voltage ( - close loop wth smooth loa - close loop wth fast varyng loa - close loop wthout loa 5 ( [ L ] Fg.. Steay state rect phase stator current as a functon of the external loa torue ( - close loop wthout loa - close loop wth smoothly startng constant loa - close loop wth fast varyng loa 4 - Open loop 4 (.9 ( [ V ] u ρ c.9 (.9 ( t [ sec] Fg. 8. Quarature phase stator voltage ( - close loop wth smooth loa - close loop wthout loa Fg.. Intal controller pole ( - close loop wthout loa - close loop wth fast varyng loa - close loop wth smoothly startng constant loa.46 ( [ V ] a.45 ( u ( Fg. 9. Quarature phase stator voltage of the close loop system wth fast varyng loa Fg.. Controller pole stance ( - close loop wthout loa - close loop wth fast varyng loa - close loop wth smoothly startng constant loa