Sensorless Speed Control of Salient Pole PMSM According to the Backstepping Observer

Transcription

1 Proceengs of he Inernaonal Conference on Recen Avances n Elecrcal Sysems, Tunsa, 26 Sensorless Spee Conrol of Salen Pole PMSM Accorng o he Backseppng Observer S. Naj S. Benacha Elecrcal Engneerng Deparmen Unversy of Sef, Algera LSPIE Laboraory, Elecrcal Engneerng Deparmen Unversy of Sef, Algera Absrac--In hs work, a novel sensorless spee conrol scheme s evelope for he esgn of salen pole permanen magne synchronous moors (PMSM). Ths esgn sraegy consss of nonlnear backseppng conrollers an an angular velocy observer o acheve he purpose of spee rackng. The PMSM esgn, n general, reures angular velocy measuremen o acheve conrol goals, bu mos PMSM sysems o no have spee measuremen evces. Therefore, a backseppng observer s use o esmae he roor spee, an hen he nonlnear conrollers base on backseppng esgn algorhm couple wh nroucon of negral acons are evelope for he PMSM sysem. The propose conrol scheme s no only o sablze he PMSM sysem, bu also o rve he spee rackng error o converge o zero asympocally. Furhermore, he complee sysem moel s smulae by usng MATLAB/Smulnk sofware, performance of he propose conroller s nvesgae exensvely a fferen ynamc operang conons such as suen loa change, comman spee change an parameer varaon. Inex Terms--Backseppng conrol, Lyapunov heorem, negral acon, salen pole permanen magne synchronous moor (PMSM), backseppng angular velocy observer, observer.. INTRODUCTION Permanen magne synchronous moors (PMSM) are wely use n hgh performance servo applcaons ue o her hgh effcency, hgh power ensy, an large orue o nera rao []. The PMSM s very smlar o he sanar woun roor synchronous machne excep ha he PMSM has no amper wnngs an he fel wnng of he roor s replace by a permanen magne [2]. Hence, he mahemacal moel of he PMSM s encal o ha of he classcal synchronous machne wh he euaons of he amper wnngs an fel curren ynamcs remove. However, he ynamc moel of a PMSM s srongly nonlnear an couple, because of he couplng beween he moor spee an he elecrcal uanes, such as he - axs currens [3]. Recenly, varous nonlnear conrol mehoologes have been apple o he conrol esgn of PMSM sysems n orer o consruc he conrollers recly by conserng he nonlnear PMSM ynamcs. Backseppng conrol s a new ype recursve an sysemac esgn mehoology for he feeback conrol of unceran nonlnear sysem, parcularly for he sysem wh mache unceranes. The mos appealng pon of s o use he vrual conrol varable o make he orgnal hgh orer sysem smple, hus he fnal oupus can be erve sep by sep sysemacally hrough suable Lyapunov funcons. The nfluence of he change some parameers an of he perurbaon of charge can be grealy reuce by nroucng negral acon n each sep, n orer o ensure hgh accuracy spee conrol [4]-[6]. In aon, he man reason why he nusral applcaon of PMSM rves has grown s ha he algorhm rvng PMSM whou spee sensors has been sealy mprove. The esgn of PMSM sysems, n general, he nformaon of roor poson an spee are necessary o acheve a hgh-performance rve, bu los of PMSM sysems o no have spee measuremen evces. Elmnang spee sensors smplfes he srucure of he rver sysem an wens he applcaon fels. Therefore, varous sensorless conrol sraeges have been nvesgae [7]-[]. In hs paper, wh he propose backseppng angular velocy observer, a nonlnear backseppng conrol esgn scheme s evelope for he spee rackng conrol of PMSM ha has exac moel knowlege. The asympoc sably of he resulng close-loop sysem s guaranee accorng o Lyapunov sably heorem. As a resul, he propose nonlnear backseppng conrol esgn whou usng spee sensor s no only o sablze he PMSM sysem, bu also o force he spee rackng error o converge o zero asympocally. The paper s organze as follows: n Secon 2, he ynamc moel of a PMSM s nrouce wh some mporan sysem properes. The sensorless backseppng conrol scheme wh negral acon conssng of an angular velocy observer an he volage npu conrollers s evelope for he purpose of spee rackng n Secon 3. The smulaon resuls are llusrae n Secon 4, an some conclung remarks are gven n Secon MATHEMATICAL MODEL The ynamc moel of a PMSM can be escrbe n he well-known - frame hrough he Park ransformaon [3], [] as follows: ISBN: (226) Eors: Tarek Boukr & Rafk Nej

2 Proceengs of he Inernaonal Conference on Recen Avances n Elecrcal Sysems, Tunsa, 26 R L p L L L R L p p f v () L L L L F P L L f TL p J J J J where v v Saor's -axs curren. Saor's -axs curren. Saor's -axs volage. Saor's -axs volage. Moor spee. R Saor ressance. L L -axs saor nucance. -axs saor nucance. p Number of pole pars. f Permanen magne flux. J Roor momen of nera. F Vscous frcon coeffcen. TL Loa orue. Roor angular poson. Here, he avalable measuremens are assume o be he roor poson an he - axs currens an. If roor poson can be obane, hen he relaonshp beween he - frame currens an he acual hree-phase curren uanes a, b an c s efne by he followng change of coornaes: a AB where b c cos A sn sn c os B 3 3 v (2) 3. SENSORLESS BACKSTEPPING CONTROL WITH INTEGRAL ACTION Ths par of he arcle sues he sraegy of spee conrol by he sensorless Backseppng echnue, can be effecvely use for lnearzng a nonlnear sysem n he presence of unceranes [4], [6]. The conrol echnue s a nonlnear Backseppng conrol havng properes of srengh. The pursu of spee akes place wh a hgh yel by he conrol volage v as long as he curren s kep eual o zero. The basc ea of he ype Backseppng conrol s o make curly euvalen subsysems of orer sysems n cascae sable whn he meanng of Lyapunov, whch gves hem he uales of robusness an a global asympoc sably [5], []. In oher wors, hs s a mul-sep meho. A each sep of he process, a vrual conrol s also generae o ensure he convergence of sysem o s eulbrum sae. Ths can be reache from Lyapunov funcons o ensure sep by sep he sablzaon of each synhess sep. The ea s o compue a conrol law o ensure ha he Lyapunov funcon s posve efne an s ervave s always negave. The calculaon of Lyapunov funcon s performe n a recursve way. I s base on he prevous sysem sae. A new Conrol Lyapunov Funcon (CLF) s consruce by he ncrease of CLF of he prevous sep [6]. Ths proceure calculaes allow us o ensure overall sysem sably. The robusness mprovemen of hs echnue by ncorporaon of nroucon n erms negraons of he conrol esgn of PMSM hereafer. The corresponng block agram of a PMSM sysem wh coornae ransformaon s shown n Fg.. I ref ref I I es es Backseppng Conrol Backseppng Observer V ref V ref Fg.. The block agram of spee rackng esgn scheme. I I Snce he spee measuremen of he PMSM s usually unavalable, an angular velocy observer mus be employe o esmae he acual moor spee. So, he spee esmaon error s efne by (3) abc abc Loa V a V b V c Ia I b PWM Inverer PMSM ISBN: (227) Eors: Tarek Boukr & Rafk Nej

3 Proceengs of he Inernaonal Conference on Recen Avances n Elecrcal Sysems, Tunsa, 26 where spee esmaon spee esmaon error k, k, k 2, k 2, k3 an k 3 posve esgn consans. Now, we employ backseppng schemes o esgn he nonlnear conrollers wh he angular velocy observer, he backseppng esgn proceure consss of hree seps: A. Curren Loop Frs of all, snce he rec axs curren mus be force o be zero, le us efne he followng rackng error: e e (4) ref where ref e k s an negral acon. Defnng he followng canae Lyapunov funcon: 2 2 V e e (5) 2 2 he me ervave s compue as: V e k e k (6), hen replacng e e n he above Snce ref euaon, we oban: R L L V e p p v L L L L k e e e e (7) B. Spee Loop To solve spee rackng problem, efne he followng spee rackng error as: 2 ref 2 ref (8) e k where k 2 ref s an negral acon erm ae o he conrol n Backseppng o ensure he convergence of he rackng error o zero n spe of he unceranes of such pecewse consan a each sep of he algorhm. The vrual conrol npu ref use o ensure he sably of he spee loop. The error ynamcs of spee from (8) s gven by: p F p e 2 ref L L ref f ref J J J T 2 ref k (9) Conser he followng Lyapunov canae funcon 2 V2 e2 s ervave s gven by he followng euaon: 2 V p F 2 e2 L L p T J J J J ref ref f ref L 2 ref k () C. Curren Loop Now o esgn he conrol npu rackng error n he curren as follows: 3 ref 3 v L we efne he e e () where 3 3 ref e k s an negral acon. Conser he followng Lyapunov canae funcon: 2 2 V3 V2 e3 e 3 (2) 2 2 By akng he me ervave of V3 s gven by: 2 ref V 3 k2e2 e3 k3 ref 3 3 ref ek (3) Replacng by: L R p p f v, hen (3) L L L L becomes: R L L V k e e p p 2 ref L L L p p v k L L L f f 3 ref 3 3 ref ek (4) Therefore, by followng he choce of (5) an (2), he complee Lyapunov funcon canae s selece as V V V2 e3 e3 (5) where s he posve aapaon gan. From (7), () an (4), he ervave of (5) s compue as follows: ISBN: (228) Eors: Tarek Boukr & Rafk Nej

4 Proceengs of he Inernaonal Conference on Recen Avances n Elecrcal Sysems, Tunsa, 26 V V V 2 e3e 3 e3 e 3 (6) R L L V e p p v L L L L p k e e e e e L L J 2 ref ref F p f ref TL k2 ref J J J R L L e p p ref 3 L L L f p p f v L L L p e e k L L J ref ref F F p f ref TL J J J J (7) A las, we are able o oban F V ke k2e2 k3e3 (8) J Clearly, V n (8) s negave efne, so mples ha he resulng close-loop sysem s asympocally sable an, hence, all he error varables e, e 2, e3 an he esmaon error wll converge o zero asympocally. Therefore, he -axs curren wll converge o zero an he angular velocy wll converge o he reference spee. In aon, because can converge o zero, he esmaon spee wll converge o he acual spee evenually. As a resul, he esre conrol objecve of sensorless spee rackng for he PMSM sysem s nee acheve by he propose non lnear backseppng conrol scheme. 4. SIMULATION RESULTS To llusrae he mahemacal analyss an o nvesgae he performance of he propose PMSM conrol of Backseppng wh negral acon accorng o he Backseppng observer, smulaons are carre ou followng he overall block agram of he conrol shown n Fg.. Table I gves he nomnal parameers of he PMSM whch s use n he smulaon ess. The smulaon has been carre ou usng Malab/ Smulnk sofware. The machne sare on he vacuum, hen a loa orue s apple of 5 (N.m) a me.4 s, an hen elmnae where he momen was.8 s. A me =.2 s, he recon of roaon s reverse a (r/s) o - (r/s). (r/s) R (Ω) TABLE I The Parameers of he PMSM. L (mh) L (mh) f (Wb) p J (kg.m²) F (N.m/r/s) Spee (r/s) Spee (r/s) Elecromagnec orue (N.m) Curren (A) Spee of roaon (r/s) Acual spee Spee ref Esmae spee Spee esmaon error (r/s) Elecromagnec orue (N.m) Curren, (A) Fg. 2. Smulaon resuls urng he varaon of saor ressance. I I ISBN: (229) Eors: Tarek Boukr & Rafk Nej

5 Proceengs of he Inernaonal Conference on Recen Avances n Elecrcal Sysems, Tunsa, 26 Fgure 2 show error of + % on he nomnal value of he saor ressance, s noe ha accorng o he shape he spee follows perfecly s reference whch s acheve very raply, wh a very fas response, we can fn ha he esmaon spee error can converge o zero asympocally, we can see ha he effec of he loa perurbaon s raply sappearng. The elecromagnec orue sablze a he loa orue value an he response of he wo curren componens an show goo ecouplng nrouce by PMSM conrol (The rec axs curren s always force o zero n orer o oren all he lnkage flux n he -axs an acheve maxmum orue per ampere), we also can see ha he curren s he couple mage. From he spee rackng smulaon resul we can fn ha he sensorless backseppng conrollers have goo performance. S p e e ( r / s ) S p e e ( r / s ) Spee of roaon (r/s) Acual spee Spee ref Esmae spee Spee esmaon error (r/s) Fg. 3. Spee behavour of he PMSM regulae by backseppng. S p e e ( r / s ) Spee of roaon (r/s) Acual spee Spee ref Esmae spee S p e e ( r / s ) Spee esmaon error (r/s) Fg. 4. Spee Behavour of he PMSM regulae by backseppng. Fgures 3 an 4 llusrae he ynamcs behavours an spee rackng error of PMSM accorng Backsppng observer a a servo-conrol of spee, loa orue s apple a 5 (N.m) from sarng. In he frs fgure an a me =.4 he spee of roaon s ncrease a 2 (r/s) o 4 (r/s), a me =.8 he recon of roaon s reverse o -4 (r/s) hen ecrease o -2 (r/s) hen a sop (r/s). In he las fgure an a me =.4 he recon of roaon s reverse a (r/s) o - (r/s), hen a sop (r/s). The smulaon resuls show he robusness of he conroller use n hgh an low spees, s also shown ha hs esmae spee follows he acual spee wh precson (spee esmaon error converge o zero) on permanen regme an he ransonal regme. 5. CONCLUSION Ths paper base on Backseppng conrol wh an negral acon for salen pole permanen magne synchronous moor accorng o he Backseppng observer, whch s aresse n a han, as a ool for a new nonlnear conrol spee PMSM, an n he oher han as a ool for suyng ynamc sably. The performance of he propose conroller has been nvesgae n smulaon. The fferen resuls show s effecveness an fas response whou overshoo a rackng a reference spee uner crcal suaon of benchmark commans whch raply change, robus performance o parameer varaons an surbances hroughou he sysem. Moreover, o overcome he problem of conrol whou mechancal sensors, we use a meho base on Backseppng echnue, for esmang spee from he measure currens. The spee elmnaon sensor reuces he consrans an gves more flexbly o conrol he machne. In he lgh of he smulaon resuls, we can conclue ha he objecve of hs suy s acheve. Thus, he Backseppng echnue o he negral acon combne wh vecor conrol offers hgh conrol performance. The nrnsc robusness Backseppng s srenghene hrough full erms ae o. ISBN: (23) Eors: Tarek Boukr & Rafk Nej

6 Proceengs of he Inernaonal Conference on Recen Avances n Elecrcal Sysems, Tunsa, 26 ACKNOWLEDGMENT Ths paper was suppore by he Algeran mnsry of hgher eucaon an scenfc research. I woul lke o express my umos graue o all he people who have recly or nrecly conrbue owars he successful compleon of hs paper. REFERENCES [] F Gr, AC Elecrc Moors Conrol, John Wley & Sons, L, 23, P. 4. [2] R. Krshnan, Permanen Magne Synchronous an Brushless DC Moor Drves, Taylor an Francs Group, 2, P. 3. [3] P. Pllay an R. Krshnan, Moelng, smulaon, an analyss of permanen-magne moor rves.. The brushless c moor rve, IEEE Transacons on Inusry Applcaons 25 (2) (989) [4] M. Krsć, I. Kanellakopoulos, an P. Kokoovć, Nonlear an Aapve Conrol Desgn, John Wlley an Sons, Inc, 995. [5] C-X Chen, Y-X Xe, an Y-H Lan," Backseppng conrol of spee sensorless permanen magne synchronous moor base on sle moel observer," Inernaonal Journal of Auomaon an Compung Aprl 25, Volume 2, Issue 2, pp [6] A. Larbaou, B. Belabbes, A. Meroufel, A. Tahour, an D. Bouguenna," Backseppng Conrol wh Inegral Acon of PMSM Inegrae Accorng o he MRAS Observer," IOSR Journal of Elecrcal an Elecroncs Engneerng, Volume 9, Issue 4 Ver. I (Jul Aug. 24), PP [7] S-S Ke an J-S Ln," Sensorless Spee Trackng Conrol wh Backseppng Desgn Scheme for Permanen Magne Synchronous Moors," Proceengs of he 25 IEEE Conference on Conrol Applcaons Torono, Canaa, Augus 28-3, 25. [8] L. Cabaon, M.L. Corran, M. Grsosom, G. Ippol, S. Longh, an G. Orlano," Aapve Exene Kalman Fler for Robus Sensorless Conrol of PMSM Drves," 2 5h IEEE Conference on Decson an Conrol an European Conrol Conference (CDC-ECC) Orlano, FL, USA, December 2-5, 2 [9] O. M. Arafa, G. A. Abel Azz, M. I. Abu El-Sebah, an A. A. Mansour," Observer-base sensorless spee conrol of PMSM: A focus onrve s sarup," Journal of Elecrcal Sysems an Informaon Technology 3, 26, [] Qan Yuan, Zhong-png Yang, Fe Ln, an Hu Sun," Sensorless Conrol of Permanen Magne Synchronous Moor wh Saor Flux Esmaon," Journal of Compuers, vol. 8, no., January 23. [] M. A. Hama, A. Umneau, an J. D. Leon,Robus Inegral Backseppng Conrol for Sensorless IPM Synchronous Moor Conroller, Journal of he Frankln Insue 349, 22, ISBN: (23) Eors: Tarek Boukr & Rafk Nej