Wol Acaemy of Scence, Engneeng an echnology 44 8 Opmal Conol Saeges fo Spee Conol of Pemanen-Magne Synchonos Moo Dves Roozbeh Molav, an Davoo A. Khab Absac he pemanen magne synchonos moo (PMSM) s vey sefl n many applcaons. Veco conol of PMSM s popla kn of s conol. In hs pape, a fs an opmal veco conol fo PMSM s esgne an hen esls ae compae wh convenonal veco conol. hen, s assme ha he measemens ae nosy an lnea aac Gassan (LQG) mehoology s se o fle he noses. he esls of nosy opmal veco conol an flee opmal veco conol ae compae o each ohe. Nonlneay of PMSM an exsence of nvee n s conol cc case ha he sysem s nonlnea an me-vaan. Wh evng aveage moel, he sysem s change o nonlnea me-nvaan an hen he nonlnea sysem s convee o lnea sysem by lneazaon of moel aon aveage vales. hs moel s se o opmze veco conol hen wo opmal veco conols ae compae o each ohe. Smlaon esls show ha he pefomance an obsness o nose of he conol sysem has been hghly mpove. Keywos Kalman fle, Lnea aac Gassan (LQG), Lnea aac eglao (LQR), Pemanen-Magne synchonos moo (PMSM). NOMECLAURE, - Sao cens v, v - Sao volages ω Eleccal spee (a/s) θ Eleccal poson (a) L Loa oe (N.m) R Phase essance (Ω) L Phase ncance (Heny) J Roo nea (kg.m ) D Dampng coeffcen (N.m.s/a) ϕ Pemanen magne flx lnkage (N.m/A) Nmbe of pole pas N I. INRODUCION ECOR conol echne, ncopoang fas sgnal V pocessng an fas powe eleconcs, have mae possble he applcaon of ac moos ves n hgh pefomance asks whee aonally only c sevo ves wee apple. A pemanen magne synchonos moo (PMSM) employng veco conol s especally favoable fo hgh pefomance sevo ve applcaons becase flflls he esgn cea of hgh pefomance sevo ve, sch as compac sce, hgh a-gap flx ensy, hgh powe o nea ao, hgh oe o nea ao an hgh oe capably [1]. As he PM ac moo s eplacng he convenonal c moos fo small op powe ang vaable spee conol sysem, he pefomance of PM ac moos whch ses veco conol an has ck ansen esponse a same me, ms be mpove []. One way o mpove he esponse of he sysem s lnea aac eglao (LQR). he lnea aac eglao (LQR) s an opmal conol mehoology ha can be employe n we ange of applcaons. he aac cos fncon poves he esgne wh los of flexbly o pefom ae off among vaos pefomance cea. he elaonshp beween cos fncon weghs an pefomance cea hol even fo hgh oe an mlple np sysems, whee classcal conol becomes cmbesome. A majo lmaon of LQR s ha he ene sae ms be mease exacly when geneang he conol. hs lmaon becomes nceasngly oblesome fo hgh oe sysems, whee measng all saes can be vey expensve. In aon no measemen s eve exac. heefoe, an opmal esgn mehoology ha esls n conolles ha lze nosy, paal sae nfomaon s esable. he lnea aac Gassan (LQG) mehoology poves a means of esgnng sch conolle [3]. he man conbon of hs pape s opmal veco conol of PMSMs ha s essenal fo hgh pecson applcaons sch as sevo ve. he es of he pape s sce as follows. he lneaze moels of PMSM ae pesene n secon II. Secon III noces he opmal saeges fo spee conol of PMSM hogh lnea aac eglao (LQR) an lnea aac Gassan (LQG) mehoologes. he compe smlaon esls ae pesene n secon IV. Fnally, secon V concles he pape. II. SAE SPACE MODELS AND LINEARIZED MODELS OF PMSM he ynamc moel fo he PMSM n he - ansfome oo efeence fame s gven n sae space as follow [4]: R/ L ϕ/ L ω R / L ω = ω 1.5 N ϕ/ J D/ J ω 1/ L v 1/ L L v N/ J (1) 48
Wol Acaemy of Scence, Engneeng an echnology 44 8 he basc pncple n conollng he PMSM s base on fel oenaon. hs s obane by leng he pemanen magne flx lnkage be algne he -axs an sao cen veco s kep along -axs econ. hs means he vale of s kep zeo n oe o acheve he fel oenaon conon. Snce he pemanen magne flx s consan, heefoe he elecomagnec oe s lnealy popoonal o he -axs cen whch s eemne by close loop conol. As a esl, maxmm oe pe ampee can be obane fom he machne n aon o achevemen of hgh ynamc pefomance. Applyng he fel oenaon concep by leng = n (1) he lneaze moel of PMSM can be escbe n sae space fom as [5]: x () = Ax() B() E() () Whee: x () = ω θ, () = v, () = (3) L RL / ϕ / L 1/ L A= 1.5 N ϕ/ J D/ J, B=, E= N/ J (4) 1 Above, he swchng effec of nvee s neglece. As a mae of fac, becase of nonlneay of PMSM an exsence of nvee, sysem s nonlnea an me-vaan (e o peocy of fncon v an v an hey have swchng fom) [6]: x () = f ( x(),) (5) v R ω L L v R ϕ ω ω (6) f = L L L 1.5N ϕ N D L ω J J J ω Afe evng aveage moel, v an v ae change o consan vale. Conseenly he sysem s change o nonlnea me-nvaan. Opmal op feeback poblem can be solve fo sysem by lneazaon aon aveage vales. In ohe wo elbm opeang pon s efne by aveage moel. Whee Δx s small changes aon aveage pon wh esle by jacoban. R / L ω ω R / L ϕ/ L Δ x () = Δx() 1.5 N ϕ / J D/ J 1 1/ L 1/ L Δ ( ) Δ( ) N / J (7) Whee: Δ x() = ω θ, Δ () = v v (8) In he nex secon, opmal saegy wll be smmaze. III. OPIMAL CONROL SRAEGY A. Lnea Qaac Reglao (LQR) he lnea aac eglao (LQR) s an opmal conol poblem whee he sae eaon of he plan s lnea, he cos fncon s aac an es conons conss of nal conon on he sae an no sbance np. he plan eaon s: x =AxB (9) y = Cx (1) A easonable cos fncon o se when he conol sysem s esgne o opeae fo long me peo s: ( ( ), ( )) 1/ ( J x = x( Qx ) ( ) ( R ) ( )) (11) One meho of fnng he opmal feeback gan max lzes a nonlnea max ffeenal eaon, known as Rcca eaon: p () = p () A A p () Q pbr () Bp () (1) P( )ms sasfy he above eaon. he solon of he opmal conol poblem can be ece o fnng he max p(), snce he opmal conol s gven [3] : () = R B px () () = kx () () (13) = plan Opmal gan max Fg. 1 Block agam of opmal conol B. Lnea Qaac Gassan (LQG) Lnea aac Gassan (LQG) conol efes o an opmal conol poblem whee he plan moel s lnea, he cos fncon s aac an he es conon conss of anom nal conons, a whe nose sbance np, an whe measemen nose. he plan s escbe by he followng [3]: x () = Ax() B () B w() (14) w whee () s he conol np an w() s a anom sbance np known as plan nose. he measemen avalable fo feeback s: m () = cx () v () (15) m whee v() s a anom sgnal known as measemen nose. x() 49
Wol Acaemy of Scence, Engneeng an echnology 44 8 w () -K() xˆ( ) v() plan Kalman fle y () Fg. A Lnea aac Gassan opmal conol sysem he sae moel fo he opmal conolle s: xˆ = [ A G() C B K()] x() G() m() m (16) () = Kx () ˆ() (17) Whee G() s Kalman gan. he sae feeback gan K() s fon by solvng he followng [3]: p () = p () A A p () Q pbr () B p () (18) K = R B p (19) () () In smmay, he solon of lnea aac Gassan opmal conol poblem can be boken no wo pas: (1) fn he lnea aac eglao feeback gans ha mnmze he cos asmmng pefec sae nfomaon. () geneae a Kalman fle o esmae he sae. hs s a emakable esl known as he sochasc sepaaon pncple [3]. IV. SIMULAION RESULS convenonal veco conol, selng me of acal spee s.5s (wh he same PI paamees). Spee oveshoo n opmal veco conol s.33% b n convenonal meho, s oveshoo s 4.58% (Fg. 4). Elecomagnec oe oveshoo has 3.73% mpovemen n compason of opmal veco conol an convenonal veco conol. A =1s, spee efeence s change fom 15a/s o 18a/s. Selng me of acal spee n opmal meho s.7s an spee oveshoo s % appoxmaely b n convenonal meho selng me of acal spee s.4s an has.8% veshoo (Fg. 5). Elecomagnec oe oveshoo whle spee s changng, has 1.94% mpovemen n compason of convenonal meho. Fg. 5 Spee of opmal an convenonal veco conol A =s, spee efeence s change fom 18a/s o 15a/s. spee selng me an spee neshoo n opmal meho ae,.1s an % especvely. B n convenonal meho hese paamees ae.8s an.83% especvely (Fg. 6). Also elecomagnec oe oveshoo has.7% mpovemen n compason of convenonal meho. Fg. 3 Block agam of opmal veco conol Fg. 6 Spee vaaon n opmal an convenonal veco conol hen s assme ha sysem has pocess an measemens noses an Kalman fle s se o ece nose effec (Fg. 7). Fg. 4 Opmal an convenonal veco conol Spee efeences s se o 15a/s an selng me of acal spee s.1s n opmal veco conol b n 43
Wol Acaemy of Scence, Engneeng an echnology 44 8 Fg. 7 Block agam of lnea aac Gassan Fg. 1 Elecomagnec oe fo LQG ne nosy conon Conseng nonlneay of PMSM an me-vaan of nvee an lneazaon of moel aon aveage vale s en o changes block agam of opmal veco conol as below: Fg. 8 Spee fo LQR an LQG ne nosy conon Fg. 8, Fg. 9 an Fg. 1 show spee an elecomagnec oe fo LQR an LQG ne nosy conon. Fg. 11 Block agam of opmal veco conol Spee efeence s se o 15a/s an selng me fo acal spee s.5s. Fg. 9 Elecomagnec oe fo LQR ne nosy conon Fg. 1 Acal spee Fg. 13 shows acal spee compason of convenonal veco conol an wo opmal veco conol mehos. 431
Wol Acaemy of Scence, Engneeng an echnology 44 8 Fg. 13 Acal spee compason V. CONCLUSION In hs pape, we nvesgae he LQR an LQG mehoologes n veco conol of PMSMs. he smlaon esls showe ha he popose conolles has bee pefomance fo he sake of esgn cea lke oveshoo an selng me of he sep esponse. Moeove, he LQG conolle shows moe obsness agans pocess an measemen noses. Conseng he nonlneay of PMSM an exsence of nvee n s conol cc ha s case he opmal gans wh convenonal opmal conol heoy can no be fon. he solon o hs poblem s sng aveage vales an lneazaon aon aveage vales an smlaon esls showe ha pefomance of sysem s mpove by hs meho. APPENDIX Moo paamee se n he smlaon: PMSM INDRAMA-MAC9B Sao essance.97 ohm Sao ncance 5.1 mh Pemanen magne flx.11 N.m/A Momen of nea.36 kg.m Fcon coeffcen.1 N.m.s/a REFERENCES [1]. H. L an C. H. L, A mlpocesso-base flly gal conol achece fo pemanen magne synchonos moo ves, IEEE ans. Powe Elecon., vol. 5, no. 4, pp. 413-43, Oc. 199. [] Raymon B. Sepe an J. H. Lang, Real-me aapve conol of he pemanen-magne synchonos moo, IEEE ans. Ins. Appl., vol.7, no.4, pp. 74-716, jly/ag. 1991. [3] Jeffey B.Bl,"Lnea opmal conol", Ason Wesley Longman, Inc, 1999. [4] P. C. Kase, Analyss of Eleccal Machnay, New Yok: McGaw Hll, 1987. [5] C. Maemls an N. Magas, Loss Mnmzaon n Veco-onolle Ineo Pemanen-Magne Synchonos Moo Dves. IEEE ansacons On Insal Eleconcs, Vol. 49, No. 6, Decembe. [6] S. Banejce, G. C. Veghese," Nonlnea Phenomena n Powe Eleconc", IEEE Pess, 1. 43