H2-TS-FUZZY POSITION CONTROL OF PMSM WITH AN AUGMENTED D-AXIS STATOR CURRENT MODEL

Transcription

1 ns o XIX Congesso seo e utomátc, C. H-S-FUZZY POSIION CONROL OF PMSM WIH N UGMENED D-XIS SOR CURREN MODEL RYMUNDO C. GRCI,, WLER I. SUEMISU, JOO O. P. PINO Lbotóo e Eetônc e Potênc, COPPE, Unvese Fee e Ro e Jneo Ih o Goveno, CEP , Ro e Jneo E-ms: [email protected], [email protected] Lbotóo e Integênc tc, Eetônc Dgt, Eetônc e Potênc, Dto. e Engenh Eétc, Unvese Fee e Mto Gosso o Su CEP , Cmo Gne, Mto Gosso o Su, s E-ms: [email protected] bstct Pemnent Mgnet Synchonous Moto (PMSM) s non-ne system use n the eveoment o sevomechnsm, wth oston oo n the -xs stto cuent oo whch contos the mgnetc ux o the moto. Fuzzy kgsugeno Moeng (S) ows the esgn o obust contoes bse on ne mtx neutes (LMI), usng combnton set o oc ne moes. he oc moes o the oston oo o PMSM hve n mct ntegto, whch euce the stey-stte eo. Howeve, the -xs cuent oo oes not hve ths ntegto. hs e esents the esgn o obust contoes o PMSM conseng the ton o n ntegto n n ntegtve gn n the -xs stto cuent oo. hs conseton s me o othe owe eectonc systems s DC-DC convetes, but ths e nyzes the use o ths technue n vecto conto o PMSM. he eect o etubtons s euce conseng the H nom technue. he oston n the -xs stto cuent oos e consee s two neenent SISO systems, whch gve moe exbty o the oe cement. Smuton esuts show the goo eomnce o ths technue. Keywos ugmente moe, H nom, Lne Mtx Ineutes, PMSM, gk-sugeno Moeng. Resumo O Moto Síncono e Ímã Pemnente (PMSM) é um sstem não ne utzo no esenvovmento e sevomecnsmos, com um mh e osção e um mh e coente no estto no exo, o u conto o uxo mgnétco o moto. moegem Fuzzykg-Sugeno (S) emte o oeto e contooes obustos bseo em esgues nees mtcs (LMI), usno um combnção e moeos nees ocs. Os moeos ocs mh e osção o PMSM ossuem um ntego, o u euz o eo em egme estáve. Não obstnte, mh coente o estto no exo não ossu uee ntego. Este tgo esent o oeto e contooes obustos PMSM conseno nseção e um ntego e um gnho e ntegção n mh e coente. Est conseção é utz no contoe e outos ccutos e otênc como os convesoes DC-DC, oém o esente tgo z um náse o uso est técnc no contoe veto o PMSM. O eeto s etubções é euzo tvés técnc nom H. s mhs e osção e coente são conses como os sstems SISO neenentes, o ue onece mo exbe n ocção os óos em mh ech. Smuções mostm o bom esemenho técnc oost. Pvs-chve Ineue Lne Mtc, Moeo umento, Moegem kg-sugeno, Nom H, PMSM. Intoucton Nowys, emnent mgnet synchonous moto (PMSM) s oweu tentve n the mementton o vbe euency ve ctons n sevomechnsm by ts bette toue/sze eton n obustness thn DC motos, n by ts owe toue e n hghe enegetc ecency thn nucton motos (Un et., ; ezc n Jc, ). On the othe hn, the conto o PMSM s moe cut thn DC moto ue to PMSM s non-ne system, whose metes cn chnge n tme, ecte by exten o etubtons (Py n Kshnn, 988; ose, 997). Vecto moe o PMSM s comose by two systems (ose, 997): the -xs ccut tht contos the ngu oston o see though the geneton o eectomgnetc toue, n the -xs ccut, tht conto the mgnetc oto ux. he -xs votges, ISN: cuents n uxes e obtne by Pk tnsomton (ose, 997). Howeve, both ccuts hve nonne comonents tht mke ne contoes s PID egutos neute o see conto o PMSM. Fuzzy kg-sugeno Moeng (S) s oweu too n the eveoment o obust contoes o non-ne nts, by eesentng them s Fuzzy combnton o ne systems (kg n Sugeno, 985; nguch et., ; nguch et., ; exe et., 3, Schute, 5; Pecu et., 7; No n Okubo, ). cton o S n see vecto conto o PMSM cn be oune n (Ln et., 7; Jung et., ; Wng et., ; Cho et., ). Howeve, thee e ew eseches bout the cton o S contoes o the ngu oston o PMSM. Geney, S moes e use wth Lne Mtx Ineutes (LMI), n oe to get cose-oo contoes whch e obust gnst vtons n the mete, conseng estctons n the conto w, 4753

2 ns o XIX Congesso seo e utomátc, C. the system oututs n o tckng obems (ssunção et., 8, ne et., ). hs e nyzes the esgn o LMI contoes o oston conto o PMSM bse on n ugmente S moe o the moto, whee n ntegto n n ntegtve gn s e to the -xs ccut. ugmente moe usng n ntegto o non-ne owe eectoncs ctons s oost convete cn be oun n (Montgne et., ; O et., 9; O et., ). he ton o n ntegto s technue use n ne systems o tckng obems (Ogt, ). hs e uses the sme methooogy o the -xs stto cuent oo. s esut, the contoe hs two comonents: stte eebck n n nteg contoe, whose metes e obtne usng LMIs. he -xs n the -xs systems e consee s SISO systems nste o n ony one MIMO system. hs conseton n the ton o stte vbe n the -xs system gve moe exbty to set the cement o the cose-oo oes usng D- stbty contons. H nom technue s use to euce the eect o etubtons. Smuton esuts show the otentty o the use o the oose S moe wth ugmente nt o oston conto o PMSM. Vecto Moe o PMSM Pk tnsom s use to exess the thee-hse votges, cuents n uxes o PMSM, [ b c ] n new othogon eeence system, conseng new vbes [ ] (ose, 997): ( λ) sn( λ k) sn( λ k) ( cos( λ cos( λ sn cos b c () Whee k π/3 n λ s the nge o the new othogon eeence system, whch s geney eu to the ngu oston o the moto sht (θ) mute by the numbe o oe s (n): λ nθ () he ynmc behvo o PMSM s escbe by the oowng eutons (Py n Kshnn, 988): (3) v s nωm t (4) v s nω m nφωm t t,. φ (5) em ( ) Whee: v, v,, b n s t em t L φ θ ω m ω m bωm tem t (6) L t θ (7) ω m t : -xs n -xs stto votges; : -xs n -xs stto cuents; : -xs n -xs nuctnces; : Coecent o cton; : Roto net; : Numbe o oe s; : Resstnce o the stto wnngs; : Eectomgnetc toue; : Lo toue; : Euvent ux by the mgnets; : ngu oston o the moto sht; : Roto see. Fo PMSM wth non-sent oes (ose, 997; Py n Kshnn, 988): (8) Recng euton (8) n euton (5), t s ove tht the eectomgnetc toue eens ony on the -xs stto cuent. t. φ (9) em On the othe hn, the -xs ux s etemne by () (ose, 997; Py n Kshnn, 988): ϕ φ () When PMSM woks beow ts nomn see (constnt toue oeton moe), s set to zeo to euce the owe consumton o the moto, ue to the mgnets oves enough ux. On the othe hn, when PMSM hs to oete bove ts nomn see (constnt owe oeton moe), s negtve n oe to euce the -xs ux ϕ (ose, 997). hs technue, ce ux wekenng, emns the conto o the -xs stto cuent. Cose-Loo Moes o PMSM. Sce-Stte Moe o ngu Poston o PMSM he stte-stte eesentton o the system tht eesents the ynmc behvo o the ngu oston o PMSM s euce om eutons (4) to (9): ISN:

3 ns o XIX Congesso seo e utomátc, C. x & x u w () w y [ ]x () ( ) mx (3) 3 3 ( ) mn (4) 3 3 Whee: x, [ θ ω ] m (3) u v (4) w t (5) L s φ n ( φ ) b [ ] [ ] w (6) (7) (8) In (), w n w e the etubton n the etubton nut vecto, esectvey. Euton (6) shows tht the stte mtx o the ngu oston system hs non-nety 3 : ( φ ) 3 n (9). S Moeng o ngu Poston Fuzzy kg-sugeno moeng (S) s technue to moe non-ne systems s uzzy vege o oc ne systems (kg n Sugeno, 985), s shown n eutons () n (): x& x u y C x () () Whee s the numbe o oc moes whe α s the uzzy membesh uncton o the non-ne nt to the -th sce-stte oc moe (,, ), eesente by the set o mtces (,, C ). α,α > () In ths e, the uzzy membesh unctons e obtne usng the methooogy esente n (nguch et., ), whch uses the mxmum n mnmum vues o evey non-nety o the sce-stte moe. In ths cse, the ony non-nety o the ngu oston system s ouce by 3, n ne moes e neee (nguch et., ). he vbes 3 n 3 e ene s: he unctons σ 3 n σ 3 e chosen to exess 3 s ne uncton o 3 n 3 : σ σ (5) σ σ (6) eng σ n σ. Fom eutons (5) n (6): σ σ s thee s ony one non-nety n : α ( t) σ3( t) ( t) σ ( t) 3 (7) (8) α (9) he S oc moes (,, C ) n (,, C ) e ene s:,, s φ s φ b b 3 3 [ ] w w C [ ] [ ] (3) (3) (3) (33) C (34) he etemnnt o n o eutons (3) n (3) s zeo, nctng tht thee s n mct ntegto n the oc moes (Cheng, 999)..3 ugmente -xs Cuent Moe he mgnetc ux eens on the -xs stto cuent (ose, 997), whose ynmc behvo s escbe n euton (3). hs euton shows tht thee s not n mct ntegto n the -xs cuent oo. ISN:

4 ns o XIX Congesso seo e utomátc, C. hs esech nsets n ntegto n n ntegtve gn k I to ths system. hs technue s use n ne systems o tckng obems (Ogt, ). Fgue shows the oose ugmente moe o the -xs stto cuent. ccong to Fgue, the nut u s: u v k k ε (35) x I Conseng tht the eeence * s constnt ung the eue tme o the stbzton o the outut, t cn be ove tht (Ogt, ): ( K ) E w w E & (4) Whee: K [ ] [ k k ] x s I (4) Fgue. ugmente -xs cuent moe he gn K cn be seecte to eect etubtons n mkes E eu to zeo n stey stte (Ogt, ). Recng eutons (8) n (35) n euton (3) n te some gebc mnutons: k t t x k ε I s s s kx On the othe hn: nωm t ki ε k x nω m ki nω ε m (36) 3 LMI Contoes 3. LMI Desgn o Feebck Contoes he eebck gns o the oston n -xs cuent contoes e esgne though ne mtx neutes (LMI) n oe to guntee obust conto, whch e bse on Lyunov stbty theoy: system wth stte mtx s symtotcy stbe exsts ostve ente mtx Q (Q > ) tht (ne et., ): Q Q < (4) ε t * ε * [ ] Fom eutons (36) n (37): (37) oc contoe K s ccute o ech oc moe, to etemne the oc conto w u (t). s esut, the conto w u(t) s (kg n Sugeno, 985; exe n Zk, 999; nguch et., ): t ε s k n ω m x ki ε * (38) Whee w ω m s the etubton o the ugmente moe, n w [n ] s the ugmente etubton nut vecto. Denng the eo vecto E s the eence between the sttes n the tme t n when they e n stey stte: E ε ( stey stte) ε( stey stte) (39) u u F x (43) Recng euton (43) n euton () n te some gebc mnutons: x& x& G α eng: ( K ) x( t) < G x α G G x (44) K,,, K, (45) < ISN:

5 ns o XIX Congesso seo e utomátc, C. ccong to Lyunov theoy e to euton (44), the cose-oo system s symtotcy stbe thee s ostve ente mtx Q > tht: G G G Q QG G G Q Q < ;,, K,,, K, < ; < (46) (47) he ugmente -xs stto cuent nt ony hs one oc moe. the LMI sttements n ths e e sove enng the vbe Y K Q (,, ) n mkng the oowng tnsomton: G Q Q K Q Q Y (48) he vbes Y n Q e ccute sovng the LMI tems. he eebck gns K cn be obtne mkng K Y Q Cose oo Contoes usng H Nom ccong to S moeng, ech oc moe eects etubtons, then, the gob non-ne nt w be obust gnst ths etubton. hs e uses H nom to estbsh LMI contons n oe to euce the eect o etubtons n the system outut. Conseng the oowng oc system: w w [ n ] (5) 3.3 Poe Pcement o the Loc Cose Loos usng D-Stbty he oston o the cose-oo oes etemnes the ynmc esonse o the system. Conseng the egon n the Lcn e sem-ne S(γ,, θ) s the set o comex numbes λ x y whch stsy the oowng contons (O et., ): S ( γ,, θ) x < γ < x y < x tn( θ) < y (53) Fgue shows the egon S(γ,, θ). I the oes o the cose-oo system beong to ths egon, the system hs mnmum ecy te γ, mnmum umng coecent ξ cos(θ) n mnmum ume euency ω.sn(θ). x& x u w (49) y C x w Whee w(t) s nose n w s ts nut vecto. he eect o the nose on the outut y shou be mnm. hs obectve cn be cheve by seectng the eebck gn K tht mnmzes the H nom between the nose n the outut, ccong to the oowng otmzton obem (ne et., ): mn < ( Z ) Q G Q QG w C Q Q > ; Z > Z w QC I Fo the oc moes o ngu oston: w w w [ ] (5) (5) Whe o the ugmente -xs stto cuent moe n ccong to euton (37): Fgue. Regon S(γ,, θ). he cose oo mtx s ce D-stbe ts egenvues beongs to D(γ,, θ) (O et., ). In conseuence, the egon D(γ,, θ) set contons o the oveshoot, sng tme n stbzng tme. he oowng eutons ene the egon D(γ,, θ) though LMIs: h Q Q γq < Q Q Q Q < ( Q Q ) hs ( Q Q ) ( Q Q) hc ( Q Q ) s sn Whee: ( θ) ; h cos( θ) c h c < hs (54) (55) (56) G,, K, (57), 5( G G ); <,, K, ISN:

6 ns o XIX Congesso seo e utomátc, C. he D-stbty contons o the ngu oston n -xs cuent contoes cn be set neenenty ue to they hve the own SISO moes. tony, the ton o the new stte o the - xs cuent system gves moe exbty to set ts ynmcs. 4 Resuts Poston ().5 ngu oston eeence Smutons tests wee one usng ML/SIMULINK, ccong to the metes n be I. he two-eve thee-hse nvete tht gves enegy to the PMSM woks wth swtchng euency o 5 khz n hs DC votge souce o V. he o toue e to the moto sht s t L.5.ω m me (s) Fgue 3. Poston eeence. be I. Smuton Pmete Pmete Vue Resstnce,78 Ω -xs nuctnce 6 mh -xs nuctnce 6 mh Euvent ux o,48 Wb PMSM mgnet Poe s 3 Roto net 4,98-4 kg.m Coecent o 5, -5 N.m.s cton D-stbty γ o oston contoes θ π/ D-stbty γ 5 o cuent 6 contoe θ π/3 he oc moes wee constucte conseng: n he obtne coseoo contoes e: Poston: K [ 3,7 x 4, ] K [ 3,7 x 4, ] D-xs stto cuent: K [ ] Fgue 3 shows the oston eeences, whe the -xs cuent eeence s set to zeo. wo tests wee one, conseng nomn nt n eucton o % n the eectc metes o PMSM. Resuts om Fgues 4 n Fgue 5 show tht the esgne oston n -xs cuent contoes hve goo ccucy, eect etubtons n e obust gnst vtons o the metes o the moto. Cuent () Poston () Cuent () Poston () D xs stto cuent me (s) x 4 Poston eo me (s) Fgue 4. Smuton esuts o nomn nt. D xs stto cuent me (s) x 4 Poston eo me (s) Fgue 5. Smuton esuts conseng eucton o % n the eectc metes o the PMSM. Concusons he esgn o cose-oo contoes o oston conto o PMSM though LMI n ugmente -xs stto cuent moe n S ngu oston moe cn eect etubtons n hs goo ccucy, even conseng vtons n the metes o the moto. he oe cement cn be one wth hgh exbty ISN:

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