Asymptotic Stabilization of an Active Magnetic Bearing System using LMI-based Sliding Mode Control

Transcription

1 Wold cde o ciece Eieei d ecolo 37 8 soic iliio o cive Meic ei se si LM-sed lidi Mode Cool l Rsid Hsi Mod No d d l Hli Mod. Yi sc is e siliio o cive Meic ei M sse wi vi oo seed si lidi Mode Cool MC eciqe is cosideed. e oscoic eec ieied i e sse is ooiol o oo seed i wic is oliei eec cses i sse isili s e oo seed iceses. lso soio o e M dic odel io ew clss o cei sse sows is oscoic eec lies i e isced o e sse i. Moeove e ce i ee is llowed o e vied i kow od s cei i e i i. MC desi eod is oosed i wic e sicie codiio ees e lol eoeil sili o e eced-ode sse is eeseed i Lie Mi eqli LM. e ew cei-ee cool lw is eslised sc e sse ses e dive o ec e swici sce d s o i eee. e eoce o e coolle lied o e M odel is deosed o silio woks de vios sse codiios. ewods cive Meic ei M lidi Mode Cool MC Lie Mi eqli LM isced cei.. NRODUCON LDNG Mode Cool MC s eceived e eio i ece es ecse o is osess is ceiies ese i sse 3 d. MC is olie cool eciqe is licle o wide e o dic sse icldi e lie olie lii/li-o discee-ie d le scle sses. ee e oces ve ee eoed d cosideed i e desi ocess o e slidi-ode cool lw sc e sse is os o eve isesiive o eic ceiies d disce. e cicl licio o MC e coolle s lso ee sccessll ded i os d lied i eos el-wold licios sc s oo ilo 5 cive ssesio sse 6 eic ssesio sse d eic. R. Hsi is wi Uivesii ekoloi Mlsi UM 83 kdi ooe Mlsi oe: ; : ; e-il: sid@ ke... M. N. d is wi Uivesii ekoloi Mlsi UM 83 kdi ooe Mlsi. He is e Hed o De. Mecoics d Rooics Fcl o Elec. E. UM e-il: o@ke.... H. M. Yi is wi e Uivesii ekoloi Mlsi UM 83 kdi ooe Mlsi.. He is e De De cdeics Fcl o Elec. E. UM e-il: li@ieee.o. eis 78. e desi o MC coolle ivolves wo ccil ses wic e cool eeed o s e eci se d e slidi se. is e e desi o o e slidi sce d cool lw sed o MC eo d is licio o ive-deee-o-eedo DOF M sse eoed i 8 e cosideed. e M sse cois e oscoic eec i wic is oliei cses e sse o e sle d e ide is ooiol o e oi seed o e oo. lso i is wok e lieied ce-oce co o e M sse is llowed o e vied % o is oil vle. is eled codiio is ioced o ccoode soe viios i e e o e vle o is ee is oiill oied o eeiel se-. e soi e sse io el o ew clss o dic sse is oed i wic i is sow e oscoic eec elos o isced cei wile e cei vle o ceoce co is ced. sed o is ew sse odel sicie codiio is eslised sc e eced-ode sse wi e ese o ese ceiies we esiced o e swici sce is eed o cieve lol sili. is slidi codiio is oled i e o Lie Mi eqli LM i wic vios owel seideiie oi eods e ville o id e desied oil solio wic coseqel ields ll eeied slidi sce. seqel ew cool lw is desied sc e sse ses e dive o e slidi sce d o ei o i eee. ode o ke e coolle o e oe cicl e od le eciqe is lso ded i e olie swici e i wic e i cei is eliied wile e osess o e sse is sill iied. e olie o is e is s ollows: ecio e odel o e M sse sed o 8 is illsed d eeseed i is deeiisic o wic seves s ool o e coolle desi. ecio coves e deil desi o MC cool loi weei o e eeiio o slidi sce i e o LMs d e develoe o olie cool lw e sow. e sili o e sse de e desied coolle is lso sow. e i ecio V e eoces o e M sse de e desied coolle e illsed o silio woks de vios sse codiios. Fill e coclsio i ecio V sies e coiio o e wok. 8

2 Wold cde o ciece Eieei d ecolo MODELNG OF N CVE MGNEC ERNG YEM ode o sesie e oosed slidi sce wi e coolle veicl s M sse odel o e licio o o olecl sse is e-deived sed o e wok doe i 8.. Meicl Model M - M. e oio o oo od e -is is slied eel divi ecis d cosideed s ie-vi ee. Fi. illses e ee-od di o e oo wic sows e ol oces oced e M d M o e sse. sed o e icile o li dics 9 e eqios o oio o e oo-eic ei sse is s ollows: ω α L L α ω L L M - M - G M - M dl e oscoic eec is eeseed e e ω α d ω wee i c e oiced is will cse e coli ewee e es o oios ooiol o e seed. e cool oces oced e M e ive e ollowi eqios: M d Fi. Veicl cive Meic ei se e oscoic eec cses e coli ewee wo es o oios ic d w. Fi. illses e ive DOF veicl eic ei i wic e veicl is is is ssed o e decoled o e sse d ece ω α l dl d M d G coolled seel. e o o e oo o e sse i Fi. is coolled civel e eic ei leled s M i wic e coil ces e e is. e oo o e oo oweve is levied o e cee o e sse si wo ses o ee es leled s dl l Fi. M ee-od di M d d L i L α wee d l e e e oces l oced e M o ec - d -is esecivel e se e oce o oo M s well. is is ossile vi e M coil wod o oce dieeil ce ode. Fo e M e e oces oced e ive e ollowi eqios: C C L L C C L α Lα Eqios d 3 c e ieed o oce e M odel i e ollowi o: wee X ω X U X α α e e ses o e sse ω R is e sse i R is e i i U e i ces. e oeo elees o e ices e sow i e edi. is sd e ce-o-oce ee i is llowed o v o % o is oil vle. is codiio is ioced sice e ee vle is oied o lieied elio weei soe icccies i occ. Wi is ewl ioced codiio e coolle desied i e ollowi secio is oled o e os owds o ol e oscoic eec lso is eic cei. ice e ee i is i e i i e ew M odel c e eeseed s ollows: X ω X U 5 i i 3 9

3 Wold cde o ciece Eieei d ecolo 37 8 wee ω d i eese e ceiies e o oscoic eec d ee viio esecivel.. M Mode i Deeiisic Fo ode o sesie e coolle e M odel is eed s cei sse i wic deeiisic oc o clssi e sse is sed sed o. e odel sow i 5 is edil i e cei clss o sse i wic ices d e e oil cos ices o e sse. e decoosiio io is deeiisic o is ossile e o e c e i d ii vles o e oiol oo seed d e od o e ee viio e kow. Fo is M odel seciicll e elees o e ω c e clcled sed o ese ville ods. e oiol seed is ive s ollows: ω. 6 LE RMEER FOR M YEM 8 ol ee Vle Ui Mss o Roo.595 k Moe o ei.6-3 k. o oiol is Moe o ei k. o dil is L Disce o e.8 M o G L Disce o lowe.83 M o G i Lieied N/ oce/ce co Lieied.8 5 N/ oce/disl. co iess coeicie. 5 N/ o M C Di 8 k/s coeicie o M ic ilce.6-3 l Disce o. lce ss o G ω Roo oiol 7 d/sec seed e si ese ods d e oe sse ee vles s sow i le ec elee o e sse i c e clcled d seciied i e ollowi o: ij ij ω ij 7 o i 8 d j 8 wee ij ω e e elee o ω i. e e d lowe s idice e i d ii vles o e elees. e elee o i d ω c e clcled sed o ese ods si e ollowi eqio: ij ij i j i j ij i j 8 o i- owj- col elees o d ω. Fo e cei e o e eic viio i s i is ssed i is wok e vle is vied % o is oil vle d is vle eeses e ies od o e cei. s is is deqe o e coolle desi.. LDNG MODE CONROL DEGN Coside clss o cei ss 9 wee R is e sse ses R is e cool i d is ie-vi scl cio. d is e sse d i ices esecivel d is o ll k. d eese e ceiies i e sse i d i ices. o colee e desciio o e cei dicl sse e ollowi ssios e ioced d ssed o e vlid. Fo eisece oses d e coios o ei es. ee eis kow osiive scls d d cio sc <. 3 e i is coollle. Deie lie swici sce s wee R is o ll ow k. eei o evios esls e swici sce ee i sold e seleced sc E e i is osil E e eced - ode sse o e sse 9 esiced o e swici sce is loll eoeill sle o ll llowle ceiies sisi ssio. 5

4 . e Desi o e lidi ce Le R e ll col k i sc d. e i c e eslised o eqivlel. Fo is i is ovios e osil soio i c e deied s ollows: sii io sse 9 ives } { is ew sse eeseio s ed e ceiies io e ced d isced cooes i wic i is ovios d elo o e isced d ced cooes esecivel. is oweve will e oe ovios s e sili lsis o e sse o e swici sce is eoed e. eoe. Coside e sse 9 d sose ssio -3 old. e ee eis i sisi E d E i ee is i > d osiive scl sc < wee is ive i ssio. Moeove e i c e oosed o e s ollows: 5 oo Fo 5 d i is ovios d eeoe codiio E is sisied. d o i is lso e 6 si e osil soio i wic ew se vile c e ioced sc sse 9 c e soed io e ollowi el o: We e sse 7 is esiced o e swici sce e o i is kow M - i.e.. Wi i is oied. e e eced-ode sse 7 o e sce is 8 wee i is el cle e cei ese i 8 ecs e sili o e sse. e ke V s Lov cdide cio e e deivive o V lo 8 is deived o e V Fo M d i c e sow. oee wi d e ls e o 9 c e eced s ollows: wee is osiive scl. s e deivive o e Lov cdide cio 9 is V Σ wee Σ } { } { ε Wold cde o ciece Eieei d ecolo

5 si 6 is is e eced o. Σ s oee wi d > ilies e eced-ode sse 8 cieves lol eoeil sili we slidi o e sce. is colees e oo o eoe. Fidi e solio o e Ricci ieqli sll ives o-iqe solio sc e codiios se e e. is oweve c e ioved i wic e solio o ol lill lso e ce o e solio is iiied. is is ove o e dveos icll o M sse sice e cool eo is lso eced. is ole c e eoled i e o LM d sed i e ollowi le. Le. e eisece solio is eqivle o solvi e ollowi LM se wic is: < V C > wee C V. oo Fo d e ollowi ieqli is oced <. < si c colee e ove LM se is oied d is colee e oo.. esis o Cool Lw o id e cool lw sc e jecoies o sse 9 e dive o e desied slidi sce d o ei o e sce o e sseqe ie e ollowi ecili codiio is sed 3 wee is osiive scl d is e slidi sce deied. Wi is eci codiio e ollowi eoe c e ioced. eoe. Coside sse 9 wi e ssio d. e LM se d s solio d e slidi sce is solvle ive 5 e ecili codiio 3 is sisied eloi e cool lw ive elow: } { wee is ive 5 d is desied osiive scl. oo o oo is eoe e ecili codiio 3 o sse 9 wi cool lw d is sed. Fo 3 i c e sow } { Fo i is kow. Wi is ieqli e ecili codiio c e e siliied o e s e ecili codiio 3 is sisied d e oo o eoe is colee. is well kow e discoios cool e i ioces i cei eec wic is desile i dic sse e o is iiie swici eqec. is i eqec swici i c ecie i-eqec Wold cde o ciece Eieei d ecolo

6 Wold cde o ciece Eieei d ecolo 37 8 odelled dics i e sse d cse sse isili. o ovecoe is cei eec e od le eciqe is sed ised o e discoios e s coveed i 3. e ew cool lw di e od le eciqes is s elow: { } s 5 Π wee П is e ickess o e od le. e deil o e selecio o e П vle d is elio owds e deo o e osess o e sse is discssed i deil i 3 d ieioll oied i is wok. e dio o e eod is el o oce cool lw is cei-ee wic is oe cicl o cl licio. V. MULON ON M YEM ND DCUON e silio wok is eoed si ML d ilik. Fo solvi e LM ole d ised o si sdd LM oolo i Ml YLM/eDMi cove ole solve o sei-deiie ole is sed 5. YLM/eDMi is o e ewl develoed cove ole solve wic is ove o oce less cosevive solio d ie coveece e. is kes i ve cive esecill o desii e coolle wic i e eced cosevis is cicll eele. e soio o e M odel io e deeiisic o e ocee o desii e coolle d is licio o e sse is olied s ollows: e : Coe e o o e sse i cei ω:.3. Noe e viio o ee i is led ssed o e % vi o is oil vle. s i is elicil kow e o o e cio wic od e cei i e i is.. Wi ese o oded vles e cos i ssio e s ollows:.3.. e : olve LM ses d o wi o eeie e slidi sce i wee idices e seicl vles. Wi is e sce ee is clcled o e s ollows: e 3: elec ee d П o coolle 5. ee. is cose. sll osiive vle sc ecili is e is deqe o e desi. Fo e od le П i does ec e eoce o e cool i wic e ie e vle is e less cei will occ i e cool ce. Howeve e de-o is less osess is oiced sice ised o eci e sle slidi sce e ses jecoies ee d oscille i e ive od 3. Fo is wok wo П vles e cose wic e П 5 d П 3 o illse e eec o e sse. e iiil vles o e ses e se α α d -. Fi. 3 sows e jecoies o ses X d Y sele o eo we e oo seed ω is se e oil vle oo seed. lso e ce-o-oce ee i is se o is oil vle wi o viio o is vle. ll oe ses lso covee o eo ol X d Y vles e sow e o ei elivel le iiil vles. Fi. e viio o % o i vle is ioced io e M odel d i is ove e sse is os de e ew cool lw s e jecoies o X d Y i wic e dieece is dl oicele coed o Fi. 3. s e oo seed is icesed o is i vle e ide o e oscillios o X d Y e eii se is isiiicl ie s sow i Fi. 5. Wi e esls sow i ese ee ies i is ove de e cool lw 5 e sse is osl sle i wide e o oo seed s well s viio occs i e i vle. si e YLM/eDMi LM solve e vle o e i is s ollows: 53

7 Wold cde o ciece Eieei d ecolo X jecoies o X d Y idel swici cool e is sed ve i cei eec c e see s sow i Fi. 8. lo is cool lw ovides e ies osess o e sse s discssed elie i is ve eesive i e o cicli. X d Y Y jecoies o X d Y X d Y ie sec Fi. 3 X d Y jecoies wi ω k.5 - X Y jecoies o X d Y X d Y Y X ie sec Fi. 6 X d Y jecoies wi ω k. d od le П.5 wi od le X d Y ie sec Fi. X d Y jecoies wi ω k X Y jecoies o X d Y ie sec Fi. 5 X d Y jecoies wi ω k. We e od vle П is eced o 5 o 3 e sse will e oe sesiive owd e ceiies d is esl is sow i Fi. 6 i wic X d Y oscille ie lides d ke loe ie o sele o eo. e cool ce o o o ese od vles e sow i Fi. 7 wee i c e oiced e cool eo o П 3 is ie i ode o i e ses o eo. Fill we e ie sec wi od le ie sec Fi. 7 Ce wi П o d П oo Cool ce wi si cio ie sec Fi. 8 Ce wi discoios e si cio 5

8 Wold cde o ciece Eieei d ecolo 37 8 Rek Noe e o o e cool i is kow o e s ollows:. d e e d wic coi e sce ee coie siiicl o e e od o e cool eo. e le sows e vles o ese es oied o solvi LMs d leic Ricci eqio RE i wic i sows eeiio wi e LM solio oce desil lowe cool eo. LE COMRON EWEEN LM ND RE Cool e LM RE V. CONCLUON is wok ew cei-ee MC coolle is oosed o siliio o ive-dof M sse i wic slidi sce is desied sed solvi se o LMs. e oosed coolle is ove o e le o cieve soic sili wide e o oiol oo seed lo wi e ese o isced d ced sse cei. e eoce o e coolle is deosed o vios silio woks. ENDX e oeo elees o i ω wee i d j idice e i- d j- e o ec elee. L L C C L L L L L LC L C L L C C L L L L L LC LC i L i Heio L-CNR olose o ioci YLML/eDMi LM solve. REFERENCE. Y. H W.. Go d. C. H Vile ce cool: ve EEE s. sil. Elecoics vol. o eo d C. Edwds lidi Mode Cool: eo d licios. Lodo: lo d Fcis X. Li d R.. DeClo Ros lidi Mode Cool o Ucei ie Del ses.. Cool vol. 76 o X. Z. Do e Feedck lidi Mode Cool o Clss o ses wi Misced Uceiies d Disce i oc. 5 Ciese Cool Coeece Heiloji M. N. d. H.. Os d M. R.. Gi ooiol- el lidi Mode cki Coolle wi licio o Roo Milo i 7 Co. o Co. o. Ro d s. CRCV ioe Y. M. d. H.. Os Modeli d Cool o cive sesio se Usi ooiol el lidi Mode oc si o. Co. vol. 7 o H. Lee.E. llie G. o.. Decke d X. Z Eeiel d o lidi Mode Cool o eck Meic ei se d iicil He sesio EEE s. o Co. s. M vol. o iviol d. Noi lidi Mode Cool Wi ie- Vi Hele o M ses EEE/ME s. o Mecoics vol. 3 o F. Ms d. Yosioo se odeli d cool desi o oiol s eic ei sse EEE s. Meics vol. MG- o M H.. Os d.d. Roes wo Level Cool e o Roo Milos. o. o Cool vol. 6 o od L. El. Goi E. Feo d V. lkis Lie Mi eqli i ses d Cool eo ildeli: M vol G. Vwe d R. D. oil o Lie d ilie Mi.. ol o ocess Cool vol D. Q. Z d.. d Cei-ee d s-esose slidi ode coolle EEoc. Co. eo l. vol. 6 o Mc Loe YLM: oolo o Modeli d Oiiio i {ML} i oc. o e CCD Co. iw. Olie. ville: ://cool.ee.e.c/~joloe/li. 5. F. Usi edmi. ML oolo o oiiio ove seic coes Oiiio Meods d owe- ecil isse o eio oi Meods Olie. ville: ://sei.cse.c. CNOWLEDGMEN e icil so Miis o ciece ecolo d ovio Mlsi MO d Uivesii ekoloi Mlsi UM de e E-ciece d de Vo. No. 79 ed Resec Mee Ui RMC is ckowleded.. R. Hsi lso wold like o k Didie 55