DISCRETE SLIDING MODE CONTROL OF PERMANENT MAGNET STEPPER MOTOR USING FLATNESS PROPERTY.

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1 DSCRETE SLDNG MODE CONTROL OF ERMANENT MAGNET STEER MOTOR USNG FLATNESS ROERTY. V. Thaar, B. Bpadha nerdscplnar rramme n Ssems Cnrl Enneern, T Bmba, Mumba 6, nda Emal: v@ee.b.ac.n ; bjnan@ee.b.ac.n hne: , Fax: 9538 Kewrds: Sepper mr, Dscree sldn mde, Reachn law apprach, Dfferenall fla ssem, Lnearzn upus Absrac The paper presens dscree me sldn mde cnrl (DSMC) fr permanen mane (M) sepper mr whch s nwn be dfferenall fla ssem. Dscrezan f cnnuus me mdel f he sepper mr has been carred u lnearzn upus are seleced. Dscree sae varables cnrl npus are baned n erms f lnearzn upus her hher rder dfferences. Lnearzed equans fr he ssem are hus baned auxlar cnrl s desned usn reachn law apprach. Then he acual cnrl s baned n erms f fla upu sae varables. s shwn ha ssem saes reach zer frm ven nal cndns. nrducn Sldn Mde Cnrl (SMC) s a rbus cnrl scheme based n he cncep f chann he cnrller s srucure, wh reference he mn f he saes f he ssem aln he predefned manfld n rder ban desred respnse. n rder ban sldn mn, hh speed, dscnnuus swchn cnrller s used chane he srucure f he ssem. n sldn mde, ssem respnse s verned b he sldn surface[. n Dscree Sldn Mde, cnrl acn can nl be acvaed a sampln nsans hence nl Quas Sldn Mde s pssble. Ga[ prpsed reachn law apprach whch ensures ha ssem rajecr wll h he swchn manfld underes zza mn abu he manfld remans whn a Quas Sldn Mde B(QSMB). The sldn mde cnrl can be appled fr he cnrl f varus ssems such as pwer cnverers, mrs drves rbs [5,. Sepper mrs are elecrmanec ncremenal mn ssems whch cnvers dal npu n anal anular mn f he rr n seps. The have ceran snfcan advanaes. The can be easl nerfaced wh sld sae elecrnc devces. The are nrmall peraed n pen lp. The are pen lp sable bu respnse s pr a hh seppn rae whch ma even lead scllar respnse lss f snchrnsm. Hence clse lp cnrl f sepper mr becmes essenal fr ceran applcans[6,. M. Zrb J. Chassn[ prpsed mehd fr psn cnrl f M sepper mr usn exac lnearzan. Marc Bdsn e al[suesed a mdel based cnrl law usn exac lnearzan mehd fr cnrl f M sepper mr n whch nnlnear bserver was used fr speed esman. M. Zrb e al[3 develped a sldn mde cnrl scheme fr M sepper mr n whch he mr s shwn psses a dfferenal flaness prper. n [8 he sldn mde cnrl fr fla ssems s frs presened. Fla ssems, as nrduced b flss hs cwrers[3, are dnamc ssems whch are lnearzable cnrllable lnear ssem b means f endenus feedbac [3. The fla ssems have fne se f dfferenall ndependen upus nwn as lnearzn upus. The ssem s dfferenall fla f all f s varables can be expressed as dfferenal funcn f lnearzn upus. s shwn n [3 ha all he sae varables cnrl npus f he sepper mr mdel can be expressed n erms f he upu her hher rder dervaves wndn curren anular psn are cnsdered as cnrl lnearzn upus. n hs paper we presen dscree sldn mde cnrl fr M sepper mr whch uses flaness prper f he mr. Ths paper s ranzed as fllws. Secn presens he nnlnear mahemacal mdel f he sepper mr smplfed nnlnear mdel baned usn drecquadraure ransfrman(dq) as ven n [, 3. Dscree mdel f he sepper mr wh lnearzn upu s nrduced n Secn 3. Represenan f dscree sae varables cnrl npus n erms f he lnearzn upus lnear mdel f he ssem are presened n secn. Desn f dscree sldn mde cnrl usn reachn law apprach has been ncluded n Secn 5. Secn 6 cnans he smulan resuls fllwed b he cncludn remars. Mahemacal mdel f he M Sepper Mr The mr mdel as ven n[,[3 s as fllws "! %$

2 C ( ) ( (! )* )+, /. /!$! ) /.3! +5%$ 6 8 Where ) 9 ) are he phase vlaes,! are he phase currens, s he anular velc f he rr, s he anular psn f he mr shaf, s he ressance f he phase wndn, s he nducance 6 f he phase wndn, s he vscus frcn ceffcen, s he nera f rr lad, s he number f rr eeh : s he mr rque cnsan. The mdel nelecs he small manec cupln beween he phase wndns, small chane n nducance as funcn f rr psn, he deen rque varan n nducance due manec sauran[. The ssem represened b Equan () ma be ransfrmed n apprprae nnlnear frm nwn as drec quadraure frm. The ransfrman s such ha he nnlnear ma be cancelled b sae feedbac[. The ransfrman ransfrmed ssem are ven as, >?! C >?RQ /.3EDF! G 3H"! JK G 3HL! M.! MK NO N. LS G 3H"! G 3H"! JG 3H"! TQ ),Q > 9? s he drec curren,? > s he quadraure curren, s he drec vlae s he quadraure vlae. s shwn n [ ha he applcan f DQ ransfrman Equan () resuls n he fllwn ssem f equans. %> >8U %>VWH?" (??XU?*WH %>Y 8 Z? X[ 6! 8 () 3 Dscree mdel f he M sepper mr Dscree mdel f he sepper mr s baned b dscrezn he ssem represened b Equan () wh sampln me dscree sae space represenan f he ssem s ven as )! () C () (3) fllws. Zc Zc Z`aNb l Z`aNb l Z`aNb n Z`aNb l %> j _ h Zc l j Zc _ h h NY, Zcde(f c j c l m h NY, /h Zc+,(f c j /h c, Zc h jda9ny(nb c jd n Zc >j h %?c j Zc Zc nzc$qp 6 n?j (5) nc 6 f!jj Nw s pssble defne lnearzn upus cnrlled upu fr he ssem asr Zc r( Zc rs r( Zc n Zc" Nw all he ssem sae varables cnrl npus can be represened n erms f her hher rder dfferences. Ssem represenan n erms f lnearzn upus The sae varables are represened n erms f lnearzn upus her hher rder dfferences as fllws. c h c c n c r r Zc r Z5 + 5aNb r r NY,nu/ Z[vNFw j9 r r Z5vNF+ j r( Zc" The cnrl npus can als be expressed n erms f lnearzn upus frm he Equan (5) as fllws. Zc Zc Z[vNFwx9NY f Zc Zc Z[vNFwx9NY _ l ` f Zc Zcde c Gh c c/ (6) ()

3 r n Q Q ƒ Subsun he expressns frm Equan (6) n Equan (), cnrl npus can be represened n erms f lnearzn upus her hher rder dfferences as fllws. j Zc r r Z`aNb/[ N Y c/ fg r r 5 w [anb Q r( r( NY, n / 5vNF+ Zc r r Q{z 5vNFw j Q r r r ` f c 5aNb+ c r r U Z[vNFw c Q r r Z` * r( 5e(" r( 9NY n [e(w Z[vNF9%$ r( r( (9) 9NY [e(w Z[vNF r r 9NY~}/ Z5vNF+ j9 Q Frm he expressns ven b he Equan (9) fr he lnearzed equans f fr he ssem are ven b r Z[W wx r Z[ W are auxlar npus. r Z`aNb* r Z`aNb* r r c c (8) () Smplfn he equans baned abve n erms f he ssem saes, we e he fllwn lnear sae equans. 5vNF h h 5vNF 5vNF ns5vnf Zcde( Zc h h 9N(nb /l c* l c l Zcd NY,nF Zc Zcd nc/ The abve ssem can be expressed as Z[vNF Zc V jdwƒ c/ c Zc () () (3) wh N Ne n m "ˆ K NY(n LŠ Œ ( A A "ˆ LŠ C NG (cž( cn/$f 5 Dscree Sldn Mde Cnrl Desn Desn f sldn mde cnrl nvlves w sep prcedure. frsl a suable swchn surface s desned hen a sldn mde cnrl s desned ban sldn mn aln he desned swchn manfld. Dfferen mehds desn swchn hperplane are dscussed n [, 9. n hs desn, mehd prpsed n [ s used desn swchn planes. 5. Reachn Law Apprach The reachn law fr he dscree sldn mde cnrl as prpsed n[ s he [anbw, Zc 8, sampln perd, A A C / szc9 N~U { () The nequal fr mus hld uaranee he mn f ssem rajecr wards he swchn plane ha wll crss he swchn plane n fne me. The presence f snum funcn ensures he sldn mn abu he swchn plane n zza paern, resrcn he rajecr n a specfed b nwn as quas sldn mde b QSMB. 5. Swchn Hper lane Desn Swchn plane equan s ven as j / Zc j j (5) Fr he desn f he swchn hperplanes, he ssem ven n Equan() s ransfrmed n suable nrmal frm b reversn he rder f he ssem equans. The ssem hus aes he fllwn frm as 5aNb Zc ƒ c (6)

4 ƒ n n wh V Le us assume, " 9 N ( NY n T "ˆ Nen š LŠ A A "ˆ C C Q œ N pgž ƒ Ÿ Kƒ Thus, he ssem ven n Equan(6) can be wren as Z5vNF Z5vNF " Zcd Zc j cdƒ The swchn planes fr he ssem (6) becmes wh S, r Zc c $ j" L 9 Zc" p () (8) Q n Q (9) () frm he Equan (6) Equan (8) he fllwn equan s baned. Then, 5aNb c s baned b assnn he eenvalues f d n a desred lcan. Thus, can be easl baned. () 5.3 Cnrl Law Desn The auxlar cnrl s baned fr he ssem represened b Equan () usn reachn law apprach ven n Equan (). Smplfed represenan f he cnrl s ven n [ whch has he fllwn frm j [ ƒ [ ƒ S he acual cnrl s c c 6 Smulan c Z Zc9 8 d $ "m Zc9 h e h c (f c c Zc az n cd Gh h _ l c* c f Zc Zc" wh he same daa same as used n [3wh Nb c NG ( }ª s c NG (} («Ë / 6, }h NF z NG n ( F j cnb (E«FD ž s (b±b «D j cnrl law are as j N c ~ () (3) The smulan sud s carred u fr he M sepper mr, The parameers fr he The bjecve s brn he ssem saes upus frm an nal cndns zer n fne me. Fure shws respnses f he phase )9² currens ) anular velc! anular psn. l f vlaes, swchn surfaces are shwn n Fure. Cnclusn The dscree mulvarable sldn mde cnrl fr M sepper s desned usn he prper f dfferenal flaness. Selecn he wndn curren anular psn f he shaf as an upu, all he sae varables npus f he dscree ssem can be expressed n erms f he hese upus,.e. lnearzn upus her hher rder dfferences. The lnear represenan f he ssem s baned whle represenn he cnrl npus n erms f lnearzn upus. The auxlar cnrl s baned usn reachn law apprach fr he lnear mdel baned usn dfferenal flaness prper f he sepper mr mdel. Fnall he acual cnrl s baned fr he sepper mr. The smulan resuls shws he effecveness f he dscree sldn mde cnrl fr M sepper mr whch brns he ssem saes zer frm nal values.

5 References a,amps ω rad/sec Curren n wndn A Anular Velc 6 b,amps θ radans.5..5 Curren n wndn B Anular psn. 6 8 [ M Bdsn, J. N. Chassn, R. T. Nvna R. B. Rews, Hh perfrmance nnlnear feedbac cnrl f a permanen mane sepper mr, EEE Transacns n Cnrl Ssem Technl, vl, n, pp. 53, (March 993) [ C. M. Drln A. S. Znber, Tw apprches hperplane desn n mulvarable srucure cnrl ssem, nernanal Jurnal f Cnrl, vl., n., pp. 658,(986) [3 M. Fless, J. Levne,. Marn. Ruchn, Flaness defec f nnlnear ssems:nrducr her example, nernanal Jurnal f Cnrl, vl6, n 6, pp. 336, (995) [ W. Ga, Y. Wan A. Hmafa, Dscree me varable srucure cnrl ssems EEE Transacn n ndusral Elecrncs, vl., n., pp., (Aprl 995). [5 J. H. Hun, W. Ga J. C. Hun, Varable srucure cnrl: A surve, EEE Transacn n ndusral Elecrncs, vl., pp., (Februar 993). Fure : Respnse f he ssem saes [6 T. Kenj A. Suawara, The seppn mrs her mcrprcessr cnrls,secnd edn, Oxfrd scence publcan,(99) Vlae n wndn A 3 Vlae n wndn B [ C. M. Saaj, B. Bpadha H. Unbehauen, A new alrhm fr dscreeme sldn mde cnrl usn fas upu sampln feedbac, EEE Transacn n ndusral Elecrncs, vl. 9, n.3, pp. 5853, (June ) swchn plane va,vls swchn plane vb,vls Fure : l f wndn vlaes swchn surfaces [8 H. SraRamrez, On he sldn mde cnrl f mulvarable nnlnear ssem, nernanal Jurnal f Cnrl, vl. 6, n., pp 565,(996) [9 S. K. Spuren Hperplane desn echnques fr dscreeme varable srucure cnrl ssem nernanal Jurnal f Cnrl, vl. 55, n, pp. 5 56,(99) [ V.. Un, Varable srucure ssems wh sldn mdes EEE Transacn n Aumac Cnrl, vl. AC, pp., (Aprl 9) [ V.. Un, Sldn mde cnrl f an elecrmechancal ssems, Francs Talr ublcans, (999). [ M. Zrb J. Chassn, sn cnrl f M sepper mr b exac lnearzan EEE Transacns n Aumac Cnrl, vl. 36, pp. 6 65, n.5, (Ma 99) [3 M. Zrb, H. SraRamrez A. Na, Sldn mde cnrl f M sepper mr frm he perspecve f dfferenall fla ssem, nved paper, F93,The Eurpean Cnrl Cnference, German,(999)

R th is the Thevenin equivalent at the capacitor terminals.

R th is the Thevenin equivalent at the capacitor terminals. Chaper 7, Slun. Applyng KV Fg. 7.. d 0 C - Takng he derae f each erm, d 0 C d d d r C Inegrang, () ln I 0 - () I 0 e - C C () () r - I 0 e - () V 0 e C C Chaper 7, Slun. h C where h s he Theenn equalen