VSC of a servo-actuated ATR90-type pantograph

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1 Proeedigs o he 44h IEEE Coeree o Deisio ad Corol ad he Europea Corol Coeree 5 Seville Spai Deeer -5 5 MoA7.5 VSC o a servo-auaed ATR9-pe paograph Beedeo Alloa Alessadro Pisao ua Pugi Elio Usai Meer IEEE Asra Oe o he proles i high-speed-rai rasporaio sses is relaed o he urre olleio quali ha a draaiall derease eause o osillaios o he paograph-aear sse. I he ree lieraure his prole has ee addressed eas o aive paographs. I order o redue he oss a oo requiree o he rai opaies is ha he orol sse ould e applied o he auall used paographs. I his paper e prese he preliiar resuls aou he possile ipleeaio o VSC ehiques o a servo-auaed seri paograph ha a e oaied odiig a passive high-speed paograph Asaldo ATR9 urrel used Ialia Railas. We osider he equivale ehaial haraerisis o he aear as ueraiies o opesae or ad he use o a rous oliear orol shee is proposed. Ree resuls aou he iluee o as auaors i seod-order slidig ode orol shees are eploied o avoid he desroig ee o a resoa auaor ad he sse perorae are veriied siulaio uder he hpohesis o oig he aual oa ore. The oa ore resuls o e ver lose o he desired value i various operaig odiios. O INTRODUCTION NE o he ai prole i high-speed rai rasporaio sses is relaed o he ieraio eee he rai paograph ad he overhead equipe he aear. Durig he rai ru he paograph deors he aear ad osillaor oios are idued. As he rai speed ireases hese osillaios a eoe larger ad larger ad he loss o oa eee he paograph head ad he olleor ire a our. Beause o is eooi relevae he ople ehavior o he aear/paograph sse gaied he aeio o a researhers ad egieers sie several deades []- [5]. These sudies poied ou ha saisaor peroraes a high speed a e ahieved eas o aive paographs ha require relaivel sall iveses. I pariular i has ee sho ha eer perorae a e ahieved i approiae odels o he aear/paograph sse are osidered o desig or ue he orollers [][6][7]. Mos auhors represeed he paograph as a liear sse [6][8][9]; urherore soe auhors assued ha he ople dais o he aear ould e ell approiaed a liear ehaial sse Mausrip reeived Marh 5 5. This or as suppored i par he Ialia Miiser or Eduaio Uiversi ad Researh (MIUR) uder he proje PRIN BS456. E. Usai ad A.Pisao are ih he Diparieo di Igegeria Eleria ed Eleroia Uiversià di Cagliari 9 Cagliari Ial (E. Usai orrespodig auhor phoe: ; a: ; e-ail: eusai@diee.uia.i pisao@diee.uia.i). B. Alloa ad. Pugi are ih he Diparieo di Eergeia S. Seo Uiversià di iree Ial. ih spae-varig luped paraeers ad used suh a approiaio i he oroller desig [7]. Auall here is a large uerai i alos all he proposed odels; hereore he use o rous orol ehiques as proposed [9]-[]. We osider he equivale odel o he aear as a uerai o opesae or eas o Variale Sruure Corol (VSC) sses ih Slidig Modes (SM) [] hih are reogied as a eeive ad siple orol approah o uerai orol sses []. We shoed ha he rous oupu-eeda approah i [] hih iplies he desig o a high-order oserver/oroller preses hard praial ipleeaio proles or he osidered paograph oaied odiig a passive high-speed paograph Asaldo ATR9. I a he auaor a e odeled as a seod order sse ih a sall dapig so ha he ipleeaio o he oupu-eeda orol shee i [] requires he esiaio o he seod derivaive o he availale oupu ih ver large values oh o isel ad o is ipshi osa ad ver large values o he orol gai. Thereore o ahieve aepale real slidig ehaviors a ver sall saplig ie is required. I order o avoid he eed o a diereiaor ree resuls aou he iluee o as auaors i seod-order slidig ode orol (-SMC) shees []-[6] are eploied i he desig o he oroller. I pariular a geeralied orulaio o he su-opial VSC (-VSC) [7] [8] ad a proper iler are used o aeuae he udesired peaig o he oa ore ad o shape he residual osillaios ha appear durig he approiae slidig regie ha is esalished. Siulaios shoed he eeiveess o he proposed oupu-eeda orol shee i diere operaig odiios. I. A MODE O THE PANTOGRAPH-CATENARY SYSTEM A ea odel o he overhead suspesio sse is ver diiul o deie eause i is a disriued sse ad is paraeers are ver hard o evaluae eal. or his reaso sipliied odels ih luped ie-varig paraeers have ee irodued [7]-[9]. The are ased o a he loal i spae haraeriaio o he equivale ehaial paraeers (i.e. ass siess ad dupig oeiie) o he oa ire. These paraeers prese a periodi ehavior varig alog eah spa ih saller variaios due o he droppers. This allos oe o represe he equivale paraeers a oiaio o haroi uios [8] /5/$. 5 IEEE 59

2 os os os os os os here is he spa legh ad he disae ro a oer. The oa eee he suppl ire ad he oa shoe is odeled ih a luped high siess ad a lo dapig ad a oliear sauraio osrai a e irodued o siulae he loss o oa. The seri paograph a e ell odeled eas o a liear luped-paraeers sse aeed he ipu ores (i.e. he osa ore applied o he loer rae he peuai posiioig sse o he paograph ad he orol ores aig o he loer or o he upper rae) ad he osrai ores (i.e. he oa ore) [8] [9]. ig.. A luped-paraeers odel o he paograph-aear sse. The dais o he sse i ig. a e represeed a se o liear dai sses possil oupled eas o he oa ore osrai hih assues ol posiive values he (i.e. he paograph shoe is i oa ih he suppl ire) > i i { } ; a (4) Durig operaio (i.e. he he oa ore is posiive) he dais o he paograph-aear sse a e represeed as a liear ie-varig sse derived oiig equaios -(4) ih he aual posiio o he rai alog he spa evaluaed iegraio o he rai speed v i.e. ( ) d v τ τ C B A (5) here [ ] T is he sae veor [ ] T is he orol veor A is a ie-varig ari B ad C are osa ari ad veors ih proper diesios ad is he oa ore i.e. A [ ] B C A sadard peuai auaor has ee used o appl he eeded pre-load ore o he paograph rae. This auaor is desiged o appl a osa ore ha opesaes or he graviaio ores ad ipose he desired osa oa ore i he ors ase odiio. I a e he assued ha os. The oa ore is regulaed eas o a ire auaor aig o he slidig os. The auaor dais a e represeed a liear seod order sse ih a sall dapig ad a large resoae pea a aou 5 rad/s; hereore u K a ω ω ζω (6) here u is he auaor oad. Taig io aou previous osideraios aou he ores aig o he paograph rae sse (5)-(6) a e redued io he olloig oral or [] ( ) ) ( u ψ γ ϕ (7) ih γk a ω / ad ψ represes he phase oupu veor hose elees are he oa ore (i.e. he sse oupu ) ad is irs ad seod ie derivaives ad R 5 is he ieral sae veor. The sae variales o he overall sse (5)- (6)[ ] T T are relaed ih hose o he oral or represeaio (7) [ ] T T he olloig aie relaioship hih is iverile i : ( ) CAB CB CB CAB A C A CB CA C P (8) 59

3 ih I P B [ B B ] I The las o ros o (8) poi ou ha he relaive degree o sse (5) depeds o he aual values o he dapig ees haraeried he values o paraeers ad. I a i resuls CB / CB CAB /( ) CAB / ( / ( )/ )/. I is lear ha a sall dapig ee orrespods o a sall orol gai ad hereore srog orol eors are usuall eeded. Disregardig he dapig ees ould ause he relaive degree o he sse o irease so leadig o a diere orol sruure hose eeiveess o he real sse should e areull aaled. Taig io aou ha he ipu-oupu relaioship i (7) is derived susiuio o (8) io he olloig C( A AA AA A ) C( A A ) B (9) C A A ω I B C A ξω I B K ω CB u a 6 ad ro he eplii or o Ψ i resuls ha he veor ield Ψ ad he uio ϕ i (7) are ie-varig sie he are aeed he aear equivale paraeers ad i eq.. These ie-varig paraeers ever vaish ad he a e assued o e suiiel sooh uio o ie. B eq. (7) i resuls ha he ieral dais o he approiae liear odel o he aear/paograph sse is BIBO sale. Thereore o he asis o he previous osideraios aou he ie-varig ers i a e laied ha all he uios ad variales i eq. (7) are ouded i.e. he olloig iequaliies are ulilled ( ) Φ( ) Ω( ) P ϕ Q sup ( ) here Φ is a o posiive deiie ove uio P ad Q are posiive o osas ad is he iiial ie isa. II. A ROBUST OUTPUT-EEDBACK CONTRO SYSTEM The required as or he orol sse is o solve a regulaio prole i.e. o aiai he aual oa ore o a desired osa value re. Cosider he oupu error e - re i is rivial o sho ha he error dais i he oral or is represeed he sae se o diereial equaios ha he orolled sse provided ha he susiuios e re (i) e (i) (i...) are perored. The regulaio prole o he oa ore eee he paograph head ad he oa ire is hereore redued o a oupu-eeda sailiaio prole or a uerai possil oliear sse. Reel soe rous oupu-eeda orol approahes ased o he use o variale sruure orollers ere preseed i he lieraure [9]-[]. The oo ai is o 6 seer o ero possil i a iie ie a proper liear oiaio o he oupu error ad is ie derivaives (he slidig variale). Oe he slidig variale is osraied o ero he error sse perors as a auooous sale liear sse o redued-order. The ai dierees are i ho he error derivaives are esiaed. Oh ad Khalil [9] solved he prole o esiaig he eeded ie derivaives o he error sigal a High-Gai Oserver (HGO) ad used a lassial VSC [] as he oroller. Dierel Barolii e al. [] ad eva [] suggesed he use o Higher-Order Slidig Modes (HOSM) o avoid oh soe diereiaio sep ad he desroig propagaio o he easuree oise durig uerial diereiaio. I a HOSM are haraeried seerig o ero he slidig variale ad is irs h- derivaives h eig he slidig order [] []. This iplies ha he slidig variale σ ould e deied as a proper liear oiaio o he error sigal ad is derivaives up o he (r-h) h -order r eig he relaive degree [] eee he sse oupu i.e. he slidig variale σ ad he disoiuous orol. The dire appliaio o he laer approah ih he use o a -VSC ih haerig avoidae eaures []-[5] o he osidered sse ould ipl he deiiio o he slidig surae as ollos: σ e e e ih oeiies o a Huri poloial. Equaio requires he esiaio o he irs ad seod derivaive o he availale sigal e - re. The are as varig durig he rasie he he oa ore is o e regulaed ad hereore he ipshi osa o e is ver large as a e evaluaed (9). urherore he seod order slidig dais hih derives ro diereiaio o aig io aou (7)-(9) a e desried he olloig relaioship σ ϕ4( u) γu ha has he dri er ϕ 4 hose eplii or is o repored or he sae o oiseess depedig aog he ohers o ers lie A ad A 4. These a aes he oud o ϕ 4 ver large. The aove osideraios ipl ha huge values o oh he diereiaor ad oroller gais should e used o iplee a oupu-eeda VSC shee as i []. Eve i heoreiall he oupu-eeda VSC approah a guaraee he aspoi regulaio o he oa ore he praial appliaio o he desried approah i his ase ould e o eeive. I a he heorei perorae oes ro he assupio ha he disoiuous orol u sihes a iiie reque. This is o possile eiher eause he oroller is digiall ipleeed or eause soe hserei rela is used o iplee he VSC. I hese ases he ivaria se o he losed-loop sse is a oudar laer o he slidig surae i.e. aer a iie ie rasie i resuls σ K τ σ Kτ T here τ ad T are he equivale sihig dela ad he 59

4 reahig ie respeivel [6]. The huge values o he osas K ad K due oh o he resoae pea o he auaor ad o he paraeers o he paograph/aear sse require alos urealisi values o τ o ahieve he desired aura. A. A approiae slidig ode oroller The orol shee e are goig o prese is ased o he hpohesis ha he presee o as auaors or sesors ould resul i a sigular peruraio o he ideal slidig ode []-[6]. Thereore e a desig he oroller disregardig he presee o he auaor ad ipleeig a haerig avoidae shee. I a suh a shee a e applied o relaive degree oe sses as i our ase ad i eeds a -SMCs ih a asade irs order iler (oe a siple iegraor) so ha he relaive degree eee he slidig variale ad he disoiuous orol is o. The he oral or or he sse ihou he auaor ould e he olloig ( ) ϕ ( ) γ ( ) (4) ψ( ) here he relaioship eee veor ad [ T ] T oes ro (8) o osiderig he las o ros ad he ipuoupu irs order dais derives ro he olloig CA CB CB (5) ih CB / ad CB. I [5] i as sho ha he use o he Geeralied Su- Opial -SMC [7] [8] ih uodeled liear as auaors o a order guaraees he reahig o a oudar laer lie ha i τ eig he as auaor s ie osa. urherore he paraeers o suh a oroller a e also ued o ouera he peaig pheoeo [8] ad o shape he reque ad agiude o he sse osillaios ihi he oudar laer [6]. The aove resuls ad he osideraios i Seio II aou he ouds o he sse dais sugges he use o he Geeralied Su-Opial -VSC ih osa paraeers guaraeeig he oooi ehavior o he slidig variale as he oroller i.e. [7] (6) V sg( e βem ) Φ V > ( ee) Ω γ β ( ) (.5] β here e M is he os ree ereal poi o he error variale e- re hose vaishig deies he slidig surae ad Φ is a upper oud o he uerai dri er o he seod order dais o e i he se hih a e esiaed he las ro o (8). Tuig he -VSC paraeers eas o he ors ase desig as saed odiios o V i (6) is usuall less eeive ha derivig he eas o preliiar siulaios. I a he laer usuall gives eer resuls sie he esiaed ouds or he uerai are ver oservaive ad ipl a ver large orol auhori. This ill resul i a larger os ad larger values o osas K ad K i. I order o osider he auaor as a sigular peruraio o he ideal sse he resoa dais o he ire auaor should e opesaed eas o a pre-iler. I he osidered ase he ire auaor has ee ideiied he haroi respose [9] dra i ig.. I is evide he large resoa pea a a high reque ha ould apli soe o he haroi opoes geeraed he iie reque sihig o he VSC suh ha he real slidig ould e desroed. The pre-iler should e desiged i suh a a ha he auaor is a lo-pass iler hose adidh is large eough o allo he orol o opesae he haroi disuraes due o he variaio o he aear paraeers. Taig io aou he sadard spa legh ( 6 ) ad he usual aial veloi o high-speed rai o Ialia raila sses (v 5 /h) he speru o he disuraes o opesae or ih he orol is ihi he rage rad/s. Thereore i he speii ase a siple irs-order lopass iler ih ie osa τ. s as he pre-iler suies (ig.). Phase (deg) Magiude (db) Phase (deg) Magiude (db) Wire-auaor haroi respose diagra -8-4 reque (rad/se) ig.. The ire auaor haroi respose aihaerig iler & auaor ideal ai haerig iler pre-opesaed auaor Bode Diagra -6-4 reque (rad/se) ig.. The pre-opesaio desig igure shos ha he pre-iler drasiall redues he iluee o he resoa ire auaor (i.e. he opesaed reque respose is alas uder he sraigh lie represeig he siple iegraor irodued or haerig avoidae purposes) hile leavig he haroi respose values alos uhaged ihi he orig reque rage rad/s (i.e. ol a ver sall gai variaio ad a aepale phase shi are prese i he osidered reque rage). The overall sse dais is he represeed he se o diereial equaios (4) (6) ih he olloig 59

5 .u u ν (9) ν V sg( ) ( ) β ( ) re hih represe he oliear dai oroller drivig he ire auaor oad sigal u. The overgee o he sse io he oudar laer is assured hoosig paraeer V suiiel high [7] hile he paraeer β a e ued o shape he reque ad he agiude o he residual osillaios o he oa ore aig io aou ha higher values o ipl saller osillaios a a higher reque u a sloer rasie as ell [6]. re M III. SIMUATION RESUTS The eeiveess o he proposed oroller has ee esed eesive siulaios o he liear odel o a ATR9 paograph or high-speed rais oied ih a liear ie varig odel o he aear. The sse paraeers i (5) have ee opued or easured o e: N g 9.5 g 4N - N - 5 N - s N - s N - s; hile he aear has ee odeled aordig o ih o 95 g g o 4 N - s o 7 N - 6 N - 6 ; ad he auaor paraeers i (6) have ee ideiied: ω rad/s ζ.77 K a. A pre-load osa ore 4 N as assued ad he ire sregh has ee ored o e alas o posiive (he ire a ol e srehed) a proper sauraio lo. Siulaios have ee arried ou eas o MATAB-Siuli usig he Ruge-Kua iegraio ehod ih iegraio sep h -4. A irs es has ee arried ou assuig ha he auaor is a pure gai. Thereore he sse dais redues o (4) ad oroller (6) has ee direl used ih paraeers V ad β.9. The ereal values are esiaed ih a irs dieree algorih ih dela -4. Wihou aeig he validi o he proposed orol shee siulaios have ee perored assuig a osa aeleraio ro o 5 /h alhough his is a urealisi aeleraio perorae. The oa ore is alos perel regulaed o he desired value re N (±.4 N) (ig. 4). Coa ore [N] Corol ore [N] Trai speed [/h] Tie [s] ig. 4. Tes o auaor dais The he auaor ih he pre-iler has ee irodued ad he siulaio as repeaed ihou hagig he - VSC paraeers (ig. 5). Also i his ase he oa ore is ell regulaed o he desired value re N (± N) ih a sall derie o he perorae. iall a easuree oise ih uior disriuio eee ± N as irodued (ig. 6). I all ess he rai speed proile as he sae as i he loer plo o ig. 4. I is evide ha he dereasig o he perorae is o draai; i a he deviaio o he oa ore ro he desired value is greaer ha he easuree error ol a he ai resoae reque he he rai speed is aou /h. The oroller uig ased o he ideal ase is sill eeive also i perured ases shoig he rousess o he -VSC ad o he proposed orol shee. Coa ore [N] Corol ore [N] Auaor ipu Coa ore [N] Corol ore [N] Tie [s] ig. 5. Tes pre-opesaed auaor Measuree oise [N] Tie [s] ig. 6. : Tes pre-opesaed auaor ih ois easurees IV. CONCUSIONS The paper preses he appliaio o a rous orol shee ased o he geeralied su-opial seod-order slidig ode oroller o he regulaio prole o he oa ore i high-speed raila sse. The auaor is pre-opesaed a properl desiged iler i asade ih he -VSC so ha he overall pre-iler ad auaor a e osidered as a sigular peruraio o he ideal seod order slidig ode. The proposed approah resuls o e siple ad eeive ad rous o he easuree oise as ell. REERENCES [] Goslig R.J. Hos A.E.W. The Ieraio o Paograph ad Overhead Equipe: Praial appliaios o a Ne Theoreial Mehod Pro Is Meh Egrs 97C pp [] Eppiger S.D. O'Coor N.D. Seerig W.P. Worle D.N. Modelig ad Eperieal Evaluaio o Aseri Paograph 594

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