Modeling of magnetic levitation system

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Horatio Terry
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1 7 s Inernonl Conference on Process Conrol (PC) June 6 7 Šrbské Pleso Slovk Modelng of gnec levon syse Peer Blko Dnc Rosnová Insue of uoove Mechroncs Slovk Unversy of Technology n Brslv Brslv Slovk dnc.rosnov@sub.sk peer.blko@sub.sk bsrc The pper s devoed o odelng nd preer denfcon for Mgnec Levon Syse (MLS) fro Ineco. MLS belongs o chllengng odellng nd conrol probles due o s unsbly nd nonlnery. We concern severl odelng dels no suffcenly descrbed n user nul correcon of nonlner odel nd presen he correspondng esureen resuls. The obned nonlner odel preers nd correspondng lnerzed odel fs he rel d uch beer hn preers provded n he reference. Keywords nonlner syse odelng unsble syse gnec levon (glev) I. INTRODUCTION Mgnec levon belongs o well-known ype of syses wh vrey of pplcon feld []-[4]. Modelng he gnec levon syse belongs o chllengng sks snce besdes nonlneres we hve o cope wh open-loop unsble syse wh fs dyncs nd very sll degree of nurl dpng. The bsc conrol s o precsely poson he levng objec whch requres dequely precse odel. There exs ny references devoed o odellng nd conrol of vrous ypes of gnec levon lborory plns [5]- [] o ls few. The lborory MLS provded by Ineco s nlyzed e.g. n [] [] [] however o he uhors bes knowledge severl porn odelng dels hve no been suffcenly repored. Bsclly hecl odel cn be obned fro bsc physcl lws (frs prncples) or by d drven denfcon ehods bsed on esureen of npu nd oupu d for dequely exced syse []. requenly boh hese pproches re cobned he srucure s deerned ccordng o heorecl (frs prncple bsed) odel nd preer vlues re esed fro esured d usng denfcon ehods. In hs pper we presen our resuls on odelng he lborory gnec levon syse [7] wh elecrognes. The conrolled oupu s bll poson beween he elecrognes where we consder only he upper one. We presen odel bsed on frs prncples wh splfed nonlneres where he respecve preers re esed fro esured d nd verfed by sulons nd he resulng odel s copred wh erenl resuls on rel lborory pln. g.. Mgnec Levon Syse wh Elecrognes EM nd EM nd levng sphere bll beween he. II. MGNETIC LEVITTION SYSTEMS: PHYSICL MODEL Mgnec levon cn be brefly chrcerzed s chevng he equlbru of n objec n he r-spce whou conc wh sold subsnce by usng elecrognec forces. In MLS g. ferrognec objec ( bll) s kep n he r-spce beween wo elecrognes where he upper one provdes vercl force overcong he bll grvy nd he lower one s used s ddonl nly for sblzon of bll horzonl poson.. Developen of Mgnec levon odel The nonlner physcl odel of he MLS cn be descrbed by se equons obned fro bsc physcl lws for bll (sphere) oon n elecrognec feld [] []. Descrpon of bll dyncs nd of elecrognec forces s bsed on Lgrnge funcon - he dfference beween knec nd poenl energy whch cn be wren s T x L( x) q Rq d gx qu where x s dsnce of he sphere fro elecrogne q s elecrc chrge s ss of he sphere R s ressnce of he elecrogne col L(x) s funcon descrbng he /7/$. c 7 IEEE 5

2 dependence of nducnce of he col on dsnce x I q s curren n he col g s grvy consn u s volge. Vrbles x( ) q( ) us ee he Lgrnge equons whch yeld d dt T d dx x d dt T d dq q d x dl I g d dx di dl dx I RI u d L dx d The frs equon fro () corresponds o Newon s second lw where n elecrognec force s gven by nd cobnng () nd (7) he resulng nonlner hecl odel s obned see [5] dx d dx d x dl( x ) x dx g dx ( ku c x) d f ( x ) pplyng he pproxon (6) wh slghly dfferen denoon fnlly we obn dx x d dx e g d dx ( ku c x) d f ( x ) dl ( ) I dx x I Dependence of col nducnce on dsnce cn be pproxed by polynol or onenl funcons x L x) L L e ( where f e x x P x. f f ep ep P x f P ep L( x) L n x n... R n n n x n... x b Below we wll use he onenl lernve (5) hvng dl dx x Le The second equon fro () s splfed by he erenlly receved pproxon [] di ( ku c I) d f ( x) where funcon f (x) hs srucure slr o (6). Inroducng se vrbles x ( ) x( ) poson of he sphere x( ) x ( ) velocy of he sphere x ( ) I( ) curren n he upper col Correspondence of he second equles fro () nd () s hrough pproxon (6) wh dl ep x L ep ep dx ep ep The rel lborory odel uses vrbles n he nervls x.6 x R x. u. n un Rerk : I should be noed h he second equon n nonlner odel () dffers fro he one presened n MLS docuenon [5]: here s coeffcen n he denonor correspondng o he frs equon fro (). Ths coeffcen ppers hen lso n he lnerzed odel below (n eleens nd n ()). B. Vldon of odel coeffcens Ths secon presens vldon of he key coeffcens fro nonlner odel () by esureens on lborory pln. Snce he pln self s unsble esureens re relzed n closed-loop wh sblzng PD conroller delvered by producer [5]. 5

3 TBLE II. PRMETERS O THE EQUTIONS ().75 esured d our pproxon pproxon fro lerure.7.65 Preers g ee ep ep f(x) fp fp c k xd MIN umin..5 Vlues.6... funcon of x nd x.75.5 funcon of x dsnce beween elecrognes nus bll deer (.-. /.4 /.54).4.4 Uns kg /s N H s- s 4.7 poson [] g.. Vldon of pproxon coeffcens obned fro esured dependence of col curren I on bll poson x..6.5 Coeffcens ep nd ep for pproxon of dl / dx correspondng o (6) cn be obned by esurng he dependence of col curren x on bll poson x n sedy se. dl / dx s hen copued fro he second equly fro ().4. esured d proxon d.. dl( x ) x g dx dl( x ) g dx x.. Poson (x) [] MESURED x x ND u OR dl/dx PPROXIMTION Curren (x) [] conrol oupu [] Mesured d x x nd u re gven n Tb.I. The correspondng g. shows coprson of pproxon for coeffcens ep nd ep obned fro esured d nd he coeffcens repored n Ineco nul. We cn confr ner ccordnce of esured nd repored pproxon coeffcens. TBLE I.. Conrol oupu (u) [] g.. Dependence of curren x on conrol oupu (provded n chne uns) for ll hree blls Preers k nd c re gven by pproxon of rel esured d see g.. We de ore erens for ll hree blls he verge preer vlues re I ku c 4.4u.4 The dfference beween rel d nd pproxon s very sll hus we cn consder he correspondng nonlner odel s ppropre for lnerzon round equlbru pons. In hs cse he obned preers k nd c re sgnfcnly dfferen fro hose repored n docuenon [5]. -dl/dx [H/] III. LINERISTION O MGNETIC LEVITTION SYSTEM The lnerzed se spce odel for () cn be obned usng Jcobn lnerzon round he deerned workng pon s ll preers of he bove equons re gven n Tble II. 54 x x Bu x u b y Cx x

4 where he eleens of he nd B rces re for he workng pon defned by [ x x ] gven s x x ep ep x f x ep ep x x ep ep b k f x Noe h snce k u c x n ny workng pon. To conver se spce odel o rnsfer funcon we use de( si ) s de( si ) s s de si s s s s s The resulng rnsfer funcon s Y ( s) G( s) C U ( s) T si B b s s The npu curren respecve o he poson of he bll s depced n g. 4 for ll hree blls. Ths dependence s used for deernng he workng pon correspondng o he requred bll poson. In he followng we presen hree lnerzed odels where preers x u vry for bg edu nd sll blls. Deernon of he workng pon The workng pon s deerned by requred bll poson x. Thus he workng pon s se by he nex seps. ) choose he bll sze nd s poson x ; ) se he vlue of x correspondng o he deerned x (see g.4.) lernvely clcule x fro he second equon fro () n sedy se for x x ; ) clcule u for he deerned x nd x fro (): x k u c.4u.4 4. g. 4. Dependence of col curren on bll poson Bg bll lnerzed odel We consder he workng pon x.[ ] x.[ ] nd u.6[ ] he se spce odel s x x Bu x u y Cx x -. nd he respecve rnsfer funcon s s s.s 4.4s G Medu bll lnerzed odel We consder he workng pon x.[ ] x.4[ ] nd u.[ ] he se spce odel s x x Bu x u y Cx x -. nd he respecve rnsfer funcon s s G Sll bll lnerzed odel s.s 4.4s 4.4 We consder he workng pon x.[ ] x.76[ ] nd u.65[ ] 55

5 x x Bu x u y Cx x edu bll rel d nonln. odel nonln. odel lnerzed odel nd he respecve rnsfer funcon s 74 G s s.s 4.4s 4.4 dsnce [].5.5 Noe h only gn vres wh he bll chnge. Coprson beween rel nd lnersed syse In hs secon sep responses round workng pons for ll hree blls re shown for: ) rel lborory MLS; ) nonlner odel () wh preer vlues gven n Tb.II denoed s nonln. odel ; ) nonlner odel wh preers fro [5] denoed s nonln. odel ; 4) lnerzed odels (6) (7) nd () for bg edu nd sll bll respecvely. The MLS syse s unsble. Therefore he coprson s relzed n closed loop wh sblzng conroller. In our cse we use PID conroller correspondng o () wh preers P=5; I=77 ; D=5.65. (Noe h splng e for MLS s s.) The resuls re llusred n he nex pcures. Coprson shows h he developed nonln. odel s well s he correspondng lnerzed odel beer fs he rel MLS response hn nonln. odel repored n [5]. (Reson cn be n slghly dfferen preers of he ndvdul MLSs.) dsnce [] sll bll rel d nonln. odel nonln. odel lnerzed odel e [s] g. 5. Coprson of he oupu responses of he developed odel lnerzed one nonlner odel fro [5] nd rel esured d for sll bll e [s] g. 6. Coprson of he oupu responses of he developed odel lnerzed one nonlner odel fro [5] nd rel esured d for edu bll dsnce [] bg bll rel d nonln. odel nonln. odel lnerzed odel e [s] g. 7. Coprson of he oupu responses of he developed odel lnerzed one nonlner odel fro [5] nd rel esured d for bg bll I s porn o noe h hough hesyse s hghly nonlner he lnerzed odel works well round he workng pon. IV. STBILITY CONDITIONS OR CLOSED LOOP WITH PID CONTROLLER In hs secon necessry sbly condons re developed for he closed loop coprsng lnerzed MLS odel wh PID conroller whch ndces he requred conroller srucure nd preers. Due o negve gn of he MLS rnsfer funcon (6) (7) () posve feedbck s consdered o obn conrol error. The PID conroller desgn s hen bsed on closed loop 56

6 G CL s sg s s G s GPID G where G s correspond o MLS rnsfer funcon (5) s preers re deerned by workng pon nd respecve bll (see (6) (7) ()); G PID s s rnsfer funcon of PID conroller n he for I G PID s P Ds s Closed-loop chrcersc polynol for () wh nd G PID s gven by (5) nd () s PID G s 4 CHP s s ( b ) s ( bp) b I Recll he preer sgns (copre (5) wh e.g. (6)) ; ; ; b ; b herefore pplyng Rouh sbly creron on () we receve closed loop sbly condons Therefore D b ; P b P D Sbly nlyss shows h nl sblzng srucure coprses proporonl P nd dervve D er. To vod sedy-se error negrl er s requred s well. Conroller desgn procedure s no ncluded n hs pper one possble conrol desgn procedure s descrbed n [4]. V. CONCLUSION We hve presened severl dels on odellng Mgnec levon syse delvered by Ineco no repored n he syse docuenon nd correced ske n heorecl odel. The erens o verfy vlues of odel preers re descrbed n dels; he respecve resuls show he jor dfference n lner pproxon of dependence of col curren on npu volge n coprson wh [5]. Modfyng he odel preer vlues ccordng o esured d provdes nonlner nd respecve lnerzed odel wh sep response very close o he rel erens. CKNOWLEDGMENT The work hs been suppored by he SRD grn No PVV 77- nd Slovk Scenfc Grn gency grn No /7/6. REERENCES [] P. Holer ser hn speedng bulle rn IEEE Specru Vol. 4 No. pp. -4. [] M. Vrvell E. Cllon L. D ore L. Mlno N. rnud esbly of gnec suspenson for second generon grvonl wve nerferoeers sroprcle Physcs Vol. No. 4 pp [] P.J. Berkeln R.L. Holls Lorenz gnec levon for hpc nercon: Devce desgn perfornce nd negron wh physcl sulons Inernonl Journl of Robocs Reserch Vol. No. 7 pp [4] M.B. Khesee N. Ko Y. Nour T. Nkur Desgn nd conrol of croroboc syse usng gnec levon EEE- SME Trnscons on Mechroncs Vol. 7 No. pp. -4. [5] Mgnec Levon Syse EM User s Mnul (Ineco Ld Krkow Polnd ). [6] Mgnec Levon Experen Qunser Consulng Inc. [7] CE 5 Mgnec Levon Model Huusof hp:// [] M 4 Lborory Seup Mgnec Suspenson r. [] P. Bn Model I serowne gneyczn lewcj Dplo hess Unversy of Scence nd Technology (GH) n Krkow [] L. Sun Y. Myke H. Ohor. Sno New Drec Closed-Loop Idenfcon Mehod for Unsble Syses nd Is pplcon o Mgnec Suspenson Syse Trns. of he Socey of Insruen nd Conrol Engneers Vol. E- () No. pp.7-. []. Turnu K. Kolek Te-opl nd PID vrble srucure conroller. Proceedngs of he Mederrnen Conference on Elecroncs nd uoc Conrol MCE Mrrkech 7- Sepeber Mroc pp [] R. C. Dvd C.. Drgoş R. G. Bulzn R. E. Precup E. M. Peru M. B. Rădc n pproch o fuzzy odelng of gnec levon syses Inernonl Journl of rfcl Inellgence vol. no. pp. -. [] C. J. Munro M. R. lho R. M. Borges S. d Slv Munreo W.T. d Cos Modelng nd observer-bsed nonlner conrol of gnec levon syse n Proceedngs of he IEEE Inernonl Conference on Conrol pplcons pp.6-67 [4] M. Hypusov. Kozkov Robus PID conroller desgn for gnec levon syse requency don pproch subed o s Inernonl Conference on Process Conrol Šrbské Pleso Slovk June 7. 57

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