Magnetic Suspension System Control Using Jacobian and Input State Linearisation. D. Giaouris, J.W. Finch

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1 Mntc Sspnson Systm Control Usn Jcobn n Inpt Stt nrston D. Gors J.W. Fnch School o Elctrcl Elctronc & Comptr Ennrn Unvrsty o Nwcstl pon Tyn Nwcstl pon Tyn NE 7RU UK mls: Dmn.Gors@ncl.c. j.w.nch@ncl.c. BSTRCT Ths ppr prsnts two rnt mthos to control Mntc Sspnson Systm MSS. Th rst control mtho ss Jcobn lnrston o th MSS ron n Eqlbrm Pont EP wth clsscl Pol Plcmnt PP controllr. Th scon mtho ss Inpt Stt nrston IS o th nonlnr mol ollow by PP controllr. Robstnss tsts on both mthos r scrb. Th vnts n th svnts o ch mtho r nct n mthos o rthr mprovmnt sst. INTRODUCTION Th bsc chrctrstc o MSS s tht t cn sspn objcts wthot ny contct. clsscl MSS F. conssts o n lctromnt n stl bll. Th rsltnt orc on th bll s th rnc btwn th lctromntc orc whch coms rom th mnt n th orc o rvty. In th ls cs whr th rsltnt orc s ro th bll wol blnc t stnc rom th mnt ths s cll th EP. Unortntly bcs o strbncs whch cnnot b moll ctly th bll wol thr mov towrs to th lctromnt n str th col or wol ll to th ron; ths th systm s nstbl th crrnt s constnt n thr s no bc. tonlly th lctromntc orc s ncton o th sqr o th crrnt n th col. Ths th systm s lso nonlnr s wll s bn nstbl. R In F. s th crrnt t th col s th volt spply to th col R s th rsstnc o th col s th nctnc o th col s th stnc o th stl bll rom th lctromnt m s th mss o th stl bll B s th rvttonl orc s th lctromntc orc ppl to th stl bll. Th orc th crrnt th volt n th stnc r ll nctons o tm. m B F. Schmtc o MSS EQUTIONS OF THE MSS Th rsltnt orc on th bll ssmn smpl Nwtonn ynmcs wth nlbl rcton cn b prss s: W F B m or m m t t whch wth lnr mntc crct cn b wrttn s: m m t

2 whr W s th nry o th lctromnt n s th cclrton to rvty. ccorn to [] th nctnc ncton cn b moll n th ollown ponntl orm: whr s lnth constnt. Ths sn qn. n qn. vs: or t m m m t n: t R 4 Usn qn. wth qn. 4 vs: t R or R 5 Th stts o th systm r: th npt o th systm s n th otpt s. lso th rto m s constnt n hnc or smplcty t cn b rplc by constnt n. Hnc th nonlnr systm s n by: [ ] C t Y R 6 Th prmtrs tht n th mol wr tn rom [] n r: col nctnc.49 H col nctnc.9 H constnt α 6.66 mm mss o th bll m.8 cclrton o rvty 9.8 m/s n col rsstnc R 4. Ohm.

3 JCOBIN INERISTION ND POE PCEMENT Th ypnov lnrston mtho llows nonlnr mol to b smpl lnrs ron n EP. Th EPs o systm cn b on by sttn hnc: R or ± ± R 7 For smplcty th prmtrs n wll b wrttn s I n rspctvly. So th lnrs mol ron th EP I s: R I I I EP [ ] C B EP 8 Th lnrs systm cn b rprsnt n th sl stt spc orm by th mtrcs B C D snc thr s no rct npt-otpt copln. Usn th vrbls rom [] n or cm ths mtrcs r: B n [ ] C 9 Ths lnrs systm cn b s to nvstt svrl control lws. Snc stt spc rprsntton s s th choc o th PP control lw ws obvos s th systm s controllbl. Th clos loop systm s BK C BF B C n C C C whr K wll b on rom th PP control lw n F rom th rror crtr. Th sr pol loctons wr chosn to b [- ]. Wth th s o Mtlb th stt n vctor ws on to b K[ ] th clos loop mtrcs wr on to b: C.44 C B. Fnlly th n mtr F sn trl n rror mthos ws on to b -.75.

4 mplt [cm] Tm [s] F. Stp rspons o th Jcobn lnrs systm wth pol plcmnt control lw Thory Bcron INPUT STTE INERISTION Th prvos mtho vn thoh t s strhtorwr hs vry b rwbc. Ths coms rom th ct tht th whol oprtn ron o th systm mst b spprss ron th EP. Th systm my not wor t ll ots ths ron snc th lnr controllr wll b nbl to compnst th nonlnrts tht wll ppr s soon s th systm movs r wy rom th EP. Thr r vros wys to ovrcom ths problm. On mtho tht hs bn s n mny clt problms s IS. Ths powrl mtho s bs on s o trnsormton o th stts n trnsormton o th npt; to scrb th systm n lnr wy. Thn smpl lnr controllr cn b ppl. For mor tl normton bot ths mtho s []. ccorn to [] th bsc stps o th IS r:. Th Snl Inpt Snl Otpt SISO systm mst b rprsnt by th stt qtons:.. to b n compnon orm. Ths mns tht th npt snl pprs only n ts ntrl orm n not v ts rvtvs lso n rprsnt smooth vctor ls.. r contnos n thr hh rvtvs st. I th systm s not n ths orm thn som n o trnsormton mst b ppl. n. Constrct th vctor ls: l whr s th brct o n : [ ] n.. Chc th controllblty n nvoltvty contons r sts. Th trm nvoltvty mns: tht rom st o vctor ls th l brct o two s tn thn th rsltnt vctor l cn b prss s lnr combnton o th ornl st o th vctor ls. 4. Fn th rst stt by n l n n 5. Compt th stt trnsormton [ ] T rvtv o wth rspct to Compt th npt trnsormton β v wth: n n β n 7. Clclt th nw lnr control lw v. n h whr s th

5 IS n MSS Th vctor ls r: R ccorn to [] th systm cn b npt stt lnrs <<. Hnc: D R D t D v v R D D v whr v s th nw lnr control lw. Snc th systm s controllbl rqrmnt or th IS pol plcmnt control lw ws chosn. Th systm now cn b scrb s Bv whr B[ ] T. Th PP control lw plc th clos loop pols t [- - -]. Typcl stp rsponss r shown n F. n 4.

6 Dstnc [m] Tm [s] Dstnc [m] Tm [s] F. Stp rspons or npt.m wth IS F. 4 Stp rspons or npt -.m wth IS Fs. -4 show tht whn th npt snl chns sn th otpt os lso not prctcl consqnc ths smply rlcts th symmtry o th ssm mol. So th systm hbts lnr bhvor. lso th rror ws vry smll n ll ths css. ROBUSTNESS Th two compnstn mthos bov hv bn chc or thr robstnss. Two tsts hv bn ppl or tht. In th rst th prmtr snstvty ws chc n n th scon th prormnc o th systm n th prsnc o nos snls vlt. Jcobn nrs Systm Th prmtr tht ws chn ws th mss. Th vrton ws % & % o th nomnl vl. Th bhvor o th systm show vry oo robstnss nst th mss chn s F. 5. mplt [cm] Tm [s] Inpt Stt nrston F. 5 Jcobn lnrs systm s snstvty to mss chns Th sm tsts wr on or th systm sn IS. Th rspons o th systm to mss chns ws not s stsctory s bor F. 6. Fnlly th systm sn IS ws chc or strbnc rjcton. wht nos snl ws to smlt th ct o n trnl nmoll orc. s cn b sn n F. 7 th otpt pns lot on th nos powr. Th o npt stt lnrston s bs on ccrt molln o th systm. Snc th strbnc orc s not moll n th systms qtons t s ntrl tht th systm s nos rjcton s poor.

7 mplt [m] Tm [s] mplt [m] Nos powr: [.] Tm [s] F. 6 Systm s snstvty to mss chns wth IS F. 7 Dstrbnc rjcton wth IS CONCUSIONS systm whch s both hhly nonlnr n opn loop nstbl ws nvstt sn two rnt compnstn mthos. Th rst ppl smpl Jcobn lnrston n n th scon n IS schm ws s; pol plcmnt control lw ollow both lnrston tchnqs. Th smlton rslts show tht th systm sn IS col hv vry w oprton ron. Hnc t ovrcm th clsscl problm tht th Jcobn lnrston my hv whr sch lnr controllr cnnot stsctorly control th systm ots smll ron nr th EP. Th problm wth th IS s tht th mol hs to b vry ccrt. In th prsnc o nos th systm wth IS cn show poor bhvor. Both compnston schms cn b vry oo pnn on th pplcton. I th pplcton rqrs smll oprtn ron n prmtr vrtons sch s mss or lctrcl rsstnc chns r lly to b smll thn th Jcobn tchnq cn b oo. I th systm ns to hv w oprtn ron thn th IS s bttr ssmn tht ts mol s ccrt. To rthr mprov th bhvor o th IS schm ptv robst tchnqs cn b s. REFERENCES [] Hrly W.G. n Wrnr H. W. Elctromntc sn o mntc sspnson systm IEEE Trnsctons on Ecton vol. 4 no. pp.4- My 997 [] Slotn E. J. n W. ppl Nonlnr Control Prntc Hll 99 ISBN 4895 [] Mttl S. n Ch-Hsn M. Prcson moton control o mntc sspnson cttor sn robst nonlnr compnston schm IEEE/SME Trnsctons on Mchtroncs vol. no. 4 pp.68-8 Dc 997