Sliding mode control of photoelectric tracking platform based on the inverse system method

Transcription

1 AEC Wb of Confrncs, 6 DOI:.5/ mtcconf/ 6 C Ownd b th uthors, publishd b EDP Scincs, 6 Sliding mod control of photolctric trcking pltform bsd on th invrs sstm mthod Zong Chn Yo,, H Zhng,b School of chnicl Enginring, Nnjing Univrsit of Scinc nd chnolog, Nnjing 9, Chin Abstrct. In ordr to improv th photolctric trcking pltform trcking prformnc, n intgrl sliding mod control strtg bsd on invrs sstm dcoupling mthod is proposd. h lctromchnicl dnmic modl is stblishd bsd on multi-bod sstm thor nd Nwton-Eulr mthod. h coupld multi-input multi-output IO nonlinr sstm is trnsformd into two psudo-linr singl-input singl-output SISO subsstms bsd on th invrs sstm mthod. An intgrl sliding mod control schm is dsignd for th dcoupld psudo-linr sstm. In ordr to limint sstm chttring phnomnon cusd b trditionl sign function in sliding-mod controllr, th sign function is rplcd b th Sigmoid function. Simultion rsults show tht th proposd dcoupling mthod nd th control strtg cn rstrin th influncs of intrnl coupling nd disturbnc ffctivl, nd hs bttr robustnss nd highr trcking ccurc. Introduction Photolctric trcking pltform is two-is gimbl sstm quippd with photolctric dtction quipmnt, cn cpturing nd rl-tim trcking of moving trgt in th ir or on th ground, hs bn widl usd in militr nd civil rs. During trcking, smll ngl rrors cn cus big dvition of trgt position. Howvr, Du to th prsnc of strong coupling nd nonlinr fctors in th pltform, th trcking prformnc nd ccurc will b ffctd. Obviousl, th problm of coupling nd nonlinr nd to b rsolvd urgntl nd th k lis in th dcoupling mthod. As nonlinr coupling sstm, prvious litrtur[ did not p nough ttntion to th coupling fctors of th sstm, it usull trtd s disturbncs nd us th disturbnc obsrvr to compnst. In rcnt rs, vrious tps of dcoupling mthod hv bn proposd. On is th Intllignt dcoupling mthod which dos not dpnd on th prcis mthmticl modl, such s nurl ntwork dcoupling[, sliding mod dcoupling[ nd fu dcoupling[ tc. Whn sstm is unbl to modling, thorticll, is mor suitbl for th us of intllignt dcoupling mthod. Howvr, ll th intllignt dcoupling mthod rquirs lot of computr rsourcs on prcticl ppliction, for mpl, nurl ntwork dcoupling rquirs lrg mount of msurmnt dt nd rptd primnts to optimi th prmtrs. Anothr kind of dcoupling mthod is linrition nd dcoupling L&D mthod, including invrs sstm mthod nd diffrntil gomtr mthod. Comprd with th diffrntil gomtr mthod, th invrs sstm mthod dos not nd compl nonlinr coordint trnsformtion, lso dos not nd to trnsform th nonlinr control problms into th "gomtric domin"[5. Bsd on ths dvntgs, th invrs sstm mthod is mor vilbl in prcticl ppliction. h trcking ccurc nd robust prformncs cn dvrsl ffctd b th ist of uncrtintis nd modl rrors[6 s wll. hrfor, th introduction of control strtg is rquird to ovrcom th shortcomings. Sliding mod control with dvntgs of whol robustnss on trnl disturbnc nd uncrtint which stisf th mtching conditions, hs bn widl usd in ir vhicl, robot control fild nd mn othr compl sstm control with disturbnc nd uncrtint[7,8. h introduction of intgrl cn dcrs th std-stt rror cusd b trditionl sliding mod, nd improv th trnsint prformnc of th sstm[9. In this ppr, sliding mod control strtg bsd on invrs sstm dcoupling mthod for th photolctric trcking pltform with nonlinr coupling, prmtr uncrtintis nd unknown disturbncs is proposd. Kinmtic rltions of th photolctric trcking pltform is nld first, nd th lctromchnicl sstm modl of th photolctric trcking pltform is stblishd b using th trditionl Nwton Eulr mthod. Bsd on th invrs sstm mthod, th coupling nd nonlinrit in sstm dnmic r rsolvd, such tht th dnmics of two gimbls cn b rgultd indpndntl. Bsd on th dcoupld psudo linr sstm, th intgrl sliding mod control is pplid to rjct th influnc of uncrtintis nd disturbncs. orovr, th smbolic function is rplcd b th proposd Sigmoid function to oongchn@gmil.com b hhng@njust.du.cn his is n Opn Accss rticl distributd undr th trms of th Crtiv Commons Attribution Licns., which prmits distribution, nd rproduction in n mdium, providd th originl work is proprl citd. Articl vilbl t or

2 AEC Wb of Confrncs rstrind th chttring phnomnon of sliding-mod controllr. At lst, th stbilit of th proposd control strtg is vrifid b th Lpunov critrion. Simultion rsults dmonstrt th ffctivnss nd rlibilit of th proposd control strtg. odling of th photolctric trcking pltform. Kinmtic rltions h structur of th photolctric trcking pltform is two-is gimbl sstm. Figur shows th coordint sstms of th photolctric trcking pltform. hr coordint frms hv bn introducd s: h bs coordint sstm {C } is fid on th bs or ground. h imuth gimbl coordint sstm {C } cn rott round th is of OZ OZ nd th rltiv imuth ngl is dfind s. h lvtion gimbl coordint sstm {C } cn rott round th is of OY OY nd th rltiv lvtion ngl is dfind s. η ξ η ξ Figur. Coordint sstms of th photolctric trcking pltform B using Eulr mtri, trnsformtion mtri from {C } to {C } bcom cos sin C sin cos h rltiv ngulr rt of imuth gimbl nd lvtion gimbl cn b dscribd s: ω [ ω [ B using trnsformtion mtri C, bsolut ngulr rt of imuth gimbl nd lvtion gimbl bcoms: ω ω sin ω Cω ω cos 5 Whr subscript stnds for th lvtion gimbl nd subscript stnds for th imuth gimbl; subscripts, nd stnd for th componnts in dirction of is, is nd is of th vribls rspctivl.. Dnmicl modl Assuming tht ch gimbl cn b rgrdd s rigid bod, bsd on th trditionl Nwton-Eulr qutions[, th dnmic modl of photolctric trcking pltform cn obtind: =ω ω +ω ω 6 Whr stnds for th trnl torqu pplid to ch gimbl; stnds for th inrti momnts; ω stnds for bsolut ngulr rt ; ω stnds for bsolut ngulr cclrtion. h qution lso cn b writtn s: 7 Bcus ch gimbl rspct to thir coordint s is ismmtric, so th product of inrti in inrti momnts mtri of ch gimbl is ro [, thn th inrti momnts bcoms: dig[ 8 h inrti momnts rspct to th rottion is of lvtion gimbl nd imuth gimbl r givn b O 9 sin cos.. Elvtion gimbl O h trnl torqu of lvtion gimbl cn b prssd s d Whr stnds for coupling torqu btwn th imuth gimbl nd th lvtion gimbl in dirctions of ch is; is th motor torqu of th lvtion gimbl; d rprsnt uncrtintis nd disturbnc including friction torqu, unmodld dnmics, wir torqu tc. B combintion of 7 nd, th dnmic modl of th lvtion gimbl cn obtind d h lvtion gimbl onl hs rottionl frdom in dirction of is, thrfor, tk its componnt of is, nd th dnmic modl of lvtion gimbl bcom d.. Aimuth gimbl Du to th rction momnt from th lvtion gimbl, th dnmic modl of th imuth gimbl bcom =ω +ω ω Whr stnds for th rction momnt from th lvtion gimbl, nd it cn b obtind b using th trnsformtion mtri C : C 5 -p.

3 ICEICE 6 h trnl torqu of imuth gimbl cn b prssd s 6 d B combintion of nd 6, th dnmic modl of th imuth gimbl cn obtind: 7 d h imuth gimbl onl hs rottionl frdom in dirction of is, thrfor, tk its componnt of is, nd th dnmic modl of imuth gimbl bcom d 8.. Driving torqu of motors h pivoting movmnt of ch gimbl wr chivd b torqu motors, thrfor, th dnmic modl of torqu motors[ for imuth gimbl nd lvtion gimbls cn b dscribd s K U nk R K 9 U nk R Whr stnd for th lctromgntic torqu gnrtd b motor; K stnd for th torqu cofficint of th motor; R stnd for th rmtur rsistnc of th motor; U stnd for th input drivn voltg of th motor. n stnd for th rduction rtio of th rducr; K stnd for bck EF constnt. Finll, th lctromchnicl dnmic modl of th photolctric trcking pltform cn obtind K U n K sin cos R d O K U n K sin cos R d O Dcoupling control Bfor dcoupling dsign, th dnmic modl should b trnsformd to form of stt spc rprsnttion. B substituting th inrti momnts nd motor prmtrs vlu from [, th diffrntil qution of th sstm cn b obtind: 6..6 sin cos.5 U d t sin cos.6 U d t Assuming tht : [ [ u [ U U [ Whr is -dimnsionl stt vctor; u stnd for th control input; stnd for th control output. h stt spc of originl sstm s cn b dscribd s: f b ug s h Whr b stnds for th input mtri, h stnds for th output mtri:.6.6sin cos f, sin cos.5 b, h [.6 It is obsrvd tht th photolctric trcking pltform is -input -output sstm with chrctristics of strong coupling nd nonlinrit.. Dcoupling nd linrition Bsd on th sstm modl, tk th drivtiv of th output with rspct to th tim t until th control vribls U nd U is plicitl includd, which cn gt th following qution:.6.6sin cos.5u sin cos.6u u 5 h cobi mtri of u with rspct to th control input vctor u cn b clcultd s follows:.5 u D 6 u.6 According to 6, dtd. orovr, th rltiv dgrs of th sstm r α = α, α =,, which stisf α + α = n, whr n is th dimnsions of th stt vctor. Bsd on th invrs sstm thor [, th originl sstm is invrtibl. Dfin th nw input vribls φ 7 B substituting 6 into nd mk [ u U U s output cn gt th invrs sstm s f b ug s 8 u D [ φ f If th invrs sstm s in sris with th originl sstm s s Figur, two psudo-linr subsstms cn b formd s Figur b. Invrsion Sstm U U Originl Sstm s s Figur dcoupling nd linrition. Controllr dsign psudo-linr subsstm psudo-linr subsstm For th psudo-linr subsstm, th sstm uncrtintis nd trnl disturbnc should b considrd, thn 7 cn b writtn s: d φ d 9 b -p.

4 AEC Wb of Confrncs Whr d nd d r th quivlnt prmtr of sstm uncrtintis nd trnl disturbnc, unknown but boundd. r Dfin th trcking rror r, in which r nd r rprsnts th rfrnc trjctor. Accordingl, r r r r d r r d hn th sliding mod surfc with intgrl ctions[5 is dsignd s follows t S c c cc d S t S c ccc d And th control input φ is dsignd s r c c cc ks ksign S r c c c c ks ksign S Whr c c c nd c r positiv constntk nd k r fid gin which stisfid with k > nd k >, k nd k r switching gin which stisfid with k > d, k > d. h control block digrm is shown in Figur. r + - Sliding mod r + - controllr. Stbilit nlsis psudo-linr sstm Figur. Control block digrm = = For th stbilit nlsis of th sstm dnmics, slct Lpunov function s V S h tim drivtiv of V is V S S S [ c c cc S [ c c c c S[ cc cc r d S[ c c c c r d Substituting into, bcoms V S S S [ k S ksign S d S[ ks ksign S d ks k d S ks k d S From Lpunov stbilit thor, it mns th dsignd control sstm is stbl, nd th sstm cn rch th sliding mod surfc in finit tim.. Chttring rstrin W cn s from tht th control lw φ contins th smbolic function sign. Although du to th istnc of it cn nsur th robustnss of sliding mod control, it cn lso ld to th chttring phnomnon of sliding mod control, ffcting th convrgnc prcision of th sstm, vn cn inspiring th unmodld dnmics nd mk th sstm unstbl. In ordr to wkn th chttring phnomnon nd strngthn th robustnss of sstm t th sm tim, th Sigmoid function[6 is dsignd to rplc th smbolic function s: Sig 5 p Whr is positiv constnt, it dtrmins th convrgnc rt of th sigmoid function. Simultion rsults o vrif th ffctivnss nd rlibilit of th dcoupling control strtg proposd in this ppr, simultions hv bn crrid out on th photolctric trcking pltform. h initil position of two gimbls is st s [, th trcking signls r givn s r sin.t, r sin.t, nd ssuming tht th disturbnc torqu of two gimbls r d.5sin t, d.sin t rspctivl. h simultion rsults r shown in Figur ~ 9: imuth gimbl position trckingrd lvtion gimbl position trckingrd rr position output trcking signl tims Figur. Position trcking of imuth gimbl position output trcking signl tims Figur 5. Position trcking of lvtion gimbl tims Figur 6. rcking rror of imuth gimbls~s -p.

5 ICEICE 6 rrrd tims Figur 7. rcking rror of lvtion gimbls~s Figur nd 5 r position trcking curvs of imuth gimbl nd lvtion gimbl rspctivl, it is obsrvd tht th sstm stt cn b rching nd prfct trcking to th dsird stt within s Figur 6 nd 7 r trcking rror curvs from s to s of imuth gimbl nd lvtion gimbl rspctivl, it is obsrvd tht th trcking rror of both gimbls r lss thn mrd. It cn b concludd tht th sstm hs fst convrgnc spd nd dnmic rspons, cn ffctivl rduc th influnc of sstm uncrtintis nd disturbnc torqu, nd stisfing th rquirmnts of control prcision. control input control input tims Figur 8 control input signl using trditionl sign tims Figur 9. Control input signl using Sig From Figur 8, it is found tht th controllr which using th trditionl smbolic function sign hs strong chttring phnomnon, nd its mplitud cn rch mor thn. Figur 8 is control input signl of which th controllr using th Sigmoid function Sig. Comprd with Figur 9, th chttring phnomnon in Figur 8 hs dcrsd significntl, nd its mplitud ws controlld t bout. It cn b concludd tht th proposd Sigmoid function is bl to rduc th chttring phnomnon ffctivl nd nsur th robustnss of th sstm. 5. Conclusions An intgrl sliding mod control strtg bsd on invrs sstm dcoupling mthod is proposd for th photolctric trcking pltform. h kinmtic rltions is nld nd th dnmic modl is dvlopd bsd on multi-bod sstm thor nd Nwton-Eulr mthod. h lctromchnicl sstm modl r linrid nd dcoupld into two psudo linr subsstms nd th strong coupling nd nonlinrit of th sstm r rsolvd. th intgrl sliding mod controllr is dsignd for th psudo linr subsstms to rjct th influnc of sstm uncrtintis nd disturbnc. B dopting th proposd Sigmoid function, h chttring phnomnon of sliding-mod controllr is rstrind. h prformnc of th proposd dcoupling control strtg is studid in simultion. It is obsrvd tht th proposd dcoupling control strtg cn ffctivl rstrin th influncs of coupling nd disturbnc, nd hs stisfctor trcking prformnc. Rfrncs.. Bhgshri, S. Kurod, P. Prkhi, th Intrntionl Confrnc on Control, Automtion nd Sstms 58. R.L. cilln,.e. Stck, K.Rokhs,. Aircrft,, S.. Dodds, A.B. Wlkr, Int.. Control, 5, Ni, IEEE. Fu. Sst. 5, Fng, Y. Rn, IEEE. Ind. Elctron. 58, 6. Fng, Y. Rn IEEE-ASE. ch. 7, 7..K. Liu, F.C. Sun, Control hor & Applictions,, W. Lu, C. Li, C. Xu, Int.. Elc. Powr, 57, 9 9..H. L, Automtic,, P.. Knnd, R.L. Knnd. IEEE. Contr. Sst..,. H. Khodddi,.R.. otlgh,. Gorji. Intrntionl Confrnc on Elctricl, Control nd Computr Enginring,. B. Kou, S. Chng, AC srvo motor nd its control Chin chin Prss, Bijing, 8. K. Li. Rsrch on Dnmics odling nd Attitud Stbilit for Floting Unmnnd Pltform Nnjing Univrsit of Scinc&chnolog, Nnjing,. X. Di, D. H, X. Zhng, t l. IEE P-Contr. 8, 5 5. V. Utkin,. Guldnr,. Shi. Sliding mod control in lctro-mchnicl sstms CRC prss, Boc Rton, 9 6. H. Kim,. Son,. L, IEEE. Ind. Elctron. 58, 69 -p.5