Adaptive control and synchronization of the Newton-Leipnik systems
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- Domenic Leonard
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1 ISSN Englnd UK Jornl o Inormtion nd Compting Scinc Vol. No. 8 pp Adptiv control nd snchronition o th Nwton-Lipni sstms Xdi Wng Cho G Nonlinr Scintiic Rsrch Cntr Fclt o Scinc Jings Univrsit Zhnjing Jings P.R.Chin. Rcivd 9 8 ccptd Agst 8 Astrct. his ppr irstl introdcs th chotic sstm Nwton---Lipni sstm which posssss two strng ttrctors.ectiv dptiv controllrs r proposd or stiliing chos to nstl qiliri. In ddition Chos snchronitions chivd mploing ctiv control schm. Nmricl simltions r providd to vri th siilit nd ctivnss so th rslt o th control is mtll vriid with th thorticl nlss nd nmricl simltions. Kwords: dptiv control snchronition Nwton-Lipni sstms nmricl simltions. Introdction Sinc th sminl wor o Ott Grogi nd Yor OGY [] thr hs n n incrsing intrst in rcnt rs in th std o controlling chotic sstms in phsics mthmtics nd nginring commnit tc. Dirnt tchniqs nd mthods [-7] hv n proposd ovr th lst dcd. orovr som o ths mthods hv n sccssll pplid to primntl sstms. hror controlling nd snchronition o th chos hv n vr importnt gols nd sjcts o mch crrnt rsrch. In 98 Nwton nd Lipni constrctd st o dirntil qtions rom Elr s rigid od qtions which wr modiid with linr dc [8]. hn in B. rlin stlishd th istnc o closd orits which wr not smptoticll stl or this sstm [9].In Chn t l. [] stdid chos control nd snchronition o th Nwton Lipni sstm or th irst tim sing stl-mnioldsd mthod. Atrwrds Richtr rthr stdid th stilition o dsird motion within on ttrctor s wll s ting th sstm dnmics rom on ttrctor to nothr ppling th Lpnov nction mthod []. or rcnt stdis Wng nd in [] showd tht this chotic sstm cn controlld to nstl priod orits nd tors with sitd linr controllr. In this ppr Nwton Lipni sstm is controlld with dptiv chos control mthod. At th sm tim w s th sm mthod to nl stilition o chotic motion to std stt s wll s snchronition twn two idnticl sstms. Comptr simltion is lso givn or th prpos o illstrtion nd vriiction.. Adptiv control o Nwton Lipni sstm h Nwton Lipni sstm is dscrid. 5 5 Whr r positiv prmtrs. h Nwton Lipni sstm is chotic sstm with two strng ttrctors. For th sstm prmtr.. 75 nd initil stts.9.6 nd.9.8 w cn otin th two strng ttrctors which r dmonstrtd in Fig.nd. his chotic sstm hs iv qiliri: E-mil ddrss: wd959@js.d.cn. Plishd World Acdmic Prss World Acdmic Union
2 8 Xdi Wng t l: Adptiv control nd snchronition o th Nwton-Lipni sstms O E.65/ / E.65/ / E.65/ / E.65/ / Fig.. Chotic ttrctor o Nwton Lipni sstm initil point in.9.6. Fig.. Chotic ttrctor o Nwton Lipni sstm initil point in.9.8. Lt s ssm tht th qtions o th controlld Nwton-lipni sstm r givn. 5 5 Whr nd r trnl control inpts. It will sitl dsignd to driv th trjctor o th sstm spciid to ch o th iv nstl qilirim points o th ncontrolld i.. sstm E E E nd E. For this prpos th gol o control is to ind O 5 [ controllr ] qiliri is smptoticll stl. nd th prmtrs stimtion pdt lw or Eq. sch tht ch o th iv.. Stiliing th qilirim O Assm th prmtrs nd r nnown constnt prmtrs w choos Lpnov nction or s ollows: V X X X Whr X nd r stimt vls o th nnown prmtrs rspctivl. h tim drivtiv o V long trjctoris is V X. 5 5 W choos th controllr s ollows: JIC mil or contrition: ditor@jic.org.
3 Jornl o Inormtion nd Compting Scinc 8 pp nd th prmtrs stimtion pdt lw s ollows With this choic th tim drivtiv o V X is givn V X. It is clr tht V X is positiv dinit nd V X is ngtiv dinit in th nighorhood o th ro soltion or th sstm. hror th qilirim soltion O o th controlld sstm is smptoticll stl... Stiliing non-ro qiliri E E E nd E h qilirim points o th controlld Nwton-Lipni sstm r dtrmind rom th soltion o th ollowing sstm: Whr. 5 5 nd i r th stt vrils nd control nctions t th qilirim points o i th controlld sstm. Now w procd to otin th prtrd qtions o th controlld sstm ot its qilirim points nd i. For this prpos w introdc th ollowing vrils: i And vi i i i 5 Sstitting Eq.5 into. W otin th prtrd qtions ot th qilirim points o th sstm in th ollowing orm: v. 5 v 6 5 v his sstm dmits th spcil soltion v i. his soltion dscris th qilirim i i points o th sstm. Assm tht som o qilirim points r nstl thn th control cn md sch tht ths stts com smptoticll stl. Lt th prmtrs nd r nnown constnt prmtrs w choos Lpnov nction or 6 s ollows: V Whr nd r stimt vls o th nnown prmtrs rspctivl. h tim drivtiv o V long trjctoris 6 is V v. 5 v W choos th controllr v v v v ] 5 v [ s ollows: JIC mil or sscription: plishing@wau.org.
4 8 Xdi Wng t l: Adptiv control nd snchronition o th Nwton-Lipni sstms v v v nd th prmtrs stimtion pdt lw 7 5 s ollows 5 With this choic th tim drivtiv o V is givn V. his trnslts to lim t t R hror th qilirim soltiono o th sst6 is smptoticll stl. According to 5 nd choosing i i th qilirim soltions o sstm r smptoticll stl i th controllr is nd th prmtrs stimtion pdt lw is 8. hror non-ro qiliri E E nd E o r smptoticll stl i th controllr is 9 nd th prmtrs stimtion pdt lw is 8.. Nmricl simltions E In this sction nmricl simltions r crrid ot. In ddition tim stp si. is mplod. W slct th prmtrs o Nwton-lipni sstm s.. 75 so tht Nwton-lipni sstm hiits chotic hvior. h initil stts o th controlld Nwton-lipni sstms r nd th initil vls o th prmtrs stimtion pdt lws r 5 5. Fig. shows tht th chos in Nwton-lipni sstm is controlld to qilirim point O in prsnc o sstm s ncrtin prmtrs with th control lw nd th prmtrs stimtion pdt lw. Fig. d shows tht th chos in Nwton-lipni sstm is controlld to qilirim point E E E nd E in prsnc o sstm s nnown ncrtin prmtrs with th control lw 9 nd th prmtrs stimtion pdt lw 8. 8 JIC mil or contrition: ditor@jic.org.
5 Jornl o Inormtion nd Compting Scinc 8 pp t Fig.. h tim rspons o stts or th controlld sstm with control lw nd th prmtrs stimtion pdt lw : stiliing th qilirim O t t t 5 t c Fig.. h tim rspons o stts or th controlld sstm with control lw 9 nd th prmtrs stimtion pdt lw 8: stiliing th qilirims E nd E rspctivl. -. d E E. Adptiv snchronition o Nwton-lipni sstm In ordr to osrv th snchronition hvior in Nwton-lipni sstm w ssm tht w hv two idnticl Nwton-lipni sstms nd tht th driv sstm with th sscript is to control th rspons sstm with sscript. h driv nd rspons sstms r dind s ollows: JIC mil or sscription: plishing@wau.org.
6 Xdi Wng t l: Adptiv control nd snchronition o th Nwton-Lipni sstms 86 whr th prmtrs nd r nnown or ncrtin nd th rspons sstm is 5 5. whr nd r prmtrs o th rspons sstm which nd to stimtd nd is th controllr w introdcd in Eq.. Sppos tht ] [ whr nd r th rror stts which r dind s ollows: And whr nd > r positiv rl constnts. horm h two sstms 9 nd cn snchronid ndr th controllr nd prmtr stimtion pdt lw i th ollowing conditions r stisid: > > > whr > nd 5 > r positiv rl constnts. Proo. It is s to s rom 9- tht th rror dnmicl sstm cn otind s ollows: Lt. Choos th ollowing Lpnov nction s ollows: V thn dirntition o V long trjctoris 5 is V choos > nd 5 > thn JIC mil or contrition: ditor@jic.org.
7 Jornl o Inormtion nd Compting Scinc 8 pp V < Whr ] nd [. 5 5 P 5. Oviosl to nsr tht th origin o rror sstm 5 is smptoticll stl th mtri P shold positiv dinit which implis tht V is ngtiv dinit. his cs is stisid i nd onl i th ollowing inqlitis hold: > 5. > 5.. Accordingl th rspons sstm is snchroniing with th driv sstm 9 ndr th controllr nd prmtr stimtion pdt lw i th conditions r stisid. h proo is compltd. 5. Nmricl simltions In this sction nmricl simltions r crrid ot. In ddition tim stp si. is mplod. W slct th prmtrs o Nwton-lipni sstm s.. 75 so tht Nwton-lipni sstm hiits chotic hvior. h initil vls o th prmtrs c r ro h initil stts o th driv nd rspons sstms 9nd r.9.6 nd >. In ordr to choos th control prmtrs > 5 nd > nd to stimtd. hrogh simltions w otin. h control prmtrs r chosn s ollows. Choos. h stt trjctoris or th 6 driv sstm 9 nd th rspons sstm with th control lws nd prmtrs stimtion pdt lw r shown in Fig.5 c. h dnmics o chnging prmtrs is shown in Fig t t JIC mil or sscription: plishing@wau.org.
8 88 Xdi Wng t l: Adptiv control nd snchronition o th Nwton-Lipni sstms t c Fig. 5. h tim rspons o stts or driv sstm 9 nd rspons sstm with th control lw nd prmtrs stimtion pdt lw ; signls nd signls nd c signls nd t 6. Conclsion Fig. 6. Chnging prmtrs o sstm with tim t. his wor ddrsss dptiv chos control nd snchronition o Nwton-lipni sstm whn th prmtrs o th driv sstm r ll ncrtin nd dirnt with thos o th rspons sstm. Bsd on Lpnov stilit thor th sicint conditions or th snchronition hv n otind nlticll. Nmricl simltions r shown to vri th proposd mthod. 7. Acnowldgmnts Rsrch ws spportd th Edction Fondtion o Jings Provinc No: 5KJB8 nd Fondtion or Advncd Rsrchrs o Jings Univrsit No: Rrncs [] E. Ott C. Grogi J.A. Yor. Controlling chos Phs. Rv. Ltt. 9 6: 96. [] G. Chn X. Dong. On dc control o chotic continos tim sstms IEEE rns. Circit Sstms. 99 : 59. [] Yssn. Chos control o Chn chotic dnmicl sstm. Chos Solitons Frctls. 5: 7. [] Yssn. Controlling chos nd snchronition or nw chotic sstm sing linr dc. Chos Solitons Frctls. 5 6: 9 JIC mil or contrition: ditor@jic.org.
9 Jornl o Inormtion nd Compting Scinc 8 pp [5] EN Snch JP Pr rtin G Chn. Chos stilition: n invrs optiml control pproch. Ltin Am Appl Rs: Int J. :. [6] Yssn. Adptiv control nd snchronition o modiid Ch_s circit sstm. Appl th Comp. 5:. [7] -L Lio S-H Lin. Adptiv control nd snchronition o Lorn sstms. J Frnlin Inst : 95. [8] RB Lipni A Nwton. Dol strng ttrctors in rigid od motion with linr dc control. Phs Ltt A 98 86:6 7. [9] A rlin Bnjmin. Priodic orits in th Nwton Lipni sstm. Int J Birc Chos. :5-. [] S Chn Q Zhng J Xi t l. A stl-mniold-sd mthod or chos control nd snchronition. Chos Solitons Frctls. : [] H Richtr. Controlling chotic sstm with mltipl strng ttrctors. Phs Ltt A. : 8-8. [] X Wng L in. Birction nlsis nd linr control o th Nwton Lipni sstm. Chos Solitons Frctls. 6 7: -8. JIC mil or sscription: plishing@wau.org.
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