Design of Fuzzy Sliding-Mode Controller for Chaos Synchronization

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1 Dsg o Fuzzy Slg-Mo Corollr or Chaos Syhrozao Chao-L Kuo Chg-Sho Shh Cha-Hug L a Shu-Pg Shh 3 No. 49 Jug-Hua Roa Hs-Shh Towshp Taa Couy 744 Tawa R.O.C. Dparm o Elral Egrg Far-Eas Uvrsy lkuo@.u.u.w Lu-Chu Hsag Kaohsug Couy 8 Tawa R.O.C. Dparm o Elral Egrg Kao-Yua Uvrsy 3 Y Chau Kaohsug Couy 84 Tawa R.O.C. Dparm o Compur a Commuao Shu-T Uvrsy Absra. Ths papr prss a uzzy orollr o solv a masr-slav haos syhrozao problm. A rs h mho o raoal slg mo orol s osr whh ulzs h souous sg uo o mak h sysm sa rahg a slg sura. N uzzy ruls ar rm aorg o h Lyapuov horm a h uzzy orollr s sg or haos syhrozao. Fally a ampl o haos syhrozao or a ura Dug-Holms sysm s prs o llusra h valy a asbly o h propos orollr. Irouo A haos syhrozao problm mas makg boh hao osllaors bhav aly h sam. Grally wo hao sysms syhrozao ar all a rv sysm a a rspos sysm rspvly. Chaos syhrozao a b appl may aras suh as hmal raos powr ovrrs sgal pross ommuao a bologal sysms [ ]. Thr ar may mhos or syhrozao o a hao sysm suh as aapv orol mho [5 6 4] baksppg orol mho [7] H orol mho [] slg mo varabl sruur orol mho [ 3 6] a uzzy orol mho [5]. Zah [7 8] a a uzzy s hory. Th uzzy log orol shms hav b wly vlop or almos 4 yars a hav b sussully appl o may applaos [3 5]. W a asly apply h uzzy log orol o orol a ll-mol sysm by prms o skll opraors. Alhough hr hav b som sussul applaos o uzzy log orol sll has som rawbaks h sg prour. For ampl h uzzy orol ruls ar o pr-or a suabl mmbrshp uos shoul b gv by m-osumg ral-a-rror prours. Bss h yam bhavor o orol sysm ao b sp prsly. I r yars som haos syhrozao bas o uzzy sysm has b propos [5]. J.-W. Park T.-G. Km a Y.-B. Km Es.: AsaSm 7 CCIS 5 pp Sprgr-Vrlag Brl Hlbrg 7

2 Dsg o Fuzzy Slg-Mo Corollr or Chaos Syhrozao 37 I hs papr w ar vo o h rsarh o sg uzzy orollr or syhrozg h sa raors o wo Dug-Holms sysms wh ral al oos sysm uras a ral surbas. Ths papr s orgaz as ollows: So srb h yams o mar-slav haos syhrozao sysm. I so 3 srb h sg approahs o uzzy orollr. Numral smulaos ha orm h valy a asbly o propos mho ar show So 4. Fally olusos ar gv So 5. Sysm Dsrpo a Problm Formulao Cosr h ollowg wo -msoal hao sysms R y y y R y y Δ y u whr u R s a orol pu s a gv olar uo o a Δ y s a ura rm rprsg h umol yams or sruural varao o h sysm a s h surba o sysm. I gral h ura rm Δ y a surba rm ar assum bou.. Δ y < α a < β 3 whr α a β ar posv. I s assum ha y a Δ y sasy all h ssary oos suh ha sysm a hav a uqu soluo h m rval [ > or ay gv al oo a y y. Th yams o sysm splay a hao moo whou orol pu u. Th orol problm osr hs papr s ha or r al oos o sysms a h wo oupl sysm.. h masr sysm a h slav sysm o b syhroz by sgg a appropra orol u whh s aah o h slav sysm suh ha lm y 4 whr s h Eula orm o a vor. 3 Fuzzy Corollr or Dsg Mhoology I hs so w wa o arss h slg-mo orol a sg prours o a uzzy orollr or haos syhrozao.

3 38 C.-L. Kuo al. 3. Slg-Mo Corol L h rror sa b y a g h rror yam quaos ar. ; u g Δ 5 Usg h op o sysms h saarz sa spa quaos o h rror sas a b oba as u g u g Δ Δ ; 6 Sysm 6 s o h orollabl aoal orm. I suh a as hr ar o ral yams []. Bas o h orol law propos by [] h slg sura a b as s 7 whr os h al sa o. Eq. 7 a also b ormula as 8 wh h al oo a h slg mo yams a b srb by h ollowg sysm o quaos: ; 9 or a mar quao orm as ; A wh h al sas bg T ] [. Th sg paramrs a b rm by hoosg h gvalus o A suh ha h orrspog harars polyomal

4 Dsg o Fuzzy Slg-Mo Corollr or Chaos Syhrozao 39 P s Hurwz. Ths gvalus ar also rlav o h sp o sysm rspos. 3. Fuzzy Corollr Dsgs Th uzzy log orol aros rom h sr o srb ompl orol wh lgus srpos. Th uzzy log orol s asy o ursa a smpl o mplm baus uzzy log mulas huma orol. Th uzzy orollr a pr sysms hav b sussully appl may ompl usral prosss. I hs so w ulz h uzzy log a uzzy proposos o sg h orollr. Th blok agram o uzzy orollr wh haos sysm s llusra Fg.. W hav h rror sa rom h masr a slav sysms. Th uzzy orollr a rm u by h rror a sysm sas. Fg.. Dagram o h uzzy orollr Th uzzy orollr has a wo-pu a a sgl-oupu. Ipu varabls ar h ormalz slg uo 7 a h rvav o slg uo s. Th ovrall orol oupu s hos as F s s. u whr F s s os h uoal hararss o h uzzy lgus so shms. Th mmbrshp uo o h pu lgus varabls s a s a h mmbrshp uos o h oupu lgus varabl u ar show Fg. rspvly. Thy ar ompos o sv uzzy paros prss as gav bg NB gav small NS zro ZE posv small PS a posv bg PB. Th uzzy rul abl s sg Tabl. Th rahg law a b hos as s k F s s. 3 whr k s a posv osa valu. From Eqs. 7 a 3 w a oba

5 4 C.-L. Kuo al. s kk k F s s. 4 k Th ral quao o orol pu u s u I h ral worl h ral surba [ g Δ y ] k F s s k k k. 5 a h sysm uray Δ y ar ukow. So h mplm orol pu s srb as u g k k k F s s. 6 k Ar grao o Eq. 6 w a oba h orol pu o h slav sysm u g kk k F s s. 7 k Fg.. Th mmbrshp uos o pu varabls a oupu varabl Tabl. Ruls-abl o uzzy orollr Fs s s NB NS ZE PS PB NB NB NB NB NS ZE s NS NB NB NS ZE PS ZE NB NS ZE PS PB PS NS ZE PS PB PB PB ZE PS PB PB PB Thorm. Cosr h masr-slav sysm a h wo sysms ar syhroz by h orollr u 7 or h slav sysm. Th h rror sa raory ovrgs o h slg sura s. Proo. W a Lyapuov uo as

6 Dsg o Fuzzy Slg-Mo Corollr or Chaos Syhrozao 4 s V. 8 Takg h m rvav o Eq. 6 w hav V ss s [ Δ y ] k F s s. 9 L [ Δ y ] s bou a [ Δ y ] k. Thr w hav V k s. Th rahg oo s s < s maa a lm s. Ths ompls h proo. 4 Numral Eampl I hs so smulao rsuls ar prs o mosra h vss o h propos uzzy slg-mo orollr or haos syhrozao problm. Cosr wo oupl Dug-Holms sysms as ollows 3 p p q os ω y y 3 y p y p y y Δ y q os ω u. Th so quao o h slav sysm s prurb by a uray rm Δ y a rr wh a surba a h orol pu u s aah o h slav sysm. L us h syhrozao rrors bw h masr sysm a slavr sysm as y a y. Subrag rom w hav h syhrozao rror yams as 3 p p y Δ u. Th h saarz sa spa quaos a b srb as p p3 3y y 3 Δ u 3 4

7 4 C.-L. Kuo al. L h slg sura b as s Th gvalus orrspog o h slg sura a b by [ ] a hs gvalus oma h ovrgg ra o h rror 3 yams. Thy a arbrarly b assg. Choos h rahg law as Eq. 7. Th orol pu s rm as k F s s } u { p p3 3y y wh h al oo u. 6 Fg. 3. Hyprhao bhavor o h Dug-Holms osllaor pla Fg. 4. Th m rspos o a y

8 Dsg o Fuzzy Slg-Mo Corollr or Chaos Syhrozao 43 Fg. 5. Th m rspos o a For h masr-slav syhrozao orol sysms a h paramrs ar p. p. 4 q. a ω. 8 h masr sysm splas hao bhavor [9]. Th masr sysm wll hb a hyprhao bhavor as show Fg. 3 whr h araor s show o h pla by gvg al oos. a.. I s suppos ha h surba. os π a h uray rm Δ y.5y h slavr sysm. Th gvalus orrspog o h slg sura ar hos as [ 3] [ 6 ] a o o h slg-mo orollr s hos as k 5. Th smulao sp sz was. s. y Fg. 6. Th rror sa rsposs

9 44 C.-L. Kuo al. Th smulao rsuls wh al oos.. y.3 a y. 3 ar show Fg. 4 a Fg. 5. Thos show ha h slav a h masr sysms a rah syhrozao wh orol oprao h slav sysm a ss. Fg. 6 shows h rror sa rsposs. Th orol pu o slav sysm was mosra Fg. 7. Fg. 7. Th orol pu o slav sysm 5 Colusos I hs papr uzzy slg-mo orol or haos syhrozao has b propos. Th uzzy orollr bas o h slg-mo a Lyapuov sably hory whh s sg or h rgulao o h rror sa vor o a sr po h sa spa. Numral smulao rsuls mosra ha h propos mho a b sussully appl o syhrozao problms o a Dug-Holms sysm. Th rv orollrs ar robus so ha h los-loop sysm s sabl h prs o uras a surba. Th harg phomo o ovoal swhg yp slg orols os o our hs suy. Akowlgm Ths work s suppor by h Naoal S Coul Tawa Rpubl o Cha ur h pro NSC96-8-E Rrs. Carroll T.L. Pora L.M.: Syhrozg hao sysm. IEEE Tras. Cr. Sys. I: Fuam. Thory Appl Ch C.L. L W.Y.: Slg mo orol or o-lar sysm wh global vara. Pro. Is. Mh. Egrs

10 Dsg o Fuzzy Slg-Mo Corollr or Chaos Syhrozao Ch G. Fog G.: Chaoao va arbrarly small bak orols: hory mho a applaos. I. J. Bur. Chaos Ch G. Dog X.: From haos o orr: mhoologs prspvs a applaos. Worl S Sgapor Lao T.-L.: Aapv syhrozao o wo Lorz sysm. Chaos Solos & Fraals La K.-Y. Lu P. Chag T.-S. Chu C.-S.: Aapv syhrozao sg or hao sysms va a salar rvg sgal. IEEE Tras. Cru Sys. I Lu J. Zhag S.: Corollg Ch s hao araor usg baksppg sg bas o paramrs ao. Phys L. A Lü J. Ch S.: Chao m srs aalyss a s applao. Wuha Uvrsy Prss Cha 9. Nayh A.H.: Appl olar yams. Wly Nw York 995. Slo J.E. L W.: Appl olar orol. Pr-Hall Eglwoo Cls 99. Suyks J.A.K. Curra P.F. Vawall J.: Robus olar syhrozao o hao Lur sysm. IEEE Tras. Cru Sys. I Tamasvus: Rproubl aalogu ru or hao syhrozao. Elro. L Taaka K. Ika T. Wag H.O.: A u approah o orollg haos va LMI-bas uzzy orol sysm sg. IEEE Tras. Cr. Sys. I Yau H.-T.: Dsg o Aapv Slg Mo Corollr or Chaos Syhrozao wh Ura. Chaos Solos & Fraals Yau H.-T. Kuo C.-L. Ya J.-J.: Fuzzy Slg Mo Corol or A Class o Chaos Syhrozao wh Uras. Iraoal Joural o Nolar Ss a Numral Smulao Y X. R Y. Sha X.: Syhrozao o sr spaomporal haos by busg varabl sruur orol. Chaos Solos & Fraals Zah L.A.: Oul o a Nw Approah o Aalyss o Compl Sysm a Dso Pross. IEEE Trasaos o Sysms Ma a Cybrs Zah L.A.: Fuzzy Log. IEEE Compur