CHAOS CONTROL VIA ADAPTIVE INTERVAL TYPE-2 FUZZY NONSINGULAR TERMINAL SLIDING MODE CONTROL

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1 CHAOS CONTROL VIA ADAPTIVE INTERVAL TYPE- FUZZY NONSINGULAR TERMINAL SLIDING MODE CONTROL Rm Hnl Far Khabr an Najb Eonbol QUERE Laboraory Engnrng Facly Unvry o S 9 S Algra CRSTIC o Rm Chamagn-Arnn Unvry IUT Troy Franc ABSTRACT In h ar a novl rob aav y- zzy nonnglar lng mo conrollr roo o ablz h nabl roc orb o ncran rrb chaoc ym wh nrnal aramr ncran an rnal rbanc. Th lr am o hav an an orm wh nknown mahmacal mol h y- zzy ym o ovrcom h conran. A global nonnglar rmnal lng mo manol roo o lmna h nglary roblm aoca wh normal rmnal lng mo conrol. Th roo conrol law can rv ym rackng rror o convrg o zro n n m. Th aav y- zzy ym o mol h nknown ynamc o ym aj on-ln by aaaon law c rom h ably analy n Lyanov n. Smlaon rl how h goo rackng rormanc an h cnly o h roo aroach. KEYWORDS Chaoc Sym Ty- Fzzy Logc Sym Nonnglar Trmnal Slng Mo Conrol Lyanov Sably.. INTRODUCTION Chao a arclar ca o nonlnar ynamc ha ha om cc characrc ch a raornary nvy o nal conon an ym aramr varaon. Th y o chao can b nroc n vral alcaon a: mcal l racal hory lcrcal crc an cr commncaon []. Nowaay h cnc commny ha n wo roblm n chao conrol: ron an ynchronzaon. Th chao ron roblm can b n a h ablzaon o nabl roc orb UPO' o a chaoc aracor n lbrm on or roc orb wh ro n mb no h chaoc aracor []. Many nonlnar conrol chn hav bn al or chao lmnaon an chao ynchronzaon ch a lnar an nonlnar conrol chn ba on back [3-6] varabl rcr conrol [7-8] nonlnar conrol [9-] acv conrol [-4] backng gn [5-7] zzy logc conrol [8-9] an aav conrol [-]. Unornaly mo o h abov aroach mnon hav no conr h ncran an nknown aramr o h chaoc ym nrnal an rnal rbanc. Thn a l an cv conrol chm o al wh ncran m varyng ror nonlnar an bon rnal rbanc h lng mo conrol SMC. Snc hn rn conrollr ba on lng mo conrol chm hav bn roo o conrol chaoc ym [-3] Howvr major rawback n raccal alcaon h charng roblm. A lo o work hav roc o olv h roblm by ng aav conrol [4-6] nllgn aroach [7-9] an hghr orr lng mo conrol [3]. In gnral h lng rac gn a a 9

2 lnar ynamc aon = c. Howvr h lnar lng rac can only garan h aymoc rror convrgnc n h lng mo.. h o rror canno convrg o zro n n m. Th rmnal lng mo TSM ha a nonlnar rac = β whl rachng h rmnal lng mo h ym rackng rror can b convrg o zro n n m. Frhrmor TSM conrollr gn mho hav a nglary roblm. Morovr h known bon o ncran rr. Ba on TSM om nonnglar rmnal lng mo NTSM conrol ym hav bn roo o avo h nglary n TSM [3-33]. Th objcv o h ar o orc h n-mnonal chaoc ym o a r a vn ha ncran ym rnal an nrnal rbanc by ncororaon h zzy y- aroach an nonnglar rmnal lng mo NTSM conrol. W nroc an aav y- zzy ym or mol h nknown ynamc o ym an w bonary layr mho o avo a charng hnomnon. Th organzaon o h ar a ollow. Ar a cron o ym an roblm ormlaon n con II h aav y- zzy nonnglar rmnal lng mo conrol chm rn n con III. Smlaon aml monra h cnly o h roo aroach n con IV. Fnally con V gv h conclon o h avoca gn mhoology.. DESCRIPTION OF SYSTEM AND PROBLEM FORMULATION Conr n-orr ncran chaoc ym whch ha an an orm: = n n = n whr = [ n ] R h marabl a vcor nknown nonlnar conno an bon ncon h rnal R conrol n o h ym bon rbanc an D rrn h ncran < F whr F an ar ov conan. Th conrol roblm o g h ym o rack an n- mnonal r vcor y whch blong o a cla o conno ncon on [ ]. L h rackng rror a; = y = [ y = [... n n y K y ] 3 n ] Thror h ynamc rror o ym can b oban a;

3 = = 3 M n = y n 4 Th conrol goal conr ha; lm = lm y 5.. Trmnal Slng Mo Conrol W conr a con orr nonlnar ym h convnonal TSM crb by h ollowng r orr rmnal lng varabl; = β 6 whr > ar ov o ngr. Th cn conon o nr h ranon rajcory o h rackng rror rom aroachng ha o h lng on : β > a gn conan an = η 7 whr η > a conan. I known an r o ncran an rnal rbanc an whn h ym rrc o h = wll b govrn by an valn conrol oban by: = y β 8 Th global conrol como o h valn conrol an conno rm ch ha; = k gn 9 whr k k > wchng gan by ang h rm o 8 w oban h global conrol: TSM = y β k gn whch nr ha TSM occr. Thn w can choo wchng gan a ollow: k = η D Whr D =. I clar ha h rackng rror wll rach h lng mo = whn h n m r whch a; r η

4 So h aanng m rom r o r =. In h ha h lng mo = rach.. h ym ynamc rmn by h ollowng nonlnar rnal aon: β = 3 By ngrang h rnal aon = r β w hav: = β 4 From TSM conrol h rm conanng may ca a nglar roblm... Non Snglar Trmnal Slng Mo Conrol In orr o ovrcom h nglary roblm n h convnonal TSM ym h roo NTSM mol crb a ollow: = 5 β whr β an < < hav bn n n 6. For ym wh h nonnglar lng mo manol 5 h conrol gn a; NTSM = y β k gn 6 Th o ay h ranon conon 7 h m rvav o : = y 7 β Ung conrol law 6 = k β Ar om manlaon w oban: gn k gn β β η Snc < < w hav > whn hn; or = η > β 9 Thror h conon or Lyanov ably a whn an h rackng rror can rach h lng mo = whn n m. Sbng h conrol 6 no ym 4 yl; β > an ar ov o ngr 8

5 = β k gn Whn = w oban = D η gn η or > η or < Thn or a mall ε ε > hr a vcny o = ch ha < ε hror concl ha h NTSM manol 5 = can b rach n h ha lan n n m. No ha n conrol law 6 h nonlnar ncon nknown. Thn h ro o h ar o aroma by nrval y- zzy logc ym an o lmna charng a araon ncon can b o rlac h gn ncon n wchng rm. Th aaaon law o ajabl aramr o h zzy ym c rom h Lyanov ably. 3. ADAPTIVE INTERVAL TYPE- FUZZY NON-SINGULAR TERMINAL SLIDING MODE CONTROL In h con h aav zzy ym o aroma h nknown ncon ha h am rcr a h o zzy ym ng h cnr o mho [34] hn w rlac by ˆ ch a: T ˆ = ξ whr ajabl vcor aramr. In orr o garan h global ably o clo loo ym wh h convrgnc o rackng rror o zro w roo h ollowng conrol law: NTSM = ˆ y β k gn To rv h aav law o w n h omal aramr vcor * = argmn ˆ Ω Ω whr n a; Ω whr = Ω an Ω ar conran o abl bon on : M Ω = { : M } M an M ar ov conan. W n h mnmm aromaon rror a; * a; an rcvly hy ar 3

6 4 ˆ * w = W can wr T M F w * ξ By ng α = M F can b aly concl ha w bon w α.. w L. To y h clo loo ably an o n h aaaon law o ajabl aramr w conr h ollowng Lyanov ncon: T V = γ ~ ~ ~ whr ~ * = an γ arbrary ov conan o h m rvav o : T V = γ ~ 3 Ung h conrol law an h m rvav o h NTSM manol 5 bcom: gn ˆ ˆ ˆ gn ˆ * * k k = = gn * k w = ξ 4 ch ha β =. Th bon o 4 n 3 wll b: = ~ gn k w V ξ γ γ 5 By choong h ollowng aaaon law: = ξ γ 6 whr = ~ hror w oban: gn V w k w k w k D = = =

7 Thn η V w α η 7 From h nvral aromaon horm c ha α wll b vry mall no al o zro n h aav zzy ym an >. So w hav V <. Th ovrall chm o h aav y- zzy nonnglar rmnal lng mo conrol n rnc o ncran rnal rbanc an h ranng aa corr wh nrnal no hown n Fgr. B IT-FLS Chaoc ym = n n = = [ ] R n n - y NTSM = y β ˆ = γ ξ kgn ξ ˆ Fgr. Ovrall chm o h aav y- zzy nonnglar rmnal lng mo conrol ym. 4. SIMULATION EXAMPLE Th abov crb conrol chm now o conrol h a o chaoc ym whch n a ollow; = 8 3 =.4..co.8 Wh nal a: = [. ] T. For r n h mlaon rl o ym ar hown n Fgr. 5

8 3 Sa ron m Fgr. Tm ron an ycal chaoc bhavor o ng ocllaor In orr o orc h a ym = o rack h rrnc rajcor y an y y = π / 3 n.3n3 h aav nrval y- n n m ch a zzy nonnglar rmnal lng mo conrol a no h ym a ollow: = 3 =.4..co.8 W choo γ = 5 = 3 = 5 an β = h TSM an NTSM manol ar lc a TSM / / an NTSM / β = β 9 = rcvly. To gn h zzy ym ˆ w n vn y- Gaan mmbrh ncon nng = lc a F l =...7 ar hown n abl. wh varanc σ =.5 an nal val = Ο 7. l Tabl. Inrval Ty- Fzzy Mmbrh Fncon For =. Man Man m m m m µ -.5 µ F F µ µ F F µ µ F µ F F In h con wo conrol law ar ao aav y- zzy nonnglar rmnal lng mo conrol AT- FNTSM crb n an aav y- zzy rmnal lng mo conrol AT-FTSM whch gn a ollow; TSM = ˆ y β k gn 6

9 Th mlaon rl ar rn n h rnc o ncran = n π n 3 π rnal rbanc = n an wh Gaan no al o h mar gnal = wh Sgnal o No Rao SNR=B. A bonary layr mho o lmna charng. 4.. Aav Inrval Ty- Fzzy Trmnal Slng Mo Conrol AT-FTSM Th rackng rormanc o a hown n Fgr 3. Th conrol n an h ha-lan rajcor o ym ar rrn n Fgr y y' m Fgr 3. Th o rajcor o..5 al lng mo raccal rajcory yy' Fgr 4. Pha-lan o rackng rror an ycal chaoc bhavor o ng ocllaor 7

10 m Fgr5. Conrol n 4.. Aav Inrval Ty- Fzzy Non-nglar Trmnal Slng Mo Conrol AT-FNTSM..5 y y' m Fgr6. Th o rajcor o al lng mo raccal rajcory yy' Fgr7. Pha-lan o rackng rror an ycal chaoc bhavor o ng ocllaor 8

11 m Fgr8. Conrol n Accorng o h abov mlaon rl w can ha boh conrollr rov a goo rackng o o ym o hr rajcor n n m. Frhrmor a nglary roblm occr n h ca o AT-FTSM conrol a hown n Fgr 5. Th roo aroach allow obanng a mooh conrol gnal Fgr 8 hn h NTSM manol 5 can lmna h nglary roblm aoca wh convnonal TSM manol. 5. CONCLUSION In h ar h roblm o ablzaon orb o nonlnar ncran chaoc ym n h rnc o rnal nrnal rbanc an rbanc olv by ncororaon o nrval y- zzy aroach an non-nglar rmnal lng mo conrol. In orr o lmna h charng hnomnon cnly a bonary layr mho an an aav nrval y- zzy ym nroc o aroma h nknown ar o ym. Ba on h Laynov ably crron h aaaon law o ajabl aramr o h y- zzy ym an h ably o clo loo ym ar nr. A mlaon aml ha bn rn o llra h cvn an h robn o h roo aroach. REFERENCES [] E. Tllo-Cal Chaoc Sym Janza Trn 9 5 Rjka Croaa. [] I. Zlnka S. Clkovky H. Rchr G. Chn Evolonary Algorhm an Chaoc Sym S n Comaonal Inllgnc vol. 67 Srngr-Vrlag Brln Hlbrg. [3] R. Fma G. Sol-Pral 8 Rob Synchronzaon o Chaoc Sym va Fback Srngr-Vrlag Brln Hlbrg. [4] X. Xao L. Zho Z. Zhang 4 Synchronzaon o chaoc Lr ym wh anz aml-aa conrollr Commn Nonlnar Sc Nmr Smla vol [5] L-X. Yang Y-D. Ch J.G. Zhang X-F. L Y-X. Chang 9 Chao ynchronzaon n aonomo chaoc ym va hybr back conrol Chao Solon an Fracal vol [6] M.T. Yan 5 Conrollng chao an ynchronzaon or nw chaoc ym ng lnar back conrol Chao Solon an Fracal vol [7] S. Vayanahan Hybr ynchronzaon o hyrchaoc l ym va lng mo conrol Inrnaonal Jornal o Chao Conrol Mollng an Smlaon IJCCMS Vol. no.. [8] F. Farvar M. A. Shoorhl M. A. Nko M. Thnhlabc Chao conrol an gnralz rojcv ynchronzaon o havy ymmrc chaoc gyroco ym va Gaan raal ba aav varabl rcr conrol Chao Solon Fracal vol [9] J. H. Park 5 Chao ynchronzaon o a chaoc ym va nonlnar conrol Chao Solon an Fracal Vol [] Q. Zhang J-a. L 8 Chao ynchronzaon o a nw chaoc ym va nonlnar conrol Chao Solon an Fracal vol [] J. H. Park 6 Chao ynchronzaon bwn wo rn chaoc ynamcal ym Chao Solon an Fracal vol

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13 Ahor Rm Hnl rcv hr ngnrng an Mar gr n Aomac rom S Unvry S Algra n 9 an rcvly. From Novmbr h Ph.D. n n h Engnrng Facly wh h QUERE laboraory a h Unvry o S. Hr rarch nr ar hghr orr lng mo conrol zzy y- an y- ym nonlnar ym Far khabr rcv h D.E.A n 99 an h Mar n 99 gr n nral conrol an h PhD n 6 rom S Unvry S Algra n aomac conrol. H crrnly a Proor n h Engnrng Facly rom h am nvry. H rarch nr ncl mlvarabl aav conrol LMI conrol an y- zzy conrol o rnwabl nrgy ym Najb Eonbol rcv h Mar rom h Unvry o Scnc an Tchnology o Marrakch FSTG n Morocco h D.E.A. n h Ph.D. n 4 an Hablaon rom Rm Unvry o Chamagn- Arnn all n Elcrcal Engnrng. From Smbr 5 o h ha bn an Aan Proor wh IUT o Troy Rm Chamagn Arnn Unvry. H a crrnly a Proor an Ha o h Mchancal Engnrng Darmn o IUT a Troy Rm Unvry. H crrn rarch nr ar n h ara o zzy logc conrol rob aav conrol rnwabl nrgy an conrol rv. 3