Reconfigurable Takagi Sugeno Fuzzy Logic Control for a Class of Nonlinear System considering Communication Time Delays on Peripheral Elements

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Cordelia Walker
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1 Reonfbe Seno Fzzy Lo Conto fo Css of onne System onsden Commnton me Deys on ee Eements enítez-éez H.*, Cádens-Foes F. nd Gí-oett F., bstt odys te stdy of fts nd te onseenes beomes n sse nto y sfety t omte neto systems. Ho to bond te effets of ft nd o to te tem nto dynm system s st n oen fed. In ee n o to te ts obem s esented. e se of Seno fzzy o stte s ven n ode to oms to enes, te esene of o fts nd te esetve tme deys tn e-tme dstbted system. s o s sed s eonfbe sttey odn to ommnton tme deys tn e-tme dstbted system. s o s sed s eonfbe sttey odn to ommnton deys. I. IRODUCIO e emeene of smt senso nd tto tenooy emoves te need fo entsed onto t feedb oos to dmb ee ttos en t t dtbs onneton []. s ves n tonomos tto nstton [] s e s o onto, sef-bton, et monton nd eonfton vbtes. Seve sttees fo mnn tme dey tn onto s ve been stded fo dffeent ese os. Fo nstne [] ooses te se of tme dey seme nteted to eonfbe onto sttey bsed on stost metodooy. On te ote nd, [4] ooses eonfton sttey bsed on efomne mese fom mete estmton ft dnoss oede. note sttey s been oosed by [5] ee tme deys e sed s netntes, modfy oe ement of obst onto. [6] esent n nteestn ve of ft toent onto o eted to tme dey on. Reonfbe onto s been stded fom te ont of ve of stt modfton sne ft ene s esented by [7] t efoms ombned modfton of system stte nd dynm systems s stded by [8, 9, ]. esent o tes tme deys de to ommnton s detemnst mesed vbes. s e s tto ft esene by modfton of mt n ode to oose Seno Fzzy onto t to ondtons oose of o ee eements nd te eted tme deys. In ee fzzy o onto [] ves tme deys s est of detemnst enítez-éez H.* s t Detmento de Ineneí de Sstems Comtones y tomtzón, IIMS,UM, do. ost De..Obeón, Méo D.F.,, Méo. F: , e: (*) Em: eto@de4.ms.nm.m Gí-oett F.., Detmento de Ineneí de Sstems Comtones y tomtzón, IIMS,UM, do. ost De..Obeón, Méo D.F.,, Méo. F: , e: (*) Em: Fbn@de4.ms.nm.m. eonfbe ommntons bsed on sedn otm. ese tme deys e stt onseene detemned by te nseton of ne eements tn ommnton nnes de to ft ene. In ft, ft esene s ten nto ont s te ost of te eted ee eement, sefy, senso o tto eements. e o of ts e s to defne sttey fo onto eonfton bsed on ommnton system eonfton. Some onsdetons need to be stted n ode to defne ts o. Fst, fts e stty o n ee eements, fts e ted by st emntn te fty eement. In ft, fts e tsto nd o. me deys e bonded nd esttve to sedn otms. s e s ed s sttey fo eonfbe systems s son n F.. In ft, ts e s fosed nto eonfbe onto de to te esene of o fts nd tme deys s onseenes. me deys e mesbe nd bonded odn e-tme sedn otm. In ts se sedn otm s te e non eest dedne fst otm. odn to F., stt eonfton tes e s est of EDF efomne nd eted se eest. Comte eto (Senso eto) Eten Fto to eest eonfton Reonfton Reest n s Contoe ode Conto L ode Yes Vdton n Vd n Dtbse (If te n s vd e eted Conto L s osen) Seeton of te Reted Conto L Conto Ls Dtbse Fst Reonfton Ste Seond Reonfton Ste o (Reeton of te oosed n) F.. Gene stte of Reonfbe System ove Comte eto e m of ts e s net, fst seton s ent ntodton, seond seton s sedn o, td seton s nt desn, fot seton s onto o fft seton esents se stdy nd fny some ondn ems e ven. II. SCHEDULIG ROCH e ommnton neto ys ey oe n ode to defne te bevo of te dynm system n tems of tme vne fom ommntons nd oessn to t esents nonne bevo. In ode to ndestnd s nonne

2 bevo, tme deys e nooted by te se of e-tme system teoy tt os tme deys to be bonded even n te se of s modftons de to eten effets. Seve otms n be sed s s Rte Monoton, Dedne Monoton o Eest Dedne Fst [,, 4]. s oede of EDF ees seve tests fom e ts s s dednes, onsmton tmes nd otes. Fo nstne, onsde tee tss t net tests (be nd F. ) fst seton nde EDF otm f ts nes ts dedne t Δt t od ve e oty tn tose tss edy defned (be ). LE. SKS USED O EXEMLIFIED EDF LGORIHM Consmton me (C) eod me () Dedne (D) oty Fo te se of dedne modfton s dsy n F., otes e modfed s son n be ee ts s te smest s tme (t s ) teefoe t s te est oty, ts s net oty nd st ts s te oest oty. eefoe te ts tt s on to be eeted fst s te one o s te sotest dedne n omson to te est of ve tss n t tme. Sne ts otm s efomed, to ondtons e stted on ts o. Conto mst be eeted fte sensos ve efomed te ts nd befoe ny tto tes n ton ove te nt, fom tese estton system efomne s to mn senos ft esent nd ft fee sttons. ese senos e eosed n F., t sos ommon onfton onsden o tme deys. τ s τ oms s ( ) C D Sensos s ( ) C D τ τ om s ( ) C D Conto Fom be ts s te smest s tme (t s ) teefoe t s te est oty, teefte, ts s net oty nd st ts s te oest oty. τ ttos F.. Reted tme deys fom onto onfton C t s C D C t' s t s ' D C D' t' s D' Reted tme deys e defned n tems of sedn otm bevo. eefoe tme deys e stted s: C t s D C t' s ' ' D' τ τ τ () s τ s oms τ () τ τ τ () om t F.. me G Reted to be odn to F., tee e to senos fo tese tee tss, fsty, ts s s tme ts, ts s s tme ts nd ts t s tme ts vn te est oty to ts. Seond seno esents dffeent oty onfomton odn to s tmes modftons. me LE. EW RIORIY ORDER FER REORGIZIO Consmton me eod me Dedne oty ee τoms nd τ om e te soe of vtons de to ommnton vton beteen eements. III. L ROCH e oosed dynm nt s bsed on te foon stte: ( ) () () () y (4) s ( ) (C) () (D) C D nn ee R, to te nt. () n n R nd R e mtes eted y e te sttes, nts nd, () nd () s ( ) C D otts esetvey. Sey s stted s s ( ) C D M τ τ ρ e ( tτ) dτ (5)

3 ee ρ nd ρ tn nto ont tt e te tot nmbe of ossbe fts nd M e te nvoved tme deys fom e ft. Cent ommnton tme deys e eessed s t nd t nd s nteted s b, ft eement b b ee b e te eements onfomed t te nt of te nt (s s ttos) nd s te ost eement de to o ft ee eesents ony one seno foon en. 5. Cent onsdes o fts nd eted tme deys of M t e t ( tτ) dτ Fo smty oses s sed n ode to det o ne nts Fom ts eesentton fzzy nt s defned s foos tn nto e tme dey nd ft ses: nd f s ee {... } nd s nd nd () () s (6) ten (7) ( ) e ent stte meses, s te nmbe of sttes, {,...,} s one of te fzzy es s te nmbe of te es s e to te nmbe of ossbe fts nd e te eted membes fntons e ssns defned s: ee nd ( ) (8) e (9) σ σ e onstnts to be tned. Fn eesentton of nt s nteted system s bsed on ente of e defzzfton metod s son n eton. () () () It s motnt to emembe eton 6 n ode to se system esonse. Fom ts eesentton of ob nonne system s te nteton of seve ne systems t s neessy to defne ob stbty s est of ts fzzy system. s eve s ven n foon seton onsden fzzy o onto o. e est of ts system eesentton os te nteton of nonne stes nd tnstons to bsy o of ne nts. IV. COROL ROCH Fom te eesentton of te nt s fzzy system [5] t s onsdeed to deveo te onto s o of bonded o ne onto s eted to e o ne system. e stte of e fzzy e s: f s nd s nd nd s ten () () () ee {,..., } te nmbe of fts to eesented, {,..., } of te nt, ee, s te nmbe of fzzy es s e ent sttes e te ssns membes fntons e: nd e ( ) σ () σ e onstnts to be tned. Ftemoe, eesents te eted onto n. Sm to fzzy system nt, fzzy onto eesentton s nteted s: nd () ( () ) () (4) () ef, (5),

4 ee ef s te efeene to be fooed by ontoe nd te vbes nd e sed de to fzzy es nteonnetons s te eesentton of dffeent ne nts nd esetve ontoes Fom ts eesentton, stbty needs to be stted s te foon ynov fnton: () ) ()( ) V( (6) nd v v (()) V (7) ee,,,, ef ef V ) V( (8) eefoe () X X ef ef V,,,, (9) ten by onsden ef (),, V () It s motnt to emembe tt () ) ( V () nd < () ee >,. s te Eden nom nd t s ossbe to defne > () s ondton s to be ven fo evey sne tme dey nd o ft ene. Ftemoe te stbty nd te onveene of sttes sod be ssed by te dete seeton of mtes nd te eted metes fom bot fzzy systems. In ts se eommendbe oede to foo s mtobetve otmzton n ode to defne tose stbe ves. V. CSE SUDY Cse stdy s bsed on tee tns o foon F. 4. I I I Seton F. 4. ee tns eesentton s s omosed by tee tns t dent oss seton. e tns e oed by to es t oss seton of. o ms e dven by DC motos o sy to nfos nd. Ld eve n e tn s mesed nd eoted s I, I nd I. s system s smn eod of 8 ms nd nomn ommnton tme dey of ms. Common eesentton of ts system s stted n en. 5 s stte se eesentton [6]. s eesentton tes nto ont te neent non-netes of te mode. U I I (4) ee [ ], [ ] y nd [ ] nd

5 ( ) μ *snm( ) (5) (6) μ Hee µ e te otfo oeffents. In tems of fzzy eesentton, tee es e ven de te ossbe nmbe of fts e to de to st to ttos e vbe. f s nd s nd s ten () () (7) f s nd s () () nd s (8) f s nd s nd s () () In ts se (9) s defned s:. (tτ) τ e d..4 (tτ) τ e d..5 (tτ) τ e d. (tτ).8 e d. (tτ).8 e d.4 (tτ).8 e d.5 τ τ τ ten ten ese tee nt eesenttons e modfed st s mt ee eonfton s efomed. In ee to ntes e esented de to motnt tme deys of eted fts. Fo nstne, s tme dey of.4 nd s tme dey of.5 seonds. In te sme y onto s e eesented, ee tee onto s e esented. f s nd s nd s () () () ten f s nd s nd s ten () () () f s nd s nd s ten () () () sed on ts eton, fn osed oo eton s () ef, ( ) (4), Sne ts eesentton s ven nd bsed on stbty oos n st seton, otmzton toobo fom ML [7] s sed to defne ent ves of mtes. VI. IMLEMEIG ROCH Foon te eve of se stdy, ts s memented ove omte neto smton [8] [8] [9] bsed on e-me sttey. Inonsstenes e dn ommnton nd onsmton tmes s e s te tte y n motnt oe. LE. COSUMIO IMES D ERIODS FROM SKS (I SECODS) Comonent Consmton tme Vton me Dednes Vton S C s. 6-8%.8 -% S C s. 6-8%.8 -% S C s. 6-8%.8 -% C C C.5 7-9%.8 -% C %.8 -% C %.8 -% be sos te onsmtons nd eods of nvoved nodes ee sensos nd ttos e onzed odn to EDF otm. s sedn modfton ffets efomne esonse of ee eements ee nd C mtes e teed s ee tme deys e defned s foos (be 5): t τ, τ tte vton nd τ. 8 t τ, τ tte vton nd τ. 8 eefoe onto eonfton beomes neessy n ode to ee etn esonse eve. Fo ts se stdy tee dffeent onto s e oosed LE 5. FUZZY LOGIC RULES COSIUDERIG HE RELED L Re If s nd s nd nd If s nd s nd nd If s nd s nd nd ( ( ( Lnezed nt ) ( ) ) ( ) ) ( ) VII. RESULS Foon te esonse of se stdy, some nteestn ests e ven eted to tee dffeent senos nmed eonfton, nd. In tese ses tee dffeent esonses e dynm esonse, onsmton nd ommnton tmes e ven. e

6 son esonse oesonds to tee tns efomne dn tee dffeent dynm ondtons t eset to ommnton tme bevo (F. 6). e o esonse esented n td seno s efomed de to modfton of te esonse nd dynm ondtons t eset to ommnton tme deys. F. 7 esents ommnton system esonse of e eement nvoved on te system. Fst esents eod tme modfton en eonfton s neessy nd seond esents tme sed to ommnte eements to te neto..5.5 Fst Seno Cent Sttes esonse.5 Seond Seno Stte Stte Stte.5 d Seno me Fst Fst Seno Seno Ve Ve Cent Ves to nt Seond Seno d Seno d Seno me Fst Seno Cent Conto esonse Seond Seno d Seno me F. 6. Dynm Resonse of Cse Stdy Foon ee Senos eod me Modftons me Commnton eto Resonse me F 7. me Dm fom sede esonse bsed on Comte eto VIII. COCLUDIG REMRKS s s been son n ts o, fzzy o onto bsed on Seno stte os te ossbty of onto eonfton s on s ne modes bsed eesentton of te nt e vbe. Deste of o fts nd bonded tme deys ene, seve ondtons sod be ff n ode to be be to foo ts oos, fo nstne, nt sod be obsevbe nd ontobe dn te oe nonne bevo s e s te sttes sod be esent dn ndesbe sttons. Seve oes n be sed s s obstness stdy dn vtons fom one to note senos s e s fte o s needed en not evey stte s vbe dn ft ondtons s est ondton. In ts se obseve desn nde ft seno need to be defned oon fo smoot tnstons beteen ft nd ft fee stts. Stbty ondtons e eed s on s system desn s sed by ens. 8-, ee fzzy o vbes nd o onto s e stbsed. CKOWLEDGEMES e tos od e to tn te fnn sot of DISC- IIMS-UM, nd UM-II (I6 nd I5) Meo n onneton t ts o. REFERECES [] enítez-éez, H., omson, H.. nd Femn,. J.; Smton of Dstbted Ft oent Heteoeneos tetes fo Re-me Conto ; 5 t IFC Woso on otms nd tetes fo Re- me Conto RC 98, , Cnún, Méo, 998. [] Msten, M. K.; Eetons: e nteene n Inteent Conto ; IFC Symosm on Inteent Comonents nd Instment fo Conto tons, nney, Fne,. -, 997. [] sson, J.; Re-me Conto t Deys ; D. ess, Detment of tomt Conto, Lnd Insttte of enooy, Seden, 998. [4] W.; Rebty of Reonfbe Conto Systems: Fzzy Set eoet esetve ; oeedns of te 6 t Confeene on Deson & Conto, IEEE, 5 5:,. 5-56, Sn-Deo, US, 997. [5] Jn J., nd Zo Q.; Reonfbe Conto sed on Imese Ft Identfton ; ooedns of te men Conto Confeene, IEEE,. 4-8, Sn Deo, Jne, 999. [6] Izd-Zmnbd R. nd ne M.; S oson System s enm fo Ft-oent Conto ; Conto Enneen te, Vo. 7,. 7-9, 999. [7] ne, M., Knnet M., Lnze J., nd Stose M.; Dnoss nd Ft oent Conto ; Sne,. [8] enítez-éez, H., nd Gí-oett, F.; Reonfbe Dstbted Conto ; Sne Ve, 5. [9] enítez-éez, H., nd Gí-oett, F.; Reonfbe Dstbted Conto bsed on Smt ee Eements ; Eoen Conto Confeene; ECC, oto, ot, ,. [] enítez-éez, H.; Smt Dstbted Systems ; D. ess C&SE Det. Unvesty of Seffed, UK, 999. [] Dnov, D., Heendoon, H., Renfn, M.; n Intodton to Fzzy Lo Conto ; Sne-Ve, 994. [] L, J.; Re-me Systems ; ente H,. [] Ksn C. M. nd Sn K. G.; Re-me Systems ; M. G H, e Yo, US, 997. [4] Ln F. Moyne J. nd by D. ; eto Desn Consdeton fo Dstbted Conto Systems ; IEEE nstons on Conto Systems enooy, Vo., o.,. 97-7, M. [5] Y, Z., nd Hen,.; Stbty of Fzzy Conto Systems t onded Unetn Deys ; IEEE nstons on Fzzy Systems, vo., o.,. 9-97,. [6] Ddne-Mon V., Hmen F., nd oe H.; Ft-oent Conto Desn nst Mo tto Fes: ton to ee-n System ; IEEE 8 t Confeene on Deson nd Conto, : , 999. [7] Mtos; System Identfton oobo Use s Gde ; ML, 998. [8] Cevn,., Hensson, D., Lnon,., Ee, J., nd zén, K.; Ho Does Conto mn ffet efomne? ; IEEE Conto Systems Mzne, Vo.,. 6-,. [9] enítez-éez, H., nd Gí-oett, F.; Reonfbe Seno Fzzy Lo Conto fo Css of onne System onsden Commnton me Deys on ee Eements ; Sbmtted to men Conto Confeene 6.

An Optimization Model for Empty Container Reposition under Uncertainty

An Optimization Model for Empty Container Reposition under Uncertainty n Omzon Mode o Emy onne Reoson nde neny eodo be n Demen o Mnemen nd enooy QM nd ene de Reee s es nsos Moné nd Mssmo D Fneso Demen o Lnd Enneen nesy o Iy o Zdds Demen o Lnd Enneen nesy o Iy Inodon. onne