Complex Neuro-Fuzzy Self-Learning Approach to Function Approximation

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1 Coplex eo-fzz Sel-Lenng ppoc o Fncon ppoxon Cnen L nd T-We Cng Loo o Inellgen Se nd pplcon Depen o Inoon ngeen onl Cenl Unve, Twn, ROC. el@g.nc.ed.w c. new coplex neo-zz el-lenng ppoc o e pole o ncon ppoxon popoed, wee coplex zz e e ed o degn coplex neo-zz e e ncon ppoxo. Pcle w opzon PSO lgo nd ecve le qe eo RLSE lgo e ed n d w o d e ee pee o e popoed coplex neo-zz e CFS. Te d PSO-RLSE lenng eod ed o e CFS pee o convege ecenl nd qcl o opl o ne-opl olon. Fo e expeenl el, e popoed CFS ow ee peonce n e donl neo-zz e FS degned w egl zz e. oeove, e PSO- RLSE d lenng eod o e CFS pove e e o lenng convegence, nd ow ee peonce n ccc. Tee enc ncon e ed. W e peonce copon own n e ppe, excellen peonce e popoed ppoc een oeved. ewod: coplex zz e, coplex neo-zz e CFS, PSO, RLSE, ncon ppoxon, cne lenng. Inodcon Teoe o zz logc nd nel newo o e dencon o e odelng ve een wdel nveged o pplcon []-[]. B oevng npop d p o n nnown e o nee, odel cn e e p o e e, ng n nellgen odelng ppoc. W e odel, e elonp o np-op evo cn e ppoxed. T poce cn e vewed e dencon, wc lo nown e odelng o ncon ppoxon. T, e dencon o odelng cn e vewed e pole o ncon ppoxon, o wc n opzon poce ll nvolved o ec o e opl olon o e pole. Howeve, de o e coplex nd nonlne n el-wold pplcon pole, ecl ppoce o e dencon e ll loo nd dcl. Snce nel newo nd zz neence e e nvel ppoxo [3]-[4], neo-zz e, wc ncopoe e dvnge o zz neence, nel ce nd lenng lexl, ve ecoe popl nd ndenl e n odelng pole. Fzz e cn e ed o elec n concep nd og, wc end o e pece, ncoplee nd

2 vge. Coplex zz e CFS [5]-[7] new developen n e eo o zz e. Te concep o CFS n exenon o zz e, wc e eep o ec eleen o coplex zz e exended o coplex-vled e. In coplex zz e, eep vle e coplex ne n e n dc o e coplex plne [5]-[6]. log e nodco eo o e CFS een peened [5], e eec on coplex zz e degn nd pplcon ng e concep o CFS ond el. In ppe, coplex neo-zz e CFS w d lenng eod popoed o e pole o ncon ppoxon. Te d lenng eod nclde e well-nown pcle w opzon PSO lgo nd e ecve le qe eo RLSE lgo o n e popoed CFS. Te popoed ppoc ow ee dpl n ppoxng cpl n donl neo-zz e, n e o ppoxon ccc nd lenng convegence e. In econ e popoed coplex neo-zz ppoc peced. In econ 3 e PSO-RLSE d lenng eod gven. In econ 4 expeenl el o ncon ppoxon e gven. Fnll, e ppe dced nd conclded. eodolog o Coplex eo-fzz Se Tee e wo eqenl ed zz neence e FIS. Te ndn pe FIS nd e oe Tg-Sgeno T-S pe FIS. Te deence eween e le n e coneqen o zz le. Fo dn zz odel, e coneqen e peced w lngc e, wc cn e dened w zz e [8]. Fo T-S zz odel [9], e coneqen e expeed polnol ncon o e np vle. In ppe, e degn o e popoed CFS n exended o e concep o donl neo-zz e FS, nd e zz T-S odel ed n e popoed CFS. Te zz eo cn e ed o epeen ncen o pece d, noon nd concep. Te vle o pplng zz e o odelng ncen, o epeenng ecve n nowledge, nd o elng n eonng pocee, ve een vlded [5]. Te concep o zz e exended o coplex zz e CFS [5]-[7], wc expnd e nge o eep o e n nevl [, ] o e n dc n e coplex plne. e ee coplex zz e S woe eep ncon gven ollow. exp Re I co n wee e e vle o e coplex zz e, e plde ncon o e coplex eep, e pe ncon. Te coplex zz e S expeed ollow. S {, U} In e ce eql o, donl zz e egded pecl ce o coplex zz e. e ee zz le w e o o I x = nd x B =B Ten, wee nd B epeen wo deen condon n e le. Te wo condon cn e deced ng wo coplex zz e, gven ollow.

3 exp 3 exp 4 B B B B B B wee nd B e e e vle; x nd x B e e lngc vle o nd B, epecvel. Ineecon o e coplex zz e nd B expeed ollow. ]exp 5 B [ B B B wee o -no opeo neecon opeo, B =, B B e neecon, nd.,. denoe e pe neecon opeon. Unon o e coplex zz e nd B expeed ollow. ]exp 6 B [ B B B wee o -no opeo non opeo, B =, B B e non o pe, nd.,. epeened e non opeo o pe. e we ve coplex zz e w T-S zz le, gven ollow. Rle : IF x nd x nd x Ten z 7 =,,...,, wee x e - np lngc vle, e - np o e vle, e coplex zz e o e - condon n e - le, z e op o e - le, nd, =, nd =,, e e coneqen pee. Fo e popoed CFS, e coplex zz e c no e ewo w x leed neo-zz newo. Te coplex zz eonng o e CFS o np o op explned ollow. Le : Te le clled e np le, wc eceve e np nd n e o e nex le decl. Te np veco gven ollow. H=[,,, ] T 8 Le : Te le clled e zz-e le. ode n e le e ed o epeen e coplex zz e o e pee p o e CFS nd o clcle e eep degee. Le : T le o e ng-eng. Te ng eng o e - le clcled ollow. exp =,,...,, wee n opeo ed o e -no clclon o e ng eng. e plde o coplex eep degee o e - zz e o e - le. Le 3: T le o e nolzon o e ng eng. Te nolzed ng eng o e - le epeened ollow. 9

4 exp exp Le 4: Te le o nolzed coneqen. Te nolzed coneqen o e - le epeened ollow. z exp exp Le 5: T le clled e op le. Te nolzed coneqen o Le 4 e congeged n e le o podce e CFS op, gven ollow. z exp exp Genell e op o e CFS epeened ollow. n co exp I Re 3 wee Re e el p o e op o e CFS, I e gn p, e ole vle o e coplex op gven n 4, nd e pe o e coplex op expeed n 5. I Re 4 n Re I 5 Bed on, e coplex neence e cn e vewed coplex ncon e, expeed ollow.,,, I Re W H F W H F W H F 6

5 wee F Re. e el p o e CFS op, F I. e gn p o e op, H e np veco o e CFS, W denoe e pee e o e CFS. Te pee e W cn e dvded no wo e, wc e e pee-p e nd e coneqen-p e, denoed W I nd W Ten, epecvel. 3 Hd PSO-RLSE Lenng o CFS Pcle w opzon PSO [] poplon-ed opzon eod, wc oved e ood ecng evo o d locng o coolng. Ec d n e w n vewed pcle. e e locon o ood vewed e opl olon n e pole pce. Ec pcle vewed poenl olon n e ec pce. Ec pcle locon cn e pped o ne o clled co w oe gven ne ncon o clled co ncon. Te pcle n e w cope o ec oe o ecoe e wnne, wc nown ge. Te e locon o pcle dng e evolon poce clled pe. Te locon nd e veloc o pcle n e w e pded ng e noon o ge nd pe. e e pole pce w Q denon. Te eod o PSO expeed ollow. V V c pe L c ge L 7 L L V 7 T,, v,,, v, Q ] V [ v 8 T,, l,,, l, Q ] L [ l 9 wee V e veloc o e - pcle on - eon, L e locon o e - pcle, {c, c } e e pee o PSO, nd {, } e ndo ne n [,]. Te RLSE [] ed o e dencon o e pee o coneqen p. Fo genel le-qe eon pole, e op o lne odel,, peced e lnel peezed expeon, gven ollow. wee e odel np,. nown ncon o nd, =,,, epeen nnown pee o e eed. Hee cn e vewed e coneqen pee o e popoed T-S zz ppoxo. To ee e nnown pee {, =,,,} o nnown ge e o ncon, e o npop d p e ed nng d, denoed ollow. TD {,,,,, } Sng d p no 7, e o lne eqon e gven ollow.

6 Te opl eon o cn e clcled ng e ollowng RLSE eqon., T T P P P P P 3 T θ P θ θ 3 =,,,-, wee [ T, ] n e - ow o [, ]. To e RLSE lgo n, we need o elec e nl vle o nd P gven ollow. P = I 4 wee lge vle nd I e den x, nd nll e o zeo. Fo e nng o e popoed CFS, e d PSO-RLSE lenng eod ppled o pde e pee pee nd e coneqen pee. In d w, e PSO ed w e RLSE o lenng convegence. Te pee pee nd e coneqen pee o e CFS e pded e PSO gven n 7 nd e RLSE gven n 3, epecvel. Te PSO ed o pde e pee pee o e CFS. Te PSO ec eod enng ccec o evolon ec lgo. Ec locon ge o e PSO povde poenl pee olon. Te RLSE ed o pde e coneqen pee, w e nolzed ng eng. W, o ee e coneqen pee, e ow veco nd e veco e nged ollow. ] [ 5 ] [ 6 ] [ θ 7 ] [ τ 8 =,,...,, nd =,,,-, ec eon o e d PSO-RLSE lenng, e op o e CFS ppoxo cn e oned n 3. Te eo eween op o e ge nd e CFS dened ollow., RSE W H F 9 Te eo ed e o dene e oo en qe eo RSE, wc ed e peonce ndex n e d. Te qe o RSE clled e en qe eo SE.

7 Co RSE 4 Expeen o e Popoed ppoc Expeen o ncon ppoxon e condced n econ o ee nd ve e peonce o e popoed ppoc. Two econ e gven n e econ. In e econ, e popoed ppoc ng e CFS ppoxo nd e d PSO-RLSE lenng eod coped o wo oe coped ppoce. Te coped ppoc e donl FS ppoxo nd e PSO lenng eod, nd e econd coped ppoc e e e CFS ppoxo nd e PSO eod lone. Te oo ncon ed n e econ. In e nd econ, e popoed ppoc coped o e ppoc n []. Two enc ncon e ed n e nd econ o peonce copon. 4. Copon o e Popoed ppoc o e PSO o CFS, nd e PSO o FS Te oo ncon gven ollow..8{. [ co3 ] [ co3 n ]} 3 7 Te geneed d p o e oo ncon e ed o n e CFS wc degned w coplex zz e nd e FS wc degned w donl zz e. Two np e ed o e CFS nd e donl FS. Ec np poee ee zz e. Tee e 9 le n CFS nd e FS, wee pee pee nd 7 coneqen pee e o e pded e PSO lgo. In e d PSO-RLSE o e CFS, PSO ed o d e pee pee o CFS nd e RLSE o pde e 7 coneqen pee. Fo e op o e CFS, we elec e el p o epeen e ppoxo op. Fo e PSO eng o e popoed CFS nd e FS, {c, c } = {, } nd e poplon ze = 635 e gven. nd, o e d PSO-RLSE eng, {c, c } = {, }, = 4, nd = zeo-vled veco e gven. Te lenng cve nd e el e popoed PSO-RLSE o e CFS e gven n Fg. o. Te peonce copon o e popoed ppoc nd e wo coped ppoc gven n Tle. Te ppoxon eo e ee ppoce e own n Fg popoed ppoc ge ncon eon

8 eo eo Fg.. Lenng cve e popoed d PSO-RLSE o e CFS o e oo ncon. Fg.. Repone e popoed CFS w PSO-RLSE o e oo ncon Hd PSO-RLSE o CFS PSO o FS PSO o CFS Tle. Peonce Copon eod RLSE PSO o FS PSO o CFS.56-3 Hd PSO-RLSE o CFS Fg. 3. ppoxon eo e PSO o FS, e PSO o CFS, nd e d PSO- RLSE o CFS 4. Copon o e Popoed ppoc o e ppoc n [] Te d PSO-RLSE eod o e popoed CFS eploed o ppoxe e wo enc ncon wc e e exponenl ncon n [-4, ] nd e peolc ngen ncon n [-5, 5]. Fo ec o e enc ncon, e eo no o ncon ppoxon ed on e en qe eo SE, dened eoe. Te nng nd eng d o ec enc ncon e 4 nd pled p. W e enc ncon, Tle ow e peonce copon n SE o expeenl l ng e popoed ppoc nd e coped ppoc []. Te ppoxon epone nd eo o e ncon e popoed ppoc e own n Fg 4 o popoed ppoc ge ncon Fg. 4. Rel e popoed CFS d lenng ppoc o e exponenl ncon Fg. 5. ppoxon eo e popoed CFS d lenng ppoc o e exponenl ncon.

9 eo.5 ge ncon popoed ppoc Fg. 6. Rel e popoed CFS d lenng ppoc o e peolc ngen ncon Fg. 7. ppoxon eo popoed CFS d lenng ppoc o e peolc ngen ncon. 5 Dcon nd Conclon Te popoed coplex neo-zz e CFS een peened o e pole o ncon ppoxon o ve e ppng peonce o e popoed CFS. Te d PSO-RLSE lenng eod een ppled o epopoed CFS o dp e pee. Te e pee e dvded no wo e o e ee e lenng poce o e opl olon o pplcon peonce. Te wo e e e pee e o pee nd e coneqen e o pee. Te oe e nclde e pee n denng e pee zz e o e CFS, nd e le e collec e coneqen pee n denng e coneqen p o e le n e CFS. Te well-nown PSO ed o pde e pee e o pee nd e RLSE o e coneqen e o pee. T d lenng eod ve ecen o nd e opl o ne opl olon o e CFS n pplcon peonce. Tle. Peonce Copon. Fncon eod Coped ppoc [] Popoed ppoc Fncon eod Coped ppoc [] Popoed ppoc exp en±d nng en±d eng ± ±. - ± ±3.8-4 n en±d nng en±d eng ± ± ± ±.77-5

10 I ond e popoed d lenng ppoc peo o e wo oe coped eod, wc e e donl FS ng e PSO nd e CFS ng e PSO. Te peonce copon o e ee ppoce own n Tle. Te popoed d lenng ppoc w RSE=.73-4 c ee e donl FS w PSO lenng eod w RSE= Te popoed ppoc lo een coped o oe eec ppoce []. Te peonce copon e own n Tle, n wc wo enc ncon e nvolved. Te coplex neo-zz e n dpve copng pdg cone e eoe o coplex zz logc nd nel newo. In ode o develop e dpl o e CFS, e newl popoed PSO-RLSE d lenng lgo een ed o ne e pee pee nd e coneqen pee n d w o ceve nd le lenng convegence. W e expeenl el, e e o e d lenng een oeved. Tog e copon expeen, e popoed ppoc own excellen peonce. Reeence. Jng, C.F., Ln, C.T.: n onlne el-concng nel zz neence newo nd pplcon. IEEE Tncon on Fzz Se, vol. 6, pp. -3, 998. Pl, S.,, S.: Seood-podc zz nel neence e. IEEE Tncon on el ewo, vol. 3, pp Hon,., Snccoe,., We, H.: lle eed owd newo e nvel ppoxo. el newo, vol., pp Wng, L.X., endel, J..: Fzz ncon, nvel ppoxon, nd oogonlle-qe lenng. IEEE Tncon on el ewo, vol. 3, pp ndel,., Ro, D., lo, R., Fedn,.: Coplex Fzz Se. IEEE Tncon on Fzz Se, vol., pp Dc, S.: Towd coplex zz logc. IEEE Tncon on Fzz Se, vol. 3, pp Ro, D., Fedn,., Lngolz, G., ndel,.: Coplex zz logc. IEEE Tncon on Fzz Se, vol., pp Fg, W.., Qnn, V.H., Le-Toe, G.: genec-ed neo-zz ppoc o odelng nd conol o dncl e. IEEE Tncon on el ewo, vol. 9, pp Tg, T., Sgeno,.: Fzz dencon o e nd pplcon o odelng nd conol. IEEE ncon on e, n, nd cenec, vol. 5, pp enned, J., Ee, R.: Pcle w opzon. In: IEEE Inenonl Coneence on el ewo 995. H, T.C.: Se dencon: Le-qe eod. D. C. He nd Copn 977. W, J.., Ln, Z.H., H, P.H.: Fncon ppoxon ng genelzed dlne. IEEE Tncon on el ewo, vol. 7, pp