Two-Input Fuzzy TPE Systems

Transcription

1 Two-Input Fuzzy TPE Sytem Joo P. Cvlho, Joe Tome, Dnel Chng-Yn IST - TU Lon, INESC-ID R. lve Redol, 9, Lo, PORTUGL {oo.vlho, oe.tome}@ne-d.pt dnlyn@gml.om tt. Sngle nput Fuzzy TPE ytem, popoed y Sudkmp nd Hmmel, llow moe effent omputtonl nfeene of ngle nput fuzzy ule e. Th ppe how tht t pole to genelze ngle nput Fuzzy TPE ytem to 2-nput ytem, theefoe extendng t nge of pole pplton. The ppe peent the detl nd poof of the extenon vldty, nd how enhmk eult ompng the 2-nput Fuzzy TPE wth ll nfeene ytem. Keywod: 2-nput fuzzy TPE, fuzzy nfeene omputtonl effeny. Intoduton Sudkmp nd Hmmell Fuzzy TPE ytem [] povde wy to elete the fuzzy nfeene poedue n -nput fuzzy ytem. It lo mke the nfeene poe ndependent fom the nume of nvolved lngut tem, ne nted of equentl pplton of ll ule n ule e, t llow the det e to the elevnt ule ung n ndexton poe nd nfe the ule eult ung pevouly omputed ontnt. Th ppoh pole due to etton mpoed on the nvolved lngut tem: ll memehp funton mut e tngul, omplementy, nd evenly ped n the unvee of doue (UoD). Howeve, Sudkmp ppoh w developed fo ngle-nput ytem, nd ould not e detly ppled n ytem wth moe nput. Theefoe t ue w lmted to few ptul polem. Though the ye TPE ytem hve een ued [2][3], ut when moe thn one nput deemed neey, ltentve nfeene method tht do not mplement pope fuzzy nfeene mehnm, lke FM, e ued nted. y pope fuzzy nfeene mehnm one men mehnm tht lthough omputtonlly fte, do not podue the ext me eult of ll fuzzy ule e nfeene poee. Th ppe how tht t pole to genelze ngle nput Fuzzy TPE ytem to 2-nput ytem extendng nge of pole Fuzzy TPE ytem pplton. Th wok w uppoted n pt y the FCT - Potuguee Foundton fo Sene nd Tehnology unde poet POSI/SRI/4788/2002

2 2 Fuzzy TPE Sytem Sudkmp nd Hmmell model ed on n even tngul ptton of ngle vle fuzzy unvee of doue (UoD), hene the nme TPE Tngul Ptton Evenly. The model e t effeny on the ymmety of the tght lne ued to defne the tngul memehp funton. Th eton ummze Sudkmp nd Hmmel fuzzy TPE ytem. In th model, the memehp funton (mf) of lngut tem n e defned y (), whee µ ( x) the memehp of p nput x n, nd µ ( ) =. ( x ) ( ) f x µ ( x) = ( x ) ( ) f x 0 othewe On n-lngut tem fuzzy vle, ll mf e equl wth the exepton of the lowe nd uppe lngut tem. The mf of thee lngut tem e epetvely tunted to the left nd to the ght of the pont,.e., µ ( ) = nd µ () n =. n ddtonl ondton tht ll mf e omplement n ll UoD,.e.: n µ ( x) =, x UoD. = Equton (2) enue the ompletene of the ule e. Fg. how even fuzzy lngut tem TPE et. µ () (2) Fg. Exmple of mf tht omply wth Fuzzy TPE ytem etton Wthn fuzzy TPE ytem, one n dvde the UoD nto et of [, ] ntevl whee µ ( x) = µ ( x), µ ( ) = nd µ ( ) = 0 fo. Theefoe, the memehp degee wthn the ntevl n e defned y the two followng tght lne equton:

3 x x = µ ( ). (3) µ x ( x) =. Wthn ngle-nput/ngle-output TPE ule ed fuzzy ytem, the fuzzy ule tht nvolve nteedent nd n e expeed : f X then Z C, f X then Z C whee C nd C e lo TPE lngut tem. Defnng nd the entl pont of C ndc, nd y ung weghted vegng defuzzfton, the eult z pefed y p nput x gven y: µ ( x) µ ( x) z =. µ ( x) µ ( x) (4) (5) y eplng (3) nd (4) n (5) nd mplfyng one otn: x( ) z = = x. (6) Equton (6) how tht only two ontnt e needed to ompute z gven ny p nput x [, ]. Thee ontnt, ( ) ( ) nd ( ) ( ), e detemned y the fuzzy ule e nd the mdpont of the tngul memehp funton, nd theefoe ndependent of x. Thu fuzzy TPE ytem n e epeented y tle ontnng the ontnt oted wth eh nput ntevl. Sne ll ntevl e equl n ze, one n detly dde the ontnt oted wth gven nput x y ung the funton 2x 2 tun( ). n On fuzzy TPE ytem one n kp the nfeene of ll ule n the ule e. Gven nput x, ll tht needed to get the ppopte ontnt fom tle nd pply (6) to fnd defuzzfed output z. Theefoe, nfeene tme on TPE fuzzy ule e ndependent of the nume of ule nd lngut tem ze, nd TPE fuzzy nfeene ptully well dpted to del wth lge fuzzy ule e. (7)

4 3 Two-nput Fuzzy TPE Sytem ngle nput vle fuzzy ule ed ytem h n ovouly lmted nteet. Multple nput Fuzzy TPE ytem e neey f one wnt to ue thee mehnm n gnfnt nge of pplton. lthough t theoetlly pole to extend the model to hghe nume of nput, n th ppe we lmt the extenon to 2-nput ytem. One mut note tht n ule ed fuzzy ytem t ometme pole to ognze n-nput ule e nto evel onneted 2-nput ule e. Th mnpulton ptul to the ytem eng modeled nd ele on the ntedependene of t nput vle. Theefoe, let u fou on the 2-nput e. ny 2-nput fuzzy ytem ule e wth n-lngut tem vle n e epeented 2 dmenonl tle. Tle, epeent n exept of gene 2- nput fuzzy ule e, whee ll, nd C x e fuzzy lngut tem defned y fuzzy memehp funton. On uh ytem, ny omnton of nput vlue tvte t mot fou dffeent ule of the ulee. Tle. n exept of 2-nput Fuzzy Rule e. Y X C C C t C u f X nd Y then Z C f X nd Y f X then Z C nd Y then Z C t f X nd Y then Z C u Sne TPE memehp funton e, y defnton, tngul nd ymmet, Tle n e epled y Nume Infeene (NI) tle, whh n ltentve epeentton whee eh lngut tem epled y t entl pont x-oodnte. Theefoe, f one ume the weghted vegng defuzzfton method, one n dedue the followng output equton fo the ule e peented n Tle, whee,, t nd u e the entl pont x-oodnte of the onequent mf: z = ( µ ( x), µ ( y)) ( µ ( x), µ ( y)) ( µ ( x), µ ( y)) ( µ ( x), µ ( y)) mn mn mn mn t u ( µ ( x), µ ( y)) ( µ ( x), µ ( y)) ( µ ( x), µ ( y)) ( µ ( x), µ ( y)) mn mn mn mn (8) In ode to fnd the 2-dmenonl equvlent of (6), onde two fuzzy vle nd defned epetvely y n nd m TPE mf. Fg.2 how ptol epeentton of nd, whee x nd y e the p nput vlue.

5 The 2-dmenonl ntevl we onde to fnd the 2-dmenonl equton equvlent of (6) e epeented y [, ] nd [, ]. The hdowed egon n Fg.2 how 2-dmenonl ntevl. 2-dmenonl ntevl ndexton ed on (7), nd eult on (9): n, f x= ntevl = ( x )( n) tun, f x 2 m, f y = ntevl = ( y )( m) tun, f y 2 (9) =n x 2 2 3=m y Fg.2 Two-dmenonl epeentton of TPE lngut tem. The hdowed egon epeent 2-dmenonl ntevl One n dvde eh 2-dmenonl ntevl n fou dffeent e, depted n Fg.3. Eh of thee e htezed y ontnt elton etween the memehp degee of x nd y n the petnng lngut tem. Fo exmple, n e, ll the followng equton hold tue: mn( µ ( x), µ ( y)) = µ ( x), mn mn ( µ ( x), µ ( y)) = µ ( x), ( µ ( x), µ ( y)) = µ ( x), ( µ ( x), µ ( y)) = µ ( x) mn.

6 x 4 3 Fg. 3 The 4 dffeent e n eh ntevl ed on the htet of eh of the 4 e, t pole to dvde the defuzzfton equton (8) nto 4 dffeent defuzzfton equton, whee the mn() funton not neey. The 4 e n e dvded y two tght lne defned y the followng equton: ( x ) ( n ) ntevl ntevl y = 2. m m n y (0) 2 ( n )( x ) ntevl ntevl y = 2. ( m) m n () Equton (0) nd () n e ued to ete Tle2, whh dvde the Unvee of Doue n ntevl, nd eh ntevl n the 4 dffeent e. Tle 2. e defnton fo eh ntevl 2 ( n )( x ) ntevl ntevl y 2 ( m) m n ( n )( x ) ntevl ntevl ( n )( x ) ntevl ntevl y < 2 y 2 ( m) m n ( m) m n e e 3 2 ( n )( x ) ntevl ntevl y < 2 ( m) m n e 2 e 4

7 Sne we e delng wth TPE ytem, (3) nd (4) e tll vld fo eh fuzzy vle. Theefoe, µ ( x), µ ( x), µ ( y), µ ( y) n e defned y: x y µ x = µ y = ( ) ( ) x µ ( x) = µ ( y) = y. (2) Condeng the 4 dffeent e defned n Fg.3 fo eh ntevl, one n eple (2) n (8) nd mnpulte the eultng expeon n ode otn the followng defuzzfton equton: e : z y ( ) ( ) ( ) x y t u t u y x x t u y y x x y x = = (3) e 2: z 2 3 ( ) ( u) ( ) y x u t x y t x y t u x y y x 2 2 y x.0 = = e 3: z 4 ( ) ( t) ( ) y x t u u y x y x t u y x y x 2 2 = = e 4: z y y x.0 ( ) t u ( ) ( ) x u t x y y x t u x y x y 2 2 y.0 = = Note tht ll 4 equton n e expeed unde the fom z = whee ll H e ontnt. yh H xh H yh H xh H x, (4) (5) (6) (7)

8 n the ngle nput e, the H ontnt e detemned y the fuzzy ule e nd y the entl pont of eh lngut tem (,, l ), nd e ndependent of nput p vlue x nd y. Theefoe, they n e omputed only one, po to the tul ule e nfeene poe. ed on (9) nd Tle 2, one n detly ndex n nfeene mtx wth dmenon [n-] [m-] [4] [8], whee n eh poton of the nfeene mtx one wll hve the pevouly defned H ontnt. Fom th nfeene mtx, one n detly ompute the defuzzfed output z. y ung th nfeene poe, one vod the equentl omputton of eh ngle ule whle multneouly mplfyng the defuzzfton poe. Theefoe, long ll H e pevouly omputed, nd wth pope ndexton of the nput vlue, t pole to otn vey ft fuzzy nfeene poe tht doe not gow exponentlly wth the nee of nume of lngut tem n eh nvolved fuzzy vle. 4 Reult In ode to tet the omputtonl effeny of 2-Input Fuzzy TPE Sytem, ompon w mde wth tdtonl fuzzy ule ed nfeene method. Sne the Fuzzy TPE method omputtonl nfeene tme doe not gow exponentlly wth the nee of nume of lngut tem n eh nvolved fuzzy vle (onty to the tdtonl method), ette eult wee to e expeted when lge nume of memehp funton wee ued. Theefoe tet wee mde nvolvng fuzzy vle wth 3 nd 7 lngut tem, eultng n omplete ule e wth 9 nd 49 ule epetvely. Tle 3. nd Tle 4. how the ued ule e. Note tht nfeene omputtonl tme ndependent of tul ule ontent. Tle 3. Fuzzy ule e fo 3 lngut tem nput vle Y X L M H L L M H M M H M H H M L Fo the tdtonl fuzzy nfeene method omputtonl mplementton, one opted to ue n ndvdul y to epeent eh lngut tem memehp funton. Th method omputtonlly fte thn ung tght lne equton, ne nfeng µ ( x) done v mple y ndexton.

9 Tle 4. Fuzzy ule e fo 7 lngut tem nput vle X VVL VL L M H VH VVH Y VVL VVL VL L M H VH VVH VL VL L M H VH VVH VH L L M H VH VVH VH H M M H VH VVH VH H M H H VH VVH VH H M L VH VH VVH VH H M L VL VVH VVH VH H M L VL VVL Ung th method, the C ode to ompute fuzzy ule lke If x nd y Then Z C, n e eumed : mu=min((,x),(,y)); //If x nd y f (mu!=0){ fo (n=0; n<nume_of_mf; n) //Then z C foutput[n] = mu*c[n]; } whee funton () mply: flot (nt LT, nt np){ //LT-Lngut Tem,np-nput etun mf[lt][np];} //mf-y of ll LT The ove mentoned ode epeted fo eh ule n the ule e. eult one otn n y epeentng fuzzy vle z. The defuzzfton method ued weghted vegng, whh lo mple to mplement: fo (n=0;n<mfze;n){ m=foutput[n]*n; e=foutput[n]; } z=m/e; lthough the defuzzfton ode mple, t not omputtonlly effent ne the nume of ule gow exponentlly wth the nee n the nume of lngut tem, nd the ode ue evel yle to un though the ule e. It pole to vtly mpove defuzzfton omputton tme, ut not wthout mpong etton to mf hpe (whh wht fuzzy TPE doe). The Fuzzy TPE method ee the tve ule though det ndexton, nd theefoe vod the neety to tet eh ngle ule. Due to the etton n the hpe of the memehp funton, t lo ue muh moe effent defuzzfton method. Tle 5. how the vege nfeene omputng tme fo the ule e n Tle 3. nd Tle 4.

10 Tle 5. vege ule e nfeene omputng tme ompon (n) Method #LT/#ule Cll Infeene Fuzzy TPE 3 / / The vege nfeene tme w otned fte omputng nput on.86ghz PentumM poeo. One n ee tht Fuzzy-TPE nfeene oughly 8 tme fte thn ll fuzzy nfeene method, whh n e ondeed ondele mpovement. Howeve, one mut not gnoe tht the nvolved omputng tme e o mll tht n often e ondeed neglgle fto n mny pplton. Theefoe, due to the etton mpoed to the lngut tem, the ue of 2-nput Fuzzy-TPE ytem n only e utfed n pplton whee lge volume of nfomton mut e poeed nd whee tme mo ue, lke eltme ontol ytem o ompute he. 5 Conluon nd Futue Development It w hown tht two-nput Fuzzy TPE nfeene ytem n e mplemented nd tht they povde gnfnt mpovement ove ll fuzzy nfeene ytem n wht onen omputng pefomne. Howeve, th mpovement n only e ueful n ytem whee tme nd pefomne e tl nd whee the etton mpoed to the lngut tem memehp funton n e epted. N-nput Fuzzy TPE ytem e theoetlly pole. Howeve, the nume of nput nee, the omplexty of the defuzzfton equton nd the dmenonlty of the omputng mtxe nee exponentlly. Theefoe, n-nput Fuzzy TPE ytem vlty ove ll nfeene ytem tll need to e poved on futue wok. Refeene. Sudkmp,T., Hmmell,R.J., Intepolton, Completon, nd Lenng Fuzzy Rule, IEEE Tnton on Sytem, Mn, nd Cyenet, Vol. 24, 2, Cmho, E., eengel, M., Ruo, F., dvned Contol of Sol Plnt, dvne n Indutl Contol, Spnge-Velg, Co,V., Sudkmp,T., Spe Dt nd Rule e Completon, Poeedng of the 2003 Confeene of the Noth men Fuzzy Infomton Soety, NFIPS 2003, Chgo, 2003