Dcson-makng wth Dstanc-basd Oprators n Fuzzy Logc Control Márta Takács Polytchncal Engnrng Collg, Subotca 24000 Subotca, Marka Orškovća 16., Yugoslava marta@vts.su.ac.yu Abstract: Th norms and conorms famly tr root s unnorm. It wll b shown, that dstanc basd oprators satsfy most of proprts of gnrally dfnd paramtrcal oprators. Basd on ths thory som nw typs of fuzzy ntgrals ar ntroducd thortcally. It can b shown, that th rason usd for fuzzy ntgral ntroducton s smlar to th rasons n dcson-makng by fuzzy control logc. Kywords:dstanc-basd oprators, fuzzy ntgrals, FLC 1 Introducton In practcal rprsntaton of th fuzzy logc thory t s vry mportant to b vsually rcognzd. If w xa an FLC modl wth trangular or trapsod mmbrshp functons, w can rcognz th rlatonshps btwn classcal ntgral calculus and th dcson makng n FLC. It s vry mportant n th cass of non-contnuous tnorms and conorms, whr th us of classcal dgr of frng s not corrct n all cass. So th nw groups of tnorms and conorms must fnd thr plac n thortcal systm of unnorms and paramtrcal norms, and basd on thortcal background stp should b mad for furthr applcatons. Th unnorms wr ntroducd, as a gnralzaton of t-norms and t-conorms. For unnorms, th nutral lmnt s not forcd to b thr 0 or 1, but can b any valu n unt ntrval. Th rcnt rsults on unnorms w can fnd n [1]. Nw trnds n nformaton aggrgaton, startng from th classcal Zadhan oprators through th group of ntropy-basd and volutonary oprators, to dstanc-basd oprators whch can found n [2] and [4], wth th proposton that ths non-classc group of norms and conorms ar gudd by unnorm thory. Papr [2] dscrbd th man da of usng ths norms n fuzzy logc control (FLC, whr th fuzzy rul output s nothng ls, but a fuzzy st wghtd wth th dgr of concdnc of th rul prms and systm nput. You can rad th sam rason at ntroducng
a nw typ of fuzzy ntgral, basd on unnorms thory n [3]. Fuzzy masur thory ndd n fuzzy ntgral calculus can b found n [5], [6] and [3]. 2 Thortcal background 2.1 Fuzzy Logc Control In control thory much of th knowldg of a controllr can b statd n th form of f-thn ruls, nvolvng som varabls. Th fuzzy logc control has bn carrd out sarchng for dffrnt mathmatcal modls n ordr to supply ths ruls. In most sourcs t was suggstd to rprsnt an IF x s A THEN y s B (1 rul n th form of fuzzy rlaton. On th othr hand, ths rlaton can b constructd as a spcal fuzzy oprator: th fuzzy mplcaton (Imp. In th dfnton of ths conncton or mplcaton l th most sgnfcant dffrncs btwn th modls of Fuzzy Logc Controllrs (FLCs. Th othr mportant part of a rul-basd systm s th nfrnc mchansm. Th nfrnc y s B s obtand whn th proposton s: th rul IF x s A THEN y s B, and th systm nput x s A. Th conncton Imp(A,B s gnrally dfnd, and t can b som typ of t-norm. Gnrally Modus Ppnns ss th ral nfluncs of th mplcaton choc on th nfrnc mchansms n fuzzy systms, of cours, whr th gnral rul consqunc s obtand by ( ( = ( ( ( ( B' y sup T A' x, Imp A x, B y. (2 x Th FLC rul bas output s constructd as a crsp valu calculatd from assocatv usng t-conorm on all rul outputs B (. 2.2. Fuzzy masurs Fuzzy masur s dfnd as a functon m : [ a, b], whr Σ s a σ algbra of fuzzy substs of ( s a non mpty st. Th ntrval [a,b] can b modfd n ntrval [0,1], as usually n FLC. Functon m must hav proprts, somtms gnralzd proprts, dscrbd as
M1. boundary condton, ( = 0 m, M2. monotoncty, for vry A and B from st of fuzzy substs, whr A B, thn m A m B. ( ( M3. contnuty, that for thr A... t s A 1 2 lm m( A = m lm A. In ordr to gnralz fuzzy masur thory w can fnd th so calld S masur typs, wth proprts MP1. m ( A B = S( m( A, m( B, for A B =,.. A and B ar sparabl, f: MP2. m ( A B = T ( m( A, m( B. (T,S s a par of t-norm and t-conorm. It s vry mportant, that th paramtrcal (T,S par, wth paramtr has furthr condtons: MP3. If =0, w hav a probablty masur, MP4. If =1, w hav a possblty masur. MP5. For paramtr m ( A B [ 0,1 ], and for vry A and B from st of fuzzy substs ( A + m( B f m( A >, m( B ( m( A, m( B othrws m >, = 2.3.Fuzzy ntgrals In [3] (S,U ntgral was ntroducd. Df.1. Lt m : [ a, b] b a Sm fathful S masur. Gvn an Sm fathful partton β = { B Bk Σ, k N}, th (S,U ntgral of a k ρ : 0,1 s dfnd by: masurabl functon [ ] ( S, U n ϕ dm = S ( ( S U a, m A Bk (4 k K = 1 whr U s a unnorm dstrbutv ovr S t-conorm.
3 (S,U Intgral as th dcson makng n FLC wth dstanc basd oprators 3.1. Dstanc basd and volutonary oprators Th mum dstanc mum oprator wth rspct to paramtr dfnd as ( x, f y > x = ( x, f y < x ( x, f y = x or y = x [ 0,1 ] Th mum dstanc mum oprator wth rspct to [ 0,1 ] s dfnd as ( x, f y > x S = ( x, f y < x. ( x, f y = x or y = x Th mum dstanc mum oprator wth rspct to [ 0,1 ] s dfnd as ( x, f y > x = ( x, f y < x ( x, f y = x or y = x Th mum dstanc mum oprator wth rspct to [ 0,1 ] s dfnd as ( x, f y > x S = ( x, f y < x ( x, f y = x or y = x Th structurs of th volutonary oprators ar llustratd n 3D, n [4]. Lmma1. Th pars (, S and (, S satsfy condtons S = 1 T ( 1 x, 1 and S = 1 T ( 1 x, 1 y. Proof. Basd on [2]. Lmma2. s
Th pars and ar commutatv, assocatv, monoton bnary opratonson th unt ntrval [0,1] and for ( x, x = Proof. Basd on [2].,.. and ar unnroms. 0 < 1 < ( x, = x From lmmas and basd on [6] w conclud that T x, S y, z = S T x, y, T x z. ( ( ( ( (, and 3.2. Dcson makng Th t-norms famly-tr root s unnorm, and ts man branch s th paramtrcal group. Usng ths norms for combnng fuzzy sts, w obtan compnsaton btwn small and larg dgrs of mmbrshps, and th pssmstc (ntrscton-typ, s MP3, and optmstc (unon-typ, MP4 conncton btwn fuzzy sts mak vn. In systm control, howvr, ntutvly qut th oppost s xpctd: lt s mak th powrful concdnc btwn fuzzy sts strongr, and th wak concdnc vn wakr. So w usd th dstanc-basd and ntropy-basd norms and conorms as volutonary oprators. Th modfd gnralsd ntropy-basd oprator satsfs volutonary condtons. Usng a novl nfrnc mchansm, n wll-known rul bas systm th gnralsd modus ponns rmans but th concdnc of th rul prms and th systm nput appars n a nw form. Bcaus of th non-monotonc proprty of ntropy-basd oprators, t was unrasonabl to us th classcal dgr of frng, to gv xprsson to concdnc of th rul prms (fuzzy st A, and systm nput (fuzzy st A, thrfor a dgr of concdnc (Doc for ths fuzzy sts has bn ntatd. It s nothng ls, but th proporton of ara undr mmbrshp functon of th modfd ntropy-basd ntrscton of ths fuzzy sts, and th ara undr mmbrshp functon of th thr unon (usng as th fuzzy unon. Ths rason has two advantags: t consdrd th wdth of concdnt of A and A', and not only th hght, andh rul output s wghtd wth a masur of concdnt of A and A' n ach rul. Th rul output fuzzy st (B s not achvd as a cut of rul consqunc (B wth Doc. or ( = T ( B( Doc B (5,
whr Doc s th dgr of concdnc, and gvs xprsson to concdnc of th rul prms (fuzzy st A, and systm nput (fuzzy st A Doc = T ( A, A dx ( A, A dx Doc, and Doc = 1 f A and A covr ovr ach othr, and Doc = 0 f A and A hav no pont of contact. It s asy to prov, that [ 0,1 ] Ths systm was tstd. Th concluson was, n smplfd rul bas systm, usng ths typ of volutonary oprators and novl Doc as th fuzzy sts concdnc xprsson, that th controlld systm obtans th dsrd stat bttr thn by classcal approach. Th FLC rul bas output s constructd as crsp valu calculatd from assocatv usng t-conorm on all rul outputs BB (. 3.3. Dcson makng wth (S,U ntgral Lt s ntroduc an m masur, for fuzzy substs A, B and A' lk n I.B, as th masur numbr rlatd to ara undr mmbrshp functon, whch dscrbs ths fuzzy st. It s asy to prov, that ths masur satsfs all proprts M1-M3. Condtons MP1-MP5. can b prov, f w for vry pont of th krnl of fuzzy st A magn th μ ( x (mmbrshp functon of A as th basc of m masur for th A. In ths cas (s MP1., MP2. ( A A = T ( m( A, m( A = Doc ( A A m, th as th wght masur for computng rul output BB' n rul n rul bas. In th on rul output s obtand as B ' p = S T j= 1 ( B(, m( A A whr w nd p S j=1 f w hav MISO systm wth p rul prmss. Ths form s n accordanc wth Eq. (5 and wth th dfnton of (S,U ntgral.
Th FLC rul bas output s constructd as crsp valu calculatd from assocatv usng (S,U ntgrals for rul outputs, basd on Eq. 4., as follows: p B ' = S S T, forallruls j= 1 ( B( m( A A It s n fact an ara, and from that ara w obtan a crsp FLC systm output wth a dfuzzfcaton mthod. As S t-conorm n ntgral, and T t-norm for calculatng masur m (s S Introductons w can us par (,, bcaus thy ar paramtrcal norms, and satsfy all condtons rqurd for fuzzy masurs and fuzzy ntgrals. 4 Concluson Usng that da, w obtan a mthod, whch s clos to th vsual dscrpton of dcson-makng n FLC, for trangular or trapzod mmbrshp functon. It s vry mportant too, that ths mthod has strong thortcal background n fuzzy ntgrals (S,U, and fnally, ths fuzzy ntgral thory was appld, and can b usd for othrs, condtonal t-norms and t-conorms. 5 Rfrncs [l] B. D Bats, J Fodor, "Rsdual Oprators of Unnorms," Soft Computng, vol. 3, no. 1, 1999, pp. 89-100. [2] Imr J. Rudas, "Masurs of fuzzynss: thory and applcatons", Hllnc Naval Acadmy, Grc, World Scntfc and Engnrng Socty Prss, 2001, p. 187. [3] Endr Pap, "Trangular norms n modlng uncrtanly, nonlnarty and dcson" n Procdngs of th 2th Intrnatonal Symposum of Hungaran rsarchrs Computatonal Intllgnc, ISBN 963 7154 06, pp. 7-18. [4] Marta Takacs, K., Lőrncz, A. Szakál, " Analyss of th Evolutonarz oprators group " n Procdngs of th 1th Intrnatonal Symposum of Hungaran rsarchrs Computatonal Intllgnc, Buadpst, Nov. 2000., pp. 7-18. [5] G.J. Klr, Tna A. Folgr, "Fuzzy Sts, Uncrtanty, and Informaton", Prntc-Hall Intrnatonal, Inc., Nw York, 1988, p. 107-134. [6] Endr Pap, "Null-Addtv St Functons ", Kluvr Acadmc Publshrs, Istr Scnc, Bratslava, Mathmatcs and Its Applcatons, Vol. 337, 2000, p. 192. [7] Marta Takacs, " Fuzzy Intgrals n FLC " n Procdngs of th INES 2002 Confrnc, Opatja, May, 2002., pp. 199-203.