Optimizing Fuzzy Membership Functions Using Particle Swarm Algorithm

Transcription

1 Proceeds of the 9 IEEE Iteratoal Coferece o Systems, Ma, ad Cyberetcs Sa Atoo, TX, USA - October 9 Optmz Fuzzy Membershp Fuctos Us Partcle Swarm Alorthm Eljah E. Omzeba ad Gbjah E. Adebayo Electrcal ad Electrocs Eeer Proramme Abubaar Tafawa Balewa Uversty Bauch, Bauch State, Nera omzebaee@yahoo.co.u ad abjah@mal.com Abstract The choce ad shape of membershp fuctos are ow to affect the performace of fuzzy systems; despte ther mportace however, MFs are eerally defed subjectvely based o eeer judmet, deser experece or chose for computatoal coveece, whch does ot ecessarly ve optmal performace whe used model or cotrol. I ths paper we preset a method for optmz membershp fuctos of a fuzzy system us partcle swarm optmzato (PSO) alorthm. The method determes the optmal shapes ad spa of membershp fucto based o a model performace measure. To demostrate ts effectveess, the proposed method was used to optmze the traular membershp fuctos of the fuzzy model of a olear system; results show that the optmzed MFs provded better performace tha a fuzzy model for the same system whe the MFs were heurstcally defed. Keywords Optmzato, Membershp Fuctos, PSO, Fuzzy Systems I. INTRODUCTION Fuzzy loc provdes a meas of process vaue, mprecse or complete formato, expressed lustc terms. Ths dffers from set theory where a value or a object ether belos to a set or s excluded from t. I fuzzy set theory a object ca belo to a set fully or partally (.e. to a certa extet). Ths vaueess s captured by membershp fuctos, MF. The extet to whch a object belo to a set (called the deree of membershp or membershp value μ) raes betwee ad 1. It s usual for MFs to be represeted raphcally; the popular MFs are the traular, trapezodal ad the smod fuctos [1]. The MFs toether wth a rule base form what s ow as the owlede base of a Fuzzy Iferec System (FIS), the complete arraemet of a FIS s show F. 1. put Fuzzfer Fure 1. Kowlede Base Rule Base Membershp Fuctos Defuzzfer Structure of a Fuzzy Iferec System output Where ad are the put ad output scal factors respectvely; the fuzzfer coverts the crsp put to fuzzy data for process by the Iferece System. After process the fuzzy output s aa coverted to a crsp value by the defuzzfer for real world applcato. Dfferet defuzzfcato methods are dscussed []. By be orazed as a ferec system, t s possble to process formato us fuzzy loc; ad has foud applcatos dfferet areas of research ad other huma edeavors, especally cotrol, model, fucto estmato ad predctve alorthms [3]. Some successful applcatos of fuzzy set theory ca be foud [4]. The results obtaed from fuzzy reaso have bee foud may cases to be superor to those of the covetoal methods; these successes have bee attrbuted to the ablty of the FIS to represet ad process mprecse formato commo to most real-lfe problems a practcal way [5]. The rest of the paper s orazed as follows: Secto I presets a bref overvew of dfferet fuzzy models, whle secto II ad III ves a bref revew of FIS ad dfferet methods of determ MFs. Sectos IV ad V descrbes the PSO ad alorthm for ts mplemetato; The ext three sectos dscusses the problem defto, how MF defto may be ecoded to the partcles of a PSO ad the parameter for determ the proress of the optmzato process.. Sectos VIII ad IX preset a umercal problem for verfy the feasblty of the proposed method, ad the results of smulatos. The last secto s the cocluso. II. LINGUISTIC VARIABLES AND MEMBERSHIP FUNCTIONS I a FIS, each put varable s dvded to overlapp fuzzy parttos such as Lare (L), Medum (M), Small (S) ad so o, to cover the etre rae (or uverse-of-dscourse) of that put. The dfferet fuzzy parttos are the descrbed by membershp fuctos, ad sce these parttos overlap, a put value ca belo to more tha oe of the parttos. For the Mamda type ad the New Fuzzy Reaso Method (NRFM) type fuzzy models, the output s also parttoed the same way as the put. For Taa-Sueo-Ka (TSK) type fuzzy model, the outputs are represeted as lear fuctos of the put ad output varables [1], [6]. From the foreo t ca observed that the fuzzy system s characterzed by ts MFs [] ad t has bee show by [7] that /9/$5. 9 IEEE 3966

2 the type ad deftos of the parameters of MFs a fuzzy set determe the performace of the FIS; For example, creased fuzzess (more parttos) results more rules ad slushess of respose because of the amout of computatos that must be carred out to resolve the rules before a output s eerated [8]. Hece the effectveess of a Fuzzy Iferec System depeds amo others factors o the umber of parttos ad the deftos of the assocated MFs. Despte ther mportace, there are o emprcal methods for determ the shape, the umber or the spa of membershp fuctos; these are decded subjectvely based o eeer judmets, tuto, requred applcato or deser experece; for cotrol purposes for stace, t s commo to use traular MFs, whle Gaussa shaped MFs seem to be preferred for fucto approxmators. III. MF DETERMINATION METHODS I [8] a teratve method of radually reduc the umber of MFs a fuzzy cotroller based o some performace crtera was proposed, the objectve was to reduce the umber of rules so as to haste the speed of respose of a flexble spacecraft atttude cotroller. Also [9] troduced a teratve procedure for alter the spread of MFs for a FIS, eve thouh the method was subjectve ad tme cosum. I other to remove the subjectvty the defto of MFs, [1] used a areato of expert opos to determe the MFs of a FIS, the fal outcome depeded larely o each experts subjectve terpretato of the varous lustc terms. O the other had, a umber of computatoal methods have bee used for the determato ad tu of MFs. Oe approach by [11] used a cluster alorthm to roup a set of raw data to clusters ad used a artfcal eural etwor (ANN) to determe the membershp fuctos from the result clusters. The method requres a lare data set for the alorthm to succeed. Also, [1] used the eetc alorthm (GA) to optmze a set of tally defed MF a fuzzy system meat to cotrol a helcopter. GA also requres a lare data set for a ood search, ad the ecod of data to the chromosomes of a GA requres ood owlede of the uderly system. I ths paper, we use the partcle swarm alorthm to optmze the membershp fucto of a NFRM used for model a olear plat; the model accuracy was compared to that of a expert desed NFRM whose MF were ot optmzed. IV. PARTICLE SWARM OPTIMIZATION The partcle swarm optmzato, PSO, alorthm le the eetc alorthm (GA) s spred by a bolocal system; whle the GA mmcs the process of atural evoluto, the PSO mmcs, for example, a floc of brds mrat search of a commo objectve ([13] ad [14]). I both alorthms, a populato of potetal soluto caddates (called a swarm PSO, each member s called a partcle) are used to see better soluto(s) over a umber of terato steps, called eeratos. The major dfferece s that whle the GA dscards the least ft members of the soluto caddates after every eerato, PSO each partcles s updated based o ts ow experece ad the experece of ts ehbors [15]. To ude the search process a performace evaluato parameter called ftess s used; ths parameter relates the problem to the search process [13]. Each partcle eeps a record of the best posto t has foud (called the persoal best, pbest), whle the swarm eeps a record of the best soluto ay partcle has attaed (called the lobal best, best or local best lbest). For the case of best, all partcles the swarm are coected to oe aother ad the search s for a lobal objectve. Whle for lbest a partcle s oly coected to those partcles ts ehborhood; ad the search s for some local solutos. The values of pbest ad best whch are updated after each terato flueces the teracto betwee partcles ad ultmately the search process [16]. V. PSO ALGORITHM The steps volved mplemet the PSO alorthm ca be summarzed as: Italze PSO parameters, ad N partcles Determe the ftess of each partcle Update pbest ad best Update veloctes ad postos (partcles) Termate o coverece or at ed of terato Go to step The N partcles toether wth ther veloctes may be talzed radomly or based o a expert put; the latter results a faster coverece. The pbest s updated by compar the curret pbest of a partcle wth that of ts prevous pbest, ad reta the better of the two values; whle the best of all the pbests ay eerato s selected as the best. Veloctes ad postos are updated us the PSO equatos (1) ad () respectvely. v = ω v + c r ( pˆ ) ( ˆ p + c r p p ) p = v + p Where, c 1 ad c are the PSO parameters to be talzed, whle r 1 ad r are ormalzed ut radom umbers ad s the terato couter. Also pˆ pbest for the th partcle ad pˆ s the best. I ths mplemetato, the best partcle s perturbed slhtly to esure that t sees better soluto ts mmedate ehborhood us (3) ad (4). v = α v + β r p = v + p I + 1 I + 1 Where r 3 s a uform radom umber, ad are parameters of the PSO. (1) () (3) (4) 3967

3 1 μ VI. PROBLEM DEFINITION I order to use the PSO to optmze MF, the problem was set up for a traular MF (The same arumet ca be exteded to other MF types) as follows: Defe the spa of the MF us a j, b j ad c j as F. such that a j x, b x m j. Where max the rae (uverse of dscourse) of x s x x ad c j s the pot of maxmum support of the fuzzy set j. Such that f c =.5( a + b ) the c j s the mdpot. j j j a c b x Fure. After ecod to PSO, each partcle has 3 N data elemets, where N s the umber of parttos the problem space. The tas s to mprove the performace of the FIS by optmz a j, b j ad c j. VII. FITNESS MEASURE The performace of a fuzzy model ca be measured by the mea-square-error,. For two model estmates for example, the oe wth the smaller wth respect to the expermetal model has a better match. The, whch s defed (5), s used as the ftess measure for the PSO alorthm mplemetato. Traular MF Parameters q ( y = 1 = q = 1 yˆ ) y Where y ad ŷ represets the actual value ad the estmated value at data pot respectvely; q s the umber of such data pots. N (a) Iput MF Curret (A) 1.5 max m (b) Output MF (5) Fure 3. The Expermetal Characterstcs TABLE I. FUZZY RELATION MATRIX N Z S M L G VIII. SIMULATION EXAMPLES To llustrate how the PSO may be used tu fuzzy membershp fuctos, the ew reaso fuzzy model (NFRM) as desed by [7] for a o-lear system whose characterstcs s show F. 3 was used. The Fuzzy Relato Matrx (FRM) ad tal MFs ( the put ad output spaces) as desed by [7] are show Table I ad F. 4 respectvely. The tas s to optmze the MFs the put ad output spaces such that the result fuzzy model s better wth the optmzed MFs tha that result from the Expert MF us the same Fuzzy Relato Matrx ad the same umber of put ad output parttos. IX. SIMULATION AND RESULTS The PSO parameters were set as Table II ad smulatos were carred out as follows: Case I: Optmze the put MFs oly (a) Wth tal expert MF I ths case the MFs of F. 6(a) were cluded as part of the tal partcles the optmzato process. Ths esures that the performace of the PSO optmzed MFs s at least equal to that of the expert desed system. (b) Wthout expert put I ths case, all the tal MFs were radomly defed. TABLE II. PSO PARAMETERS Parameter Value c 1.5 c No of Partcles 1 teratos - 5 ftess Fure 4. Expert MF for Iput ad Output 3968

4 The tred of over the terato perod for case I(a) s show F. 5(a), whle the fuzzy model for the best tal ad fal (or optmzed) MFs s preseted F. 5(b). The best perform tal partcle s the expert defed MF cluded as oe of the tal partcles. Smlar outcome for case I(b) s show F. 5(c) ad 5(d). Case II: Optmze the output fucto oly (a) Wth a expert put I ths case the output MF of F. 6(b) were cluded as part of the tal partcles the optmzato process. Aa esur that the performace of the PSO optmzed MFs s at least equal to that of the expert desed system (b) Wthout a expert put I ths case, all the tal MFs were radomly defed. Aa, for Case II(a) the tred of over the terato perod s show F. 5(e) whle the Fuzzy models wth the best tal MFs ad PSO optmzed MFs are show F 5(f). Smlar results for Case II(b) are show Fs. 5() ad 5(h) I both cases, a expert put esured that the started lower ad sped-up the coverece of the lear process as expected. The fal MFs eerated after the lear process s show F. 6. (a) Tred of for Case I (a) Fal Geeratos (e) Tred of for case II (a) Best tal eeratos Best tal Fal Curret(A) (b) Fuzzy Geerated Models for Case I(a) Curret (A) (f) Fuzzy Geerated Models for Case II (a) (c) Tred of for Case I (b) Geeratos eeratos () Tred of for Case II (b) Fal Best tal Fal Best tal Curret(A) (d) Fuzzy Geerated Models for Case I(b) Curret (A) (h) Fuzzy Geerated Models for Case II (b) Fure 5: Smulato Results for Cases I ad II 3969

5 Membershp Grade,μ N Z S M L G Curret (A) (a) Fal MFs for Case I(a) N Curret (A) (b) Fal MFs for Case I(b) (c) Fal MFs for Case II(a) REFERENCES [1] Babusa R ad Verbrue H. B.: A Overvew of Fuzzy model for Cotrol. Cotrol Eeer Practce, vol 4 No 11, 1996 pp [] Lee Chue Che: Fuzzy Loc Cotrol Systems: Fuzzy Loc Cotroller Part I. IEEE Trascatos o Systems, Ma ad Cyberetcs. Vol No, March/Aprl, 199. [3] Su Halo ad Lu Lxa: A lear output structure for fuzzy loc cotrollers. Fuzzy Sets ad Systems, No 131,, pp [4] accessed Jauary, 9. [5] Precup R.-L, Tomescu M. L. ad Pred S: Lorez System Stablzato Us Fuzzy Cotrollers. Iteratoal Joural of Computers, Commucatos ad Cotrol, Vol 1 No 3 7, pp [6] Taa Tomohro ad Mcho Sueo: Fuzzy Idetfcato of Systems ad ts Applcato to Model ad Cotrol. IEEE Trasacto o Systems, Ma ad Cyberetcs, vol SMC-15 No, February, 1985, pp [7] Par D., Kedel A. ad Laholtz G.: Geetc-Based New Fuzzy Reaso Models wth Applcato to Fuzzy Cotrol. IEEE Trasactos o Systems, Ma ad Cyberetcs, vol 9 o 1, Jauary. 1994, pp39 46 [8] Eljah E. Omzeba: Rule Optmsato Alorthm for PID type Fuzzy Cotroller. The Nera Joural of Research ad Producto, Vol. 6 No 1 Aprl, (5) pp [9] Procy, T. S ad Mamda, E. H.: A Lustc Self- Oraz Process Cotroller. Automatca Vol 15, 1979, pp 15 3 [1] Wataabe N: Statstcal method for estmat Membershp Fuctos. Japaese Joural of Fuzzy Theory ad Systems, Vol 5 No 4, [11] Taa ad Yahash NN-Drve fuzzy Reaso. Iteratoal Joural of Approxmate Reaso. Vol 5, 1991, pp [1] Meredth D. L., Kar C. L. ad Kamur K. K.: The use of Geetc Alorthm the Des of Fuzzy Loc Cotrollers. 3 rd Worshop o Neural Networs WNN9, 199, pp [13] Paquet U. ad Eelbrecht A. P.: Tra Support Vector Maches wth Partcle Swarms. I Proceeds of Iteratoal Jot Coferece o Neural Networs (IJCNN) Coferece, 3, pp [14] Veu G. G. ad Gaesh K. V.: Evolv Dtal Crcuts Us Partcle Swarm. I Proceeds of Iteratoal Jot Coferece o Neural Networs (IJCNN) Coferece, 3, pp [15] Motes de Oca ad Marco A Partcle Swarm Optmzato 7 Retreved d Aprl, 8 from [16] L Xaodo: Partcle Swarm Optmzato: A troducto ad ts recet developmets. I SEAL 6. (d) Fal MFs for Case II(b) Fure 6: Smulato Results for Cases I ad II X. CONCLUSION It s clear from the results that a expert put helps the PSO search process ad reduces the umber of teratos requred for coverece. Also t s observed that ths case, the output membershp fuctos have more fluece o the performace of the fuzzy model, optmz the output membershp fuctos yelded a result comparable to that of [7] where the GA was used to optmze both the fuzzy relato matrx ad put membershp fuctos. The results also show a ood promse for pso-fuzzy systems. 397