Nonlinear System Modeling Using GA-based B-spline Membership Fuzzy-Neural Networks

Transcription

1 nd Inernaonal Conference on Auonomous Robos and Agens December 3-5, 4 Palmerson Nor, New Zealand Absrac Nonlnear Sysem Modelng Usng GA-based B-slne Members Fuzzy-Neural Newors Y-Guang Leu Dearmen of Elecronc Engneerng, Hwa-Hsa Insue of Tecnology No., Gong Juan Rd., Cung Ho Cy, Tae, Tawan, 35, R.O.C. leuy@cc.w.edu.w In s aer, we nvesgae e omzaon roblem of B-slne members fuzzy-neural newors. Wen e B-slne members fuzzy-neural newors are used for comlex nonlnear sysem modelng, ere are some roblems, suc as ow o selec e arorae no osons, and ow o coose conrol ons omally. Tese roblems are sgnfcanly moran n acevng good aroxmaon. Te unsuable no osons and e unsuable conrol ons ofen cause e oor erformance of B-slne members fuzzy-neural newors. So far, ere s less eory abou ow bo no ons and conrol ons can be cosen omally. Snce e wegng facors, e no osons, and e conrol ons are consdered o be varables, becomes a gly nonlnear omzaon roblem. Terefore, we roose a genec algorm (GA) o smulaneously omze ese varables. Also, s algorm can ossess e caably of escang from local mnma. For e urose of llusrang effecveness of e roosed meod, an examle of nonlnear sysems s smulaed. Keywords: B-slne funcons, Fuzzy-neural newors, Nonlnear sysem modelng, Genec algorm Inroducon Snce neural newors and fuzzy logc sysems are unversal aroxmaors [,], nonlnear sysem modelng va ese aroxmaors as wdely been develoed for many raccal alcaons [3,4]. Moreover, many researces [4-7] combnng fuzzy logc w neural newors ave been develoed o mrove e effcency of nonlnear sysem modelng. Te aroxmaors can be exressed as a lnear combnaon of bass funcons. An arorae coce of bass funcons s e B-slne. Te B-slne funcon s a ecewse olynomal mang. In B-slne fuzzy-neural newor srucure [3,7], e B-slne members funcons are assumed o be fxed and only conrol ons are adused durng e learnng rocess. Before e learnng rocess, e desgner as o secfy e nos of e B-slne members funcons. Because e selecon of e aroraed nos of e B-slne members funcons s crucal o obanng good aroxmaon, s an moran ssue for engneerng roblem. Terefore, e nos mus be reaed as varables. Ten e roblem becomes a comlex nonlnear and mulvarable omzaon roblem w many local oma. Tus, s dffcul o oban a global omum. Recenly, some researcers ave been ryng o use socasc aroaces o solvng suc roblems. For examle, smulaed annealng [8] and genec algorm [9] are socasc aroaces. Tese algorms ossess e caably of fndng e global omum soluons. Snce B-slne funcons ossess caracerscs of easy local adusng, smle calculaon and mlemenaon, ey ave wdely been used n grac rocess [,], conrol [,3], modelng [3,7,8,4,5], and so on. A deermnsc erave aroac o adave esmaon of aramerc deformable conours based on B-slne reresenaons as been develoed n []. In [], e acual exermens for a DC moor seed conrol sysem ave been resened. In [3], e B-slne members funcons ave been consruced and aled successfully e fuzzy-neural modelng. Ten, e B-slne members fuzzy -neural newors ave been exended o on-lne nonlnear conrol n [3]. Wen e B-slne newor s no aroxmang a gven funcon, e auors n [4,5] add more no ons unformly rougou e doman of neres unl e aroxmaon s sasfacory. In [8], a novel no-omzng B-slne newor as been roosed o aroxmae nonlnear sysem beavor. Snce radonal omzaon meods are dffcul o searc a global mnmum of a mullocal mnma nonlnear funcon, some socasc aroaces ave araced muc aenon. In [6,7], e auors resened er algorms mxed by some socasc aroaces n order o ncrease e effcency of algorms. An annealng-genec algorm for solvng NP-Hard roblems as been roosed n [6]. In [7], a socasc aroac mxed by smulaed annealng algorm, genec algorm, and cemoaxs algorm as been resened for solvng comlex omzaon roblems.. 9

2 nd Inernaonal Conference on Auonomous Robos and Agens December 3-5, 4 Palmerson Nor, New Zealand In s aer, we nvesgae e omzaon roblem of B-slne members fuzzy-neural newors. Mang e roblem-omzed srucure consss of wo arameer searc roblems. Te frs s o oban e wegng facors of fuzzy-neural newors. Te second s o consruc B-slne funcons, ncludng e no osons, and a se of conrol ons. Snce e wegng facors, e no osons, and e conrol ons are consdered o be varables, becomes a gly nonlnear omzaon roblem. Tus, e obecve s o roose a socasc omzaon algorm o smulaneously omze ese varables. Also, s algorm can ossess e caably of escang from local mnma. Because genec algorms ave been eorecally and emrcally roven o rovde e effcen searc for many gly nonlnear roblems, ey offer a good cance of success. Here, e genec algorm s used as e socasc omzaon algorm.. Fuzzy-Neural Newors Te basc confguraon of fuzzy logc sysems consss of some fuzzy IF-THEN rules and a fuzzy nference engne. Te fuzzy nference engne uses e fuzzy IF-THEN rules o erform a mang from an T n nu lngusc vecor x [ x x x n ] o an ouu lngusc varable y. Te fuzzy IF- THEN rule s wren as were If x s Ten y s A B A, A,, A and... and x n s and n B A n are fuzzy ses. Le be e number of e fuzzy IF-THEN rules. By usng roduc nference, cener-average and sngleon fuzzfer, e ouu of e fuzzy logc sysem can be exressed as n w x ) A () ( A n ( x ) ( x ) A were x ) s e members funcon value of e fuzzy varable x, s e number of e oal IF THEN rules. Te wegng facors w,,,, are adusable arameers. Fg. sows e confguraon of fuzzy-neural newors. Te sysem as a oal of four layers. Nodes a layer I are nu nodes a reresen nu lngusc varables. Nodes a layer II are erm nodes wc ac as B-slne members funcons o reresen e erms of e resecve lngusc varables. Eac node a layer III s a fuzzy rule. Layer IV s e ouu layer. x x... x n A A A n... w... w w Layer I Layer II Layer III Layer IV y ( x) Fg.. Te confguraon of fuzzy-neural newors. B-slne members funcons A B-slne funcon s a ecewse olynomal. Le T {,,, } be e no vecor, were e r are calls nos w. Te B- r slne bass funcon of order, denoed by defned as and f N, () oerwse N, ( ( ) ( ) N ) N, ( ), ( ) (3) N,, s For r conrol ons, e B-slne funcon s() s defned as r c s( ) N (. (4) ), Accordng o [3], e B-slne members funcon x ) s defned as A ( c r ( x ) N ( x ) A, (5) were x s e nu daa and A s a fuzzy se. Fg. sows llusraon of B-slne members funcon. Te B-slne members funcon as 7 conrol ons, 9 no ons, and order. 3

3 nd Inernaonal Conference on Auonomous Robos and Agens December 3-5, 4 Palmerson Nor, New Zealand c c c3 c c Fg.. Illusraon of e B-slne members funcons 3 Te roosed genec algorm In order o solve e omzed B-slne members fuzzy-neural newors, we assume e rule and B-slne members funcon x ) as r+ c4 c5 ( A conrol ons. Ten, e adused varables nclude e wegng facors w,,,,, e conrol ons ons c,, q,,, r,,,, n, and e no q,,, r,,,, n. Noe a because e equaly s allowed, e q ( q ) redundan nos wll be avoded. Hence, e obecve of e searc algorm s o mnmze e error funcon E( w, c, ), were c {,, r,,,..., n}, c { q,, r,,,..., n} q,and w { w,,, }. Te error funcon s defne as * E ( w, c, ) y l y l ) (6) m ( m l were m s e number of e ranng daa ars, and * y and y reresen e ouus and e desred l l ouus resecvely. Defne a cromosome as l T T T l l l l z [ w c ] [ z z z z ], were a se of e wegng facors w range from wn e nerval D [ w, w ] R, a se of e conrol ons c mn max range from wn e nerval D [ c, c ] R, and a se of e no ons range from wn e nerval D [, ] R,. Te and c 3 are e nal values of e no ons and e conrol ons. Besdes, defne a fness funcon as fness E ( w, c, ) (7) Te roosed genec algorm erformed sown as Fg. 3. Te deal descron s as follows: Genec_Algorm() { Inalze e Poulaon_of_Cromosomes; Calculae e Fness_Funcon; Wle (no ermnae-condon) { Perform Selecon w Sorng; Perform Crossover Accordng o e Sored Poulaon; Perfom Muaon Accordng o e Sored Poulaon ; Calculae e Fness_Funcon; } Fg. 3. Te roosed genec algorm. Te nalzaon rocedure begns w e nalzaon of e cromosomes n e oulaon. Eac cromosome s coded as an adusable vecor w floang on ye comonens. Durng e nalzaon se, e nal values of cromosomes are randomly creaed n some nervals. Durng e selecon rocess, e oulaon s frs sored by ranng e fness of cromosomes. In arcular, e frs cromosome of e sored oulaon as e ges fness value (or smalles error). Ten, based on e sored oulaon, e selecon rocess reans a bes ndvdual n e curren generaon uncanged for e nex generaon. Afer e selecon rocess, e crossover rocedure selecs randomly subars from wo aren cromosomes and creaes a new offsrng cromosome. Here, e wo aren cromosomes are seleced accordng o e sored oulaon. In arcular, a ar of crossover cromosomes s frs seleced f ey ave beer fness values. Durng e muaon rocess, ceran comonens n some randomly seleced cromosomes may be randomly relaced by new comonens. Moreover, some cromosomes w worse fness values may be comleely relaced, and e robably of e comlee relacemen s based on e sored oulaon. 4 Smulaon resuls An examle s llusraed o sow e effecs of e B- slne members fuzzy-neural newor o aroxmae nonlnear sysems usng e roosed genec algorm. Eac nu of e fuzzy-neural newor as 5 B-slne members funcons. All of e B-slne members funcons are w order α= 3

4 nd Inernaonal Conference on Auonomous Robos and Agens December 3-5, 4 Palmerson Nor, New Zealand and eac B-slne members funcon as 7 conrol ons and 9 no ons. Consder a nonlnear sysem governed by e dfference equaon ).3 ) u ( )(. y ( )) 3 u ( ) were u( ).6sn( / 5).4sn( /). 49 ranng daa ars are gven. I s assumed a e number of e cromosome s and -)=. Te nal arameers w are random values n e nervals D w, w ] [ 5,5], and Te nal [ mn max arameers c and q of e B-slne members funcons for u() or -) are values accordng o no ons and conrol ons sown n Fg. 4. Fg. 5 and Fg. 6 sow e B-slne members funcons of e fuzzy-neural newor for -) and u() afer 4 eraons, resecvely. Te error curve of e roosed meod afer 4 eraons s sown n Fg. 7. As demonsraed n Fgs. 7, e roosed genec algorm successfully aroxmaes e nonlnear sysem Fg. 6. B-slne members funcons of e fuzzyneural newor for u() afer 4 eraons..4 error eraons Fg. 7. Error curve of e fuzzy-neural newor w resec o eraons Fg. 4. B-slne members funcons of e fuzzyneural newor for -) or u() before eraons Fg. 5. B-slne members funcons of e fuzzyneural newor for -) afer 4 eraons. 5 Conclusons In s aer, snce e selecon of e wegng facors, e no osons, and e conrol ons of e B-slne members fuzzy-neural newors s crucal o obanng good aroxmaon for comlex nonlnear sysems, we develo a genec algorm w an effcen searc sraegy o omze ese varables and escae from local mnma. For e urose of llusrang effecveness of e roosed meod, an examle w g nu dmensons as been smulaed. 6 Acnowledgemens Ts wor was suored arally by e Naonal Scence Councl, Tawan, under Gran NSC 9-3- E References [] K. Horn, M. Snccombe, and H. We, Mullayer feedforward newors are unversal aroxmaors, Neural Newors, no., ,989. [] L.X. Wang and J.M. Mendel, Fuzzy bass funcons, unversal aroxmaon, and orogonal leas squares learnng, IEEE Trans. 3

5 nd Inernaonal Conference on Auonomous Robos and Agens December 3-5, 4 Palmerson Nor, New Zealand Neural Newors, vol. 3, no. 5,.87-84, 99. [3] C.H. Wang, W.Y. Wang, T.T. Lee, and P.S. Tseng, Fuzzy B-slne members funcon (BMF) and s alcaons n fuzzy-neural conrol, IEEE Trans. Sys. Man, Cyber., vol. 5, no. 5,.84-85, 995. [4] L.X. Wang, Adave fuzzy sysems and conrol: desgn and sably analyss, Englewood Clffs,NJ: Prence-Hall, 994. [5] S. Horawa, T. Furuas, and Y. Ucawa, On fuzzy modelng usng fuzzy neural newors w e bac-roagaon algorm, IEEE Trans. Neural Newors, vol. 3, no.5, Seember 99. [6] C. T. Ln and C. S. George Lee, Neuralnewor-based fuzzy logc conrol and decson sysem, IEEE Trans. Comuer, vol. 4, no.,.3-336, December 99. [7] W.Y. Wang, T.T. Lee, andc.l. Lu, Funcon aroxmaon usng fuzzy-neural newors w robus learnng algorm, IEEE Trans. Sys. Man, Cyber. Par B, vol. 7, no. 4, , 997. [8] K.F. C. Yu, S. Wang, K. L. Teo, and A. C. Tso, Nonlnear Sysem Modelng va Kno- Omzng B-Slne Newors, IEEE Trans. NEURAL NETWORKS, vol., no. 5,.3-,. [9] J. N. Aaral, K. Tumer, and J. Gos, Desgnng genec algorms for e sae asgnmen roblem, IEEE Trans. Sys., Man, Cybern., vol. 5, , 995. [] M. Fgueredo, J. Leão, and A. K. Jan, "Unsuervsed conour reresenaon and esmaon usng B-slnes and a mnmum descron," IEEE Transacons on Image Processng, vol. 9, no. 6,.75-87,. [] P. San-Marc, H. Rom, and G. Medon, Bslne conour reresenaon and symmery deecon, IEEE Trans. Paern Anal. Macne In-ell., vol. 5,. 9 97, Nov [] S. Cong and G. L, "Te desgn of adave B- slne fuzzy neural newor conroller," Comuer Engneerng and alcaon, 35(9),.66-68, 999 [3] Y.G. Leu, T.T. Lee, and W.Y. Wang, On-lne unng of fuzzy-neural newor for adave conrol of nonlnear dynamcal sysems, IEEE Trans. Sys. Man, Cyber. ar B: Cybernecs, vol. 7, no. 6,.34-43, December 997. [4] T. Kavl, ASMOD An algorm for adave slne modelng of observaon daa, In. J. Conr., vol. 58, no. 4, , 993. [5] K. Hlavacova and M. Verleysen, Placng slne nos n neural newors usng slnes as acvaon funcons, Neurocomu., vol. 7,.59 66, 997. [6] F.-T. Ln e al., Alyng e genec aroac o smulaed annealng n solvng some NP-ard roblems, IEEE Trans. Sys., Man, Cybern., vol. 3, , Nov./Dec [7] B. L and W. Jang, A novel socasc omzaon algorm, IEEE Trans. Sys. Man, Cyber. Par B, vol. 3, no.,.93-98,. 33