An improved algorithm for online identification of evolving TS fuzzy models

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1 Poceedngs of the 8th WSEAS Intenatonal Confeence on Fuzzy Systems, Vancouve, Btsh Columba, Canada, June 9-, 7 3 An mpoved algothm fo onlne dentfcaton of evolvng S fuzzy models (Invted Pape) ESMAEEL BANYSAEED, MOHAMMADEZA AFIEI, and MOHAMMAD HADDAD Depatment of Electcal & obotcs Engneeng Shahood Unvesty of echnology, Shahood IAN afe@eee.og Abstact In ths pape an mpoved algothm fo onlne dentfcaton of aag-sugeno fuzzy ule-based models fom I/O data steams s poposed. he S model has evolvng stuctue.e the fuzzy ules can be added, modfed o deleted automatcally. Both pats of dentfcaton algothm (unsupevsed fuzzy ule-base antecedent leanng by a ecusve, non-teatve clusteng, and the supevsed lnea sub-model paametes leanng by LS estmaton) ae developed fo the MIMO case. he adus of nfluence of each fuzzy ule s calculated as an adaptve vecto nstead of beng fxed vecto, allowng dffeent aeas of data space to be coveed. he centes and wdths of membeshp functons ntally detemned by onlne clusteng ae optmzed contnuously usng a gadent descent method. hs featue enables the dentfcaton algothm to deal wth tme-vayng systems and non-statonay data steams. Smulaton studes (usng two benchma poblems ) and compasons wth some othe onlne leanng algothms demonstate that a moe compact stuctue wth hghe pefomance s acheved by the poposed appoach. Keywods onlne system dentfcaton, evolvng fuzzy model, paamete optmzaton Intoducton System Modelng plays an mpotant ole n many engneeng felds such as pedcton, communcaton and contol. S fuzzy models ae poweful pactcal engneeng tools fo modelng and contol of complex systems. hey have a quas-lnea natue and utlze the dea of appoxmaton of a nonlnea system by a collecton of fuzzly mxed local lnea sub-models. hs featue enables them to appoxmate sevee nonlneaty, multple opeatng modes, and sgnfcant paamete o stuctue vaaton []. he methods fo leanng S models fom data have two mao pats: stuctue dentfcaton (estmaton the numbe of equed fuzzy ules and the centes and wdths of the membeshp functons) and paamete dentfcaton (leanng the fee paametes of lnea sub-models of the consequents). Wth fxed antecedent paametes, the S model s tansfomed to a lnea model wth some paametes whch ae calculated by LS algothm. eal-tme mplementaton of offlne dentfcaton methods s vey dffcult o even mpossble due to the tme-consumng e-tanng pocedues. Hghvolume and non-statonay data steams (poduced fo example n lage ndustal pocesses ) can not be analyzed n batch mode methods such as bacpopagaton o genetc algothm. eal-wold engneeng poblems such as ntellgent systems eque mpotant featues such as fast and onlne ncemental adaptve leanng, open stuctue oganzaton, ablty fo memozng nfomaton, nowledge acquston and self-mpovement [7]. hese equements lead to development of evolvng fuzzy neual netwos (EfuNN), evolvng connectonst systems and evolvng ule-based taagsugeno (es) models. he es leanng algothm s based on a ecusve evaluaton of the nfomatve potental of new data ponts and the focal ponts of the ules[]. he algothm updates the numbe of ules and the antecedent and consequent paametes contnuously and dynamcally. Outles have no chance to become

2 Poceedngs of the 8th WSEAS Intenatonal Confeence on Fuzzy Systems, Vancouve, Btsh Columba, Canada, June 9-, 7 33 ule centes. It s mpotant to note that leanng could stat wthout a po nfomaton and only one data sample. heefoe ths appoach s vey useful n ealtme contol, adaptve contol, obotc, dagnostc systems and data acquston []. In [], an appoach fo onlne leanng of MISO (Mult-Input Sngle-Output)S fuzzy models s poposed. the extenson of ths algothm nto MIMO (Mult-Input Mult-Output) case s demonstated n []. In some stuatons the numbe of fuzzy ules gows damatcally because the ctea do not consde the condtons whee some neffcent fuzzy ules need to be deleted. efeence [3] pesents an adaptve fuzzy modelng and contol scheme. Both [] and [3] consde the adus of nfluence of each fuzzy ule as a constant vecto that s detemned ntally by the use. o the authos' expeence, the adus of each cluste should not ept constant, because the dstbuton of data ponts sgnfcantly changes n dynamc systems. hs concept s consdeed n ths pape. Even f the adus of nfluence of some clustes can be constant, due to the lac of po nfomaton about the dstbuton of data ponts n clustes t s not easy to detemne t at the stat of algothm wthout clusteng. Most of stuctue dentfcaton methods ae based on data clusteng such as fuzzy C-means clusteng, mountan clusteng, and subtactve clusteng. hese appoaches eque all nput-output pas of data to be eady befoe statng the dentfcaton pocess. A few onlne clusteng methods can be found such as one pesented n [6], but t s sutable fo a specal neual fuzzy system only (self-constuctng neual fuzzy nfeence netwo). In [4] a nonlnea tansvesal fuzzy flte wth onlne clusteng s poposed. he clusteng method needs some constants to be pedefned. hese constants sgnfcantly affect the numbe of ules, but thee s no staghtfowad way to detemne them. In [5] and [6], dynamc fuzzy neual netwos (DFNN) ae pesented. Howeve, n these methods all past tanng data must be stoed. heefoe, heavy memoy and computaton buden ae unavodable. In ths pape, a modfed onlne subtactve clusteng s used to detemne the numbe of ules and the cente and adus of nfluence of each new fuzzy ule. o each a pecse and effcent model, the modelng appoach needs to be able to handle all the necessay opeatos as addng, emovng, and modfyng. But, most of the exstng methods ethe do not consde all the afoementoned opeatos (especally the emovng one) o suffe fom lac of a good mechansm to handle the opeatos. In ths pape, heustc cteons ae poposed to add, modfy o delete the ules. he centes and wdths of all membeshp functons ae updated usng the gadent descent (GD) method. hs pape s oganzed as follows. In secton we fomulate the MIMO S model. An onlne dentfcaton method fo models wth some new aspects s pesented n secton 3. Numecal examples ae povded n secton 4 to llustate the pefomance of the pesented method. Concludng emas ae gven n secton 5. Geneal descpton of MIMO S Model a evew S models can be descbed as a set of fuzzy ules of the followng fom: : IF ( x s A ) AND... AND ( x n s A n) () HEN y = x e π ; = {, } s the numbe of fuzzy ules, x= [x, x,, x n ] denotes the nput vecto, x e = [;x] s extended nput vecto, to consde a fee paamete of consequent pat. A denote the antecedent fuzzy sets, y s the multdmensonal output vecto of the -th lnea submodel and π ae ts paametes(assumng m output vaables ): y = [ y,,..., ] y y m () a a a m a a am π (3) = an an anm We use Gaussan antecedent fuzzy sets because t ensues geatest possble genealzaton of the descpton ( t assumes nomal dstbuton of data n a clustes ): xc (4) µ ( x ) = exp ; = : ; = : n σ Whee c and σ ae the cente and wdth of the Gaussan functon that epesents the degee of belongness of x to the -th cluste. Lneaty of consequent pat sub-models s a vey useful chaactestc, especally fo contolle desgn. Fo example, n the model pedctve contolles

3 Poceedngs of the 8th WSEAS Intenatonal Confeence on Fuzzy Systems, Vancouve, Btsh Columba, Canada, June 9-, 7 34 whch ae extensvely beng used n pocess contol, an optmzaton poblem should be solved n each samplng peod. If we use the nonlnea model fo the pocess, the optmzaton poblem becomes nonlnea and complcated, whch has no explct soluton and should be solved by tme-consumng teatve pocedues. By usng lnea model of the pocess, thee ae seveal explct solutons n the fom of pogam codes fo a lnea optmzaton poblem whch s encounteed. Lewse, contolle desgn fo a lnea system s ease than a nonlnea ones. Actvaton level of the ules ae defned as Catesan poduct of a espectve fuzzy sets fo ths ule: ( ) n ϕ x = µ ( x ) (5) = he nomalzed actvaton level of the -th ule s calculated as follows: ϕ (6) λ = ϕ = he level of contbuton of the -th lnea sub-model to the oveall output of the S model s popotonal to λ he -th output vaable of S model s calculated by weghted aveagng of ndvdual ules contbuton : y = λ y ; = : m (7) = 3 Impoved onlne dentfcaton of es model Essental pats of onlne leanng of es models ae ecusve clusteng, devaton of a gadually evolvng ule-base and weghted LS algothm. Basc stages of the pocedue ae descbed. 3. ule-base ntalzaton At the stat of leanng, the fst data pont s consdeed as the focal pont of the fst cluste. Its potental s assumed equal to one. he covaance matx C s ntalzed wth lage values, Ω, n ts man dagonal: = ; = ; * x = x ; p ( c) = (8) θ = π = ; C = ΩI Whee c s the cente of fst cluste and * x s the focal pont of fst ule( a poecton of c on the axs x) 3. Calculatng potental of new data pont (stat the loop) At the next tme step ( = + ), the potental of new data pont (z )s calculated by a cauchy type functon of fst ode: n+ m ( ) = ( ) ;,3,... d = + l l= = dl = z z. p z whee l Statng fom (9) and expessng the poecton of dstances n an explct fom fo the tme, ecusve fomula of the potental s deved as follows, whch s vey mpotant fo onlne mplementaton of the leanng algothm: ( ) p ( ) z = ( )( ω + ) + τ v Whee l ( ) ( ) (9) () n + m ; n + m ; n + m ; l = z = z = z β β = z = l= = = l= ω τ ν It s seen that β and τ ae elated to the pevous data samples. n ode to educe the computatonal load, these vaables ae calculated ecusvely: n m ( ) ; τ = τ + + z β = β + () z = 3.3 Updatng the potentals of the centes Defnton of potental depends on the dstance to all data ponts, ncludng new data pont. heefoe the new data pont z affects the potental of the centes of exstng clustes. Usng (9), the potentals of the focal ponts of the exstng clustes ae updated ecusvely: ( ) p ( ) ( ) cl p cl = () [ + ( ) ( )* p c + p c c z l l 3.4 ule base evoluton he potental of z s compaed to the potental of the centes of exstng clustes and a decson whethe to add, modfy o delete a ule s made as follows: o add o modfy(eplace) the ule: IF the potental of the new data pont s hghe than the potental of exstng cluste centes:

4 Poceedngs of the 8th WSEAS Intenatonal Confeence on Fuzzy Systems, Vancouve, Btsh Columba, Canada, June 9-, 7 35 ( ) ( ) ; : p z > p c = (3) AND z s close to the th cluste cente: δ m n <.5 (4) c Whee δ mn = mn = * x x s the dstance fom z to the closest exstng ule cente,hen z eplaces ths cente: * ; ( ) ( x = x p ) c = p z (5) By ths method, we eplace a less nfomatve ule wth a moe nfomatve one. Note that the defnton of δ mn s done aound x nstead of z. heefoe, t wll be mpossble fo a ule wth smla antecedents to exst n ule-base. Such ules could exst accodng to the defnton of δ mn used n [-], f the consequents ae dffeent. such ules lead to contadcton n the lngustc ntepetaton. IF only condton(3) s satsfed, HEN z s added to the ule-base as a new cente and a new ule s fomed wth a focal pont based on a poecton of ths cente on the axs x : = + ; * ; ( ) ( x = x p c = P z ) (6) o delete the edundant ule At ths stage, suppose that we have ule. let d as : mn c cl dmn = mn exp ; = : ; l = : (7) σ + σ l and p = ( ) max max p c. Suppose c a and c b ae = the two centes wth the closest dstance and p ( a) ( b) c < p c. IF d ( ) mn p ca + < HEN the a-th ule s deleted σ a pmax fom the ule-base. 3.5 Adustng consequent paametes Stuctue dentfcaton needs an ntal set of paametes fo model vefcaton. heefoe, stuctue and paamete dentfcaton can not be completely sepaated. It s possble to detemne the ntal consequent paametes usng offlne tanng o chose them abtaly. fo a fxed ule-base and antecedent paametes, the ule consequent fom a set of lnea equatons leadng to a lnea egesson poblem, whch can be solved by ecusve least squae estmaton. Dung contnuous leanng, nomalzed actvaton levels of ules change, whch affects past and new data. heefoe, staghtfowad applcaton of LS and WLS s not coect. o solve ths poblem, Covaance matces and paametes ae eset once an exstng ule s modfed/deleted o a new ule s added. Vecto fom of the output s pesented as yˆ + = ψ θ (8) Whee ψ s a vecto of nputs that ae weghted by a nomalzed actvaton levels of the ules and θ s a vecto composed of the lnea sub-model paametes: θ [,,..., ] = π π π (9) ψ = [,,..., ] λ xe λ xe λxe () Fo global optmzaton, we have: J N = y ψ θ () G = ( ) Consequent paametes vecto θ that mnmze J G can be estmated by the followng LS algothm: ˆ θ = ˆ θ + C ψ ( y ψ ˆ θ ) () C C ψ ψ C = C ; = [, N] + ψ C ψ (3) Whee θ ˆ = and C = ΩΙ.. When a new ule s added, the LS algothm s eset n the followng fom:. Consequent paametes ae calculated as: ˆ [ θ = ˆ π ( ), ˆ π ( ),..., ˆ π ( ), ˆ π ( + ) ] (4) whee ˆ π ( + ) = λ ˆ π ( ).. Covaance matces eset as follows: ρα ρα... ( n+ m) ρα ( n m)... ρα ( ) ( )... n m n m C = Ω Ω (5) α s an element of pevous covaance matx and + ρ =.

5 Poceedngs of the 8th WSEAS Intenatonal Confeence on Fuzzy Systems, Vancouve, Btsh Columba, Canada, June 9-, 7 36 When a ule s eplaced wth anothe ule, the covaance matx and consequent paametes of the new ule ae taen fom pevous ule wthout change. It s qute staghtfowad how to eset C and θ when a ule s deleted. In the case of local optmzaton, θ should mnmze the followng cost functon: L = Y π Λ Y π = J ( X ) ( X ) he local paamete estmaton s based on WLS: ( ) ( )( C xe x y xe ) ( ) λ x C( ) xe xe C( ) = C( ) + ( ) λ x xe C x e (6) ˆ π = ˆ π + λ ˆ π (7) C (8) Intal condtons ae ˆπ = ; C =ΩI. the covaance matces ae sepaate fo each ule n ths case: ( n m) ( n m C + + ) ; = [, ] (9) When a new ule s added, ts consequent paametes ae detemned by (4). Also we have: C (+) =ΩI. Paametes of the pevous ules ae nheted (π =π (-), C =C (-) ; =[,]). Paametes of all ules eman unchanged f a ule s eplaced. When a ule s deleted, ts paametes ae deleted too. 3.6 Calculatng adus of new ule he wdth of fuzzy model membeshp functons s sgnfcant fo ts genealzaton. f the wdth s less than the dstance between the adacent nputs whch means undelappng, the neuon does not genealze well. Howeve, f the wdth s too lage, the output of the sub-model may always be lage (nea ) espectve of nputs and the patton whch maes no sense n ths case. heefoe, the wdth must be caefully selected so as to ensue pope and suffcent degee of ovelappng [6]. In ou method, he wdth of the -th membeshp functon of newly geneated ule s computed as follows : { z ca z cb } max, σ ( + ) * n S = (3) Whee c a and c b ae the two neaest neghbong centes of the clustes adacent to the eceptve feld, whee the newly aved patten s located,. Also S s an obtay constant (.5 < S <.5) and n s the nput dmenson. G. Optmzaton of membeshp functon paametes Contay to the conventonal fuzzy-neual netwo based appoaches, consequent paametes of the es model can be computed wthout the BP based algothms [4]. Because the es model s lnea afte the coespondng centes and the wdths ae allocated. he scheme of paametes detemnaton consst of two phases, dung the fowad pass, paametes of membeshp functons ae assumed to be fxed and the fee paametes of lnea sub-models ae detemned. dung the bacwad pass, the centes and wdths wll be optmzed by a gadent descent method.[3] o optmze the centes and wdths, we adopt the followng cost functon N N = e ( ) = ˆ = = ( ( ) ( )) E y y (3) Whee y( ) and yˆ( ) ae the eal and estmated outputs.he centes wll be updated as follows: c = c + c (3) E E y λ c = lc = lc c y λ c x ( y y )( y y ) m c = l cλ ˆ ˆ ˆ = σ (33) And the wdths ae updated as follows: σ = σ + σ (34) E E y λ σ = lσ = lσ σ y λ σ (35) m ( ) ( )( ) x c = lσ λ y ˆ ˆ ˆ y y y 3 = σ Whee l c and l σ ae the leanng ates of centes and wdths espectvely. Fo the lnea egesson model n the consequent pats, the second ode statstcs of the nput sgnals n (8) s not only decded by the model nput, but also the nonlnea mappng whch depends on the membeshp functons. In othe wods, the nput of the lnea egesson models ae non-statonay whch s made possble by adustng of the centes and wdths of membeshp functons. o optmze the model, the ecusve algothm has to see the optmal weghts θ and eep tac of changng poston of the optmal pont as well. o pevent deteoatng the

6 Poceedngs of the 8th WSEAS Intenatonal Confeence on Fuzzy Systems, Vancouve, Btsh Columba, Canada, June 9-, 7 37 pefomance, the paametes of the membeshp functons ae updated n each p sample peod, not n each samplng peod. p s a constant whch depends on the numbe of fee paametes n the consequent pat. Nomally p s set to µ(n+) because the LS algothm s convegent n the mean value fo evey tme-step p geate than µ(n+) [4].heefoe, the centes and wdths ae updated as follows: E p E p c = ; σ = (36) c σ + p whee ( ) E p = y( ) yˆ ( ). = he algothm then etuns to 3. 4 Smulaton esults Example : Nonlnea dynamc system dentfcaton: the nonlnea dynamc plant s descbed as: y ( t ) y ( t )[ y ( t ) +.5] y ( t + ) = + u ( t ) (37) + y ( t ) + y ( t + ) he nput/output data of ths plant ae wdely used to vefy the pefomance of system dentfcaton pocedues. he coespondng fuzzy model can be epesented by yˆ( t + ) = f ( y( t), y( t ), u( t)) (38) Whee f (.) s unnown. Ou obectve s to fnd an evolvng S fuzzy model of f(.) by usng the algothm poposed n secton 3. he nput sgnal s chosen as u( ) =.5sn( π / 5) +.5sn( π ).the esults and the compason wth some othe methods ae shown n fgue and table espectvely. able confms that ou model s of lowe eo (MSE), whle ts numbe of paametes s much lowe than the numbe of paametes of the othe onlne models. Example : Macey-glass tme sees pedcton: hs sees s a well-nown benchma poblem fo testng system dentfcaton algothms. It s geneated by the followng equaton. x( t τ ) xɺ ( t) =. x( t) (39) + x ( t τ ) We assume x() =., τ = 7. he am s to dentfy the followng pedcton model: x( t + 85) = f [ x( t), x( t 6), x( t ), x( t 8)] (4) Fgue shows the ule-base evoluton and estmaton eo. In ths secton, non-dmensonal eo ndex (NDEI) s defned as a ato of MSE ove the standad devaton of taget data. he compason of the poposed model wth othe models based on dffeent onlne dentfcaton methods s shown n table. It s clealy seen n table that ou model s moe accuate wth least numbe of paametes. Although the NDEI of some othe models (such as Denfs wth 883 fuzzy ules) s elatvely close to the NDEI of ou model, but the numbe of paametes of these models ae much hghe. In ode to test the obustness of the es model a 5 % andom nose has been added to the standad Maceyglass tme-sees. Ou model has evolved toonly 4 ules and NDEI=.589 whch means that the nose 3 able compason of stuctue and pefomance Algothm/authos neuons / ules MSE Paametes DFNN[5] GDFNN[6] OLS (s.chu,.,99) BF-AFS (cho,..,996) Adaptve model[3] Ou method Numbe of ules Fg.. samples Numbe of ules & Estmaton eo Fg.. Numbe of ules Estmaton eo Estmaton eo samples Numbe of ules & Estmaton eo..5..5

7 Poceedngs of the 8th WSEAS Intenatonal Confeence on Fuzzy Systems, Vancouve, Btsh Columba, Canada, June 9-, 7 38 able compason of stuctue and pefomance Algothm/authos ules/ NDEI unts AN(asabov&song,) ESOM(asabov&song,) 4.3 ESOM(asabov&song,).44 EfuNN (asabov&song,) 93.3 EfuNN (asabov&song,) 5.94 Denfs (asabov&song,) Denfs (asabov&song,) es[] 3.95 es (vcto&duado, 3 ) 9.38 Fuzzy ansvesal flte[4] SONFIN [c.uang & c.ln 9.796,998] Neual gas.6 DFNN[5] OLS BF-AFS Ou method 4.7 has no effect on the numbe of ules (clustes). he ablty of poposed algothm to eect the nose and fndng the appopate numbe of clustes despte the nose s an mpotant chaactestc especally n classfcaton poblems and patten ecognton. 5 Concluson An mpoved appoach to onlne dentfcaton of evolvng S fuzzy models s poposed. Heustc cteons ae used n onlne potental-based subtactve clusteng to evolve the ule-base. he centes and wdths of membeshp functons ae updated usng dentfcaton eo and GD method. he dentfcaton algothm can wo wthout a po nfomaton of a pocess and/o any pedefned constants. Futhemoe, the model s obtaned usng I/O data of nomal opeaton modes and thee s no need to excte the system wth pesstent exctng sgnals (such as PBS), whch can be pactcally dffcult, expensve o dangeous. Its eason s that the new opeatng modes (aeas) ae easly and apdly epesented by some new ules. Ou evolvng fuzzy model s a pomsng canddate fo many eal-tme contol, adaptve & model pedctve contol, as well as pedcton, sgnal pocessng, dagnostc systems, data acquston, and atfcal ntellgence due to ts hghe pecson, compact stuctue (wth low numbe of needed paametes) and adaptve natue. EFEENCES [] P. Angelov and D. Flev, An Appoach to Onlne Identfcaton of aag-sugeno Fuzzy Models, IEEE ans.syst.,man, and Cyben., Vol.34, No., pp , Febuay 4. [] P. Angelov, C. Xydeas, and D.Flev, On-lne Identfcaton of MIMO Evolvng aag-sugeno Fuzzy Models, IEEE Conf. pp 55-6, July 4, Budapest. Hun. [3].Q and M. Bdys, Adaptve Fuzzy Modelng and Contol fo Dscete-me Nonlnea Uncetan Systems, Amecan Contol Conf., June 8-, 5. Potland, O, USA. [4] Z. L and M.J. E, A Nonlnea ansvesal Fuzzy Flte Wth Onlne Clusteng, pp , 5 th Asan Contol Confeence,4. [5] S. Wu & M.J.E, Dynamc Fuzzy Neual Netwos-A Novel Appoach to Functon Appoxmaton,IEEE ans. on Syst.,Man, and Cyben.,Vol. 3, No., pp , Febuay 4. [6] S. Wu, M.J. E and Y. Gao, A fast appoach fo automatc geneaton of fuzzy ules by genealzed dynamc fuzzy neual netwos, IEEE ans. On fuzzy systems, Vol. 9, No., pp ,. [7] N. Kasabov and Q. Song, DENFIS: dynamc evolvng Neual-Fuzzy Infeence System and ts Applcaton fo me-sees Pedcton, IEEE ans. On fuzzy systems, Vol., No., pp 44-54, [8] W.J. Hao, W.Y. Qang, Q.Cha, J.L.ang, Onlne Data-Dven Fuzzy Modelng fo Nonlnea Dynamc Systems, 4 th ntenatonal confeence on machne leanng, Guangzhou, 8- August 5 [9] K.H. Quah and C. Que, Onlne Local Leanng Wth Genec Fuzzy Input aag-sugeno-ang Fuzzy Famewo fo Nonlnea System Estmaton, IEEE ans.syst.,man Cyben.,vol.36, no., Febuay 6 [] M. Sugneo and M.Y asuawa, A fuzzy logc based appoach to qualtve modelng IEEE ans. Fuzzy Syst., Vol., pp 7-3,993 [] C.F. Juang and X.. Ln, A ecuent Self- Oganzng Neual Fuzzy Infeence Netwo, IEEE ans. On Neual Netwos, Vol., pp , 999 [] M.F.Azeem and M.Hanmandlu, Stuctue Identfcaton of Genealzed Adaptve Neuo-Fuzzy Infeence Systems, IEEE ans. Fuzzy Syst., Vol., No.5, pp , Octobe 3 [3] S. Wu, M.J. E and Y.Gao, A Fast Appoach fo Automatc Geneaton of Fuzzy ules by Genealzed Dynamc Fuzzy Neual Netwos, IEEE ans. Fuzzy Syst., Vol. 9, No. 4, pp , August.