An asymmetry-similarity-measure-based neural fuzzy inference system

Transcription

1 Fuzzy Sets and Systems 15 (005) An asymmetry-smlarty-measure-based neural fuzzy nference system Cheng-Jan Ln, Wen-Hao Ho Department of Computer Scence and Informaton Engneerng, Chaoyang Unversty of Technology, No. 168, Jfong E. Rd., Wufong Townshp, Tachung County 41349, Tawan Receved 10 December 00; receved n revsed form 5 March 004; accepted 5 November 004 Avalable onlne 8 December 004 Abstract In ths paper, a new asymmetry-smlarty-measure-based neural fuzzy nference system (ASM-NFIS) s proposed. A pseudo-gaussan membershp functon can provde a neural fuzzy nference system whch has a hgher flexblty and can approach the optmzed result more accurately. An on-lne self-constructng learnng algorthm s proposed to automatcally construct the ASM-NFIS. It conssts of structure learnng and parameter learnng that would create adaptve fuzzy logc rules. The structure learnng s based on the smlarty measure of asymmetrc Gaussan membershp functons, and the parameter learnng s based on a supervsed gradent descent method. Computer smulatons were conducted to llustrate the performance and applcablty of the proposed model. 004 Elsever B.V. All rghts reserved. Keywords: Asymmetrc fuzzy smlarty measure; Pseudo-Gaussan; Backpropagaton; Predcton 1. Introducton The man purpose of a fuzzy system s to acheve a set of local nput output relatonshps that descrbe a process. As s well known, the problem of system modelng requres two man stages: structure dentfcaton and parameter optmzaton. Structure dentfcaton deals wth the problem of determnng the nput output space partton and how many rules must be used by the fuzzy system. Parameter optmzaton fnds the optmum value of all the parameters nvolved n the fuzzy system; that s, t locates the membershp functons n the premse and consequent of each rule [1 4,6 8,10]. Correspondng author. Fax: E-mal address: cln@mal.cyut.edu.tw (C.-J. Ln) /$ - see front matter 004 Elsever B.V. All rghts reserved. do: /.fss

2 536 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) To prevent a newly generated membershp functon from beng too smlar to an exstng membershp functon, the smlarty measure has been wdely researched and broadly appled [,7,8,10]. They adopt the tradtonal symmetrc Gaussan membershp functons. Recently, many researchers [11,1] use the asymmetrc Gaussan membershp functon, whch s called Pseudo-Gaussan (PG), to act as the nput term node. Because the asymmetrc Gaussan membershp functon s varablty and malleablty are hgher than those of the tradtonal membershp functon, the PG membershp functon can provde a neural fuzzy nference system whch has a hgher flexblty and can approach the true result more easly. In [11], the parttonng of nput space s performed n advance; namely the number of fuzzy rules wll be pre-set by the users. Then, an onlne learnng algorthm s proposed to construct the fuzzy systems dynamcally n order to overcome the aforementoned drawback. We also develop a new asymmetrc smlarty measure to check the smlartes between a new membershp functon and exstng ones. Therefore, n ths paper the proposed asymmetrc smlarty measure method s dfferent from the tradtonal symmetrc smlarty measure method [,7,8,10]. In ths paper, we present an asymmetry-smlarty-measure-based neural fuzzy nference system (ASM- NFIS). It s a standard four-layer feedforward neural network. An on-lne learnng algorthm s proposed to automatcally construct the ASM-NFIS. It conssts of structure learnng and parameter learnng. We wll add a new node to satsfy the fuzzy parttonng of the tranng data n structure learnng. The smlarty measure of asymmetrc Gaussan membershp functons s proposed to estmate the rule s smlarty degree. The back-propagaton learnng s then used for tunng nput/output membershp functons. Ths method has the advantage of not requrng an expert s assstance snce the nput output characterstcs of the ASM-NFIS and ts structure are obtaned from the tranng examples. Ths s n contrast to [11] and [1], whch the nput space needs to be dvded properly n advance. The proposed model has been used to dentfy the dynamc system and predct the chaotc tme-seres. The smulaton results show that the proposed ASM-NFIS model has a better learnng performance than other learnng systems.. The structure of ASM-NFIS The th fuzzy f-then rule shown below s used by the ASM-NFIS: R : IF x 1 s A 1 and...and x n s A n, THEN y = b, (1) where x and y are the nput and output varables, respectvely; A s the lngustc term of the precondton part wth membershp functon μ A ; b s the constant consequent; and n s the number of nput dmensons. The membershp functon of the precondton part dscussed n ths paper s dfferent from the typcal Gaussan membershp functon. We adopt the Pseudo-Gaussan (PG) membershp functon [11] to approxmate desred results. The defnton of PG membershp functon s as follows: ( μ A (x ) = exp (x ) ( m ) σ U(x ;,m ) + exp (x ) m ), σ U(x ; m, ), (),+ { 1 f a x <b where U(x ; a,b) = 0 otherwse,

3 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg. 1. (a) One-dmensonal PG, (b) two-dmensonal PG. where m s the mean of the PG membershp functon; σ, s the negatve devaton of the PG membershp functon; and σ,+ s the postve devaton of the PG membershp functon. The PG membershp functon s asymmetrc and has great flexblty. Fg. 1 shows the one-and two-dmensonal PG membershp functons. A typcal network conssts of nodes wth a fnte number of fan-n connectons from other nodes represented by weght values and a fnte number of fan-out connectons to other nodes. Assocated wth the fan-n of a node s an ntegraton functon whch combnes nformaton, actvaton, or evdence from other nodes and provdes the net nput,.e., net nput = f(z (k) 1,z(k),...,z(k) p ; w(k) 1,w(k),...,w(k) p ), (3) where z (k) s the th nput varable to a node n layer k and w (k) s the weght of the assocated lnk. The superscrpt n the above equaton ndcates the layer number. Ths notaton wll be also used n the followng equatons. Each node also outputs an actvaton value as a functon of ts net nput output = a[f( )], (4) where a( ) denotes the actvaton functon. The ASM-NFIS s a standard four-layer network [8], as shown n Fg., where the functons of the nodes n each layer are descrbed as follows: Layer 1: The nodes n ths layer are nput nodes (.e., nput-lngustc nodes) whch represent nputlngustc varables and whch pass nput sgnals to the next layer drectly: f(x (1) ) = x (1) (5) and a[f( )] =f( ), (6) where x (1) s the th nput varable to a node n layer 1.

4 538 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg.. The structure of ASM-NFIS. Layer : The nodes n ths layer are term nodes that act as the PG membershp functon. They can react to the terms of the respectve nput-lngustc varables. For the th rule node, ( ) f(z () ) = exp (z() m ) σ U(z () ;,m ), ( ) + exp (z() m ) σ U(z () ; m, ), (7),+ and { where U(z () () 1 f a z ; a,b) = <b 0 otherwse a[f( )] =f( ). (8) The m, σ,, and σ,+ are the mean, negatve devaton, and postve devaton, respectvely of the PG membershp functons of th term assocated wth th nput varable x.

5 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Layer 3: The nodes n ths layer are compensatory fuzzy nodes. They represent the precondton part of the fuzzy logc rules whch can nput the multple ncomng sgnals and output the product result. For the th rule node, and f(z (3) ) = n z (3) (9) a[f( )] =f( ) (10) where n s dmenson number. Layer 4: The nodes n ths layer are denoted by Σ. That s, t receves the multple ncomng sgnals and outputs the result of summaton. For the output y, f(z (4) ) = M w (3) z (4) (11) and a[f( )] =f( ) (1) where M s the rule number and w (3) s the lnk weght. 3. The on-lne learnng algorthm In ths secton, we propose an onlne learnng algorthm whch conssts of the structure learnng algorthm and the parameter learnng algorthm. The structure learnng algorthm s used to fnd proper fuzzy parttons n the nput space and create fuzzy logc rules. The asymmetrc smlarty measure method s used to prevent the newly generated membershp functon from beng too smlar to the exstng membershp functon. The parameter learnng algorthm s the most general supervsed learnng scheme and s used to adust the PG membershp functons n the precondton part and to modfy the lnk weght n the consequent part. As a result, the parameter learnng algorthm s based on the back-propagaton algorthm, whch mnmzes the cost functon to approxmate the desred results. The procedure of the structure and parameter learnng algorthms s through nputtng the tranng pattern to learn successvely The structure learnng algorthm The proposed structure learnng algorthm decdes the proper fuzzy parttons by usng the nput patterns. The procedure of the structure learnng algorthm uses the PG membershp functons to fnd the fuzzy logc rules. However, the structure learnng algorthm determnes whether to add a new node n layer va the nput pattern data and whether to add the assocated fuzzy logc rule n layer 3. After the nput pattern s entered n layer, the frng strength of the PG membershp functon wll be obtaned from Eq. (), whch s used to calculate the degree measure μ A. In layer 3, the frng

6 540 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg. 3. An asymmetrc trangle wth the unty heght and the length of bottom edge. strength of the fuzzy logc rule s obtaned from Eq. (4), used to obtan the degree measure of the precondton part P = n =1 μ A (x ), = 1,,...,M(t), (13) where M(t) s the number of exstng rules at tme t. Accordng to the degree measure, we can obtan the exstng maxmum P of the frng strength of the fuzzy logc rule P max = max P. 1 M(t) (14) IF P th >P max, the structure learnng needs to add a new node n the ASM-NFIS. The P th s the preset threshold, whch should be decreased when the structure learnng algorthm lmts the rule of ASM-NFIS. The P th s an mportant parameter and ts value s set between zero and one. A low P th value leads to the learnng of coarse clusters, whereas a hgh P th value leads to the learnng of fne clusters. The new mean, postve devaton, and negatve devaton of the PG membershp functon are preset values accordng to the nput pattern or heurstc. The process of how the node ncreases s shown as follows: m (new) = x, (15) σ (new), = σ(new),+ = σ, where x s the new nput pattern; σ s the preset constant. To prevent the newly generated membershp functon from beng too smlar to the exstng membershp functon, the smlartes between the new membershp functon and the exstng functons must be checked. If the new fuzzy rule s dfferent from the exstng fuzzy rule, we confrm that the new fuzzy rule wll be added n the ASM-NFIS. It can cause the neural fuzzy nference system to perform better. Therefore, we use the smlarty measure of asymmetrc Gaussan membershp functons to estmate the rule s smlarty degree. Snce the area of the asymmetrc Gaussan membershp functon, calculated from Eq. (), s between σ + π and σ π, the heght s always 1, and the center of the bottom-lne at m s on the x-axs. We can approxmate t by an asymmetrc trangle Δ(m, σ,, σ,+ ) wth a unty heght and wth the length of bottom edge σ + π + σ π (see Fg. 3). Assume that the two end-ponts of the bottom lne of Δ(m 1, σ 1,, σ 1,+ ) are a and b on the x-axs, and another end-ponts of Δ(m, σ,, σ,+ ) are c and d on the x-axs. That s, a = m 1 σ 1, π,b = m1 + σ 1,+ π,c = m σ, π, and d = m + σ,+ π. (16)

7 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg. 4. The fve possble stuatons between two asymmetrc trangles: (a) Case 1: a d, (b) Case : b d > a c (c) Case 3: b > d and c > a (d) Case 4: d > b and a > c (e) Case 5: d > b and c > a. Frst, f m 1 = m, when a c and b d, A B = 1 (σ,+ + σ, ) π, c a and d b, A B = 1 (σ 1,+ + σ 1, ) π,a c and b d, A B = 1 (σ,+ + σ 1, ) π,a c and b d, A B = 1 (σ 1,+ + σ, ) π. In the followng dscusson, we assume that m 1 >m. Let us consder the followng fve possble stuatons (see Fg. 4): Case 1: If a d, then A B =0, snce the two membershp functons do not overlap. Case : If b d >a c, then A B = 1 (d a)y = 1 (m + σ,+ π m1 + σ 1, π). (17) σ 1, π + σ,+ π Case 3: If b>dand c>a, then A B = 1 (x c)y + 1 (y 1 + y ) (x 1 x ) + 1 (d x 1)y 1 = 1 (m σ, π m1 + σ 1, π) + 1 σ 1, π + σ, π (m + σ,+ π m1 + σ 1, π) σ 1, π + σ,+ π. (18)

8 54 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Case 4: If d>band a>c, then A B = 1 (x a)y + 1 (y 1 + y ) (x 1 x ) + 1 (b x 1)y 1 = 1 (m + σ,+ π m1 + σ 1, π) + 1 σ 1, π + σ,+ π Case 5: If d>band c>a, then ( m σ,+ π + m1 + σ 1,+ π) σ 1,+ π σ,+ π. (19) A B = 1 (x 3 c)y (y + y 3 ) (x x 3 ) + 1 (y 1 + y )(x 1 x ) + 1 (b x 1)y 1 = 1 (m σ, π m1 + σ 1, π) + 1 σ 1, π + σ, π (m + σ,+ π m1 + σ 1, π) σ 1, π + σ,+ π + 1 ( m σ,+ π + m1 + σ 1,+ π). (0) σ 1,+ π σ,+ π We can conclude a general formula for A B : A B = 1 h (m + σ,+ π m1 + σ 1, π) + 1 h (m σ, π m1 + σ 1, π) σ 1, π + σ,+ π σ 1, π + σ, π + 1 h (m + σ,+ π m1 σ 1,+ π), (1) σ 1,+ π σ,+ π where h(x) = max{0,x}. Thus, the approxmate smlarty measure of fuzzy sets s E(A,B) = = A B A B 1 σ 1,+ A B π + 1 σ 1, π + 1 σ,+ π + 1. () σ, π A B The smlarty measure E between the new membershp functon and all exstng membershp functons are calculated, and the maxmum E,E max, s calculated as follows: E max = max 1 M(t) E(μ(m(new) 1, σ (new) 1,+, σ(new) 1, ), μ(m 1, σ 1,+, σ, )). (3) If E max E th, where E th (0, 1) s a prespecfed threshold, then the new fuzzy logc rule s adopted and the rule number s ncremented. M = M + 1. (4) Therefore, the new mean, devaton, and lnk weght are generated randomly. 3.. The parameter learnng algorthm After the structure network has been accordngly adusted to the current tranng pattern, the network enters the parameter learnng stage. The parameter learnng algorthm adusts the parameter of ASM- NFIS optmally wth the same tranng pattern. The back-propagaton phase s used for ths supervsed

9 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) learnng to fnd the output errors of the node n each layer and to analyze the error n order to adust the parameter. The goal s to mnmze the error functon E = 1 (yd (t) y(t)), (5) where y d (t) s the desred output and y(t) s the ASM-NFIS output (Fg. 5). The parameter learnng algorthm based on back-propagaton s then as follows: Assumng that w s the adustable parameter n a node, the generally-used learnng rule s w(t + 1) = w(t) η E w = E f = E a f w, a f f w, ( E w ), where η s the learnng rate. To show the learnng rules, we derve the parameter learnng layer by layer. Layer 4: There s no parameter to be adusted n ths layer. Layer 3: Usng (13) and (14), the lnk weght s adusted by the amount ( ) w (3) (t + 1) = w (3) (t) η w (3) E w (3) = E y (4) y (4) w (3) = (y (4) y d )p (3). E w (3) (6) (7), (8) (9) Layer : The m of the PG membershp functon s adusted by the amount ( ) E m (t + 1) = m (t) η m m [ ] [ ] E E y (4) P (3) μ () A = m y (4) P (3) μ () m A = (y (4) y d )w (3) n l=1 l = [ (x (1) m ) σ, + (x(1) m ) σ,+ μ () A l ( ) exp (x(1) m ) σ, ( ) exp (x(1) m ) σ,+ U(x (1) ;,m ) U(x (1) ; m, ) ] (30) (31)

10 544 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg. 5. The flow dagram of the structure/parameter learnng for the ASM-NFIS. The σ, of the PG membershp functon s adusted by the amount ( ) E σ, (t + 1) = σ, (t) η σ σ, (3)

11 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) [ ] [ ] E E y (4) P (3) = σ, y (4) P (3) μ () A n = (y (4) y d )w (3) l=1 l = [ (x (1) m ) μ () A l μ () A σ, ( ) σ 3 exp (x(1) m ), σ, The σ,+ of the PG membershp functon s adusted by the amount ( ) E σ,+ (t + 1) = σ,+ (t) η σ+ σ,+ [ ] [ ] E E y (4) P (3) μ () A = σ,+ y (4) P (3) μ () σ,+ A n = (y (4) y d )w (3) l=1 l = μ () A l U(x (1) ;,m ) ]. (33) [ ( ) ] (1) (x m ) σ 3 exp (x(1) m ),+ σ U(x (1) ; m, ), (35),+ where η m, η σ+ and η σ represent the learnng rate parameters of the PG membershp functon respectvely. (34) 4. Illustratve examples In ths secton, we smulate some popular problems. The frst example s to dentfy a dynamc system [6,7,9]. The second example s to predct tme-seres [1,3,4,6]. Example 1. Identfcaton of the dynamc system The plant to be dentfed s guded by the dfference equaton y(k + 1) = y(k) 1 + y (k) + u3 (k), (36) u(k) = sn(πk/100). (37) The output of the plant depends nonlnearly on both ts pass output values and nput values, but the effects of the nput and output values are addtve. Two hundred tranng patterns are generated by the plant Eqs. (36) and (37). The ntal parameters are η = 0.01, σ = 0.,P th = 0.15, and E th = 0.6. Startng at zero, the numbers of clusters grow dynamcally for the ncomng tranng data. The tranng process s contnued

12 546 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg. 6. Smulaton results of the ASM-NFIS on the PG membershp functons of each nput varable n Example 1. (a) The nput tranng patterns and the fnal assgnment of rules. (b) The dstrbuton of the membershp functons on the u(k) and y(k) dmensons. 500 tmes. After tranng, the fnal root mean square (rms) error of the dentfcaton output approxmates There are fve fuzzy logc rules generated n ASM-NFIS. Fg. 6 llustrates the dstrbuton of the tranng patterns and the fnal assgnment of the rules (.e., dstrbuton of the membershp functons) n [u(k), y(k)] plan (nput space). Fg. 7 shows the output of the plant and the dentfcaton model after the 500 tranng steps are fnshed. In ths fgure, the output of the ASM-NFIS model are represented by whle the plant output values are represented as o. The results show the good dentfcaton capablty of the traned ASM-NFIS model. The selecton of P th wll crucally affect the smulaton results. A low P th value leads to the learnng of coarse clusters, whereas a hgh P th value leads to the learnng of fne clusters. In ths smulaton, we adopt the dfferent P th values to perform the dentfcaton problem. The smulaton results are tabulated

13 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg. 7. Smulaton results of the ASM-NFIS model n Example 1. Table 1 Performance comparson of varous P th values on the dentfcaton problem P th values Rule numbers RMS error n Table 1. Table 1 shows f we choose a hgh P th value, the numbers of fuzzy rules wll be ncreased but the rms errors wll be decreased. Example. Predcton of the chaotc tme-seres Let P(k),k = 1,,..., be a tme seres. The problem of the tme-seres predcton can be formulated n the followng way: gven P(k m + 1), P (k m + ),...,P(k), determne P(k+ l), where m and l are fxed postve ntegers. (.e., determne a mappng from [P(k m + 1), P (k m + ),...,P(k)] R m to [P(k+ l)] R). To llustrate the onlne learnng ablty, the ASM-NFIS model s used to predct the Mackey Glass chaotc tme-seres. The Mackey Glass chaotc tme-seres s generated from the followng delay dfferental equaton: dx(t) dt = 0.x(t τ) 1 + x x(t), (38) (t τ) where τ > 17. In our smulaton, we chose the seres wth τ = 30. Fg. 8 shows 1000 ponts of ths chaotc seres used to test the ASM-NFIS model. We chose m = 9 and l = 1 (.e., nne pont values n the seres are used to predct the value of the next tme pont). The 00 ponts of the seres from x(501) x(700) are used as

14 548 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg. 8. The Mackey Glass chaotc tme seres. Fg. 9. Learnng curve of the proposed ASM-NFIS model. tranng data, and the fnal 300 ponts from x(701) x(1000) are used as test data. The ntal parameters are η = 0.01, σ = 0.,P th = 0.15, and E th = 0.6. After the structure and parameter learnng, three fuzzy logc rules were generated n our model. The learnng curve s shown n Fg. 9. The rms error of the predcton output approxmates Fg. 10 shows the predcton results of the traned ASM-NFIS. We compared the performance of our system wth that of other exstng methods that can generate fuzzy rules from numercal data automatcally. The comparson results are tabulated n Table. The smulaton results show that our system has better learnng performance than other learnng systems. In [13], Wang and Mendel tred to mprove the predcton accuracy by usng an updatng fuzzy rule base procedure and dvdng the doman nterval nto fner regons n ther system. In the end, ther system

15 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) Fg. 10. The sold lne denotes the output of ASM-NFIS and the dotted lne denotes the true output. Table Performance comparson of varous rule generaton methods on the tme-seres predcton problem ASM-NFIS FNN SONFIN FALCON-ART Wang and Mendel Kosko (AVQ) [5] [8] [] [6] [13] UCL DCL Rule numbers RMS error acheved perfect predcton capablty when the doman nterval was dvded nto 9 regons. Km and Kasabov [4] proposed a hybrd neural fuzzy nference system (HyFIS) for buldng fuzzy models. In the HyFIS model, fuzzy rules were extracted by usng fuzzy technques proposed by [13] and parameters were adusted by usng a gradent descent learnng algorthm. In [4] and [13], the nput space needs to be dvded properly n advance. Jang [1] proposed an adaptve-network-based fuzzy nference system (ANFIS) model for learnng and tunng a fuzzy predctor. By usng a hybrd learnng procedure, the proposed ANFIS can construct an nput/output mappng based on both human knowledge and stpulated nput/output pars. The ANFIS also has perfect predcton capablty of the chaotc tme seres predcton problem. However, the proper fuzzy rules and space partton must be gven n advance by experts. 5. Dscusson In ths secton, we summarze the features of the proposed ASM-NFIS model. Frst, dstrbuted representaton s used to represent the nput patterns n the ASM-NFIS model. The nput space s dvded nto overlappng smaller regons and the parttonng s not performed n advance, but s dynamcally and approprately adusted durng the learnng process. As a result, each regon vares n sze, and the degree of overlap between regons s also adustable. Ths s n contrast to [1,4,5,13] n whch the nput space needs

16 550 C.-J. Ln, W.-H. Ho / Fuzzy Sets and Systems 15 (005) to be dvded properly n advance. The second feature of the proposed ASM-NFIS model prevents the newly generated membershp functon from beng too smlar to the exstng membershp functon. The asymmetrc smlarty measure between the new membershp functon and the exstng functons must be checked. Ths s n contrast to the symmetrc smlarty measure [,4,8,10]. The thrd feature of the proposed ASM-NFIS model s that t flexbly parttons the nput space. Ths s n contrast to the grd-type space partton [1]. In [1], as the number of nput varables ncreases, the number of the parttoned grds wll grow exponentally. As a result, the requred sze of memory or hardware may become mpractcally huge. 6. Conclusons In ths paper, we ntroduced a neural fuzzy system called ASM-NFIS. An on-lne learnng algorthm was proposed to automatcally construct the ASM-NFIS. We addressed the automatc determnaton of the structure of the ASM-NFIS and the smultaneous optmzaton of both membershp functons and fuzzy rule conclusons. The PG membershp functon was used to construct the general neural fuzzy system and to make the varablty and the flexblty of the ASM-NFIS hgher. The smlarty measure of asymmetrc Gaussan membershp functons was proposed to estmate the rule s smlarty degree. The proposed ASM-NFIS can obtan a smaller RMS error and generate less fuzzy logc rules. Acknowledgement Ths work was supported by the Natonal Scence Councl, R.O.C., under grant NSC 9-13-E References [1] J.S.R. Jang, ANFIS: adaptve-network-based fuzzy nference systems, IEEE Trans. on Syst. Man, and Cybern. 3 (3) (1993) [] C.-F. Juang, C.-T. Ln, An on-lne self-constructng neural fuzzy nference network and ts applcatons, IEEE Trans. on Fuzzy System 6 (1) (1998) 1 3. [3] N. Kasabov, Q. Song, DENFIS: dynamc evolvng neural-fuzzy nference system and ts applcaton for tme-seres predcton, IEEE Trans. on Fuzzy Systems 10 () (00) [4] J. Km, N. Kasabov, HyFIS: adaptve neuro-fuzzy nference systems and ther applcaton to nonlnear dynamcal systems, Neural Networks 1 (1999) [5] B. Kosko, Neural Networks and Fuzzy Systems, Prentce-Hall, Englewood Clffs, NJ, 199. [6] C.-J. Ln, C.-T. Ln, An ART-based fuzzy adaptve learnng control network, IEEE Trans. Fuzzy System 5 (4) (1997) [7] C.-T. Ln, C.-J. Ln, C.-S.G. Lee, Fuzzy adaptve learnng control network wth on-lne neural learnng, Fuzzy Sets and Systems 71 (1995) 45. [8] F.-J. Ln, C.-H. Ln, P.-H. Shen, Self-constructng fuzzy neural network speed controller for permanent-magnet synchronous motor drve, IEEE Trans. Fuzzy System 9 (5) (001) [9] K.-S. Narendra, K. Parthasarathy, Identfcaton and control of dynamcal systems usng neural networks, IEEE Trans. on Neural Networks 1 (1) (1990) 4 7. [10] S. Paul, S. Kumar, Subsethood-product fuzzy neural nference system (SuPFuNIS), IEEE Trans. on Neural Networks 13 (3) (00)

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