Sparse Fuzzy Model Identification Matlab Toolox RuleMaker Toolbox

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1 Johanyák, Z. C.: Sparse Fuzzy Model Identfcaton Matlab Toolbox - RuleMaker Toolbox, IEEE 6th Internatonal Conference on Computatonal Cybernetcs, November 27-29, 2008, Stara Lesná, Slovaka, pp Sparse Fuzzy Model Identfcaton Matlab Toolox RuleMaker Toolbox Z.C. Johanyák Kecskemét College/Insttute of Informaton Technologes, Kecskemét, Hungary ohanyak.csaba@gamf.kefo.hu Abstract Fuzzy systems applyng a sparse rule base and a fuzzy rule nterpolaton based reasonng method are popular solutons n cases wth partal knowledge of the modeled area or cases when the full coverage of the nput space by rule antecedents would requre too many rules. In several practcal applcatons there s no human expert based knowledge; the fuzzy model has to be dentfed from sample data. Ths paper presents a freely avalable Matlab toolbox called RuleMaker that supports the automatc generaton of a fuzzy model wth low complexty. The mplemented model dentfcaton methods are also revewed. Fgure 1. Antecedent space of a sparse fuzzy rule base I. INTRODUCTION Fuzzy systems applyng sparse rule bases are wdely used when ether one does not possesses the necessary knowledge for the full coverage of the nput space or the complexty of the resultng system would be too hgh owng to the ncreased number of rules. Due to the ncreasng popularty of the fuzzy rule nterpolaton (FRI) based reasonng methods (e.g. KH [20], IMUL [28], FIVE [22], GM [1], IRG [5], and LESFRI [14]) startng from the early 90s several methods amng the automatc generaton of sparse fuzzy systems have been developed. Ths paper reports the development of a Matlab toolbox called RuleMaker that s freely avalable under GNU GPL [12] and mplements the fuzzy model dentfcaton methods RBE-DSS [15], RBE-SI [15] and ACP [13]. The rest of ths paper s organzed as follows. Secton II gves a bref survey on the man tendences n sparse fuzzy model dentfcaton and ntroduces shortly the mplemented technques. Secton III presents the RuleMaker toolbox. II. SPARSE FUZZY RULE BASE GENERATION A. Sparse Rule Base The rule base of a fuzzy system s consdered as sparse when there s at least one pont of the nput space for whch there s no rule wth an actvaton degree greater than a prescrbed ε 0 value. In mathematcal terms the degree of coverage (c) of the nput space s descrbed by c = arg max mn ε ( A X N = 1 n * { max { t( A, A ) } } = 1, ε, [ 0,1], ε, (1) where X s the th dmenson of the antecedent space, A s the fuzzy set descrbng the observaton n the th antecedent dmenson, A s the th lngustc term of the th antecedent dmenson, t s an arbtrary t-norm, n s the number of the lngustc terms of the th antecedent dmenson, N s the number of the antecedent dmensons, argmax. calculates that ε value for whch the ε and ( ) expresson n the parentheses takes ts maxmum. If c > ε 0 the rule base s called ε 0 coverng (dense) otherwse t s consdered as sparse. Fg. 1 llustrates the antecedent space of a sparse rule base. The nput of the system s three dmensonal. Each rule s represented by a cubod defned by the supports of the fuzzy sets referred n the antecedent part of the rule. The bg cubod contanng the small ones defnes the multdmensonal nput unverse of dscourse correspondng to the allowed nput ranges of the system n each antecedent dmenson. B. Automatc Sparse Fuzzy Model Identfcaton In case of automatc fuzzy model dentfcaton one possesses a tranng data set that descrbes the expected behavor of the system n several ponts,.e. contans a lst of matng nput-output values. The model dentfcaton algorthm seeks for those values of the system parameters for whch the dfference between the output values of the tranng data set and the output provded by the current fuzzy system n case of the same nput s small. In most of the cases especally when the tranng data set contans a large amount of data an exact match cannot be reached. Therefore the fuzzy system gves only a more or less good approxmaton of the modeled nput-output relaton descrbed by the sample data.

2 Usually the approxmaton can be mproved by ncreasng the number of lngustc terms and rules. The stuaton s smlar to the neural networks based modelng. However, the advantage of the applcaton of a fuzzy system s gven by ts self explanng capablty,.e. one can extract from the tuned system IF-THEN rules that are easly nterpretable and understandable by humans. Sparse fuzzy systems wth low complexty can be obtaned followng two dfferent approaches. The frst one starts wth a dense rule base ( c > ε0 n (1)) [21],.e. as step 0 a coverng rule base should be generated and then the rules consdered as non relevant ones are elmnated. The methods proposed by Botzhem, Cabrta, Kóczy and Ruano [3], Botzhem, Hámor and Kóczy [4], as well as by Kóczy, Botzhem and Gedeon [19] follow ths way. The second approach tres to create a sparse rule base drectly from the avalable data. These methods can be categorzed nto three groups. 1. Methods tryng to dentfy the so called optmal fuzzy rules (Kosko and Dckerson [23], Kosko [24]). 2. Methods based on fuzzy clusterng (e.g. Chu [6], Chong [9], Klawonn and Kruse [18], Sugeno and Yakusawa [25], Tkk, Gedeon, Kóczy and Bíró [26], Wong, Kóczy, Gedeon, Chong and Tkk [29] as well as Johanyák [13]). 3. Methods followng the concept of rule base extenson (e.g. Johanyák and Kovács [15]). The current verson of the RuleMaker Matlab ToolBox contans the mplementaton of the clusterng based ACP method [13] and the mplementaton of the technques RBE-DSS and RBE-SI [15] that follow the concept of rule base extenson. All three methods consst of two steps. Frst a raw (startng) fuzzy system s generated followed by an teratve parameter dentfcaton process (second step). C. Automatc Fuzzy System Generaton based on Fuzzy Clusterng and Proecton (ACP) The ACP method [13] was developed usng some elements and concepts of the methods [7], [9], [25], [26], and [29]. The method conssts of two man steps. Frst a startng (raw) fuzzy system s generated usng clusterng and proecton. Next the quaz-optmal parameters of the fuzzy system are dentfed by an teratve process based on a hll clmbng approach. Further on n course of the method descrpton we suppose a multple nput sngle output (MISO) fuzzy system. If the modeled phenomenon s a multple output (MIMO) one a separate model s generated for each output dmenson. The frst step of the ACP method starts wth the determnaton of the optmal cluster number for a fuzzy c-mean (FCM) type clusterng of the output sample data. It s calculated by the help of a cluster valdty ndex (CVI). One can fnd several CVIs n the lterature. A survey and comparson of the avalable technques was made by Wang and Zhang n [27]. ACP adapted the hybrd approach proposed by Chong, Gedeon and Kóczy [8] enhanced by the applcaton of an upper lmt for the cluster number. It was necessary because n case of some data sets the method resulted n a very hgh optmal Fgure 2. Creatng a Ruspn partton from output clusters usng a core estmaton wth an α=0.85 cuttng level cluster number, whch was nacceptable for practcal applcatons. The hybrd technque frst calculates the Fukuyama- Sugeno [11] ndex followed by the examnaton of the mergeablty of the adacent clusters usng the cluster mergng ndex proposed by Chong, Gedeon and Kóczy [8]. After the determnaton of the optmal output cluster number one does a one-dmensonal FCM clusterng on the output data. The appled technque s based on the FCM proposed by Bezdek [2] extended wth several modfcatons proposed n [13] n order to meet some specal requrements of the automatc fuzzy model dentfcaton. The output partton and ts fuzzy sets are estmated from the prevously dentfed clusters. ACP uses the most smple and straghtforward technque for ths task, whch was orgnally proposed by Chong n [7]. Ths approach produces a Ruspn partton wth trapezodal shaped fuzzy sets by cuttng the clusters at an α-level (usually α=0.85) and creatng the set cores from the wdth values of the cuts. Owng to the Ruspn character of the partton one determnes the supports of the sets automatcally from the correspondng endponts of the adacent lngustc terms. Fg. 2. llustrates the creaton of the consequent lngustc terms (bold lnes) from the output clusters (thn lnes). Next the consequent lngustc terms are proected nto the antecedent space. The proecton means that for each output fuzzy set one selects the data rows whose output falls nto the support of the set. Thus some data correspondng to the overlappng parts of the lngustc terms wll be taken nto consderaton twce. Afterwards one does a one dmensonal fuzzy clusterng n each antecedent dmenson on the current subset of the sample data. The cluster centers wll be used later as reference ponts of the estmated antecedent fuzzy sets. A, k An dentfer wth three ndces (, ) s assgned to each cluster, where s the ordnal number of the antecedent dmenson, s the ordnal number of the consequent fuzzy set n ts partton, and k s the ordnal number of the cluster n the current clusterng. One can generate two or more rules for each output fuzzy set n the form R m : a = A1,, AND... AND a = A,, 1 k1 THEN b = B, N N kn where m s the ordnal number of the rule, N s the number of antecedent dmensons. Thus one obtans... (2)

3 several groups of cluster centers n each nput dmenson. Next they are nterleaved and renumbered n order to get one cluster partton n each dmenson. Some of the cluster centers wll be dentcal or nearly dentcal. They are unted applyng a prescrbed dstance threshold whose value s a parameter of the method. After a cluster unon the related rules are also corrected or unted. One estmates the antecedent parttons and lngustc terms usng the same methods as n the case of the consequent partton. The resultng fuzzy model s further mproved n the second step of ACP by a parameter dentfcaton (tunng) process. The algorthm uses an teratve hll clmbng approach. After each step (each parameter modfcaton) the resultng fuzzy system s evaluated by the means of a performance ndex, whch gves nformaton about the dfference between the prescrbed output gven by the sample data set and the calculated data set usng the fuzzy system. The toolbox contans seven optons for the selecton of the performance ndex. Probably the most used and straghtforward one s the relatve value of the mean square error expressed n percentage of the output range (RMSEP) M ( y ˆ y ) 2 = 1 1 PI RMSEP = DRN , (3) M where M s the number of data ponts n the sample, s the th prescrbed output value, y ŷ s the th calculated output value, and DR N+ 1 s the range of the output. The consequents n (2) can be constant or affne functons of the nputs. Applcatons can be found n fuzzy control systems [18], [30]. The algorthm nvestgates the parameters one-by-one n course of each teraton cycle. Frst the system s evaluated ( PI RMSEP ) wth the current parameter set. Next one calculates two new values for the actual parameter one by decreasng the orgnal value and one by ncreasng the orgnal value. The step ( st ) s calculated n functon of the range ( DR ) n the actual () dmenson st = C, (4) DR where C s a constant that apples to each dmenson. After the evaluaton of the system wth he two new parameter values one keeps that value from the avalable three ones (orgnal and two new values), whch results the best system performance. An teraton cycle contans the nvestgaton of all parameters once. At the end of the cycle one compares the actual performance ( PI k ) wth the performance of the system measured at the end of the prevous cycle ( PI k 1 ). If the ameloraton of PI s greater than a threshold the value of C s doubled otherwse C s decreased to ts half. When t becomes smaller than a threshold ( C mn ) for the frst tme the process receves a second chance by ncreasng C to ts orgnal value. The algorthm stops when ether C reaches ts mnmum for the second tme or the prescrbed number of teraton cycles s reached or the performance ndex of the system becomes better than a threshold value. D. Automatc Fuzzy Model Identfcaton by Rule Base Extenson The concept of rule base extenson (RBE) [15] starts the fuzzy model dentfcaton wth the creaton of two rules, one descrbng the mnmum output and one descrbng the maxmum output,.e. the reference ponts of the consequent sets wll be dentcal wth the mentoned extreme ponts. If the mnmum (maxmum) output s reached n case of several dfferent nput values one chooses those values as reference ponts for the antecedent sets that are closer to the lower or upper bounds of the correspondng nput doman. In case of trapezodal shaped membershp functons one determnes a default core and support wdth for each dmenson dependng on the range of the doman. Smlar to ACP the RBE based methods (RBE-DSS and RBE-SI) [15] also have a second step amng the mprovement of the orgnal fuzzy system n order to reach a quas-optmal performance ndex. The algorthm s smlar to the one appled n the second step of ACP. It dffers manly n the handlng of the local optmum of PI. In case of RBE when the coeffcent (C) of the step ( st ) reaches ts mnmum one generates a new rule fttng the output value where the dfference between the sample output and the calculated output s maxmal. Rule Base Extenson usng Default Set Shapes (RBE- DSS) creates the fuzzy sets of the new rule wth the same core and support wdth values (supposng trapezodal shaped membershp functons) as the ones appled for the frst two rules. However, the nserton of the new rule sometmes results n a temporary deteroraton of the performance ndex. In order to avod ths problem Rule Base Extenson usng Set Interpolaton (RBE-SI) calculates the shape of the antecedent and consequent lngustc terms based on set nterpolaton. It s not mandatory but t could be expedent to choose the set nterpolaton method conform to the appled fuzzy rule nterpolaton based reasonng technque. III. RULEMAKER TOOLBOX A. General descrpton The RuleMaker toolbox s mplemented n Matlab and s avalable for download under GNU General Publc Lcense [12] from the webste [10]. It was developed and tested usng Matlab 2006b. RuleMaker ams the generaton of a low complexty fuzzy system usually wth a sparse rule base. Its current verson contans the mplementaton of the methods ACP, RBE-DSS and RBE- SI. The toolbox s a collecton of Matlab functons that can be called from command lne and from other Matlab programs. The package also contans software wth graphcal user nterface ensurng an easy-to-use access to

4 Input data Sample output Method based on extremes (RBE) System generaton Method based on clusterng (ACP) Model dent. parameters Method selecton Iteratve param. dentfcaton Raw systems Raw fuzzy system Graphcal repr. of the parameters Instant graphcal and text dsplay of the system performance Graphcal representaton of the prescrbed and calculated output Fnal fuzzy system all the servces offered by the toolbox. Its nternal structure s presented n fg. 3. B. Usage of the GUI The graphcal user nterface of the toolbox can be started by typng n the Matlab Command Wndow the command RuleMaker. Conform to the mplemented fuzzy model dentfcaton methods one can dstngush two man areas n the functonalty of the GUI program. The frst one deals wth the generaton of the raw fuzzy system from the sample data. The tranng data should be loaded n two separate text fles, one contanng the output values and one contanng the nput values. In both cases each row corresponds to one data pont. If the nput or output s mult-dmensonal the co-ordnates n the dfferent dmensons are separated by tabulators. One can generate the raw fuzzy system ether by clusterng or by the extreme values based approach. The system can be saved n a text fle usng the same format as the Fuzzy Rule Interpolaton Matlab toolbox [17]. The program generates a separate raw system for each output dmenson n case of sample data wth mult-dmensonal Fgure 3. Internal structure of the program output. The current verson (1.0.0) of the toolbox supports only the trapezodal shaped membershp functons, whch ncludes wth correspondng parameterzaton the trangle and the sngleton shaped fuzzy sets as well. The default core and support wdth values and some other parameters can be set by the user through dalog boxes. They are also dsplayed n a textbox n the man wndow (see fg. 4). The program makes possble the graphcal representaton of the nput and output parttons (see e.g. fg. 5) separately, as well as a three dmensonal representaton of the rule base or antecedent space f the number of antecedent dmensons does not exceed three. If the fuzzy model has only one nput dmenson the program offers a three dmensonal representaton of the rule base each rule beng ndcated by a pyramd defned by the antecedent and consequent membershp functons (see e.g. fg. 6). If the number of nput dmensons s two one can use the pyramd-type representaton for the vsualzaton of the antecedent space. Here the pyramds are defned by the antecedent fuzzy sets. However, the user can choose the cubod-type representaton (e.g. fg. 1) where the rule consequents stll can be drawn nto the vsualzaton. In ths case the edges of a cubod represent the supports of the antecedent and consequent lngustc µ 1 A 1;1 A 1;6 0.5 A 1;2 A A 1;3 1;4 A 1;5 0 x Fgure 4. Parameters of the raw fuzzy system generaton Fgure 5. Antecedent partton of a fuzzy system

5 Fgure 6. 3D representaton of a SISO fuzzy system. Each pyramd represents a rule and s defned by the antecedent and consequent membershp functons terms of a rule. The drawback of ths type s that other parts of the sets shape cannot be vsualzed. In case of fuzzy systems havng three nput dmensons only the cubod-type representaton s avalable. It can be used for the vsualzaton of the rule antecedents. The second man area of the program functonalty deals wth the parameter dentfcaton. After each step (modfcaton of a parameter) the system s evaluated usng a performance ndex. The user can choose from seven PIs. They are the mean square of the error (MSE), the relatve value of the MSE compared to the range of the output varable (RMSE), the RMSEP (3), the correlaton factor between the prescrbed output and the calculated one (R), the transformed value of R (RR) [13], the mean of the dfferences n absolute value between the prescrbed output and the calculated one (AD), as well as the relatve value of AD compared to the range of the output varable (ADP). The fuzzy nference s done by the help of the Fuzzy Rule Interpolaton (FRI) Matlab toolbox [17]. Ten fuzzy nference methods can be selected. The RuleMaker toolbox offers four fuzzy set parameterzaton types (breakponts, relatve dstances, reference ponts, and a core endponts based one that always ensures the Ruspn character of the partton). Besde the determnaton of the performance ndex the calculated and prescrbed output values can be compared n a dagram. Fg. 7 llustrates the graphcal representaton of the momentary output. The abscssa ndcates the ordnal number of the data ponts. The output from the sample data s represented by (blue) crcles and the output calculated by the fuzzy system for same nput s represented by (red) pentagrams. Fgure 8. Varaton of the performance ndex (RMSEP) n course of the parameter dentfcaton represented n functon of the number of system evaluatons (SE) One can also get a comprehensve vew of the tunng from a dagram (see fg. 8) showng the varaton of the performance ndex (RMSEP on fg. 8). Here the abscssa shows the number of system evaluatons (SE). The fuzzy model can be saved after each teraton step. The parameter dentfcaton usually s a tme and computatonal resources consumng task. Therefore the user can pause or stop the process at any tme by the help of push buttons. C. Further development plans The RuleMaker Toolbox s under contnuous development. We plan ts development n three man drectons. Implementaton of new fuzzy model dentfcaton methods and technques. Extendng the exstent mplementatons n order to support all knds of polygonal membershp functons. Implementaton of new performance ndex types. IV. CONCLUSIONS Automatc fuzzy model dentfcaton from sample data s one of the key ssues n practcal applcaton of fuzzy systems. After revewng the methods ACP, RBE-DSS and RBE-SI ths paper presented the RuleMaker toolbox a software tool that mplements these methods. All the methods produce a fuzzy system wth low complexty applyng usually a sparse rule base. Therefore the created fuzzy systems should use a fuzzy rule nterpolaton based nference technque. The software seamlessly nteroperates wth the FRI Matlab toolbox. Thus the produced fuzzy systems are generated for one of the FRI nference technques avalable n the FRI toolbox. The software s avalable under GNU GPL. One of the future researches wll be focused on extendng the approach to automatc fuzzy model dentfcaton methods for nonlnear plants control [31]. REFERENCES Fgure 7. Prescrbed (crcles) and calculated (pentagrams) output n functon of the ordnal number of the data ponts [1] Barany, P., Kóczy, L. T. and Gedeon, T. D.: A Generalzed Concept for Fuzzy Rule Interpolaton, n IEEE Transacton on Fuzzy Systems, Vol. 12, No. 6, pp [2] Bezdek, J. C.: Pattern Reconton wth Fuzzy Obectve Functon Algorthms, Plenum Press, New York, [3] Botzhem, J., Cabrta, C., Kóczy, L. T. and Ruano, A. E.: Fuzzy rule extracton by bacteral memetc algorthms, n Proceedngs of

6 the 11th World Congress of Internatonal Fuzzy Systems Assocaton, IFSA 2005, Beng, Chna, July 2005, pp [4] Botzhem, J., Hámor, B., and Kóczy, L.T.: Extractng trapezodal membershp functons of a fuzzy rule system by bacteral algorthm, 7th Fuzzy Days, Dortmund 2001, Sprnger-Verlag, pp [5] Bouchon-Meuner, B., Marsala, C.; Rfq, M.: Interpolatve reasonng based on gradualty. In Proc. FUZZ-IEEE 2000, 2000, pp [6] Chu, S. L.: Fuzzy Model Identfcaton Based on Cluster Estmaton, Journal of Intellgent and Fuzzy Systems, Vol. 2, 1994, pp [7] Chong, A.: Constructng Sparse and Herarchcal Fuzzy Rulebases, PhD Thess, Murdoch Unversty, Perth, W.A., [8] Chong, A., Gedeon, T. D. and Kóczy, L. T.: A Hybrd Approach for Solvng the Cluster Valdty Problem, n Proceedngs of the 14th Internatonal Conference on Dgtal Sgnal Processng, Santorn, Greece, 2002, pp [9] Chong, A., Gedeon, T. D. and Kóczy, L. T.: Proecton Based Method for Sparse Fuzzy System Generaton, n Proceedngs of the 2nd WSEAS Internatonal Conference on Scentfc Computaton and Soft Computng, Crete, Greece, 2002, pp [10] A webste dedcated to fuzzy rule nterpolaton and sparse fuzzy model dentfcaton ssues. [11] Fukuyama, Y. and Sugeno, M.: A new method of choosng the number of clusters for fuzzy c-means method, n Proceedngs of the 5 th Fuzzy Systems Symposum, 1989, pp [12] GNU General Publc Lcence: [13] Johanyák, Z. C.: Fuzzy rule nterpolaton methods and automatc system generaton based on sample data (n hungaran), PhD thess, Unversty of Mskolc, [14] Johanyák, Zs. Cs. and Kovács, Sz.: Fuzzy Rule Interpolaton by the Least Squares Method, n Proceedngs of the 7th Internatonal Symposum of Hungaran Researchers on Computatonal Intellgence (HUCI 2006), Budapest, Hungary, 2006, pp [15] Johanyák, Zs. Cs. and Kovács, Sz.: Sparse Fuzzy System Generaton by Rule Base Extenson, n Proceedngs of the 11th IEEE Internatonal Conference of Intellgent Engneerng Systems (IEEE INES 2007), Budapest, Hungary, 2007, pp [16] Johanyák, Zs. Cs. and Kovács, Sz.: The effect of dfferent fuzzy partton parameterzaton strateges n gradent descent parameter dentfcaton, n Proceedngs of the 4th Internatonal Symposum on Appled Computatonal Intellgence and Informatcs (IEEE SACI 2007), Tmsoara, Romana, 2007, pp [17] Johanyák, Zs. Cs., Tkk, D., Kovács, Sz. and Wong, K. W: Fuzzy rule nterpolaton Matlab toolbox FRI toolbox, n Proceedngs of the IEEE World Congress on Computatonal Intellgence (WCCI'06), 15th Int. Conf. on Fuzzy Systems (FUZZ-IEEE'06), 2006, Vancouver, BC, Canada, pp [18] Klawonn, F. and Kruse, R.: Constructng a fuzzy controller from data, Fuzzy Sets and Systems, Vol. 85, Issue 2, 1997, pp [19] L. T. Kóczy, J. Botzhem, and T. D. Gedeon. Fuzzy models and nterpolaton. In M. Nkravesh, J. Kacprzyk, and L. A. Zadeh, edtors, Forgng New Fronters: Fuzzy Poneers I & II, volume 217 of Studes n Fuzzness and Soft Computng. Sprnger, [20] Kóczy, L. T. and Hrota, K.: Approxmate reasonng by lnear rule nterpolaton and general approxmaton, n Internatonal Journal of Approxmatve Reasonng, Vol. 9, 1993, pp [21] Kóczy, L. T. and Hrota, K.: Sze reducton by nterpolaton n fuzzy rule bases, n IEEE Transactons on System, Man and Cybernetcs, Vol. 27, 1997, pp [22] Kovács, Sz.: Extendng the Fuzzy Rule Interpolaton "FIVE" by Fuzzy Observaton, Theory and Applcatons, Sprnger Berln Hedelberg, 2006, pp [23] Kosko, B. and Dckerson, J. A.: Functon approxmaton wth addtve fuzzy systems, n Theoretcal Aspects of Fuzzy Control, John Wley & Sons, [24] Kosko, B.: Optmal fuzzy rules cover extrema, Internatonal Journal of Intellgent Systems, Vol. 10, 1995, pp [25] Sugeno, M. and Yasukawa, T.: A fuzzy-logc-based approach to qualtatve modelng, n IEEE Transactons on Fuzzy Systems, Vol. 1, 1993, pp [26] Tkk, D., Gedeon, T. D. Kóczy, L. T. and Bíró, G.: Implementaton detals of problems n Sugeno and Yasukawa's qualtatve modelng, Research Workng Paper RWP-IT , School of Informaton Technology, Murdoch Unversty, Perth, W.A., [27] Wena Wang, Yune Zhang: On fuzzy cluster valdty ndces. Fuzzy Sets and Systems 158(19): (2007) [28] Wong, K. W., Gedeon, T. D. and Tkk, D.: An mproved multdmensonal α-cut based fuzzy nterpolaton technque, In Proc. Int. Conf. Artfcal Intellgence n Scence and Technology (AISAT 2000), Hobart, Australa, 2000, pp [29] Wong, K. W., Kóczy, L. T., Gedeon, T. D., Chong, A. and Tkk, D. :Improvement of the Cluster Searchng Algorthm n Sugeno and Yasukawa's Qualtatve Modelng Approach, n Lecture Notes n Computer Scence, Vol. 2206, 2001, pp [30] Precup, R.-E. and Pretl, S.: PI-Fuzzy controllers for ntegral plants to ensure robust stablty, Informaton Scences, vol. 177, no. 20, Oct ,, pp [31] Precup, R.-E., Pretl, S., Tar, J. K., Tomescu, M. L., Takács, M., Korond, P. and Barany, P.: Fuzzy control system performance enhancement by teratve learnng control. IEEE Transactons on Industral Electroncs, Vol. 55, Nr. 9, 2008, pp