Fuzzy modeling based on generalized conjunction operations

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Fuzzy modeling based on generalized conjunction operations
  678 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 5, OCTOBER 2002 Fuzzy Modeling Based on GeneralizedConjunction Operations Ildar Batyrshin, Okyay Kaynak   , Senior Member, IEEE  , and Imre Rudas  , Fellow, IEEE   Abstract— A novel approach to fuzzy modeling based on thetuning of parametric conjunction operations is proposed. First,somenovelmethodsfortheconstructionofparametricgeneralizedconjunction operations simpler than the known parametric classesof conjunctions are considered and discussed. Second, severalexamples of function approximation by fuzzy models, based onthe tuning of the parameters of the new conjunction operations,are given and their approximation performances are comparedwith the approaches based on a tuning of membership functionsand other approaches proposed in the literature. It is seen that thetuning of the conjunction operations can be used for obtainingfuzzy models with a sufficiently good performance when thetuning of membership functions is not possible or not desirable.  Index Terms— Conjunction, disjunction, functions approxima-tion, fuzzy modeling, t-norm. I. I NTRODUCTION F UZZY inference systems are known as universal approx-imators [13], [16], [24]. This property enables one toconstruct optimal fuzzy models of processes and systems. Theoptimization of such a fuzzy model is usually based on a tuningof the membership functions used in the model. However, wecan point out at least two situations when such tuning maynot be desirable or effective. The first situation usually ariseswhen the initial shapes of the membership functions are basedon some expert knowledge available about the system to bemodeled. In such a case, after tuning of membership func-tions, this knowledge can be lost. The second situation ariseswhen there exist limitations on the number of membershipfunctions and rules used in the fuzzy model. These limitationsmay have an influence on the performance of fuzzy models.In such situations, the construction of optimal fuzzy modelsmay also be based on the tuning of fuzzy connectives usedin fuzzy model, if these connectives are given parametrically.Such tuning may be used instead of or additionally to tuningof the parameters of membership functions. Manuscript received October 23, 2001; revisedFebruary 22, 2002. This work was supported in part by the Foundation for Promotion of Advanced Automa-tion Technology, Japan, and by the Academy of Sciences of the Republic of Tatarstan, Russia.I. Batyrshin is with the Institute of Problems of Informatics, Academy of Sciences of the Republic of Tatarstan, and also with the Department of Infor-matics and Applied Mathematics, Kazan State Technological University, Kazan420015, Russia (e-mail: batyr@emntu.kcn.ru).O. Kaynak is with the Electrical Engineering Department, Bogazici Univer-sity, 80815 Istanbul, Turkey (e-mail: kaynak@boun.edu.tr).I. Rudas is with the Budapest Polytechnic, H-1081 Budapest, Hungary(e-mail: rudas@zeus.banki.hu).Digital Object Identifier 10.1109/TFUZZ.2002.803500. Thesimplestofthefuzzymodelsmayconsistofthefollowingtypes of fuzzy rules:If is and is and andis then Mamdani modelIf is and is and andis thenSugeno modelwhere and are fuzzy input and output variables (e.g., PRESSURE ,  VOLUME ,  TEMPERATURE , etc), and are lin-guistic terms (e.g.,  SMALL ,  POSITIVE LARGE ,  ZERO ), andare the real values of input and output variables and are realvalued functions. For given crisp input values , the firingvalues of the rules may be calculated as follows:where are some conjunction operations (usually identical-norms for all and ) used as the connective  and   andare the membership values of in fuzzy sets .After aggregation and defuzzification of conclusions of rules(these methods usually differ for different types of modelsand methods [11], [13], [16], [19], [22], [24]), the output of  fuzzy model will be obtained as some real value . As aresult, fuzzy models determine some real valued functionswhich may be used as an approxima-tion of the given experimental data or as an approximation of traditional mathematical model of some system or process.The membership functions in fuzzy models are often givenparametrically and a tuning of these parameters is used for theminimization of the approximation error. The use of parametricconjunction operations in a fuzzy model gives rise tothe possibility of tuning the parameters of this operation.Unfortunately, the known parametric classes of -norms and-conorms, which can be used in fuzzy models, are generallytoo complicated for tuning and for hardware realization. Theconstruction of simpler parametric classes of these operations,therefore, looks very attractive. One possible way of suchsimplification may be based on the deletion of the associativityproperty and, perhaps, the commutativity property from thedefinition of conjunction and disjunction operations [2], [3].It is clear that the associativity property is not necessary for 1063-6706/02$17.00 © 2002 IEEE Authorized licensed use limited to: ULAKBIM UASL - BOGAZICI UNIVERSITESI. Downloaded on February 19, 2009 at 04:55 from IEEE Xplore. Restrictions apply.  BATYRSHIN  et al. : FUZZY MODELING BASED ON GENERALIZED CONJUNCTION OPERATIONS 681 Fig. 1. The surfaces of the conjunction operation           fordifferent values of the parameters    and    . Theorem 4:  Suppose and are -conjunctions, andare functions satisfying (2) and(3), , , are nondecreasing functions suchthat ; then, the following functions:are -conjunctions. Proof:  The monotonicity of in all expressions followsfrom the monotonicity of functions used in the right-hand sidesof the expressions. If or , then and,hence, . If , then, all and equal to 1, from (5) it follows also thatand, hence, in all expressions we have .By the use of Theorems 3 and 4, the simplest parametric-conjunction operations can be obtained(14)(15)where . The surfaces of these conjunctions are shownin Figs. 1–3. The last -conjunction is commutative and thefirst two -conjunctions satisfy the property of generalizedcommutativity.It is seen that the proposed definition of conjunction and dis- junction operations enables one to build the simplest parametricclasses of conjunction and disjunction operations. It is to benotedthatanothersystemofaxiomsforgeneralizedconnectivesis considered in [7] where the fuzzy conjunction is con-sidered and its application to fuzzy modeling is discussed. Thisconjunctionbelongstotheparametricclassofconjunctions(14) Fig. 2. The surfaces of the conjunction operation           fordifferent values of the parameters    and    .Fig. 3. The surfaces of the conjunction operation                0     for different values of the parameters    and    . with . In the following sections, we discuss two examplesoffunctionapproximationsbyfuzzymodelsbasedonthetuningof new conjunctions.IV. A PPROXIMATION OF  T WO -I NPUT  sinc F UNCTION First, we consider the example of approximation of a two-input sinc function sinc , with Authorized licensed use limited to: ULAKBIM UASL - BOGAZICI UNIVERSITESI. Downloaded on February 19, 2009 at 04:55 from IEEE Xplore. Restrictions apply.
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