T-NORMS vs. IMPLICATION FUNCTIONS AS IMPLICATION OPERATORS IN FUZZY CONTROL

A comparative study of the implication functions and t-norms behavior when they are used as implication operators in the fuzzy control inference process is carried out in this paper. In order to do it, the widely studied application of the Inverted Pendulum is considered. Moreover, a comparison methodology of the different fuzzy logic controllers designed is defined with the purpose of get more information on the problem of the selection of the best implication operator in fuzzy control. 1.Introduction Finally, when the rules of the Knowledge Base have more than one variable in the antecedent (that is, they present the generical form If X1 is A1 and X2 is A2 ... and Xn is An then Y is B), x0 = (x1,..., xn) and μA (x0) = T (μA1(x1), μA2(x2), ... , μAn(xn)), being T a conjunctive operator. In the specialized literature it is proposed a very big amount of operators that can be used as implication operators in the inference process of fuzzy logic controllers (FLCs). Many authors have presented and analyzed several implication operators such as: implications introduced from many-valued logic systems [13], implication functions [12,17], t-norms [5,6,15] and a wide range of other kind implications [1,2,9]. Analyzing these works, it is possible to draw that the selection of the best implication operator has become in one of the principal problems of inference in fuzzy control as we are going to see in the following. It is found that the final expression of the CRI in fuzzy control depends directly on the implication operator selected. Many studies adding some information in order to select this operator have been developed in the specialized literature. In [13], there are presented some implication operators of many-valued logic systems and study their behavior in fuzzy reasoning based on the GMP and on the Generalized Modus Tollens when the inputs to the Inference System are fuzzy concepts. In a later work, [14], the accuracy of several of these implication operators in the fuzzy control of a plant model is analyzed. In [9], thirty six implication operators are collected and a study of their accuracy in the fuzzy modeling of a d.c. series motor is developed. In [1,2], a new comparison methodology is defined and the behavior of the operators employed in [9] is analyzed using them in the fuzzy modeling of different mathematical functions. Finally, in [6] is presented a study of the behavior of several implication operators based on t-operators in the same problem considered in [14]. Other studies have been carried out in [3,4,10,16]. As it is known, the inference process in fuzzy control is based on the use of the Generalized Modus Ponens (GMP) proposed by Zadeh in the way If X is A then Y is B X is A'