Searching for basic properties obtaining robust implication operators in fuzzy control

This paper deals with the problem of searching basic properties for robust implication operators in fuzzy control. We use the word “robust” in the sense of good average behavior in different applications and in combination with different defuzzification methods. We study the behavior of the two main families of implication operators in the fuzzy control inference process. These two families are composed by those operators that extend the boolean implication (implication functions) and those ones that extend the boolean conjunction (t-norms and forceimplications). In order to develop the comparative study, we will build different fuzzy controllers by means of these implication operators and will apply them to the fuzzy modeling of the real function Y=X and two three-dimensional surfaces. We analyze whether one of these two properties, extension of the boolean implication and extension of the boolean conjunction, is sufficient for obtaining a good implication operator or whether some complementary properties are necessary. Next, we analyze whether we can get basic properties for good implication operators, presenting three basic properties for the so called robust implication operators.