New Types of Generalized Operations

Abstract

New methods for constructing generalized triangular operators, using a minimum and maximum fuzziness approach are outlined. Based on the entropy of a fuzzy subset, defined by using the equilibrium of the generalized fuzzy complement, the concept of elementary entropy function and its generalizations are introduced. These functions assign a value to each element of a fuzzy subset that characterizes its degree of fuzziness. It is shown that these functions can be used to construct the entropy of a fuzzy subset. Using this mapping, the generalized intersections and unions are defined as mappings, that assign the least and the most fuzzy membership grade to each of the elements of the operators’ domain, respectively. Next further classes of new generalized T-operators are introduced, also defined as minimum and maximum entropy operations. It is shown that they are commutative semigroup operations on [0,1] with identity elements but they are not monotonic. Simulations have been carried out so as to determine the effects of these new operators on the performance of the fuzzy controllers. It is concluded that the performance of the fuzzy controller can be improved by using some sets of generalized T-operations for a class of plants.