On choosing models for linguistic connector words for Mamdani fuzzy logic systems

We examine ten antecedent connector models in the framework of a singleton or nonsingleton fuzzy logic system (FLS), to establish which models can be used. In this work, a usable connector model must lead to a separable firing degree that is a closed-form and piecewise-differentiable function of the membership function parameters and also the parameter characterizing that connector model. Our analysis shows that: for a singleton FLS where the Mamdani-product or Mamdani-minimum implication method is used, all ten antecedent connector models are usable; for a nonsingleton FLS where the Mamdani-product implication method is used, only one antecedent connector model is usable; and for a nonsingleton FLS where the Mamdani-minimum implication method is used, none of the ten antecedent connector models is usable. We also show, by examples, that the parameter of the antecedent connector model provides additional freedom in adjusting a FLS, so that the FLS has the potential to achieve better performance than a FLS that uses the traditional product or minimum t-norm for the antecedent connections.