Theoretical analysis of crisp-type fuzzy logic controllers using various t-norm sum-gravity inference methods

The input-output parametric relationship of a class of crisp-type fuzzy logic controllers (FLCs) using various t-norm sum-gravity inference methods is studied. Four most important t-norms are used to calculate the matching level of each control rule and the explicit mathematical forms of reasoning surfaces obtained by using these four t-norms are addressed. Reasoning surfaces of these crisp-type FLCs are proved to be composed of a two-dimensional multilevel relay no matter which t-norm is used and a local position-dependent nonlinear compensator with output pattern influenced by the t-norms is selected. By analyzing the intrinsic operation of the four t-norms, the authors find that both standard intersection and algebraic product are suitable operators to perform the inference of the FLC. However, bounded difference and drastic intersection are disqualified because they cannot satisfy some important criteria. A measure of relative degree-of-nonlinearity is defined to examine the output figures of these crisp-type FLCs. The ultimate behavior of these crisp-type FLCs as the number of linguistic terms approaches infinity is also explored. The local stability criteria for the proportional-integral (PI)-type fuzzy control systems and the natural global stability characteristic for the proportional-derivative (PD)-type fuzzy control systems are also examined