Mathematical models of the simplest fuzzy two-term (PI/PD) controllers using minimum inference

This paper reveals mathematical models of the simplest Mamdani PI/PD controllers which employ two fuzzy sets (N-negative and P-positive) on the universe of discourse (UoD) of each of two input variables (displacement and velocity) and three fuzzy sets (N-negative, Z-zero, P-positive) on the UoD of output variable (control output in the case of PD, incremental control output in the case of PI). The basic constituents of these models are algebraic product / minimum AND, bounded sum / algebraic sum / maximum OR, minimum inference, three linear fuzzy rules, and center of sums (CoS) defuzzification. Properties of all these models are investigated. It is shown that all these controllers are different nonlinear PI/PD controllers with their proportional and derivative gains changing with the inputs. The proposed models are significant and useful to control community as they are completely new and qualitatively different from the reported ones in the literature.