Mathematical modeling and computational study of the simplest fuzzy two-term controllers via height defuzzification

In this paper, mathematical models of Mamdani type simplest fuzzy Proportional Integral (PI)/Proportional Derivative (PD) controllers via Height (Ht) defuzzification are presented. Minimal number of fuzzy sets are chosen for the two inputs and output of the fuzzy controller. L - type and Gamma - type membership functions and different Universes of Discourse (UoDs) are considered for the input variables. Membership functions of output variable are chosen in such a way that the sum of all the membership functions at any point is unity. Three linear fuzzy control rules relating all four input fuzzy sets to three output fuzzy sets are chosen. Two triangular norms namely Algebraic Product (AP) and Minimum (Min), and three triangular co - norms (also sometimes called s - norms) such as Bounded Sum (BS), Algebraic Sum (AS) and Maximum (Max) are used. Properties of the fuzzy controller models are studied. Since digital controllers are implemented on the digital processors, finally the computational and memory requirements of these fuzzy controllers and conventional (nonfuzzy) controllers are compared. A rough estimate of the computational time taken by the digital computer while implementing these discrete - time fuzzy controllers is given.