Minimum t-norm leads to unrealizable fuzzy PID controllers

In this paper, we show that Minimum triangular norm (Min t-norm) always leads to unrealizable fuzzy PID (FPID) controllers. To show this, mathematical models of two FPID controllers are obtained. Two fuzzy sets, Negative (N) and Positive (P), for fuzzification of each of the three input variables (error, change of error, and double change of error), and four fuzzy sets N, Negative Small (NS), Positive Small (PS), and P on the output variable (change of control effort) are considered. Min t-norm, Maximum (Max) s-norm, triangular membership functions, Larsen Product (LP)/ Mamdani Minimum (MM) inference, and Centre of Area (Centre of Gravity) (CoA (CoG)) defuzzification are used for mathematical modeling of the controllers. Then upon thoroughly analyzing the properties of the mathematical models of the controllers, we show that Min t-norm leads to unrealizable FPID controllers.