The simplest fuzzy two-term controllers: mathematical models and stability analysis

This paper reveals mathematical models for some simplest fuzzy two-term (PI/PD) controllers which employ two symmetric fuzzy sets for each of the two input variables and three symmetric fuzzy sets for the output variable.
The basic constituents of these models are -function type and L-function type membership functions for the input variables and triangular membership functions
for the output variable, minimum triangular norm, algebraic sum triangular conorm, different inference methods and Centre of Sums (COS) defuzzification
method. The properties of these models are studied to examine their suitability for control application. Using the well-known small-gain theorem, sufficient condition for Bounded-Input Bounded-Output (BIBO) stability of a feedback
system, containing any fuzzy controller as a subsystem, is established. Finally, some numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PI/PD controllers.