Analytical Structure of the Typical Fuzzy Controllers Employing Trapezoidal Input Fuzzy Sets and Nonlinear Control Rules

Fuzzy controllers are generally nonlinear controllers, and as such, it is important toreveal analytical structure of fuzzy controllers relative to conventional nonlinear controllers so that the well-developed nonlinear control theory can be utilized to effectively analyze and design fuzzy control systems. In our previous papers, we have shown that the analytical structure of the fuzzy controllers using triangular input fuzzy sets and linear fuzzy control rules is the sum of a global two-dimensional multilevel relay and a local nonlinear PI controller. In this paper, we present the analytical structure of much more general and typical fuzzy controllers - the ones that employ trapezoidal input fuzzy sets and arbitrary nonlinear control rules. We analytically prove that the structure of such controllers is the sum of a global nonlinear controller and a local nonlinear PI-like controller whose proportional-gain and integral-gain vary both globally and locally with change of the controllers' inputs. This result significantly generalizes our previous ones and contains them only as special cases. Analytical results are presented, which reveal the characteristics and properties of the fuzzy controllers. We show that the global nonlinear controller is characterized solely by fuzzy control rules, that is, different nonlinear control rules result in different types of global nonlinear controllers. When the number of input fuzzy sets approaches oo, the global nonlinear controller approaches a control rules-dependent nonlinear controller whereas the local nonlinear PI-like controller disappears. The results presented in this paper can directly be used to generate analytical structure of any specific fuzzy controllers and we demonstrate how by illustrative examples.