General analytical structure of typical fuzzy controllers and their limiting structure theorems

Abstract

The fuzzy controllers studied in this paper have r input variables (error, rate change of error of process output, etc.). N members of input fuzzy sets are employed to fuzzify the inputs. N r nonlinear control rules represented by an arbitrary function f are used, which are evaluated by any type of fuzzy logic in conjunction with any inference method. The center of gravity algorithm is utilized for defuzzification. The author proves that the general analytical structure of the fuzzy controllers is the sum of a global f-dependent nonlinear controller and a local nonlinear controller which locally adjusts control action of the global controller. As N → +∞, the global nonlinear controller approaches a global f-dependent nonlinear controller while the local nonlinear controller disappears. A necessary and sufficient condition is obtained for judging convergence of limiting structure of the fuzzy controllers. If linear control rules are used, the global controller is a global r-dimensional multilevel relay which approaches a global linear controller as N → +∞. Two illustrative examples, which deal with linear, product, maximum and minimum control rules, are given.