Conditions on general Mamdani fuzzy controllers as nonlinear, variable gain state feedback controllers with stability analysis

Abstract

At present, most Mamdani fuzzy controllers must be designed and analyzed individually using the trial-and-error approach coupled with computer simulation. This is largely due to the lack of knowledge on the structural relationship between the fuzzy controllers and conventional controllers, making effective and systematic utilization of analytical analysis and design tools in control system theory very difficult. The author attempts to partially resolve these issues. The Mamdani fuzzy controllers covered are general; they use arbitrary continuous input fuzzy sets, arbitrary fuzzy rules, arbitrary inference methods, either Zadeh or the product fuzzy logic AND operator, singleton output fuzzy sets and the centroid defuzzifier. The author has established conditions for these fuzzy controllers as nonlinear state feedback controllers with variable gains, which he calls fuzzy state feedback controllers for terminological convenience. The necessary and sufficient condition on the fuzzy controllers using Zadeh fuzzy logic AND operator is that the input fuzzy sets must be linear or piecewise linear (e.g., trapezoidal or triangular). This condition, however, becomes only a necessary one for the controllers using the product fuzzy logic AND operator. These structural conditions have enabled the development of a necessary and sufficient local stability condition that can be used not only for stability determination, but also for practically designing stable fuzzy state feedback control systems even when the system models are mathematically unknown.