Three-dimensional min-max-gravity based fuzzy PID inference analysis and tuning

This paper proposes a novel algorithm to produce an analytical solution for three-input fuzzy PID systems. The analysis is based on the most common inference method, Zadeh-Mamdani's min-max-gravity fuzzy reasoning. Two types of fuzzy systems are investigated. Both types use unevenly distributed triangular shaped fuzzy input memberships. In the first type, the output fuzzy sets are considered to be uniformly distributed symmetrical triangular type memberships and in the second type the output fuzzy sets are considered unevenly distributed singleton functions. A new input transformation technique is developed to reduce the number of input conditions required in defining the fuzzy output. The multi-phase complexity that exists in conventional fuzzy reasoning analysis is thus avoided. An efficient solution algorithm is outlined to produce the general fuzzy output solution using a minimum number of nonlinear expressions. With the proposed method, the fuzzy output solution for a three-input fuzzy system having uniform output membership functions requires only two nonlinear expressions. When the output fuzzy sets are unevenly distributed, the general three-input based solution requires only four nonlinear expressions. The fuzzy controller output is finally decomposed into linear and nonlinear parts. The output decomposition is useful in defining tuning gains for fuzzy PID systems.