Fuzzy control on the basis of equality relations with an example from idle speed control

The way engineers use fuzzy control in real world applications is often not coherent with an understanding of the control rules as logical statements or implications. In most cases fuzzy control can be seen as an interpolation of a partially specified control function in a vague environment, which reflects the indistinguishability of measurements or control values. In this paper the authors show that equality relations turn out to be the natural way to represent such vague environments and they develop suitable interpolation methods to obtain a control function. As a special case of our approach the authors obtain Mamdani´s model and can justify the inference mechanism in this model and the use of triangular membership functions not only for the reason of simplified computations, and they can explain why typical fuzzy partitions are preferred. The authors also obtain a criterion for reasonable defuzzification strategies. The fuzzy control methodology introduced in this paper has been applied successfully in a case study of engine idle speed control for the Volkswagen Golf GTI