On choosing the fuzziness parameter for identifying TS models with multidimensional membership functions

Fuzzy clustering is a well-established method for identifying the structure/fuzzy partitioning of Takagi-Sugeno (TS) fuzzy models. The clustering algorithms require choosing the fuzziness parameter m. Prior work in the area of pattern recognition shows, that a suitable choice of m is application- dependent. Yet, the default of m=2 is commonly chosen. This paper examines the suitable choice of m for identifying TS models. The focus is on models that use the classifiers resulting from fuzzy clustering as multi-dimensional membership functions or their projection and approximation. At first, the differentiability and grouping properties of the fuzzy classifiers are analyzed to make a general recommendation of choosing m(1;3). Besides, the effect of the cluster number c on the classification fuzziness is examined. Finally, requirements that are specific to TS modeling are introduced, which narrow down the suitable range for m. Building on algorithm analysis and four case studies (function approximation, a vehicle engine and an axial compressor application for nonlinear regression), it is demonstrated that choosing m2(1;1.3) for local and m2(1;1.5) for global estimation will typically provide for good results.