Transformation Of A Hierarchical Mamdani Fuzzy System To A Single Fuzzy System Representation

As of late, hierarchical Fuzzy Inference Systems (hFIS) have received considerable attention because of their ability to control complex decision-making procedures. Although the performance of hFISs appears to be extremely effective in complex scenarios, there is a question about the differences between the hierarchical and single FISs (sFIS) and the relationship between the two. In this work, a transformation is derived which converts a Mamdani type hFIS into its equivalent sFIS representation. This is accomplished by equating their output membership functions. The equivalency between the outputs of the two representations is proven using a Satisfiability Modulo Theories (SMT) solver approach. Furthermore, we provide a detailed development procedure and results are presented which demonstrate the ease of implementation of the transformation as well as a wide applicability to meaningful large-scale problems.