Conditions for Equivalence of Hierarchical Fuzzy Systems and Zero-Order TS Fuzzy Systems

For conventional fuzzy systems, Mamdani type or TS type, it is; well known that the number of fuzzy rules and adjustable parameters exponentially increases with the number of input variables. This problem is often referred as curse of dimensionality. Hierarchical fuzzy systems have been proposed to resolve this problem. At present, however, it is unclear whether or not hierarchical fuzzy systems can truly resolve this issue. In this paper, we investigate the zero-order TS fuzzy systems and their corresponding hierarchical fuzzy systems. We rigorously prove that to approximate any continuous functions with the same degree of accuracy, the number of parameters used by the hierarchical fuzzy systems are exactly the same as those employed by the TS systems. This result implies that the hierarchical fuzzy systems do not offer a real solution, at least for the zero-order TS fuzzy systems