Approximation capability analysis of hierarchical Takagi-Sugeno fuzzy systems

Based on the approach of hierarchical structure analysis of continuous functions, this paper discusses the approximation capacities of hierarchical Takagi-Sugeno fuzzy systems. By first introducing the concept of the natural hierarchical structure, it is proved that continuous functions with the natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and the number of parameters. Then, based on the Kolmogorov's theorem, it is shown that any continuous function can be represented as a superposition of functions with natural hierarchical structure and can then be approximated by hierarchical Takagi-Sugeno fuzzy systems to achieve the universal approximation property.