Approximation properties and continuous differentiability of a class of SISO fuzzy systems

The goal of the present paper is to investigate approximation and smoothness properties of Larsen type single input single output (SISO) fuzzy systems, that is, fuzzy logic systems using the maximum as aggregation for the individual rule outputs, product (Goguen) t-norm as the conjunctive operator and center of gravity defuzzification. We prove that the function providing the output of the above considered Larsen type fuzzy system is capable of approximating any continuous function. Also, it is continuously differentiable under very relaxed conditions as e.g. continuous differentiability of the antecendents except at their core, and continuous differentiability of the consequences of fuzzy rules except at their core and the enpoints of their support. We show practical examples regarding the approximation and smoothness of the Larsen type operators, showing also by an example that the conditions on the antecedent part cannot be in general weakened further without loosing continuous differentiability. The present paper provides a theoretical background for claims in literature stating that the output of a fuzzy control system of this type is smooth.

I. Introduction

The study of approximation capability of fuzzy systems started from B. Kosko's paper [7]. Most of the papers available in literature use the Takagi-Sugeno approach [13], [11], [2] or they are using sum as the aggregation method for the fuzzy rules [9], [14], [6]. In this situation smoothness properties have been investigated in e.g., [3]. Also, min→ aggregation was considered [4], [5], however, smoothness properties were not investigated for this case.