Approximation properties of higher order Takagi-Sugeno fuzzy systems

We study higher order Takagi-Sugeno fuzzy systems from the point of view of their approximation capability. Higher order Takagi-Sugeno fuzzy systems have the output of each individual rule defined as polynomials of a certain degree. In the present paper the individual rule outputs are Taylor polynomials. Under very relaxed conditions both on the target function and the fuzzy sets in the antecedent part, we obtain an approximation theorem and an error estimate in terms of the modulus of continuity, higher order derivatives of the target function, and properties of the antecedents in the fuzzy rule base considered. This theorem is then used to prove a result on the approximation by higher order Takagi-Sugeno fuzzy systems with Gaussian antecedents. An example is constructed in order to exemplify the approach of the present paper.