Approximation properties of ELM-fuzzy systems for smooth functions and their derivatives

In this paper, we utilize generalized Bernstein polynomials to construct fuzzy system. Different from traditional Bernstein polynomials, partition of interval on input variable can be chosen as non-equidistant division. We prove that generalized Bernstein fuzzy systems are universal approximators to a given continuous function and its high-order derivatives. Further, ELM method is used to tune the parameters of generalized Bernstein fuzzy system and Spline fuzzy system. It is proved that ELM-Spline fuzzy system can approximate a function and its derivative. Simulation examples show that the proposed ELM-Bernstein fuzzy system and ELM-spline fuzzy system can achieve high approximation capability for nonlinear model.