A system identifier generating the Fourier series for periodic dynamical systems

An algorithm for computing the Fourier expansion for an arbitrary periodic dynamical system is described. One period data from the system output are sampled and are used to compute the cubic spline interpolation function for the system output. A weighted sum of the corresponding terms of the Fourier series of each cubic B-spline is computed with the interpolation coefficients as weights. The resulting Fourier series turns out to be a very accurate approximation of the Fourier expansion for the periodic dynamical system. An analysis of the error estimation for the computed Fourier coefficients is presented and two examples of simulation results are included to verify that the algorithm works properly. A fuzzy system is designed to represent the spline interpolation function using the cubic B-splines as input fuzzy sets and the Fourier coefficients as the support boundaries of the output fuzzy sets as weights. Thus, our algorithm can be viewed as a fuzzy identifier which does not require any training.