Function approximation with polynomial membership functions and alternating cluster estimation

Abstract Nonlinear functions are often approximated using local linear models. Sets of local linear models can be represented as a first-order Takagi Sugeno (TS) system. triangular and trapezoidal left-hand side membership functions are not compatible with TS systems because they lead to non-differentiable input-output characteristics. We develop a method to determine parameters of piecewise quadratic membership functions to obtain characteristics which exactly match the rule centers and the corresponding slopes. The right-hand side parameters are obtained using (i) fuzzy c -elliptotypes alternating optimization and (ii) alternating cluster estimation. In our experiments the smoothest and most accurate approximations are obtained with piecewise quadratic membership functions and alternating cluster estimation.