Function approximation with polynomial membership functions and alternating cluster estimation (Q1299616)

Nonlinear functions can be approximated by local linear models, for instance by a first-order Takagi-Sugeno model with membership function \(\mu_i(x)\) and linear right-hand side functions \(f_i(x)= m_i(x- x_i)+ y_i\) \((i= 1,\dots, c)\). Since piecewise linear membership functions lead to nondifferentiable input-output characteristics, the authors compute piecewise quadratic membership functions \(\mu_i(x)\) for a differentiable, minimal-order piecewise polynomial approximation. The parameters \(x_i\), \(y_i\) and \(m_i\) are obtained by fuzzy \(c\)-elliptotypes alternating optimization or alternating cluster estimation. In numerical tests the best approximation results are obtained with piecewise quadratic membership functions and alternating cluster estimation.