Gradient descent method for optimizing various fuzzy rule bases

The authors derive the gradient descent optimization equations for Takagi-Sugeno fuzzy rule bases with symmetric and asymmetric triangular membership functions, minimum and multiplication operators, and constant and affine output functions. A new type of affine output Takagi-Sugeno rules called centered Takagi-Sugeno rules is proposed. It makes it possible to avoid a class of local minima. The gradient descent method is systematically tested for the approximation of a one-input, one-output analytical function including a discontinuity and a high curvature point, and for the approximation of a two-input function