A Novel Neuro-fuzzy Learning Algorithm for First-Order Takagi–Sugeno Fuzzy Model: Caputo Fractional-Order Gradient Descent Method

As an essential tool for processing fuzzy or chaotic information, the main feature of the first-order Takagi–Sugeno (T–S) neuro-fuzzy model is utilizing a set of IF-THEN fuzzy rules to represent non-linear systems, showcasing commendable non-linear approximation ability and significant interpretability. However, the coexistence of linear rules and the affiliation function of fuzzy sets makes the integer-order gradient descent method (IOGDM), commonly used in training the first-order T–S neuro-fuzzy model, fail to accurately capture the intricate relationships among weights, resulting in the error function struggling to converge rapidly to low values. To enhance the convergence speed and training accuracy of the first-order T–S neuro-fuzzy model during the training process, a fractional-order gradient descent method (FOGDM) is proposed to update the fuzzy rule parameters and neural network weights of the model in this paper. By subdividing the gradient into fractional orders, FOGDM exhibits heightened flexibility in gradient adjustments, thus better capturing the complex non-linear relationships among parameters during the optimization process. The weak and strong convergence of the proposed approach is meticulously demonstrated in this paper, ensuring that the weight of error functions converges to a constant value and that the gradient of the error functions tends toward zero, respectively. Simulation results analysis indicates that, compared to IOGDM, FOGDM exhibits faster convergence speed and more significant generalization capabilities.