IT2ANFIS with Dual Uncertainty in Membership Function: A Gradient-Based Learning Approach

Abstract

Modeling under uncertainty remains a fundamental challenge in engineering and nonlinear system identification, particularly when both interpretability and computational efficiency are required. Interval type-2 adaptive neuro-fuzzy inference systems (IT2ANFIS) have demonstrated strong capabilities in handling uncertainty; however, the considered approaches typically introduce uncertainty either in the center or in the width of membership functions, limiting their representational flexibility. To address this problem, this paper proposes a novel IT2ANFIS model that simultaneously incorporates uncertainty in both the centers and widths of Gaussian membership functions, enabling a more expressive yet compact representation of uncertainty. First, a formulation of the proposed membership function is developed. This allows the application of gradient-based learning with explicitly derived update rules for both premise and consequent parameters. Second, the proposed approach avoids explicit type-reduction procedures by directly aggregating lower and upper firing strengths. This eliminates the need for computationally intensive iterative algorithms and improves inference efficiency. Third, a local first-order gradient analysis is used to define a practical reference for learning-rate scaling. In addition, multiple learning-rate strategies, including fixed, switching, and moment-based strategies, are investigated to evaluate their influence on convergence and model performance. Furthermore, the proposed approach is validated through benchmark systems and experimental evaluation on a DC motor system. The results demonstrate improved modeling accuracy and robustness compared to conventional ANFIS-based approaches. The proposed framework provides a practical and computationally efficient approach to uncertainty-aware nonlinear system modeling and related engineering applications.