Training Error Approximation Through the State-Space Representation of the Fuzzy Model

Various algorithms for fuzzy model identification have been introduced in the literature, and the focus is mostly on proper rule generation and parameter updating. However, irrespective of the optimization algorithm in the design step, the structural properties of fuzzy models, e.g., the number of fuzzy rules, highly affects the quality of the nonlinear function approximation to reach the balance between the accuracy in model training and the complexity of the model for real application. In this paper, we focus on the issue of error approximation in smooth fuzzy systems, upon which the required number of fuzzy rules can be determined. In the proposed approach, the designer will not need to assign the optimum parameters of the fuzzy model to estimate the error in the approximation of the nonlinear systems with the smooth fuzzy model. Therefore, the designer will skip the iteration of optimal parameter seeking and fuzzy model training before finding the fuzzy model's approximation error. Instead, through the state-space representation of the fuzzy model, one would estimate the error in the fuzzy approximation of the nonlinear system using the extended Kalman filter first. In the next step and after estimating the error in the approximation of the fuzzy model, the designer will endeavor to find the fuzzy model's optimum parameters straightforward. The proposed method has been illustrated in an example.