Tuning of Fuzzy Systems as an Ill-Posed Problem

This paper is concerned with the data-driven construction of fuzzy systems from the viewpoint of regularization and approximation theory, where we consider the important subclass of Sugeno controllers. Generally, we obtain a nonlinear constrained least squares approximation problem which is ill-posed. Therefore, nonlinear regularization theory has to be employed. We analyze a smoothing method, which is common in spline approximation, as well as Tikhonov regularization, along with rules how to choose the regularization parameters based on nonlinear regularization theory considering the error due to noisy data.For solving the regularized nonlinear least squares problem, we use a generalized Gauss-Newton like method. A typical numerical example shows that the regularized problem is not only more robust, but also favors solutions that are easily interpretable, which is an important quality criterion for fuzzy systems.