Regularized Optimization of Fuzzy Controllers

This paper is devoted to the mathematical analysis and the numerical solution of the problem of designing fuzzy controllers. We show that for a special class of controllers (so-called Sugeno controllers), the design problem is equivalent to a nonlinear least squares problem, which turns out to be ill-posed. Therefore we investigate the use of regularization methods in order to obtain stable approximations of the solution. We analyze a smoothing method, which is common in spline approximation, as well as Tikhonov regularization with respect to stability and convergence. In addition, we develop an iterative method for the regularized problems, which uses the special structure of the problem and test it in some typical numerical examples. We also compare the behavior of the iterations for the original and the regularized least squares problems. It turns out that the regularized problem is not only more robust but also favors solutions that are interpretable easily, which is an important criterion for fuzzy systems.