10. Sequential Adaptive Fuzzy Inference System for Function Approximation Problems

In the classic approaches to design a fuzzy inference system, the fuzzy rules are determined by a domain expert a priori and then they are maintained unchanged during the learning. These fixed fuzzy rules may not be appropriate in real-time applications where the environment or model often meets unpredicted disturbances or damages. Hence, poor performance may be observed. In comparison to the conventional methods, fuzzy inference systems based on neural networks, called fuzzy-neural systems, have begun to exhibit great potential for adapting to the changes by utilizing the learning ability and adaptive capability of neural networks. Thus, a fuzzy inference system can be built using the standard structure of neural networks. Nevertheless, the determination of the number of fuzzy rules and the adjustment of the parameters in the if-then fuzzy rules are still open issues. A sequential adaptive fuzzy inference system (SAFIS) is developed to determine the number of fuzzy rules during learning and modify the parameters in fuzzy rules simultaneously. SAFIS uses the concept of influence of a fuzzy rule for adding and removing rules during learning. The influence of a fuzzy rule is defined as its contribution to the system output in a statistical sense when the input data is uniformly distributed. When there is no addition of fuzzy rules, only the parameters of the “closest” (in a Euclidean sense) rule are updated using an extended Kalman filter (EKF) scheme. The performance of SAFIS is evaluated based on some function approximation problems, via, nonlinear system identification problems and a chaotic time-series prediction problem. Results indicate that SAFIS produces similar or better accuracies with lesser number of rules compared to other algorithms.