Two major approaches to neuro-fuzzy systems are distinguished in the paper. The previous one refers to fuzzy neural networks, which are neural networks with fuzzy signals, and/or fuzzy weights, as well as fuzzy transfer functions. The latter approach concerns neuro-fuzzy systems in the form of multilayer feed-forward networks, which differ from standard neural networks, because elements of particular layers conduct different operations than standard neurons. These structures are neural network representations of fuzzy systems and they are also called connectionist models of fuzzy systems, adaptive fuzzy systems, fuzzy inference neural networks, etc. Two different defuzzifiers, applied to fuzzy systems, are in focus of the paper. Center-of-sums method is an example of parametric defuzzification. Standard neural networks a defuzzifier presents nonparametric approach to defuzzification. For both cases learning algorithms of neuro-fuzzy systems are proposed. These algorithms take a form of recursions derived based on the momentum back-propagation method. Computer simulation demonstrates a comparison between performance of neuro-fuzzy systems with the parametric and nonparametric defuzzifier. Truck backer-upper control problem has been used to illustrate the systems performance. Conclusions concerning the simulation results are summarized. The paper pertains many references on neuro-fuzzy systems, especially selected publications of Czogala, whom it is dedicated.
A Learning Algorithm of Fuzzy Neural Networks with Fuzzy Weights