The paper presents the operation of two neuro-fuzzy systems of an adaptive type, intended for solving problems of the approximation of multi-variable functions in the domain of real numbers. Neuro-fuzzy systems being a combination of the methodology of artiﬁcial neural networks and fuzzy sets operate on the basis of a set of fuzzy rules “if-then”, generated by means of the self-organization of data grouping and the estimation of relations between fuzzy experiment results. The article includes a description of neuro-fuzzy systems by Takaga-Sugeno-Kang (TSK) and Wang-Mendel (WM), and in order to complement the problem in question, a hierarchical structural self-organizing method of teaching a fuzzy network. A multi-layer structure of the systems is a structure analogous to the structure of “classic” neural networks. In its ﬁnal part the article presents selected areas of application of neuro-fuzzy systems in the ﬁeld of geodesy and surveying engineering. Numerical examples showing how the systems work concerned: the approximation of functions of several variables to be used as algorithms in the Geographic Information Systems (the approximation of a terrain model), the transformation of coordinates, and the prediction of a time series. The accuracy characteristics of the results obtained have been taken into consideration.
The development of digital signal processors and the increase in their computing capabilities bring opportunities to employ algorithms with multiple variable parameters in active noise control systems. Of particular interest are the algorithms based on artificial neural networks. This paper presents an active noise control algorithm based on a neural network and a nonlinear input-output system identification model. The purpose of the algorithm is an active noise control system with a nonlinear primary path. The algorithm uses the NARMAX system identification model. The neural network employed in the proposed algorithm is a multilayer perceptron. The error backpropagation rule with adaptive learning rate is employed to update the weight of the neural network. The performance of the proposed algorithm has been tested by numerical simulations. Results for narrow-band input signals and nonlinear primary path are presented below.