In this article, we present a new method for the synthesis of almost and quasiorthogonal polynomials of arbitrary order. Filters designed on the bases of these functions are generators of generalised quasi-orthogonal signals for which we derived and presented necessary mathematical background. Based on theoretical results, we designed and practically implemented generalised first-order (k = 1) quasi-orthogonal filter and proved its quasi-orthogonality via performed experiments. Designed filters can be applied in many scientific areas. In this article, generated functions were successfully implemented in Nonlinear Auto Regressive eXogenous (NARX) neural network as activation functions. One practical application of the designed orthogonal neural network is demonstrated through the example of control of the complex technical non-linear system -laboratory magnetic levitation system. Obtained results were compared with neural networks with standard activation functions and orthogonal functions of trigonometric shape. The proposed network demonstrated superiority over existing solutions in the sense of system performances.

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This paper presents a new method for designing adaptive neuro-fuzzy inference systems (ANFIS). Improvements are made by introducing specially developed orthogonal functions into the very structure of ANFIS, specifically, into the layer that imitates Sugeno stile defuzzification. These functions are specially tailored for analysis and synthesis of dynamic systems and they also contain an adaptive measure of the variability of the systems operating in a real environment, which can be implemented inside the ANFIS as hormonal effect.

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In this paper, we present analysis of different train ing types for nonlinear autoregressive neural network, used for simulat ion of magnetic levitation system. First, the model of this h ighly nonlinear system is described and after that the Nonlinear Auto Regressive eXogenous (NARX) of neural network model is given. Also, numerical optimization techniques for improved network training are described. It is verified that NARX neural network can be successfully used to simulate real magnetic levitation system if suitable train ing procedure is chosen, and the best two training types, obtained from experimental results, are described in details.