Application of neural networks with orthogonal activation functions in control of dynamical systems

Abstract

In this article, we present a new method for the synthesis of almost and quasi-orthogonal 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.

Keywords:

• orthogonal polynomials
• orthogonal filters
• Legendre-type filter
• neural network
• magnetic levitation system

Additional information

Funding

This article was realised as a part of the projects [III 43007], [III 44006], and [TR 35005] financed by the Ministry of Education and Science of the Republic of Serbia for the period 2011–2015.