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 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.

Orthogonal Hybrid-Fuzzy Controllers

“…Quasi-orthogonal functions and especially quasi-orthogonal polynomials as well as their numerous applications are discussed in many papers [11,12,[14][15][16]. It is important to notice that classical orthogonal filters and orthogonal signal generators have transfer functions with the order of numerator polynomial for one less then denominator.…”

“…On the other hand, many systems in practice are imperfect where we neglect some dynamics which can represents with constants ε and δ [9,12,17] in transfer function. This is a reason of formulation of theory about almost orthogonal and quasi-orthogonal functions and polynomials.…”

“…Complete mathematical background for designing generalized quasi-orthogonal filters based on polynomials given by (4) can be found in [12].…”

“…One of the most important applications of orthogonal polynomials is designing orthogonal filters. Today, these filters become an efficient tool for identification, modelling and control of dynamical systems [9][10][11][12][13].…”

“…The first few members of the first order (k=1) quasi-orthogonal polynomial over interval (0, 1) with weight function w(x)=1 sequence are [12]:…”