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A new class of the almost orthogonal filters is described in this article. These filters are a generalization of the classical orthogonal filters commonly used in the circuit theory, control system theory, signal processing, and process identification. Almost orthogonal filters generate the series of almost orthogonal Legendre functions over the interval (0, 1). It is well known that all real systems suffer from some imperfections, so the models of these systems should reflect this fact. A new method for obtaining an imperfect system model is proposed. This method uses an almost orthogonal filter, which is based on almost orthogonal functions. Experiments with modular servo drive were performed to validate theoretical results and demonstrate that the method described in the article is suitable for modelling of imperfect systems. IntroductionThe history of orthogonal polynomials is very old [1]. Legendre polynomials and their orthogonal properties were established during the eighteenth century. The problems of solving ordinary differential equations and obtaining the expansion of arbitrary functions became popular among mathematicians in the following century and, as a result, Hermite polynomials were introduced. The theory of continued fractions gave rise to all orthogonal polynomials, and Laguerre polynomials were discovered. After these breakthroughs, the theory of orthogonal polynomials went on constantly developing. The last 20 years have seen a great deal of progress in the field of orthogonal systems (orthogonal algebraic and trigonometric polynomials [2,3], orthogonal rational functions [4][5][6], Müntz [7-9] and Malmquist orthogonal systems [10,11], etc.). Many papers have dealt with Chebyshev and Legendre orthogonal systems and their applications in electronics, circuit theory, signal processing, and control system theory [9,[11][12][13][14][15].One of the most important applications of the orthogonal functions is designing orthogonal filters [11][12][13][14][15][16][17]. These filters are useful for designing orthogonal signal generators, least square approximations, and practical realizations of the optimal and adaptive systems. However, since the components of those systems cannot be manufactured quite exactly, filters made by those components are not quite orthogonal, but rather almost orthogonal. The signals, obtained by these filters, are almost orthogonal, too. The measure of nearness between the obtained and the regular orthogonal signals depends on the exactness of the components manufactured. Such almost orthogonal filters are imperfect filters. Therefore,

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In this paper we define a new class of the almost orthogonal filters. These filters are a generalization of the classical orthogonal filters commonly used in the circuit theory, control system theory, signal processing and process identification. Almost orthogonal filters generate the series of almost orthogonal Legendre functions on the interval (0, ∞). They can be successfully used for the analysis, synthesis and designing of imperfect technical systems. They can also be used for approximation of arbitrary functions on (0, ∞). A new method for obtaining the models of imperfect systems is presented as well. Simulations were performed in order to validate the theoretical results and the comparison of new filters with classical filters is given.

Abstract-This paper presents a new method for modelling dynamic systems based on quasi-orthogonal functions. First, we defined a new class of Legendre type quasi-orthogonal functions that can be used for signal approximation as well as for modelling, analysis, synthesis, and simulation of dynamic systems, especially systems that suffer from some imperfections. In this paper, functions have been applied in modelling of cascade-connected dynamic system, typical for tire industry. Considered rubber cooling system is a represent of complex, nonlinear, and stochastic systems with imperfections. Developed quasi-orthogonal adjustable models can be used for modelling of arbitrary dynamic systems. Optimal model parameters in the sense of the mean-squared error were obtained using genetic algorithm. For experimental purposes, simplified cascade-connected system with four transporters has been practically realized. The experimental results proved the accuracy, simplicity, and quality of realized quasi-orthogonal model.