Modelling of dynamical systems based on almost orthogonal polynomials

Abstract

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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“…Moreover, we have proved that they are very useful tool for modelling [12], control of dynamical systems [10], and approximation of signals [11]. Also, these filters can be used for sensitivity analysis of continuous systems [21].…”

“…In these papers, almost orthogonal filters contain a measure of imperfection labelled as ε (constant close to zero), which describes cumulative impacts of all imperfect elements [12]. These coefficients can be determined using several experiments.…”

“…These coefficients can be determined using several experiments. In [11] we considered new filters, which represent the further improvement of filters designed in [12], in the sense of simplicity, higher accuracy, lesser approximation time, and even a possibility to approximate signals generated by systems with built-in imperfections. These filters can be used for generating of sequences of functions, approximations, modelling, control and analysis of imperfect systems.…”

“…Parameters c i are being adjusted until the minimization of the function (17) is achieved, ie, as long as necessary to obtain the best model of unknown system in the sense of mean square error. The complete process of modelling is given in [11,12].…”

“…In [10,11] we presented a new method for designing orthogonal rational functions using specific transformations in complex domain and derived a necessary mathematical background. We also presented a new method for obtaining models of continuous dynamical systems based on these orthogonal functions [12]. Practical realization of derived orthogonal filters is very simple due to factorized form of orthogonal functions and they are very fast, robust and precise.…”