Асимптотичний аналіз нестаціонарних систем автоматичного ке-рування
Journal Title: Математичне моделювання - Year 2018, Vol 1, Issue 2
Abstract
ASYMPTOTIC ANALYSIS OF NON-STATIONARY OF AUTOMATIC CONTROL SYSTEMS Rashevs’kyi M.O. Abstract Models of non-stationary automatic control systems are differential equations with variable coefficients. Such equations do not integrate in quadratures in the general case. Asymptotic methods are methods of approximate integration of differential equations with variable coefficients. In the article the non-stationary automatic control system with slowly variable parameters is considered. To study this system it is necessary to construct an asymptotic representation of its solution. In the theory of asymptotic integration exist a problem to construction of the asymptotic solution of a system in the presence of a turning point. Special methods have been developed to construct a solution to such systems. The most common methods is the method of reference equations, the method of the Maslov’s canonical operator, the multiphase Kucherenko method , the method of W. Wasow. The purpose of the article is to construct an asymptotic solution of a linear system of differential equations with available a turning point. In this paper we consider a system with an almost diagonal matrix and a turning point. Methods for the integration of almost diagonal systems and for systems with Jordan structure of the matrix are significantly different. For solving a system, the method of W. Wasow is used. An asymptotic representation of the solution of the system is constructed and an error estimate is given. An approximate solution is used to find a matrix of impulse transitive functions; for this matrix an asymptotic image is written. The case of the presence of a Jordan cell in the main matrix of the system is also researched. For n = 2, the asymptotic solution is constructed without using the multi-scale method. For construction, the S.A. Lomov's regularization method and the method of successive approximations are used. Further research may be aimed at finding a unified approach to solving such problems and to ascertain the physical meaning of the turning point in specific systems of automatic control. References [1] Abgaryan K.A. Matrichniye i asimptoticheskiye metody v teorii lineinyh system [Matrix and asymptotic methods in the theory of linear systems]. Moscow, 1973, 432 p. [2] Leifura V.N. “On One Problem of Automatic Control with Turning Points.” Proc. of the Second International Conference “Symmetry in Nonlinear Mathematical Physics”, Kyiv, 1997, vol. 2, p. 488–491. [3] Lomov S.A. Vvedenie v obshchuyu teoriyu singulyarnyh vozmushchenij [Introduction to the general theory of singular perturbations]. Moscow, 1981, 400 p. [4] Rashevs’kyi M.O. Asymptotychni rozvyazky linijnyh dyferentsial’nyh rivnyan’ drugogo poryadku z vidhylennyam argumentu [Asymptotic solutions of the second order linear differential equations with delay]. Available at: http://enpuir.npu.edu.ua/bitstream/123456789/13935/1/rashevskyi179-187.pdf (accessed 2012) [5] Samoilenko A.M., Shkil’ M.I., Yakovets V.P. Linijni Systemy dyferentsial’nuh rivnyan’ z vyrodzhennyamy [Linear systems of differential equations with degenerations]. Kyiv, 2004, 294 p. [6] Shkil’ M.I., Voronoj A.N., Leifura V.N. Asimptoticheskiye metody v differentsyal’nyh I integro-differentsyal’nyh uravneniyah [Asymptotic methods in differential and integro-differential equations]. Kyiv, 1985, 248 p. [7] Wasow W. “On a Turning Point Problems for Systems with Almost Diagonal Coefficient Matrix.” Funkc. Ekv., 1966, vol. 8, no 3, p. 143–171.
Authors and Affiliations
М. О. Рашевський
Keywords
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- EP ID EP444590
- DOI 10.31319/2519-8106.2(39)2018.154210
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