DOI: 10.17586/2226-1494-2020-20-3-364-370

Omorov R.O. 

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Omorov R.O. Robustness research of interval dynamic systems by algebraic method. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2020, vol. 20, no. 3, pp. 364–370 (in Russian). doi: 10.17586/2226-1494-2020-20-3-364-370

The paper considers robust stability study of continuous and discrete interval dynamic systems by algebraic method. The original robustness results obtained for continuous and discrete linear interval dynamic systems within the algebraic direction of robustness stability are presented. The author formulated and proved the basic theorem on the robustness of linear continuous dynamic system with interval elements of the right-hand part matrix, which is determined through the separate angular coefficients of characteristic polynomial of the system. The basic theorem is proved on the basis of a lemma on the separative coefficients of the characteristic polynomial obtained by optimization methods of nonlinear programming on multiple interval elements of the system matrix. Their possible values can be the upper or lower limits of the corresponding interval or zero. A clarification note to the basic theorem for continuous systems is formulated. The idea lies in the need for a complete set of four angular polynomials for the robustness stability of the system, excluding multiple cases of the characteristic polynomial, when the set of Kharitonov polynomials degenerates and will consist of the less required four different polynomials. The theorem is obtained on the necessary and sufficient conditions of robustness stability for the polyhedron of interval matrices. A discrete analogue of the Kharitonov theorem is obtained for discrete systems. The algorithm of robustness stability determination for discrete interval dynamic systems is presented. Comparative characteristics of the results obtained in the works of well-known authors having studied the algebraic trend of robust stability problem are considered. They show the distinctive feature of this method, which consists in consideration of interval matrices of general type. The validity of the method is tested on the known counterexamples to Bialas’s theorem, as well as the other researchers studying robustness problems of interval dynamic systems.

Keywords: interval dynamic system, robust stability, algebraic direction of robust stability, interval characteristic polynomial, Kharitonov’s angular polynomials, interval matrix, separate slopes, polyhedron of matrixes, discrete analog of Kharitonov’s theorems, intermittency point and interval, counterexamples to Bialas’s theorem

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