MSC: 11C08, 52B11, 93D05
DOI: 10.21538/0134-4889-2021-27-3-246-255
Full text
In this paper we investigate the Schur stability region of the $n$th order polynomials in the coefficient space. Parametric description of the boundary set is obtained. We show that all the boundary can be obtained as a multilinear image of three $(n-1)$-dimensional boxes. For even and odd $n$ these boundary boxes are different. Analogous properties for the classical multilinear reflection map are unknown. It is shown that for $n \geq 4$, both two parts of the boundary which are pieces of the corresponding hyperplanes are nonconvex. Polytopes in the nonconvex stability region are constructed. A number of examples are provided.
Keywords: Schur stability, stability region, polytope, boundary set
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Received March 25, 2021
Revised June 1, 2021
Accepted June 15, 2021
Vakif Dzhafarov, Prof., Department of Mathematics, Faculty of Science, Eskisehir Technical University, 26470 Eskisehir, Turkey, e-mail: vcaferov@eskisehir.edu.tr
Taner Büyükköroğlu, Department of Mathematics, Faculty of Science, Eskisehir Technical University, 26470 Eskisehir, Turkey, e-mail: tbuyukkoroglu@eskisehir.edu.tr
Handan Akyar, Department of Mathematics, Faculty of Science, Eskisehir Technical University, 26470 Eskisehir, Turkey, e-mail: hakyar@eskisehir.edu.tr
Cite this article as: V. Dzhafarov, T. Büyükköroğlu, H. Akyar. Stability Region for Discrete Time Systems and Its Boundary, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 246–255.
Русский
В. Джафаров, T. Бююккёроглу, Х. Акьяр. Область устойчивости и ее граница для многошаговых систем
Исследуется область устойчивости по Шуру многочленов порядка $n$ в пространстве коэффициентов. Получено параметрическое описание граничного множества. Показано, что вся граница может быть получена как мультилинейный образ трех $(n-1)$-мерных параллелепипедов, которые различны для четных и нечетных $n$. Аналогичные свойства для классического отображения отражения неизвестны. При $n \geq 4$ показана невыпуклость обеих частей границы, которые являются кусками соответствующих гиперплоскостей. Построены многогранники в невыпуклой области устойчивости. Приведено несколько примеров.
Ключевые слова: устойчивость по Шуру, область устойчивости, многогранник, граничное множество
