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http://elar.urfu.ru/handle/10995/93134
Название: | The Limits of Applicability of the Linearization Method in Calculating Small–Time Reachable Sets |
Авторы: | Gusev, M. I. |
Дата публикации: | 2020 |
Издатель: | N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of Russian Academy of Sciences Ural Federal University named after the first President of Russia B.N. Yeltsin |
Библиографическое описание: | Gusev M. I. The Limits of Applicability of the Linearization Method in Calculating Small–Time Reachable Sets / M. I. Gusev. — DOI 10.15826/umj.2020.1.006. — Text : electronic // Ural Mathematical Journal. — 2020. — Volume 6. — № 1. — P. 71-83. |
Аннотация: | The reachable sets of nonlinear systems are usually quite complicated. They, as a rule, are non-convex and arranged to have rather complex behavior. In this paper, the asymptotic behavior of reachable sets of nonlinear control-affine systems on small time intervals is studied. We assume that the initial state of the system is fixed, and the control is bounded in the L2-norm. The subject of the study is the applicability of the linearization method for a sufficiently small length of the time interval. We provide sufficient conditions under which the reachable set of a nonlinear system is convex and asymptotically equal to the reachable set of a linearized system. The concept of asymptotic equality is defined in terms of the Banach-Mazur metric in the space of sets. The conditions depend on the behavior of the controllability Gramian of the linearized system – the smallest eigenvalue of the Gramian should not tend to zero too quickly when the length of the time interval tends to zero. The indicated asymptotic behavior occurs for a reasonably wide class of second-order nonlinear control systems but can be violated for systems of higher dimension. The results of numerical simulation illustrate the theoretical conclusions of the paper. |
Ключевые слова: | NONLINEAR CONTROL SYSTEMS SMALL-TIME REACHABLE SETS ASYMPTOTICS INTEGRAL CONSTRAINTS LINEARIZATION |
URI: | http://elar.urfu.ru/handle/10995/93134 |
Условия доступа: | Creative Commons Attribution License |
Текст лицензии: | https://creativecommons.org/licenses/by/4.0/ |
ISSN: | 2414-3952 |
DOI: | 10.15826/umj.2020.1.006 |
Сведения о поддержке: | This work is supported by the Research and Education Center of IMM UB of RAS in the framework of the Ural Mathematical Center (project “Set-Valued Dynamics in Control and Estimation Problems for Dynamical Systems with Uncertainty”). |
Источники: | Ural Mathematical Journal. 2020. Volume 6. № 1 |
Располагается в коллекциях: | Ural Mathematical Journal |
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umj_2020_6_1_71-83.pdf | 323,22 kB | Adobe PDF | Просмотреть/Открыть |
Лицензия на ресурс: Лицензия Creative Commons