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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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