Estimation of the Accuracy of Approximation of Reachable Sets of a Linear System with Geometric Constraints for Various Types of Discretization
Goal: Solve the problem of constructing reachability sets of a linear continuous system with geometric control constraints. Methods: The discretization of the initial continuous system is considered as a solution method. Results: For piecewise constant, piecewise linear, and spline-like controls, eq...
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| Vydáno v: | Modelling and Data Analysis Ročník 15; číslo 2; s. 110 - 126 |
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| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
16.07.2025
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| ISSN: | 2219-3758, 2311-9454 |
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| Abstract | Goal: Solve the problem of constructing reachability sets of a linear continuous system with geometric control constraints. Methods: The discretization of the initial continuous system is considered as a solution method. Results: For piecewise constant, piecewise linear, and spline-like controls, equivalent discrete systems are obtained explicitly. A theorem on the rate of convergence in the Hausdorff metric of sets of reachability of auxiliary systems to the set of reachability of the initial system is proved, depending on the type of approximation. Numerical calculations have been performed. Conclusions: The obtained theoretical results can be used in solving analysis problems and for designing linear systems. Depending on the type of continuous control chosen, the corresponding discrete model may have greater accuracy in terms of reachability areas, which, however, will complicate its description.
Цель: Решить задачу построения множеств достижимости линейной непрерывной системы с геометрическими ограничениями на управление. Методы: В качестве метода решения рассматривается дискретизация исходной непрерывной системы. Результаты: Для кусочно-постоянных, кусочно-линейных и сплайн-подобных управлений эквивалентные дискретные системы получены явно. Доказана теорема о скорости сходимости в метрике Хаусдорфа множеств достижимости вспомогательных систем к множеству достижимости исходной системы в зависимости от типа аппроксимации. Проведены численные расчеты. Выводы: Полученные теоретические результаты могут быть использованы при решении задач анализа и для проектирования линейных систем. В зависимости от выбранного типа непрерывного управления, соответствующая дискретная модель может обладать большей точностью в смысле областей достижимости, что, однако, усложнит ее описание. |
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| AbstractList | Goal: Solve the problem of constructing reachability sets of a linear continuous system with geometric control constraints. Methods: The discretization of the initial continuous system is considered as a solution method. Results: For piecewise constant, piecewise linear, and spline-like controls, equivalent discrete systems are obtained explicitly. A theorem on the rate of convergence in the Hausdorff metric of sets of reachability of auxiliary systems to the set of reachability of the initial system is proved, depending on the type of approximation. Numerical calculations have been performed. Conclusions: The obtained theoretical results can be used in solving analysis problems and for designing linear systems. Depending on the type of continuous control chosen, the corresponding discrete model may have greater accuracy in terms of reachability areas, which, however, will complicate its description.
Цель: Решить задачу построения множеств достижимости линейной непрерывной системы с геометрическими ограничениями на управление. Методы: В качестве метода решения рассматривается дискретизация исходной непрерывной системы. Результаты: Для кусочно-постоянных, кусочно-линейных и сплайн-подобных управлений эквивалентные дискретные системы получены явно. Доказана теорема о скорости сходимости в метрике Хаусдорфа множеств достижимости вспомогательных систем к множеству достижимости исходной системы в зависимости от типа аппроксимации. Проведены численные расчеты. Выводы: Полученные теоретические результаты могут быть использованы при решении задач анализа и для проектирования линейных систем. В зависимости от выбранного типа непрерывного управления, соответствующая дискретная модель может обладать большей точностью в смысле областей достижимости, что, однако, усложнит ее описание. |
| Author | Vuksanovic, S.M. |
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| Cites_doi | 10.21538/0134-4889-2018-24-1-93-105 10.1007/BF01145876 10.21538/0134-4889-2021-27-3-141-151 10.21538/0134-4889-2021-27-2-162-168 10.1134/S0005117919030019 |
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