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Optimization-based subdivision algorithm for reachable sets

DOI zum Zitieren der Version auf EPub Bayreuth: https://doi.org/10.15495/EPub_UBT_00005054
URN to cite this document: urn:nbn:de:bvb:703-epub-5054-7

Title data

Riedl, Wolfgang ; Baier, Robert ; Gerdts, Matthias:
Optimization-based subdivision algorithm for reachable sets.
Mathematisches Institut, Universität Bayreuth, Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr in Neubiberg/München
Bayreuth , 2020 . - 38 S.

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Project information

Project title:
Project's official title
Project's id
European Union's Seventh Framework Programme
338508
Project financing: 7. Forschungsrahmenprogramm für Forschung, technologische Entwicklung und Demonstration der Europäischen Union

Abstract

Reachable sets for nonlinear control systems can be computed via the use of solvers for optimal control problems. The paper presents a new improved variant which applies adaptive concepts similar to the framework of known subdivision techniques by Dellnitz/Hohmann. Using set properties of the nearest point projection, the convergence and rigorousness of the algorithm can be proved without the assumption of diffeomorphism on a nonlinear mapping. The adaptive method is demonstrated by two nonlinear academic examples and for a more complex robot model with box constraints for four states, two controls and five boundary conditions. In these examples adaptive and non-adaptive techniques as well as various discretization methods and optimization solvers are compared. The method also offers interesting features, like zooming into details of the reachable set, self-determination of the needed bounding box, easy parallelization and the use of different grid geometries. With the calculation of a 3d funnel in one of the examples, it is shown that the algorithm can also be used to approximate higher dimensional reachable sets and the resulting box collection may serve as a starting point for more sophisticated visualizations or algorithms.

Further data

Item Type: Preprint, postprint
Additional notes (visible to public): erschienen in:
In: Journal of Computational Dynamics. (20 Oktober 2020) . - S. 1-32.
DOI: https://doi.org/10.3934/jcd.2021005

Contents:
1. Introduction and preliminaries
2. Grid construction via subdivision
3. Implementation
4. Numerical examples
5. Advantages of the algorithm
5.1 Transformed grids
5.2 Zooming
5.3 Determination of a bounding box
5.4 Parallelization
5.5 Solution funnel in 3d
6. Conclusions
Keywords: reachable sets; subdivision; optimal control; direct discretization; nonlinear
systems; nonlinear optimization
Subject classification: Mathematics Subject Classification Code: 93B03 49M37 (49M25 49J53 93C10)
DDC Subjects: 500 Science > 510 Mathematics
Institutions of the University: Faculties
Faculties > Faculty of Mathematics, Physics und Computer Science
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics)
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) > Chair Mathematics V (Applied Mathematics) - Univ.-Prof. Dr. Lars Grüne
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Scientific Computing
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Scientific Computing > Chair Scientific Computing - Univ.-Prof. Dr. Mario Bebendorf
Profile Fields
Profile Fields > Advanced Fields
Profile Fields > Advanced Fields > Nonlinear Dynamics
Language: English
Originates at UBT: Yes
URN: urn:nbn:de:bvb:703-epub-5054-7
Date Deposited: 15 Sep 2020 06:31
Last Modified: 30 Apr 2021 10:46
URI: https://epub.uni-bayreuth.de/id/eprint/5054

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