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Optimization-based subdivision algorithm for reachable sets

URN to cite this document: urn:nbn:de:bvb:703-epub-3157-8

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 , 2016 . - 33 S.

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Project information

Project title:
Project's official titleProject's id
European Union's Seventh Framework Programme338508

Project financing: Andere


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): 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 > Lehrstuhl Wissenschaftliches Rechnen
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-3157-8
Date Deposited: 30 Jan 2017 07:33
Last Modified: 20 Mar 2019 10:56
URI: https://epub.uni-bayreuth.de/id/eprint/3157


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