Optimization-based subdivision algorithm for reachable sets

Wolfgang Riedl; Robert Baier; Matthias Gerdts

doi:10.3934/jcd.2021005

Article Contents

2021, Volume 8, Issue 1: 99-130. Doi: 10.3934/jcd.2021005

This issue Previous Article A general framework for validated continuation of periodic orbits in systems of polynomial ODEs Next Article

Optimization-based subdivision algorithm for reachable sets

1.
Lehrstuhl für Ingenieurmathematik, Universität Bayreuth, 95440 Bayreuth, Germany
2.
Lehrstuhl für Angewandte Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany
3.
Institut für Angewandte Mathematik und Wissenschaftliches Rechnen, Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr, 85577 Neubiberg/München, Germany

^* Corresponding author.
^* Corresponding author.

Received: November 30, 2016

Revised: August 31, 2020

Published: January 2021

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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.

Keywords:

Mathematics Subject Classification: Primary: 93B03, 49M37; Secondary: 49M25, 49J53, 93C10.

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Figure 1. Reachable set of the bilinear problem for a full $ 33 \times 33 $ grid

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Figure 2. Reachable set of the bilinear problem, generated with (right) and without (left) the subdivision algorithm

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Figure 3. First step of the subdivision algorithm. Choose an initial bounding box (left), solve the optimization problems to approximate the map $ \mathcal{P} $ (middle) and select the boxes for the next step (right)

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Figure 4. Selection of boxes for the next subdivision steps

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Figure 5. Inner and outer approximation of the reachable set

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Figure 6. Demonstration of reusability of the results within the subdivision algorithm

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Figure 7. $ B = [-1,1] $ covered by boxes (left) or by balls (right)

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Figure 8. $ B = [-1,1]^2 $ covered by boxes (left) or by balls (right)

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Figure 9. Kenderov problem using different discretization methods for the ODE-constraints with 16 timesteps each: Explicit Euler (left), implicit Euler (middle), implicit trapezoidal rule (right)

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Figure 10. Kenderov problem using explicit Euler (left), implicit Euler (middle) and the implicit trapezoidal rule (right) with 261 timesteps each

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Figure 11. Illustration of the robot from Ex. 4.3

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Figure 12. Rough approximation of the reachable set of the robot problem

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Figure 13. Finer approximation of the reachable set of the robot problem

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Figure 14. Reachable set of the robot without the transformation in the objective function (left) and the transformed set using the same results and colorcode (right)

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Figure 15. Objective function of the non-transformed robot problem (left) and the transformed version (right)

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Figure 16. Approximated transformed reachable set of the robot with a less elaborate strategy for initial guesses

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Figure 17. Reachable set of the robot problem, generated with Ipopt (left) and WORHP (right)

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Figure 18. Robot model using a circular grid

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Figure 19. Graphical (left) and computational (right) zoom on the example of the Kenderov Problem

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Figure 20. Self-finding of the bounding box

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Figure 21. Piecewise construction of the whole reachable set, by choosing new bounding boxes at locations containing target points from previous runs or cut off looking edges

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Figure 22. Solution funnel of the bilinear problem for $ T \in [0, 1] $. The plot only shows the slices at $ t_i = \frac{1}{8} \cdot i,\; i\in \{0, 1, \dots, 8\} $ to make it easier to read (cpu-time 103 534s)

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Figure 23. Final box collections of the solution funnel of the bilinear problem for $ T \in [0, 1] $, rendered with POV-Ray

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Table 1. Comparison of the computational effort of the algorithm with (S) and without (N) using the subdivision algorithm for a few different grids

Number of optimization problems
Grid	N	S	S/N
3 × 3	9	9	1
5 × 5	25	18	0.72
9 × 9	81	41	0.506
17 × 17	289	92	0.318
33 × 33	1 089	231	0.212
65 × 65	4 225	635	0.150
129 × 129	16 641	1948	0.119
257 × 257	66 049	6822	0.103
513 × 513	263 169	25477	0.097
1025 × 1025	1 050 625	98040	0.093
CPU time
Grid	N	S	S/N
3 × 3	0.13s	0.13s	1
5 × 5	0.35s	0.25s	0.714
9 × 9	1.2s	0.57s	0.475
17 × 17	4.3s	1.24s	0.288
33 × 33	16.12s	3.12s	0.194
65 × 65	62.74s	8.55s	0.136
129 × 129	253.53s	26.29s	0.104
257 × 257	1128.09s	92.63s	0.082
513 × 513	8234.45s	355.29s	0.043
1025 × 1025	94221.4s	1741.58s	0.018

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Table 2. Computational times for generating parts of the reachable set with the shown boxes

Box	Times (with subdivision)	Times (without subdivision)
[-1.1, 0.4]×[-0.5, 0.5]	7.49 s	18.21 s
[ 0.4, 1.9]×[-0.5, 0.5]	7.00 s	17.58 s
[-1.1, 0.4]×[ 0.5, 1.5]	9.35 s	18.12 s
[ 0.4, 1.9]×[ 0.5, 1.5]	10.55 s	18.69 s

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