Document Open Access Logo

An Interval Constraint Programming Approach for Quasi Capture Tube Validation

Authors Abderahmane Bedouhene, Bertrand Neveu, Gilles Trombettoni, Luc Jaulin, Stéphane Le Menec

PDF
Thumbnail PDF

File

LIPIcs.CP.2021.18.pdf
  • Filesize: 1.97 MB
  • 17 pages

Document Identifiers

Author Details

Abderahmane Bedouhene
  • LIGM, Ecole des Ponts ParisTech, Université Gustave Eiffel, CNRS, Marne-la-Vallée, France
Bertrand Neveu
  • LIGM, Ecole des Ponts ParisTech, Université Gustave Eiffel, CNRS, Marne-la-Vallée, France
Gilles Trombettoni
  • LIRMM, Université de Montpellier, CNRS, France
Luc Jaulin
  • Lab-STICC, ENSTA-Bretagne, Brest, France
Stéphane Le Menec
  • MBDA, Le Plessis Robinson, France

Funding

This work was supported by the French Agence Nationale de la Recherche (ANR) [grant number ANR-16-CE33-0024].

Acknowledgements

We also thank our colleagues, Alexandre Goldsztejn and Alessandro Colotti, for the exchange of ideas and their kind help on the experiments.

Cite As Get BibTex

Abderahmane Bedouhene, Bertrand Neveu, Gilles Trombettoni, Luc Jaulin, and Stéphane Le Menec. An Interval Constraint Programming Approach for Quasi Capture Tube Validation. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CP.2021.18

Abstract

Proving that the state of a controlled nonlinear system always stays inside a time moving bubble (or capture tube) amounts to proving the inconsistency of a set of nonlinear inequalities in the time-state space. In practice however, even with a good intuition, it is difficult for a human to find such a capture tube except for simple examples. In 2014, Jaulin et al. established properties that support a new interval approach for validating a quasi capture tube, i.e. a candidate tube (with a simple form) from which the mobile system can escape, but into which it enters again before a given time. A quasi capture tube is easy to find in practice for a controlled system. Merging the trajectories originated from the candidate tube yields the smallest capture tube enclosing it. 
This paper proposes an interval constraint programming solver dedicated to the quasi capture tube validation. The problem is viewed as a differential CSP where the functional variables correspond to the state variables of the system and the constraints define system trajectories that escape from the candidate tube "for ever". The solver performs a branch and contract procedure for computing the trajectories that escape from the candidate tube. If no solution is found, the quasi capture tube is validated and, as a side effect, a corrected smallest capture tube enclosing the quasi one is computed. The approach is experimentally validated on several examples having 2 to 5 degrees of freedom.

Subject Classification

ACM Subject Classification
  • Applied computing → Operations research
  • Mathematics of computing → Ordinary differential equations
  • Mathematics of computing → Differential algebraic equations
  • Mathematics of computing → Interval arithmetic
  • Theory of computation → Constraint and logic programming
Keywords
  • Constraint satisfaction problem
  • Interval analysis
  • Dynamical systems
  • Contractor

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. A. Akkouche, J.-B. Bénéfice, Q. Bréfort, B. Desrochers F. Carbonera, T. Issautier, M. Laranjeira-Moreira, V. Le Doze, D. Monnet, and A. Oubelhaj. BUBBIBEX with IBEX. Engineering internship report, ENSTA Bretagne, 2014. URL: https://www.ensta-bretagne.fr/jaulin/archirob.html#bm_2013_14.
  2. J.-P. Aubin. Viability Kernels and Capture Basins of Sets Under Differential Inclusions. SIAM Journal on Control and Optimization, 40(3):853-881, 2001. URL: https://doi.org/10.1137/S036301290036968X.
  3. F. Benhamou, F. Goualard, L. Granvilliers, and J.F. Puget. Revising Hull and Box Consistency. In D. De Schreye, editor, Logic Programming: The 1999 International Conference, Las Cruces, New Mexico, USA, November 29 - December 4, 1999, pages 230-244. MIT Press, 1999. Google Scholar
  4. F. Blanchini and S. Miani. Set Theoretic Methods in Control. Birkhauser, 2008. Google Scholar
  5. J.C. Butcher. Numerical Methods for Ordinary Differential Equations. Wiley, 2004. URL: https://books.google.fr/books?id=okzpIwEX8aEC.
  6. G. Chabert. IBEX - an Interval-Based EXplorer, 2020. URL: http://www.ibex-lib.org/.
  7. H. Collavizza, F. Delobel, and M. Rueher. Comparing Partial Consistencies. Reliable Computing, 5(3):213-228, 1999. Google Scholar
  8. Julien Alexandre dit Sandretto and Alexandre Chapoutot. Validated Explicit and Implicit Runge-Kutta Methods. Reliable Computing, 22(1):79-103, July 2016. Google Scholar
  9. A. Djaballah, A. Chapoutot, M. Kieffer, and O. Bouissou. Construction of Parametric Barrier Functions for Dynamical Systems using Interval Analysis. Automatica, 78:287-296, 2017. URL: https://doi.org/10.1016/j.automatica.2016.12.013.
  10. F. Domes. GLOPTLAB: a Configurable Framework for the Rigorous Global Solution of Quadratic Constraint Satisfaction Problems. Optimization Methods & Software, 24:727-747, October 2009. URL: https://doi.org/10.1080/10556780902917701.
  11. A. Girard, C. Le Guernic, and O. Maler. Efficient Computation of Reachable Sets of Linear Time-invariant Systems with Inputs. Hybrid Systems: Computation and Control, 3927:257-271, 2006. Google Scholar
  12. E. R. Hansen. Global Optimization using Interval Analysis. Marcel Dekker, New York, NY, 1992. Google Scholar
  13. L. Jaulin, D. Lopez, V. Le Doze, S. Le Menec, J. Ninin, G. Chabert, M. S. Ibnseddik, and A. Stancu. Computing Capture Tubes. In Marco Nehmeier, Jürgen Wolff von Gudenberg, and Warwick Tucker, editors, Scientific Computing, Computer Arithmetic, and Validated Numerics, pages 209-224, Cham, 2016. Springer International Publishing. Google Scholar
  14. T. Kapela, M. Mrozek, D. Wilczak, and P. Zgliczynski. CAPD: : Dynsys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems. CoRR, abs/2010.07097, 2020. URL: http://arxiv.org/abs/2010.07097.
  15. F. Le Bars, J. Sliwka, L. Jaulin, and O. Reynet. Set-membership State Estimation with Fleeting Data. Automatica, 48(2):381-387, 2012. URL: https://doi.org/10.1016/j.automatica.2011.11.004.
  16. S. Le Menec. Linear Differential Game with Two Pursuers and One Evader. Advances in Dynamic Games, 11:209—226, 2011. Google Scholar
  17. Y. I. Lee and B. Kouvaritakis. Constrained Robust Model Predictive Control Based on Periodic Invariance. Automatica, 42:2175–2181, 2006. Google Scholar
  18. O. Lhomme. Consistency Techniques for Numeric CSPs. In R. Bajcsy, editor, Proceedings of the 13th International Joint Conference on Artificial Intelligence. Chambéry, France, August 28 - September 3, 1993, pages 232-238. Morgan Kaufmann, 1993. URL: http://ijcai.org/Proceedings/93-1/Papers/033.pdf.
  19. M. Lhommeau, L Jaulin, and L. Hardouin. Inner and Outer Approximation of Capture Basins using Interval Analysis. ICINCO 2007, 2007. Google Scholar
  20. R. Lohner. Enclosing the Solutions of Ordinary Initial and Boundary Value Problems. In E. Kaucher, U. Kulisch, and Ch. Ullrich, editors, Computer Arithmetic: Scientific Computation and Programming Languages, pages 255-286. BG Teubner, Stuttgart, Germany, 1987. Google Scholar
  21. F. Messine. Méthodes d'Optimisation Globale basées sur l'Analyse d'Intervalle pour la Résolution des Problèmes avec Contraintes. PhD thesis, LIMA-IRIT-ENSEEIHT-INPT, Toulouse, 1997. Google Scholar
  22. R. E. Moore. Interval Analysis, volume 4. Prentice-Hall Englewood Cliffs, 1966. Google Scholar
  23. N. S. Nedialkov, K. R. Jackson, and J. D. Pryce. An Effective High-Order Interval Method for Validating Existence and Uniqueness of the Solution of an IVP for an ODE. Reliable Computing, 7(6):449-465, 2001. URL: https://doi.org/10.1023/A:1014798618404.
  24. S. Olaru, J.A. De Dona, M.M. Seron, and F. Stoican. Positive Invariant Sets for Fault Tolerant Multisensor Control Schemes. International Journal of Control, 83(12):2622-2640, 2010. Google Scholar
  25. S. V. Rakovic, E. C. Kerrigan, K. I. Kouramas, and D. Q. Mayne. Invariant approximations of the minimal robust positively invariant set. IEEE Trans. Autom. Control, 50(3):406-410, 2005. Google Scholar
  26. S. Ratschan and Z. She. Providing a Basin of Attraction to a Target Region of Polynomial Systems by Computation of Lyapunov-like Functions. SIAM J. Control and Optimization, 48(7):4377—4394, 2010. Google Scholar
  27. N. Revol, K. Makino, and M. Berz. Taylor Models and Floating-point Arithmetic: proof that arithmetic operations are validated in COSY. Journal of Logic and Algebraic Programming, 64:135-154, 2005. Google Scholar
  28. S. Rohou, A. Bedouhene, G. Chabert, A. Goldsztejn, L. Jaulin, B. Neveu, V. Reyes, and G. Trombettoni. Towards a Generic Interval Solver for Differential-Algebraic CSP. In Proc. CP, Constraint Programming, Springer, LNCS 12333, pages 864-879. Springer, 2020. Google Scholar
  29. S. Rohou et al. The Tubex Library - Constraint-programming for robotics, 2021. URL: http://simon-rohou.fr/research/tubex-lib/.
  30. Simon Rohou, Luc Jaulin, Lyudmila Mihaylova, Fabrice Le Bars, and Sandor M. Veres. Guaranteed Computation of Robot Trajectories. Robotics and Autonomous Systems, 93:76-84, 2017. URL: https://doi.org/10.1016/j.robot.2017.03.020.
  31. S. Romig, L. Jaulin, and A. Rauh. Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System. Algorithms, 12(262), 2019. Google Scholar
  32. P. Saint-Pierre. Hybrid Kernels and Capture Basins for Impulse Constrained Systems. In C.J. Tomlin and M.R. Greenstreet, editors, in Hybrid Systems: Computation and Control, volume 2289, pages 378-392. Springer-Verlag, 2002. Google Scholar
  33. F. Tahir and M. Jaimoukha. Low-Complexity Polytopic Invariant Sets for Linear Systems Subject to Norm-Bounded Uncertainty. IEEE Trans. Autom. Control, 60:1416-1421, 2015. Google Scholar
  34. G. Trombettoni and G. Chabert. Constructive Interval Disjunction. In Proc. CP, Constraint Programming, LNCS 4741, pages 635-650. Springer, 2007. Google Scholar
  35. J. Wan, J. Vehi, and N. Luo. A Numerical Approach to Design Control Invariant Sets for Constrained Nonlinear Discrete-time Systems with Guaranteed Optimality. Journal of Global Optimization, 44:395-407, 2009. Google Scholar
  36. J. A. Yorke. Invariance for Ordinary Differential Equations. Mathematical System Theory, 1(4):353-372, 1967. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact