Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
167 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

State-Augmented Linear Games with Antagonistic Error for High-Dimensional, Nonlinear Hamilton-Jacobi Reachability (2403.16982v2)

Published 25 Mar 2024 in eess.SY, cs.SY, and math.OC

Abstract: Hamilton-Jacobi Reachability (HJR) is a popular method for analyzing the liveness and safety of a dynamical system with bounded control and disturbance. The corresponding HJ value function offers a robust controller and characterizes the reachable sets, but is traditionally solved with Dynamic Programming (DP) and limited to systems of dimension less than six. Recently, the space-parallelizeable, generalized Hopf formula has been shown to also solve the HJ value with a nearly three-log increase in dimension limit, but is limited to linear systems. To extend this potential, we demonstrate how state-augmented (SA) spaces, which are well-known for their improved linearization accuracy, may be used to solve tighter, conservative approximations of the value function with any linear model in this SA space. Namely, we show that with a representation of the true dynamics in the SA space, a series of inequalities confirms that the value of a SA linear game with antagonistic error is a conservative envelope of the true value function. It follows that if the optimal controller for the HJ SA linear game with error may succeed, it will also succeed in the true system. Unlike previous methods, this result offers the ability to safely approximate reachable sets and their corresponding controllers with the Hopf formula in a non-convex manner. Finally, we demonstrate this in the slow manifold system for clarity, and in the controlled Van der Pol system with different lifting functions.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (36)
  1. I. M. Mitchell, A. M. Bayen, and C. J. Tomlin, “A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games,” IEEE Transactions on automatic control, vol. 50, no. 7, pp. 947–957, 2005.
  2. S. Bansal, M. Chen, S. Herbert, and C. J. Tomlin, “Hamilton-Jacobi reachability: A brief overview and recent advances,” in 2017 IEEE 56th Annual Conference on Decision and Control (CDC).   IEEE, 2017, pp. 2242–2253.
  3. A. Lin and S. Bansal, “Generating formal safety assurances for high-dimensional reachability,” in 2023 IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2023, pp. 10 525–10 531.
  4. M. R. Kirchner, R. Mar, G. Hewer, J. Darbon, S. Osher, and Y. T. Chow, “Time-optimal collaborative guidance using the generalized Hopf formula,” IEEE Control Systems Letters, vol. 2, no. 2, pp. 201–206, apr 2018. [Online]. Available: https://doi.org/10.11092Flcsys.2017.2785357
  5. D. Lee and C. J. Tomlin, “Iterative method using the generalized Hopf formula: Avoiding spatial discretization for computing solutions of Hamilton-Jacobi equations for nonlinear systems,” in 2019 IEEE 58th Conference on Decision and Control (CDC).   IEEE, 2019, pp. 1486–1493.
  6. M. Chen, S. Herbert, and C. J. Tomlin, “Exact and efficient Hamilton-Jacobi guaranteed safety analysis via system decomposition,” in 2017 IEEE International Conference on Robotics and Automation (ICRA).   IEEE, 2017, pp. 87–92.
  7. C. He, Z. Gong, M. Chen, and S. Herbert, “Efficient and guaranteed hamilton-jacobi reachability via self-contained subsystem decomposition and admissible control sets,” IEEE Control Systems Letters, 2023.
  8. J. Darbon and S. Osher, “Algorithms for overcoming the curse of dimensionality for certain Hamilton–Jacobi equations arising in control theory and elsewhere,” Research in the Mathematical Sciences, vol. 3, no. 1, p. 19, 2016.
  9. Y. T. Chow, J. Darbon, S. Osher, and W. Yin, “Algorithm for overcoming the curse of dimensionality for time-dependent non-convex Hamilton–Jacobi equations arising from optimal control and differential games problems,” Journal of Scientific Computing, vol. 73, pp. 617–643, 2017.
  10. ——, “Algorithm for overcoming the curse of dimensionality for state-dependent Hamilton-Jacobi equations,” Journal of Computational Physics, vol. 387, pp. 376–409, 2019.
  11. W. Sharpless, N. Shinde, M. Kim, Y. T. Chow, and S. Herbert, “Koopman-Hopf Hamilton-Jacobi reachability and control,” arXiv preprint arXiv:2303.11590, 2003.
  12. W. Sharpless, Y. T. Chow, and S. Herbert, “Conservative linear envelopes for high-dimensional, Hamilton-Jacobi reachability for nonlinear systems via the Hopf formula,” 2024.
  13. Y. Igarashi, M. Yamakita, J. Ng, and H. H. Asada, “Mpc performances for nonlinear systems using several linearization models,” in 2020 American Control Conference (ACC).   IEEE, 2020, pp. 2426–2431.
  14. M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data–driven approximation of the koopman operator: Extending dynamic mode decomposition,” Journal of Nonlinear Science, vol. 25, pp. 1307–1346, 2015.
  15. E. Yeung, S. Kundu, and N. Hodas, “Learning deep neural network representations for Koopman operators of nonlinear dynamical systems,” in 2019 American Control Conference (ACC).   IEEE, 2019, pp. 4832–4839.
  16. Y. Li, H. He, J. Wu, D. Katabi, and A. Torralba, “Learning compositional Koopman operators for model-based control,” 2020.
  17. S. L. Brunton, B. W. Brunton, J. L. Proctor, and J. N. Kutz, “Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control,” PloS one, vol. 11, no. 2, p. e0150171, 2016.
  18. I. Abraham and T. D. Murphey, “Active learning of dynamics for data-driven control using Koopman operators,” IEEE Transactions on Robotics, vol. 35, no. 5, pp. 1071–1083, 2019.
  19. M. Haseli and J. Cortés, “Modeling nonlinear control systems via koopman control family: Universal forms and subspace invariance proximity,” 2024.
  20. M. Korda and I. Mezić, “On convergence of extended dynamic mode decomposition to the Koopman operator,” Journal of Nonlinear Science, vol. 28, pp. 687–710, 2018.
  21. F. Nüske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann, “Finite-data error bounds for Koopman-based prediction and control,” Journal of Nonlinear Science, vol. 33, no. 1, p. 14, 2023.
  22. J. J. Bramburger and G. Fantuzzi, “Auxiliary functions as koopman observables: Data-driven polynomial optimization for dynamical systems,” arXiv e-prints, pp. arXiv–2303, 2023.
  23. B. O. Koopman, “Hamiltonian systems and transformation in Hilbert space,” Proceedings of the National Academy of Sciences, vol. 17, no. 5, pp. 315–318, 1931.
  24. L. C. Evans and P. E. Souganidis, “Differential games and representation formulas for solutions of Hamilton-Jacobi-isaacs equations,” Indiana University mathematics journal, vol. 33, no. 5, pp. 773–797, 1984.
  25. A. F. Filippov, “Differential equations with discontinuous right-hand side,” Matematicheskii sbornik, vol. 93, no. 1, pp. 99–128, 1960.
  26. J. K. Scott and P. I. Barton, “Bounds on the reachable sets of nonlinear control systems,” Automatica, vol. 49, no. 1, pp. 93–100, 2013.
  27. A. B. Kurzhanski, “Dynamics and control of trajectory tubes. theory and computation,” in 2014 20th International Workshop on Beam Dynamics and Optimization (BDO).   IEEE, 2014, pp. 1–1.
  28. I. Rublev, “Generalized Hopf formulas for the nonautonomous Hamilton-Jacobi equation,” Computational Mathematics and Modeling, vol. 11, no. 4, pp. 391–400, 2000.
  29. Y. T. Chow, J. Darbon, S. Osher, and W. Yin, “Algorithm for overcoming the curse of dimensionality for certain non-convex Hamilton–Jacobi equations, projections and differential games,” Annals of Mathematical Sciences and Applications, vol. 3, no. 2, pp. 369–403, 2018.
  30. D. Bruder, B. Gillespie, C. D. Remy, and R. Vasudevan, “Modeling and control of soft robots using the Koopman operator and model predictive control,” arXiv preprint arXiv:1902.02827, 2019.
  31. X. Chen, E. Abraham, and S. Sankaranarayanan, “Taylor model flowpipe construction for non-linear hybrid systems,” in 2012 IEEE 33rd Real-Time Systems Symposium.   IEEE, 2012, pp. 183–192.
  32. M. Korda and I. Mezić, “Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control,” Automatica, vol. 93, pp. 149–160, 2018.
  33. S. Bogomolov, M. Forets, G. Frehse, K. Potomkin, and C. Schilling, “JuliaReach: a toolbox for set-based reachability,” in Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control, 2019, pp. 39–44.
  34. StanfordASL, “Hamilton-Jacobi reachability analysis in jax.” 2021. [Online]. Available: https://github.com/StanfordASL/hj_reachability
  35. UCSD-SASLab, “Julia package for Hamilton-Jacobi reachability via Hopf optimization,” 2023. [Online]. Available: https://github.com/UCSD-SASLab/HopfReachability
  36. S. Pan, E. Kaiser, B. M. de Silva, J. N. Kutz, and S. L. Brunton, “Pykoopman: A python package for data-driven approximation of the Koopman operator,” 2023.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets