Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
162 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 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

A Novel State-Centric Necessary Condition for Time-Optimal Control of Controllable Linear Systems Based on Augmented Switching Laws (Extended Version) (2404.08943v2)

Published 13 Apr 2024 in math.OC, cs.SY, and eess.SY

Abstract: Most existing necessary conditions for optimal control based on adjoining methods require both state and costate information, yet the unobservability of costates for a given feasible trajectory impedes the determination of optimality in practice. This paper establishes a novel theoretical framework for time-optimal control of controllable linear systems with a single input, proposing the augmented switching law (ASL) that represents the input control and the feasibility in a compact form. Given a feasible trajectory, the perturbed trajectory under the constraints of ASL is guaranteed to be feasible, resulting in a novel state-centric necessary condition without dependence on costate information. A first-order necessary condition is proposed that the Jacobian matrix of the ASL is not full row rank, which also results in a potential approach to optimizing a given feasible trajectory with the preservation of arc structures. The proposed necessary condition is applied to high-order chain-of-integrator systems with full box constraints, contributing to some theoretical results challenging to reason by costate-based conditions.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (27)
  1. E. Trélat, “Optimal control and applications to aerospace: some results and challenges,” Journal of Optimization Theory and Applications, vol. 154, pp. 713–758, 2012.
  2. Y. Wang, C. Hu, Z. Wang, S. Lin, Z. Zhao, W. Zhao, K. Hu, Z. Huang, Y. Zhu, and Z. Lu, “Optimization-based non-equidistant toolpath planning for robotic additive manufacturing with non-underfill orientation,” Robotics and Computer-Integrated Manufacturing, vol. 84, p. 102599, 2023.
  3. Y. Wang, C. Hu, Z. Wang, S. Lin, Z. Zhao, and Y. Zhu, “Slice extension for high-quality hybrid additive-subtractive manufacturing,” in IECON 2023-49th Annual Conference of the IEEE Industrial Electronics Society.   IEEE, 2023, pp. 1–6.
  4. G. Zhao and M. Zhu, “Pareto optimal multirobot motion planning,” IEEE Transactions on Automatic Control, vol. 66, no. 9, pp. 3984–3999, 2020.
  5. Y. Wang, J. Wang, Y. Li, C. Hu, and Y. Zhu, “Learning latent object-centric representations for visual-based robot manipulation,” in 2022 International Conference on Advanced Robotics and Mechatronics (ICARM).   IEEE, 2022, pp. 138–143.
  6. S. Güler, B. Fidan, S. Dasgupta, B. D. Anderson, and I. Shames, “Adaptive source localization based station keeping of autonomous vehicles,” IEEE Transactions on Automatic Control, vol. 62, no. 7, pp. 3122–3135, 2016.
  7. R. F. Hartl, S. P. Sethi, and R. G. Vickson, “A survey of the maximum principles for optimal control problems with state constraints,” SIAM review, vol. 37, no. 2, pp. 181–218, 1995.
  8. Y. Wang, C. Hu, Z. Li, S. Lin, S. He, and Y. Zhu, “Time-optimal control for high-order chain-of-integrators systems with full state constraints and arbitrary terminal states (extended version),” arXiv preprint arXiv:2311.07039, 2024.
  9. R. Gamkrelidze, “Optimal processes with bounded phase coordinates,” Izv. Akad. Nauk, USSR Sec. Mat, vol. 24, pp. 315–356, 1960.
  10. S. Chang, “Optimal control in bounded phase space,” Automatica, vol. 1, no. 1, pp. 55–67, 1963.
  11. S. E. Dreyfus, “Dynamic programming and the calculus of variations,” Journal of Mathematical Analysis and Applications, vol. 1, no. 2, pp. 228–239, 1960.
  12. D. H. Jacobson, M. M. Lele, and J. L. Speyer, “New necessary conditions of optimality for control problems with state-variable inequality constraints,” Journal of mathematical analysis and applications, vol. 35, no. 2, pp. 255–284, 1971.
  13. K. Makowski and L. W. Neustadt, “Optimal control problems with mixed control-phase variable equality and inequality constraints,” SIAM journal on control, vol. 12, no. 2, pp. 184–228, 1974.
  14. A. E. Bryson Jr, W. F. Denham, and S. E. Dreyfus, “Optimal programming problems with inequality constraints i: Necessary conditions for extremal solutions,” AIAA journal, vol. 1, no. 11, pp. 2544–2550, 1963.
  15. E. Kreindler, “Additional necessary conditions for optimal control with state-variable inequality constraints,” Journal of Optimization theory and applications, vol. 38, pp. 241–250, 1982.
  16. Y. Wang, C. Hu, Z. Li, Y. Lin, S. Lin, S. He, and Y. Zhu, “Chattering phenomena in time-optimal control for high-order chain-of-integrators systems with full state constraints,” arXiv preprint arXiv:2403.17675, 2024.
  17. E. B. Lee and L. Markus, “Foundations of optimal control theory,” Minnesota Univ Minneapolis Center For Control Sciences, Tech. Rep., 1967.
  18. R. Bellman, “On the theory of dynamic programming,” Proceedings of the national Academy of Sciences, vol. 38, no. 8, pp. 716–719, 1952.
  19. J. Moon, “Generalized risk-sensitive optimal control and hamilton–jacobi–bellman equation,” IEEE Transactions on Automatic Control, vol. 66, no. 5, pp. 2319–2325, 2020.
  20. F. Tedone and M. Palladino, “Hamilton–jacobi–bellman equation for control systems with friction,” IEEE Transactions on Automatic Control, vol. 66, no. 12, pp. 5651–5664, 2020.
  21. Z. Li, C. Hu, Y. Wang, Y. Yang, and S. E. Li, “Safe reinforcement learning with dual robustness,” arXiv preprint arXiv:2309.06835, 2023.
  22. L. Evans and M. James, “The hamiltonian–jacobi–bellman equation for time-optimal control,” SIAM journal on control and optimization, vol. 27, no. 6, pp. 1477–1489, 1989.
  23. P. R. Wolenski and Y. Zhuang, “Proximal analysis and the minimal time function,” SIAM journal on control and optimization, vol. 36, no. 3, pp. 1048–1072, 1998.
  24. B. Sun and B.-Z. Guo, “Convergence of an upwind finite-difference scheme for hamilton–jacobi–bellman equation in optimal control,” IEEE Transactions on automatic control, vol. 60, no. 11, pp. 3012–3017, 2015.
  25. M. Zelikin and V. Borisov, “Optimal chattering feedback control,” Journal of Mathematical Sciences, vol. 114, no. 3, pp. 1227–1344, 2003.
  26. S. He, C. Hu, Y. Zhu, and M. Tomizuka, “Time optimal control of triple integrator with input saturation and full state constraints,” Automatica, vol. 122, p. 109240, 2020.
  27. L. Berscheid and T. Kröger, “Jerk-limited real-time trajectory generation with arbitrary target states,” in Robotics: Science and Systems, 2021.

Summary

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