Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 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

Closed-Loop Finite-Time Analysis of Suboptimal Online Control (2312.05607v3)

Published 9 Dec 2023 in eess.SY and cs.SY

Abstract: Suboptimal methods in optimal control arise due to a limited computational budget, unknown system dynamics, or a short prediction window among other reasons. Although these methods are ubiquitous, their transient performance remains relatively unstudied. We consider the control of discrete-time, nonlinear time-varying dynamical systems and establish sufficient conditions to analyze the finite-time closed-loop performance of such methods in terms of the additional cost incurred due to suboptimality. Finite-time guarantees allow the control design to distribute a limited computational budget over a time horizon and estimate the on-the-go loss in performance due to suboptimality. We study exponential incremental input-to-state stabilizing policies and show that for nonlinear systems, under some mild conditions, this property is directly implied by exponential stability without further assumptions on global smoothness. The analysis is showcased on a suboptimal model predictive control use case.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (36)
  1. L. S. Pontryagin, Mathematical theory of optimal processes. CRC press, 1987.
  2. R. Bellman, “Dynamic programming,” Science, vol. 153, no. 3731, pp. 34–37, 1966.
  3. Athena Scientific, 2022.
  4. Athena scientific, 2012.
  5. R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction. MIT press, 2018.
  6. N. Hovakimyan and C. Cao, ℒ⁢1ℒ1\mathcal{L}1caligraphic_L 1 adaptive control theory: Guaranteed robustness with fast adaptation. SIAM, 2010.
  7. K. S. Narendra and A. M. Annaswamy, Stable adaptive systems. Courier Corporation, 2012.
  8. P. O. Scokaert, D. Q. Mayne, and J. B. Rawlings, “Suboptimal model predictive control (feasibility implies stability),” IEEE Transactions on Automatic Control, vol. 44, no. 3, pp. 648–654, 1999.
  9. B. Kouvaritakis and M. Cannon, “Model predictive control,” Switzerland: Springer International Publishing, vol. 38, 2016.
  10. Y. Li, X. Chen, and N. Li, “Online optimal control with linear dynamics and predictions: Algorithms and regret analysis,” Advances in Neural Information Processing Systems, vol. 32, 2019.
  11. N. M. Boffi, S. Tu, and J.-J. E. Slotine, “Regret bounds for adaptive nonlinear control,” in Learning for Dynamics and Control, pp. 471–483, PMLR, 2021.
  12. G. Belgioioso, D. Liao-McPherson, M. H. de Badyn, S. Bolognani, R. S. Smith, J. Lygeros, and F. Dörfler, “Online feedback equilibrium seeking,” arXiv preprint arXiv:2210.12088, 2022.
  13. Z. He, S. Bolognani, J. He, F. Dörfler, and X. Guan, “Model-free nonlinear feedback optimization,” arXiv preprint arXiv:2201.02395, 2022.
  14. D. Angeli, “A Lyapunov approach to incremental stability properties,” IEEE Transactions on Automatic Control, vol. 47, no. 3, pp. 410–421, 2002.
  15. D. N. Tran, B. S. Rüffer, and C. M. Kellett, “Convergence properties for discrete-time nonlinear systems,” IEEE Transactions on Automatic Control, vol. 64, no. 8, pp. 3415–3422, 2018.
  16. W. Lohmiller and J.-J. E. Slotine, “On contraction analysis for non-linear systems,” Automatica, vol. 34, no. 6, pp. 683–696, 1998.
  17. H. Tsukamoto, S.-J. Chung, and J.-J. E. Slotine, “Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview,” Annual Reviews in Control, vol. 52, pp. 135–169, 2021.
  18. F. Bullo, Contraction Theory for Dynamical Systems. Kindle Direct Publishing, 1.1 ed., 2023.
  19. A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos, “The explicit linear quadratic regulator for constrained systems,” Automatica, vol. 38, no. 1, pp. 3–20, 2002.
  20. A. Davydov, S. Jafarpour, and F. Bullo, “Non-euclidean contraction theory for robust nonlinear stability,” IEEE Transactions on Automatic Control, vol. 67, no. 12, pp. 6667–6681, 2022.
  21. A. Zanelli, Q. T. Dinh, and M. Diehl, “A Lyapunov function for the combined system-optimizer dynamics in nonlinear model predictive control,” arXiv preprint arXiv:2004.08578, 2020.
  22. D. Liao-McPherson, M. M. Nicotra, and I. Kolmanovsky, “Time-distributed optimization for real-time model predictive control: Stability, robustness, and constraint satisfaction,” Automatica, vol. 117, p. 108973, 2020.
  23. A. Karapetyan, E. C. Balta, A. Iannelli, and J. Lygeros, “On the finite-time behavior of suboptimal linear model predictive control,” arXiv preprint arXiv:2305.10085, 2023.
  24. B. P. Demidovich, “Lectures on stability theory (in Russian),” 1967.
  25. A. B. Taylor, J. M. Hendrickx, and F. Glineur, “Exact worst-case convergence rates of the proximal gradient method for composite convex minimization,” Journal of Optimization Theory and Applications, vol. 178, no. 2, pp. 455–476, 2018.
  26. H. Khalil, “Nonlinear systems, third edition, vol. 115,” Upper Saddle River, NJ, USA: Patience-Hall, 2002.
  27. W. M. Haddad and V. Chellaboina, Nonlinear dynamical systems and control: a Lyapunov-based approach. Princeton university press, 2008.
  28. Z.-P. Jiang, Y. Lin, and Y. Wang, “Nonlinear small-gain theorems for discrete-time feedback systems and applications,” Automatica, vol. 40, no. 12, pp. 2129–2136, 2004.
  29. Z.-P. Jiang and Y. Wang, “A converse Lyapunov theorem for discrete-time systems with disturbances,” Systems & control letters, vol. 45, no. 1, pp. 49–58, 2002.
  30. W. J. Rugh, Linear system theory. Prentice-Hall, Inc., 1996.
  31. A. Pavlov, A. Pogromsky, N. van de Wouw, and H. Nijmeijer, “Convergent dynamics, a tribute to boris pavlovich demidovich,” Systems & Control Letters, vol. 52, no. 3-4, pp. 257–261, 2004.
  32. D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, vol. 36, no. 6, pp. 789–814, 2000.
  33. D. Liao-McPherson, T. Skibik, J. Leung, I. Kolmanovsky, and M. M. Nicotra, “An analysis of closed-loop stability for linear model predictive control based on time-distributed optimization,” IEEE Transactions on Automatic Control, vol. 67, no. 5, pp. 2618–2625, 2021.
  34. D. Limón, T. Alamo, F. Salas, and E. F. Camacho, “On the stability of constrained mpc without terminal constraint,” IEEE transactions on automatic control, vol. 51, no. 5, pp. 832–836, 2006.
  35. J. Leung, D. Liao-McPherson, and I. V. Kolmanovsky, “A computable plant-optimizer region of attraction estimate for time-distributed linear model predictive control,” in 2021 American Control Conference (ACC), pp. 3384–3391, IEEE, 2021.
  36. E. D. Sontag and Y. Wang, “New characterizations of input-to-state stability,” IEEE transactions on automatic control, vol. 41, no. 9, pp. 1283–1294, 1996.
Citations (3)
View on Semantic Scholar

Summary

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

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