Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
47 tokens/sec
Gemini 2.5 Pro Pro
43 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Automated Stability Analysis of Piecewise Affine Dynamics Using Vertices (2307.03868v3)

Published 8 Jul 2023 in eess.SY and cs.SY

Abstract: This paper presents an automated algorithm to analyze the stability of piecewise affine (PWA) dynamical systems due to their broad applications. We parametrize the Lyapunov function as a PWA function, with polytopic regions defined by the PWA dynamics. Using this parametrization, Stability conditions can be expressed as linear constraints restricted to polytopes so that the search for a Lyapunov function involves solving a linear program. However, a valid Lyapunov function might not be found given these polytopic regions. A natural response is to increase the size of the parametrization of the Lyapunov function by dividing regions and solving the new linear program. This paper proposes two new methods to divide each polytope into smaller ones. The first approach divides a polytope based on the sign of the derivative of the candidate Lyapunov function, while the second divides it based on the change in the vector field of the PWA dynamical system. In addition, we propose using Delaunay triangulation to achieve automated division of regions and preserve the continuity of the PWA Lyapunov function. Examples involving learned models and explicit MPC controllers demonstrate that the proposed method of dividing regions leads to valid Lyapunov functions with fewer regions than existing methods, reducing the computational time taken for stability analysis

Definition Search Book Streamline Icon: https://streamlinehq.com
References (33)
  1. T. Marcucci, R. Deits, M. Gabiccini, A. Bicchi, and R. Tedrake, “Approximate hybrid model predictive control for multi-contact push recovery in complex environments,” in 2017 IEEE-RAS 17th International Conference on Humanoid Robotics (Humanoids).   IEEE, 2017, pp. 31–38.
  2. X. Sun, H. Zhang, Y. Cai, S. Wang, and L. Chen, “Hybrid modeling and predictive control of intelligent vehicle longitudinal velocity considering nonlinear tire dynamics,” Nonlinear Dynamics, vol. 97, no. 2, pp. 1051–1066, 2019.
  3. A. Bemporad, F. Borrelli, and M. Morari, “Piecewise linear optimal controllers for hybrid systems,” in Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No. 00CH36334), vol. 2.   IEEE, 2000, pp. 1190–1194.
  4. J. Qiu, W. Ji, H.-K. Lam, and M. Wang, “Fuzzy-affine-model-based sampled-data filtering design for stochastic nonlinear systems,” IEEE transactions on fuzzy systems, vol. 29, no. 11, pp. 3360–3373, 2020.
  5. A. Alessio and A. Bemporad, “A survey on explicit model predictive control,” Nonlinear Model Predictive Control: Towards New Challenging Applications, pp. 345–369, 2009.
  6. Í. Elguea-Aguinaco, A. Serrano-Muñoz, D. Chrysostomou, I. Inziarte-Hidalgo, S. Bøgh, and N. Arana-Arexolaleiba, “A review on reinforcement learning for contact-rich robotic manipulation tasks,” Robotics and Computer-Integrated Manufacturing, vol. 81, p. 102517, 2023.
  7. I. Jebellat, H. N. Pishkenari, and E. Jebellat, “Training microrobots via reinforcement learning and a novel coding method,” in 2021 9th RSI International Conference on Robotics and Mechatronics (ICRoM).   IEEE, 2021, pp. 105–111.
  8. P. Samanipour and H. A. Poonawala, “Stability analysis and controller synthesis using single-hidden-layer relu neural networks,” IEEE Transactions on Automatic Control, 2023.
  9. A. Abate, D. Ahmed, M. Giacobbe, and A. Peruffo, “Formal synthesis of lyapunov neural networks,” IEEE Control Systems Letters, vol. 5, no. 3, pp. 773–778, 2020.
  10. M. Korda, “Stability and performance verification of dynamical systems controlled by neural networks: algorithms and complexity,” IEEE Control Systems Letters, 2022.
  11. H. Ravanbakhsh and S. Sankaranarayanan, “Learning lyapunov (potential) functions from counterexamples and demonstrations,” arXiv preprint arXiv:1705.09619, 2017.
  12. M. Fazlyab, M. Morari, and G. J. Pappas, “Safety verification and robustness analysis of neural networks via quadratic constraints and semidefinite programming,” IEEE Transactions on Automatic Control, 2020.
  13. S. Chen, M. Fazlyab, M. Morari, G. J. Pappas, and V. M. Preciado, “Learning lyapunov functions for hybrid systems,” in Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control, 2021, pp. 1–11.
  14. H. Dai, B. Landry, L. Yang, M. Pavone, and R. Tedrake, “Lyapunov-stable neural-network control,” arXiv preprint arXiv:2109.14152, 2021.
  15. R. Zhou, T. Quartz, H. De Sterck, and J. Liu, “Neural lyapunov control of unknown nonlinear systems with stability guarantees,” arXiv preprint arXiv:2206.01913, 2022.
  16. J. Anderson and A. Papachristodoulou, “Advances in computational lyapunov analysis using sum-of-squares programming,” Discrete & Continuous Dynamical Systems-B, vol. 20, no. 8, p. 2361, 2015.
  17. R. Iervolino, D. Tangredi, and F. Vasca, “Lyapunov stability for piecewise affine systems via cone-copositivity,” Automatica, vol. 81, pp. 22–29, 2017.
  18. R. Iervolino, S. Trenn, and F. Vasca, “Asymptotic stability of piecewise affine systems with filippov solutions via discontinuous piecewise lyapunov functions,” IEEE Transactions on Automatic Control, vol. 66, no. 4, pp. 1513–1528, 2020.
  19. M. Johansson, “Piecewise linear control systems,” Ph.D. dissertation, Ph. D. Thesis, Lund Institute of Technology, Sweden, 1999.
  20. H. A. Poonawala, “Stability analysis via refinement of piece-wise linear lyapunov functions,” in 2019 IEEE 58th Conference on Decision and Control (CDC), 2019, pp. 1442–1447.
  21. ——, “Stability analysis of conewise affine dynamical systems using conewise linear lyapunov functions,” in 2021 American Control Conference (ACC).   IEEE, 2021, pp. 2406–2411.
  22. H. A. Poonawala, N. Lauffer, and U. Topcu, “Training classifiers for feedback control with safety in mind,” Automatica, vol. 128, p. 109509, 2021.
  23. M. Henk, J. Richter-Gebert, and G. M. Ziegler, “Basic properties of convex polytopes,” in Handbook of discrete and computational geometry.   Chapman and Hall/CRC, 2017, pp. 383–413.
  24. M. Della Rossa, A. Tanwani, and L. Zaccarian, “Smooth approximation of patchy lyapunov functions for switched systems,” IFAC-PapersOnLine, vol. 52, no. 16, pp. 364–369, 2019.
  25. R. Baier, L. Grüne, and S. Freyr Hafstein, “Linear programming based lyapunov function computation for differential inclusions,” 2012.
  26. M. El Ghami, Z. Guennoun, S. Bouali, and T. Steihaug, “Interior-point methods for linear optimization based on a kernel function with a trigonometric barrier term,” Journal of Computational and Applied Mathematics, vol. 236, no. 15, pp. 3613–3623, 2012.
  27. J. Zago, E. Camponogara, and E. Antonelo, “Vertex-based reachability analysis for verifying relu deep neural networks,” arXiv preprint arXiv:2301.12001, 2023.
  28. M. Rubagotti, S. Trimboli, D. Bernardini, and A. Bemporad, “Stability and invariance analysis of approximate explicit mpc based on pwa lyapunov functions,” IFAC Proceedings Volumes, vol. 44, no. 1, pp. 5712–5717, 2011.
  29. V. T. Rajan, “Optimality of the delaunay triangulation in rd,” in Proceedings of the seventh annual symposium on Computational geometry, 1991, pp. 357–363.
  30. M. ApS, “Mosek optimizer api for python,” Version, vol. 9, no. 17, pp. 6–4, 2022.
  31. M. Kvasnica, P. Grieder, M. Baotić, and M. Morari, “Multi-parametric toolbox (mpt),” in Hybrid Systems: Computation and Control: 7th International Workshop, HSCC 2004, Philadelphia, PA, USA, March 25-27, 2004. Proceedings 7.   Springer, 2004, pp. 448–462.
  32. M. Farsi, Y. Li, Y. Yuan, and J. Liu, “A piecewise learning framework for control of unknown nonlinear systems with stability guarantees,” in Learning for Dynamics and Control Conference.   PMLR, 2022, pp. 830–843.
  33. Y.-C. Chang, N. Roohi, and S. Gao, “Neural lyapunov control,” Advances in neural information processing systems, vol. 32, 2019.
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