Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash 97 tok/s
Gemini 2.5 Pro 39 tok/s Pro
GPT-5 Medium 29 tok/s
GPT-5 High 28 tok/s Pro
GPT-4o 93 tok/s
GPT OSS 120B 462 tok/s Pro
Kimi K2 215 tok/s Pro
2000 character limit reached

Learning Dissipative Neural Dynamical Systems (2309.16032v2)

Published 27 Sep 2023 in cs.LG, cs.SY, eess.SY, math.DS, and math.OC

Abstract: Consider an unknown nonlinear dynamical system that is known to be dissipative. The objective of this paper is to learn a neural dynamical model that approximates this system, while preserving the dissipativity property in the model. In general, imposing dissipativity constraints during neural network training is a hard problem for which no known techniques exist. In this work, we address the problem of learning a dissipative neural dynamical system model in two stages. First, we learn an unconstrained neural dynamical model that closely approximates the system dynamics. Next, we derive sufficient conditions to perturb the weights of the neural dynamical model to ensure dissipativity, followed by perturbation of the biases to retain the fit of the model to the trajectories of the nonlinear system. We show that these two perturbation problems can be solved independently to obtain a neural dynamical model that is guaranteed to be dissipative while closely approximating the nonlinear system.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (33)
  1. M. Gevers, “Identification for control: From the early achievements to the revival of experiment design,” European journal of control, vol. 11, no. 4-5, pp. 335–352, 2005.
  2. L. Ljung, “Estimating linear time-invariant models of nonlinear time-varying systems,” European Journal of Control, vol. 7, no. 2-3, pp. 203–219, 2001.
  3. A. Mauroy and J. Goncalves, “Linear identification of nonlinear systems: A lifting technique based on the koopman operator,” in 2016 IEEE 55th Conference on Decision and Control (CDC).   IEEE, 2016, pp. 6500–6505.
  4. M. Korda and I. Mezić, “Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control,” Automatica, vol. 93, pp. 149–160, 2018.
  5. R. T. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, “Neural ordinary differential equations,” Advances in neural information processing systems, vol. 31, 2018.
  6. R. Wang and R. Yu, “Physics-guided deep learning for dynamical systems: A survey,” arXiv preprint arXiv:2107.01272, 2021.
  7. J. C. Willems, “Dissipative dynamical systems part i: General theory,” Archive for rational mechanics and analysis, vol. 45, no. 5, pp. 321–351, 1972.
  8. ——, “Dissipative dynamical systems part ii: Linear systems with quadratic supply rates,” Archive for rational mechanics and analysis, vol. 45, pp. 352–393, 1972.
  9. D. Hill and P. Moylan, “The stability of nonlinear dissipative systems,” IEEE transactions on automatic control, vol. 21, no. 5, pp. 708–711, 1976.
  10. D. J. Hill and P. J. Moylan, “Dissipative dynamical systems: Basic input-output and state properties,” Journal of the Franklin Institute, vol. 309, no. 5, pp. 327–357, 1980.
  11. S. Coogan and M. Arcak, “A dissipativity approach to safety verification for interconnected systems,” IEEE Transactions on Automatic Control, vol. 60, no. 6, pp. 1722–1727, 2014.
  12. P. J. Antsaklis, B. Goodwine, V. Gupta, M. J. McCourt, Y. Wang, P. Wu, M. Xia, H. Yu, and F. Zhu, “Control of cyberphysical systems using passivity and dissipativity based methods,” European Journal of Control, vol. 19, no. 5, pp. 379–388, 2013.
  13. E. Agarwal, S. Sivaranjani, V. Gupta, and P. J. Antsaklis, “Distributed synthesis of local controllers for networked systems with arbitrary interconnection topologies,” IEEE Transactions on Automatic Control, vol. 66, no. 2, pp. 683–698, 2020.
  14. E. Agarwal, S. Sivaranjani, V. Gupta, and P. Antsaklis, “Sequential synthesis of distributed controllers for cascade interconnected systems,” in 2019 American Control Conference (ACC).   IEEE, 2019, pp. 5816–5821.
  15. M. Arcak, “Compositional design and verification of large-scale systems using dissipativity theory: Determining stability and performance from subsystem properties and interconnection structures,” IEEE Control Systems Magazine, vol. 42, no. 2, pp. 51–62, 2022.
  16. A. Lavaei and M. Zamani, “From dissipativity theory to compositional synthesis of large-scale stochastic switched systems,” IEEE Transactions on Automatic Control, vol. 67, no. 9, pp. 4422–4437, 2022.
  17. E. Agarwal, M. J. McCourt, and P. J. Antsaklis, “Dissipativity of hybrid systems: Feedback interconnections and networks,” in 2016 American Control Conference (ACC).   IEEE, 2016, pp. 6060–6065.
  18. A. U. Awan and M. Zamani, “From dissipativity theory to compositional abstractions of interconnected stochastic hybrid systems,” IEEE Transactions on Control of Network Systems, vol. 7, no. 1, pp. 433–445, 2019.
  19. J. Liu, L. Wu, C. Wu, W. Luo, and L. G. Franquelo, “Event-triggering dissipative control of switched stochastic systems via sliding mode,” Automatica, vol. 103, pp. 261–273, 2019.
  20. S. Sivaranjani, E. Agarwal, V. Gupta, P. Antsaklis, and L. Xie, “Distributed mixed voltage angle and frequency droop control of microgrid interconnections with loss of distribution-pmu measurements,” IEEE Open Access Journal of Power and Energy, vol. 8, pp. 45–56, 2020.
  21. S. Sivaranjani, E. Agarwal, L. Xie, V. Gupta, and P. Antsaklis, “Mixed voltage angle and frequency droop control for transient stability of interconnected microgrids with loss of pmu measurements,” in 2020 American Control Conference (ACC).   IEEE, 2020, pp. 2382–2387.
  22. K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural networks, vol. 2, no. 5, pp. 359–366, 1989.
  23. S. Sivaranjani, E. Agarwal, and V. Gupta, “Data-driven identification of dissipative linear models for nonlinear systems,” IEEE Transactions on Automatic Control, vol. 67, no. 9, pp. 4978–4985, 2022.
  24. ——, “Data-driven identification of approximate passive linear models for nonlinear systems,” in Learning for Dynamics and Control.   PMLR, 2020, pp. 338–339.
  25. K. Hara, M. Inoue, and N. Sebe, “Learning koopman operator under dissipativity constraints,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 1169–1174, 2020.
  26. M. Khosravi and R. S. Smith, “Kernel-based identification with frequency domain side-information,” Automatica, vol. 150, p. 110813, 2023.
  27. T. X. Nghiem, J. Drgoňa, C. Jones, Z. Nagy, R. Schwan, B. Dey, A. Chakrabarty, S. Di Cairano, J. A. Paulson, A. Carron et al., “Physics-informed machine learning for modeling and control of dynamical systems,” arXiv preprint arXiv:2306.13867, 2023.
  28. Y. D. Zhong, B. Dey, and A. Chakraborty, “Dissipative symoden: Encoding hamiltonian dynamics with dissipation and control into deep learning,” arXiv preprint arXiv:2002.08860, 2020.
  29. J. Drgoňa, A. Tuor, S. Vasisht, and D. Vrabie, “Dissipative deep neural dynamical systems,” IEEE Open Journal of Control Systems, vol. 1, pp. 100–112, 2022.
  30. 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, vol. 67, no. 1, pp. 1–15, 2020.
  31. C. Verhoek, P. J. Koelewijn, S. Haesaert, and R. Tóth, “Convex incremental dissipativity analysis of nonlinear systems,” Automatica, vol. 150, p. 110859, 2023.
  32. J. Lofberg, “Yalmip: A toolbox for modeling and optimization in matlab,” in 2004 IEEE international conference on robotics and automation (IEEE Cat. No. 04CH37508).   IEEE, 2004, pp. 284–289.
  33. J. Fiala, M. Kočvara, and M. Stingl, “Penlab: A matlab solver for nonlinear semidefinite optimization,” arXiv preprint arXiv:1311.5240, 2013.
Citations (1)
View on Semantic Scholar
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Summary

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

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

Paper Prompts

Sign up for free to create and run paper prompts using GPT-5.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
Dice Question Streamline Icon: https://streamlinehq.com

Follow-up Questions

We haven't generated follow-up questions for this paper yet.

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