Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
158 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

Observability for Nonlinear Systems: Connecting Variational Dynamics, Lyapunov Exponents, and Empirical Gramians (2402.14711v3)

Published 22 Feb 2024 in eess.SY and cs.SY

Abstract: Observability quantification is a key problem in dynamic network sciences. While it has been thoroughly studied for linear systems, observability quantification for nonlinear networks is less intuitive and more cumbersome. One common approach to quantify observability for nonlinear systems is via the Empirical Gramian (Empr-Gram) -- a generalized form of the Gramian of linear systems. In this technical note, we produce three new results. First, we establish that a variational form of nonlinear systems (computed via perturbing initial conditions) yields a so-called Variational Gramian (Var-Gram) that is equivalent to the classic Empr-Gram; the former being easier to compute than the latter. Via Lyapunov exponents derived from Lyapunov's direct method, the technical note's second result derives connections between existing observability measures and Var-Gram. The third result demonstrates the applicability of these new notions for sensor selection/placement in nonlinear systems. Numerical case studies demonstrate these three developments and their merits.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (38)
  1. Y. Kawano and T. Ohtsuka, “Observability at an initial state for polynomial systems,” Automatica, vol. 49, no. 5, pp. 1126–1136, 2013.
  2. R. E. Kalman, “Mathematical Description of Linear Dynamical Systems,” Journal of the Society for Industrial and Applied Mathematics Series A Control, vol. 1, no. 2, pp. 85–239, 1963.
  3. R. Hermann and A. J. Krener, “Nonlinear Controllability and Observability,” IEEE Transactions on Automatic Control, vol. 22, no. 5, pp. 728–740, 1977.
  4. A. J. Whalen, S. N. Brennan, T. D. Sauer, and S. J. Schiff, “Observability and controllability of nonlinear networks: The role of symmetry,” Physical Review X, vol. 5, no. 1, pp. 1–40, 2015.
  5. A. J. Krener and K. Ide, “Measures of unobservability,” Proceedings of the IEEE Conference on Decision and Control, pp. 6401–6406, 2009.
  6. B. C. Moore, “Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction,” IEEE Transactions on Automatic Control, vol. 26, no. 1, pp. 17–32, 1981.
  7. S. Lall, J. E. Marsden, and S. Glavaški, “Empirical model reduction of controlled nonlinear systems,” IFAC Proceedings Volumes, vol. 32, no. 2, pp. 2598–2603, 1999.
  8. ——, “A subspace approach to balanced truncation for model reduction of nonlinear control systems,” International Journal of Robust and Nonlinear Control, vol. 12, no. 6, pp. 519–535, 2002.
  9. L. Kunwoo, Y. Umezu, K. Konno, and K. Kashima, “Observability Gramian for Bayesian Inference in Nonlinear Systems with Its Industrial Application,” IEEE Control Systems Letters, vol. 7, pp. 871–876, 2023.
  10. A. Haber, F. Molnar, and A. E. Motter, “State Observation and Sensor Selection for Nonlinear Networks,” IEEE Transactions on Control of Network Systems, vol. 5, no. 2, pp. 694–708, 2018.
  11. M. H. Kazma, S. A. Nugroho, A. Haber, and A. F. Taha, “State-Robust Observability Measures for Sensor Selection in Nonlinear Dynamic Systems,” 2023 62nd IEEE Conference on Decision and Control (CDC), no. Cdc, pp. 8418–8426, 2023. [Online]. Available: http://arxiv.org/abs/2307.07074
  12. Y. Y. Liu and A. L. Barabási, “Control principles of complex systems,” Reviews of Modern Physics, vol. 88, no. 3, pp. 1–58, 2016.
  13. Aleksandr Mikhailovich Lyapunov, “General Problem Of the Stability of Motion,” 1892.
  14. A. Czornik, A. Konyukh, I. Konyukh, M. Niezabitowski, and J. Orwat, “On Lyapunov and Upper Bohl Exponents of Diagonal Discrete Linear Time-Varying Systems,” IEEE Transactions on Automatic Control, vol. 64, no. 12, pp. 5171–5174, 2019.
  15. K. Krishna, S. L. Brunton, and Z. Song, “Finite Time Lyapunov Exponent Analysis of Model Predictive Control and Reinforcement Learning,” arXiv, pp. 1–22, 2023.
  16. J. Cortés, A. Van Der Schaft, and P. E. Crouch, “Characterization of gradient control systems,” SIAM Journal on Control and Optimization, vol. 44, no. 4, pp. 1192–1214, 2005.
  17. Y. Kawano and J. M. Scherpen, “Empirical differential Gramians for nonlinear model reduction,” Automatica, vol. 127, p. 109534, 2021.
  18. M. Balcerzak, A. Dabrowski, B. Blazejczyk-Okolewska, and A. Stefanski, “Determining Lyapunov exponents of non-smooth systems: Perturbation vectors approach,” Mechanical Systems and Signal Processing, vol. 141, p. 106734, 2020.
  19. L. S. Young, “Mathematical theory of Lyapunov exponents,” Journal of Physics A: Mathematical and Theoretical, vol. 46, no. 25, 2013.
  20. K. E. Atkinson, W. Han, and D. Stewart, “Numerical Solution of Ordinary Differential Equations,” Numerical Solution of Ordinary Differential Equations, pp. 1–252, 2011.
  21. S. C. Shadden, F. Lekien, and J. E. Marsden, “Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows,” Physica D: Nonlinear Phenomena, vol. 212, no. 3-4, pp. 271–304, 2005.
  22. S. Hanba, “Existence of an observation window of finite width for continuous-time autonomous nonlinear systems,” Automatica, vol. 75, pp. 154–157, 2017.
  23. ——, “On the “Uniform” Observability of Discrete-Time Nonlinear Systems,” IEEE Transactions on Automatic Control, vol. 54, no. 8, pp. 1925–1928, 2009.
  24. S. Smale, “Mathematical problems for the next century,” The Mathematical intelligencer, vol. 20, no. 2, pp. 7–15, 1998.
  25. N. D. Powel and K. A. Morgansen, “Empirical observability Gramian rank condition for weak observability of nonlinear systems with control,” Proceedings of the IEEE Conference on Decision and Control, vol. 54th IEEE, no. Cdc, pp. 6342–6348, 2015.
  26. A. Mesbahi, J. Bu, and M. Mesbahi, “Nonlinear observability via Koopman Analysis: Characterizing the role of symmetry,” Automatica, vol. 124, p. 109353, 2021.
  27. D. Martini, D. Angeli, G. Innocenti, and A. Tesi, “Ruling Out Positive Lyapunov Exponents by Using the Jacobian’s Second Additive Compound Matrix,” IEEE Control Systems Letters, vol. 6, pp. 2924–2928, 2022.
  28. T. H. Summers, F. L. Cortesi, and J. Lygeros, “On Submodularity and Controllability in Complex Dynamical Networks,” IEEE Transactions on Control of Network Systems, vol. 3, no. 1, pp. 91–101, mar 2016.
  29. P. J. Nolan, M. Serra, and S. D. Ross, “Finite-time Lyapunov exponents in the instantaneous limit and material transport,” Nonlinear Dynamics, vol. 100, no. 4, pp. 3825–3852, 2020.
  30. M. Tranninger, R. Seeber, S. Zhuk, M. Steinberger, and M. Horn, “Detectability Analysis and Observer Design for Linear Time Varying Systems,” IEEE Control Systems Letters, vol. 4, no. 2, pp. 331–336, 2020.
  31. J. Frank and S. Zhuk, “A detectability criterion and data assimilation for nonlinear differential equations,” Nonlinearity, vol. 31, no. 11, pp. 5235–5257, 2018.
  32. N. Barabanov, “Lyapunov exponent and joint spectral radius: Some known and new results,” Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC ’05, vol. 2005, no. 3, pp. 2332–2337, 2005.
  33. G. Rota and W. Gilbert Strang, “A note on the joint spectral radius,” Indagationes Mathematicae (Proceedings), vol. 63, no. 638, pp. 379–381, 1960.
  34. S. Joshi and S. Boyd, “Sensor selection via convex optimization,” IEEE Transactions on Signal Processing, vol. 57, no. 2, pp. 451–462, 2009.
  35. K. Manohar, J. N. Kutz, and S. L. Brunton, “Optimal Sensor and Actuator Selection Using Balanced Model Reduction,” IEEE Transactions on Automatic Control, vol. 67, no. 4, pp. 2108–2115, 2022.
  36. L. Zhou and P. Tokekar, “Sensor Assignment Algorithms to Improve Observability while Tracking Targets,” IEEE Transactions on Robotics, vol. 35, no. 5, pp. 1206–1219, 2019.
  37. N. N. Smirnov and V. F. Nikitin, “Modeling and simulation of hydrogen combustion in engines,” International Journal of Hydrogen Energy, vol. 39, no. 2, pp. 1122–1136, 2014.
  38. N. Powel and K. A. Morgansen, “Empirical Observability Gramian for Stochastic Observability of Nonlinear Systems,” arXiv, jun 2020.
Citations (2)
View on Semantic Scholar

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