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$k$-Contraction in a Generalized Lurie System (2309.07514v4)

Published 14 Sep 2023 in eess.SY and cs.SY

Abstract: We derive a sufficient condition for $k$-contraction in a generalized Lurie system~(GLS), that is, the feedback connection of a nonlinear dynamical system and a memoryless nonlinear function. For $k=1$, this reduces to a sufficient condition for standard contraction. For $k=2$, this condition implies that every bounded solution of the GLS converges to an equilibrium, which is not necessarily unique. We demonstrate the theoretical results by analyzing $k$-contraction in a biochemical control circuit with nonlinear dissipation terms.

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References (53)
  1. Z. Aminzare and E. D. Sontag, “Contraction methods for nonlinear systems: A brief introduction and some open problems,” in Proc. 53rd IEEE Conf. on Decision and Control, Los Angeles, CA, 2014, pp. 3835–3847.
  2. V. Andrieu and S. Tarbouriech, “LMI conditions for contraction and synchronization,” IFAC-PapersOnLine, vol. 52, no. 16, pp. 616–621, 2019, 11th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2019).
  3. D. Angeli, J. E. Ferrell, and E. D. Sontag, “Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems,” Proceedings of the National Academy of Sciences, vol. 101, no. 7, pp. 1822–1827, 2004.
  4. D. Angeli, D. Martini, G. Innocenti, and A. Tesi, “A small-gain theorem for 2-contraction of nonlinear interconnected systems,” 2023. [Online]. Available: https://arxiv.org/abs/2305.03211
  5. E. M. Aylward, P. A. Parrilo, and J.-J. E. Slotine, “Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming,” Automatica, vol. 44, no. 8, pp. 2163–2170, 2008.
  6. E. Bar-Shalom, O. Dalin, and M. Margaliot, “Compound matrices in systems and control theory: a tutorial,” Math. Control Signals Systems, vol. 35, pp. 467–521, 2023.
  7. N. Boffi, S. Tu, N. Matni, J.-J. Slotine, and V. Sindhwani, “Learning stability certificates from data,” in Proc. 2020 Conf. on Robot Learning, vol. 155, 2021, pp. 1341–1350.
  8. N. M. Boffi and J.-J. E. Slotine, “Implicit regularization and momentum algorithms in nonlinearly parameterized adaptive control and prediction,” Neural Computation, vol. 33, no. 3, pp. 590–673, 2021.
  9. V. Centorrino, A. Gokhale, A. Davydov, G. Russo, and F. Bullo, “Euclidean contractivity of neural networks with symmetric weights,” IEEE Control Systems Letters, vol. 7, pp. 1724–1729, 2023.
  10. C.-Y. Cheng, K.-H. Lin, and C.-W. Shin, “Multistability in recurrent neural networks,” SIAM J. Applied Math., vol. 66, no. 4, pp. 1301–1320, 2006.
  11. S.-J. Chung and J.-J. E. Slotine, “Cooperative robot control and concurrent synchronization of Lagrangian systems,” IEEE Trans. Robot., vol. 25, no. 3, pp. 686–700, 2009.
  12. M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IEEE Trans. Systems, Man, and Cybernetics, vol. SMC-13, no. 5, pp. 815–826, 1983.
  13. O. Dalin, R. Ofir, E. B. Shalom, A. Ovseevich, F. Bullo, and M. Margaliot, “Verifying k𝑘kitalic_k-contraction without computing k𝑘kitalic_k-compounds,” IEEE Trans. Automat. Control, vol. 69, no. 3, pp. 1492–1506, 2024.
  14. A. Davydov, S. Jafarpour, and F. Bullo, “Non-Euclidean contraction theory for robust nonlinear stability,” IEEE Trans. Automat. Control, vol. 67, no. 12, pp. 6667–6681, 2022.
  15. A. Fradkov, “Early ideas of the absolute stability theory,” in 2020 European Control Conference (ECC), 2020, pp. 762–768.
  16. M. Giaccagli, V. Andrieu, S. Tarbouriech, and D. Astolfi, “Infinite gain margin, contraction and optimality: an LMI-based design,” Euro. J. Control, p. 100685, 2022.
  17. S. Jafarpour, A. Davydov, A. Proskurnikov, and F. Bullo, “Robust implicit networks via non-Euclidean contractions,” in Advances in Neural Information Processing Systems, M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, Eds., vol. 34.   Curran Associates, Inc., 2021, pp. 9857–9868.
  18. L. Kozachkov, M. Ennis, and J.-J. Slotine, “RNNs of RNNs: Recursive construction of stable assemblies of recurrent neural networks,” Advances in Neural Information Processing Systems, vol. 35, pp. 30 512–30 527, 2022.
  19. L. Kozachkov, M. Lundqvist, J.-J. Slotine, and E. K. Miller, “Achieving stable dynamics in neural circuits,” PLoS computational biology, vol. 16, no. 8, p. e1007659, 2020.
  20. L. Kozachkov, J. Tauber, M. Lundqvist, S. L. Brincat, J.-J. Slotine, and E. K. Miller, “Robust and brain-like working memory through short-term synaptic plasticity,” PLOS Computational Biology, vol. 18, no. 12, p. e1010776, 2022.
  21. L. Kozachkov, P. Wensing, and J.-J. Slotine, “Generalization as dynamical robustness–the role of Riemannian contraction in supervised learning,” Transactions on Machine Learning Research, 2023. [Online]. Available: https://openreview.net/forum?id=Sb6p5mcefw
  22. A. Lakshmanan, A. Gahlawat, and N. Hovakimyan, “Safe feedback motion planning: A contraction theory and ℒ1subscriptℒ1\mathcal{L}_{1}caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-adaptive control based approach,” in Proc. 59th IEEE Conf. on Decision and Control, 2020, pp. 1578–1583.
  23. D. Lee, L. Aolaritei, T. L. Vu, and K. Turitsyn, “Robustness against disturbances in power systems under frequency constraints,” IEEE Trans. Control Network Systems, vol. 6, no. 3, pp. 971–979, 2019.
  24. M. Y. Li and J. S. Muldowney, “On R. A. Smith’s autonomous convergence theorem,” Rocky Mountain J. Math., vol. 25, no. 1, pp. 365–378, 1995.
  25. M. R. Liberzon, “Lur’e problem of absolute stability - a historical essay,” IFAC Proc. Volumes, vol. 34, no. 6, pp. 25–28, 2001, 5th IFAC Symposium on Nonlinear Control Systems 2001, St Petersburg, Russia, 4-6 July 2001.
  26. W. Lohmiller and J.-J. E. Slotine, “On contraction analysis for non-linear systems,” Automatica, vol. 34, pp. 683–696, 1998.
  27. K. A. Lurie, “Some recollections about Anatolii Isakovich Lurie,” IFAC Proc. Volumes, vol. 34, no. 6, pp. 35–38, 2001, 5th IFAC Symp. Nonlinear Control Systems, St. Petersburg, Russia.
  28. I. R. Manchester and J.-J. E. Slotine, “Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design,” IEEE Trans. Automat. Control, vol. 62, no. 6, pp. 3046–3053, 2017.
  29. ——, “Combination properties of weakly contracting systems,” 2015. [Online]. Available: https://arxiv.org/abs/1408.5174
  30. M. Margaliot, E. D. Sontag, and T. Tuller, “Entrainment to periodic initiation and transition rates in a computational model for gene translation,” PLoS ONE, vol. 9, no. 5, p. e96039, 2014.
  31. M. Margaliot, “Stability analysis of switched systems using variational principles: An introduction,” Automatica, vol. 42, no. 12, pp. 2059–2077, 2006.
  32. M. Margaliot, E. D. Sontag, and T. Tuller, “Contraction after small transients,” Automatica, vol. 67, pp. 178–184, 2016.
  33. J. S. Muldowney, “Compound matrices and ordinary differential equations,” Rocky Mountain J. Math., vol. 20, no. 4, pp. 857–872, 1990.
  34. H. D. Nguyen, T. L. Vu, J.-J. Slotine, and K. Turitsyn, “Contraction analysis of nonlinear DAE systems,” IEEE Trans. Automat. Control, vol. 66, no. 1, 2021.
  35. R. Ofir and M. Margaliot, “Multiplicative and additive compounds via Kronecker products and Kronecker sums,” 2024, submitted. [Online]. Available: https://arxiv.org/abs/2401.02100
  36. R. Ofir, M. Margaliot, Y. Levron, and J.-J. Slotine, “Serial interconnections of 1-contracting and 2-contracting systems,” in Proc. 60th IEEE Conf. on Decision and Control, 2021, pp. 3906–3911.
  37. ——, “A sufficient condition for k𝑘kitalic_k-contraction of the series connection of two systems,” IEEE Trans. Automat. Control, vol. 67, no. 9, pp. 4994–5001, 2022.
  38. R. Ofir, A. Ovseevich, and M. Margaliot, “Contraction and k-contraction in Lurie systems with applications to networked systems,” Automatica, vol. 159, p. 111341, 2024.
  39. Q.-C. Pham and J.-J. Slotine, “Stable concurrent synchronization in dynamic system networks,” Neural Networks, vol. 20, no. 1, pp. 62–77, 2007.
  40. A. V. Proskurnikov, A. Davydov, and F. Bullo, “The Yakubovich S-Lemma revisited: Stability and contractivity in non-Euclidean norms,” SIAM J. Control Optim., vol. 61, no. 4, pp. 1955–1978, 2023.
  41. H. Qiao, J. Peng, and Z.-B. Xu, “Nonlinear measures: a new approach to exponential stability analysis for Hopfield-type neural networks,” IEEE Trans. Neural Networks, vol. 12, no. 2, pp. 360–370, 2001.
  42. M. Revay, R. Wang, and I. R. Manchester, “Recurrent equilibrium networks: Flexible dynamic models with guaranteed stability and robustness,” IEEE Trans. Automat. Control, 2023, to appear.
  43. ——, “Recurrent equilibrium networks: Flexible dynamic models with guaranteed stability and robustness,” IEEE Trans. Automat. Control, pp. 1–16, 2023.
  44. G. Russo, M. di Bernardo, and E. D. Sontag, “Global entrainment of transcriptional systems to periodic inputs,” PLOS Computational Biology, vol. 6, p. e1000739, 2010.
  45. G. Russo, M. di Bernardo, and E. Sontag, “A contraction approach to the hierarchical analysis and design of networked systems,” IEEE Trans. Automat. Control, vol. 58, no. 5, pp. 1328–1331, 2013.
  46. G. Russo, M. di Bernardo, and J. J. Slotine, “Contraction theory for systems biology,” in Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology, H. Koeppl, G. Setti, M. di Bernardo, and D. Densmore, Eds.   New York, NY: Springer, 2011, pp. 93–114.
  47. S. Singh, B. Landry, A. Majumdar, J.-J. Slotine, and M. Pavone, “Robust feedback motion planning via contraction theory,” Int. J. Robotics Research, 2023.
  48. D. Sun, S. Jha, and C. Fan, “Learning certified control using contraction metric,” in Proc. 2020 Conf. on Robot Learning, vol. 155, 2021, pp. 1519–1539.
  49. H. Tsukamoto, S.-J. Chung, and J.-J. E. Slotine, “Neural stochastic contraction metrics for learning-based control and estimation,” IEEE Control Systems Letters, vol. 5, no. 5, pp. 1825–1830, 2021.
  50. W. Wang and J. J. E. Slotine, “On partial contraction analysis for coupled nonlinear oscillators,” Biological Cybernetics, vol. 92, no. 1, 2005.
  51. P. M. Wensing and J.-J. Slotine, “Beyond convexity—contraction and global convergence of gradient descent,” PLOS ONE, vol. 15, no. 8, pp. 1–29, 08 2020.
  52. C. Wu, I. Kanevskiy, and M. Margaliot, “k𝑘kitalic_k-contraction: theory and applications,” Automatica, vol. 136, p. 110048, 2022.
  53. C. Wu, R. Pines, M. Margaliot, and J.-J. Slotine, “Generalization of the multiplicative and additive compounds of square matrices and contraction theory in the Hausdorff dimension,” IEEE Trans. Automat. Control, vol. 67, no. 9, pp. 4629–4644, 2022.
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