Spectral Koopman Method for Identifying Stability Boundary (2312.06885v1)
Abstract: The paper is about characterizing the stability boundary of an autonomous dynamical system using the Koopman spectrum. For a dynamical system with an asymptotically stable equilibrium point, the domain of attraction constitutes a region consisting of all initial conditions attracted to the equilibrium point. The stability boundary is a separatrix region that separates the domain of attraction from the rest of the state space. For a large class of dynamical systems, this stability boundary consists of the union of stable manifolds of all the unstable equilibrium points on the stability boundary. We characterize the stable manifold in terms of the zero-level curve of the Koopman eigenfunction. A path-integral formula is proposed to compute the Koopman eigenfunction for a saddle-type equilibrium point on the stability boundary. The algorithm for identifying stability boundary based on the Koopman eigenfunction is attractive as it does not involve explicit knowledge of system dynamics. We present simulation results to verify the main results of the paper.
- P. A. Parrilo, “Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization,” Ph.D. dissertation, California Institute of Technology, Pasadena, CA, 2000.
- P. Bevanda, M. Beier, S. Kerz, A. Lederer, S. Sosnowski, and S. Hirche, “Diffeomorphically learning stable Koopman operators,” IEEE Control Systems Letters, vol. 6, pp. 3427–3432, 2022.
- H.-D. Chiang, M. Hirsch, and F. Wu, “Stability regions of nonlinear autonomous dynamical systems,” IEEE Transactions on Automatic Control, vol. 33, no. 1, pp. 16–27, 1988.
- B. Hou, S. Bose, and U. Vaidya, “Sparse learning of kernel transfer operators,” Asilomar Conference on Signals, Systems & Computers, 2021.
- S. Klus, I. Schuster, and K. Muandet, “Eigendecompositions of transfer operators in reproducing kernel hilbert spaces,” Journal of Nonlinear Science, vol. 30, no. 1, pp. 283–315, 2020.
- X. Ma, B. Huang, and U. Vaidya, “Optimal quadratic regulation of nonlinear system using koopman operator,” in 2019 American Control Conference (ACC), 2019, pp. 4911–4916.
- M. Korda and I. Mezić, “On convergence of extended dynamic mode decomposition to the Koopman operator,” Journal of Nonlinear Science, vol. 28, no. 2, pp. 687–710, 2018.
- ——, “Optimal construction of Koopman eigenfunctions for prediction and control,” IEEE Transactions on Automatic Control, vol. 65, no. 12, pp. 5114–5129, 2020.
- A. Kumar, B. Umathe, U. Vaidya, and A. Kelkar, “Safe Operating Limits of Vehicle Dynamics Under Parameter Uncertainty Using Koopman Spectrum1,” ASME Letters in Dynamic Systems and Control, vol. 3, no. 2, p. 021008, 10 2023. [Online]. Available: https://doi.org/10.1115/1.4063479
- A. Lasota and J. Yorke, “On the existence of invariant measure for piecewise monotonic transformations,” Trans. A.M.S., vol. 186, pp. 481–488, 1973.
- I. Mezić, “Spectrum of the Koopman operator, spectral expansions in functional spaces, and state-space geometry,” Journal of Nonlinear Science, vol. 30, no. 5, pp. 2091–2145, 2020.
- ——, “Koopman operator, geometry, and learning of dynamical systems,” Not. Am. Math. Soc, vol. 68, no. 7, pp. 1087–1105, 2021.
- S. A. Deka, S. S. Narayanan, and U. Vaidya, “Path-integral formula for computing Koopman eigenfunctions,” arXiv preprint arXiv:2307.06805, Accpted for publication in IEEE CDC, 2023.
- S. P. Nandanoori, S. Sinha, and E. Yeung, “Data-driven operator theoretic methods for phase space learning and analysis,” Journal of Nonlinear Science, vol. 32, no. 6, p. 95, 2022.
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