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Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control

Author

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  • Steven L Brunton
  • Bingni W Brunton
  • Joshua L Proctor
  • J Nathan Kutz

Abstract

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.

Suggested Citation

  • Steven L Brunton & Bingni W Brunton & Joshua L Proctor & J Nathan Kutz, 2016. "Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control," PLOS ONE, Public Library of Science, vol. 11(2), pages 1-19, February.
  • Handle: RePEc:plo:pone00:0150171
    DOI: 10.1371/journal.pone.0150171
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    Cited by:

    1. Nassir Cassamo & Jan-Willem van Wingerden, 2020. "On the Potential of Reduced Order Models for Wind Farm Control: A Koopman Dynamic Mode Decomposition Approach," Energies, MDPI, vol. 13(24), pages 1-21, December.
    2. Riccardo Colantuono & Riccardo Colantuono & Massimiliano Mazzanti & Michele Pinelli, 2023. "Aviation and the EU ETS: an overview and a data-driven approach for carbon price prediction," SEEDS Working Papers 0123, SEEDS, Sustainability Environmental Economics and Dynamics Studies, revised Feb 2023.
    3. Jamiree Harrison & Enoch Yeung, 2021. "Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators," Mathematics, MDPI, vol. 9(23), pages 1-25, December.
    4. Oster, Matthew R. & King, Ethan & Bakker, Craig & Bhattacharya, Arnab & Chatterjee, Samrat & Pan, Feng, 2023. "Multi-level optimization with the koopman operator for data-driven, domain-aware, and dynamic system security," Reliability Engineering and System Safety, Elsevier, vol. 237(C).
    5. Keita Hara & Masaki Inoue, 2021. "Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design," Mathematics, MDPI, vol. 9(9), pages 1-18, April.
    6. Chen, Zhong & Chen, Xiaofang & Liu, Jinping & Cen, Lihui & Gui, Weihua, 2024. "Learning model predictive control of nonlinear systems with time-varying parameters using Koopman operator," Applied Mathematics and Computation, Elsevier, vol. 470(C).
    7. Mallen, Alex T. & Lange, Henning & Kutz, J. Nathan, 2024. "Deep Probabilistic Koopman: Long-term time-series forecasting under periodic uncertainties," International Journal of Forecasting, Elsevier, vol. 40(3), pages 859-868.
    8. Camilo Garcia-Tenorio & Alain Vande Wouwer, 2022. "A Matlab Toolbox for Extended Dynamic Mode Decomposition Based on Orthogonal Polynomials and p-q Quasi-Norm Order Reduction," Mathematics, MDPI, vol. 10(20), pages 1-18, October.

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