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Data-driven predictor of control-affine nonlinear dynamics: Finite discrete-time bilinear approximation of koopman operator

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  • Iman, Sara
  • Jahed-Motlagh, Mohammad-Reza

Abstract

This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models.

Suggested Citation

  • Iman, Sara & Jahed-Motlagh, Mohammad-Reza, 2025. "Data-driven predictor of control-affine nonlinear dynamics: Finite discrete-time bilinear approximation of koopman operator," Applied Mathematics and Computation, Elsevier, vol. 487(C).
  • Handle: RePEc:eee:apmaco:v:487:y:2025:i:c:s0096300324005290
    DOI: 10.1016/j.amc.2024.129068
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    References listed on IDEAS

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    1. Gholaminejad, Tahereh & Khaki-Sedigh, Ali, 2022. "Stable deep Koopman model predictive control for solar parabolic-trough collector field," Renewable Energy, Elsevier, vol. 198(C), pages 492-504.
    2. Bethany Lusch & J. Nathan Kutz & Steven L. Brunton, 2018. "Deep learning for universal linear embeddings of nonlinear dynamics," Nature Communications, Nature, vol. 9(1), pages 1-10, December.
    3. 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).
    4. 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.
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