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Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

Author

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  • Zlatko Drmač

    (Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia)

  • Igor Mezić

    (Department of Mechanical Engineering and Mathematics, University of California, Santa Barbara, CA 93106, USA)

  • Ryan Mohr

    (AIMdyn, Inc., Santa Barbara, CA 93101, USA)

Abstract

Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems.

Suggested Citation

  • Zlatko Drmač & Igor Mezić & Ryan Mohr, 2021. "Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method," Mathematics, MDPI, vol. 9(17), pages 1-29, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:17:p:2075-:d:623371
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    References listed on IDEAS

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    1. Jan-Hendrik Niemann & Stefan Klus & Christof Schütte, 2021. "Data-driven model reduction of agent-based systems using the Koopman generator," PLOS ONE, Public Library of Science, vol. 16(5), pages 1-23, May.
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