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Data-Driven Approximations of Dynamical Systems Operators for Control

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The Koopman Operator in Systems and Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 484))

Abstract

The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.

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Acknowledgements

EK gratefully acknowledges support by the “Washington Research Foundation Fund for Innovation in Data-Intensive Discovery” and a Data Science Environments project award from the Gordon and Betty Moore Foundation (Award #2013-10-29) and the Alfred P. Sloan Foundation (Award #3835) to the University of Washington eScience Institute, and funding through the Mistletoe Foundation. SLB and JNK acknowledge support from the Defense Advanced Research Projects Agency (DARPA contract PA-18-01-FP-125). SLB acknowledges support from the Army Research Office (W911NF-17-1-0306 and W911NF-17-1-0422). JNK acknowledges support from the Air Force Office of Scientific Research (FA9550-19-1-0011).

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  1. Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195, USA

    Eurika Kaiser & Steven L. Brunton

  2. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195, USA

    J. Nathan Kutz

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  1. Department of Mathematics, University of Namur, Namur, Belgium

    Alexandre Mauroy

  2. Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, CA, USA

    Igor Mezić

  3. Department of Electrical and Information Systems, Osaka Prefecture University, Sakai, Japan

    Yoshihiko Susuki

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Kaiser, E., Kutz, J.N., Brunton, S.L. (2020). Data-Driven Approximations of Dynamical Systems Operators for Control. In: Mauroy, A., Mezić, I., Susuki, Y. (eds) The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-030-35713-9_8

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