Data-driven discovery of Koopman eigenfunctions for control

By Eurika Kaiser, J. Nathan Kutz, Steven L. Brunton. Machine Learning: Science and Technology, 2017

407 citations recorded by the source corpus.

Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Previous studies have used finite-dimensional approximations of the Koopman operator for model-predictive control approaches. In this work, we illustrate a fundamental closure issue of this approach and argue that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigenfunctions. These coordinates form a Koopman-invariant subspace by design and, thus, have improved predictive power. We show then how the control can be formulated directly in these intrinsic coordinates and discuss potential benefits and caveats of this perspective. The resulting control architecture is termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). It is further demonstrated that these eigenfunctions can be approximated with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. Validating discovered eigenfunctions is crucial and we show that lightly damped eigenfunctions may be faithfully extracted from EDMD or an implicit formulation. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. KRONIC is then demonstrated on a number of relevant examples, including (a) a nonlinear system with a known linear embedding, (b) a variety of Hamiltonian systems, and (c) a high-dimensional double-gyre model for ocean mixing.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b6030388f7bfb6dec208db9b08e80a6c3e968e79

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Citations in the corpus

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control (2019, methodology): “The relationship between conservation laws and Koopman eigenfunctions has recently been exploited in [19, 20], where conserved quantities are learned from data.”
Resurrecting Recurrent Neural Networks for Long Sequences (2023, methodology): “Several options exist for extending Koopman operator theory to systems with inputs (Surana, 2016; Proctor et al., 2018; Kaiser et al., 2021; Korda & Mezić, 2020).”
Koopman Lyapunov‐based model predictive control of nonlinear chemical process systems (2019, methodology): “This is critical because, in practice, EDMD models are known to have closure issues as there is no guarantee that the selected observables form Koopman invariant subspaces.(13,14,23) This is especially true for the systems with multiple steady states (as a linear system can only characterize a singl”
Error bounds for kernel-based approximations of the Koopman operator (2023, methodology): “Koopman methods have found widespread application in system identification [5], control [29, 49, 30, 22, 56], sensor placement [38], molecular dynamics [57, 51, 42, 43, 23, 64], and many other fields.”