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

By Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, J. Nathan Kutz. PLoS ONE, 2015

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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.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a3c279828af3621d2c16ac26e5900b970383f60e

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Citations in the corpus

Variational Approach for Learning Markov Processes from Time Series Data (2017, background): “An approach that can be applied to deterministic systems and for stochastic systems with additive white noise is to set g (x) = x, and then choose f as the transformation with smallest modeling error (Brunton et al. 2016a, b).”
Deep learning for universal linear embeddings of nonlinear dynamics (2017, methodology): “Identifying regression models based on nonlinear measurements will generally result in closure issues, as there is no guarantee that these measurements form a Koopman invariant subspace (38).”
Multi-scale proper orthogonal decomposition of complex fluid flows (2018, methodology): “An introduction to the Koopman framework is proposed by Brunton et al. (2016a), while extensive reviews on its connection to the DMD are proposed by Mezić (2013), Budǐsić et al. (2012) and Tu et al. (2014).”
Higher Order Dynamic Mode Decomposition (2017, background): “2) implies that the snapshots lie in the invariant subspace under the action of the Koopman group [58, 11], generated by R.”
Machine Learning Control – Taming Nonlinear Dynamics and Turbulence (2016, background): “This is a vibrant area of research, with many promising connections to dynamical systems via the Koopman operator [162, 163, 187, 47, 170, 188, 41], although this is beyond the scope of this book.”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, neutral): “[22] proposed the use of observables selected in a data-driven manner [23] from a function dictionary.”
Chaos as an intermittently forced linear system (2016, background): “A linear model, however detailed, cannot capture multiple fixed points or the unpredictable behavior characteristic of chaos with a positive Lyapunov exponent [15].”
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control (2016, methodology): “For a related work on extending Koopman operator methods to controlled dynamical systems, see [2, 18, 19, 24].”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “In a related modeling framework, we have developed an algorithm for the sparse identification of nonlinear dynamics (SINDY) from data (Brunton et al., 2016b), relying on the fact that most dynamical systems of interest have relatively few nonlinear terms in the dynamics out of the family of…”
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”
Data-driven spectral analysis of the Koopman operator (2017, neutral): “For example, global stability is analyzed in [27], whereas [28] deals with the so-called isostables and isochrones; ergodic partition and mixing properties are analyzed in [7], and [18, 6] utilize the Koopman operator approximations for control whereas [29] for model reduction.”
Linear Model Regression on Time-series Data: Non-asymptotic Error Bounds and Applications (2018, methodology): “In the context of data-driven analysis of dynamical systems, Koopman analysis has also been used for operatortheoretic identification of nonlinear systems and their spectral properties [9]–[11].”