Generalizing Koopman Theory to Allow for Inputs and Control

@article{Proctor2018GeneralizingKT,
  title={Generalizing Koopman Theory to Allow for Inputs and Control},
  author={Joshua L. Proctor and Steven L. Brunton and J. Nathan Kutz},
  journal={SIAM J. Appl. Dyn. Syst.},
  year={2018},
  volume={17},
  pages={909-930}
}
We develop a new generalization of Koopman operator theory that incorporates the effects of inputs and control. Koopman spectral analysis is a theoretical tool for the analysis of nonlinear dynamical systems. Moreover, Koopman is intimately connected to Dynamic Mode Decomposition (DMD), a method that discovers spatial-temporal coherent modes from data, connects local-linear analysis to nonlinear operator theory, and importantly creates an equation-free architecture allowing investigation of… 
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References

SHOWING 1-10 OF 62 REFERENCES
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control
TLDR
This work presents 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 and demonstrates the usefulness of nonlinear observable subspaces in the design of Koop man operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
TLDR
This approach is an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes, and if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation.
Dynamic Mode Decomposition with Control
TLDR
This work develops a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models.
Applied Koopmanism.
TLDR
The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice, and the paper highlights its strengths in applied and numerical contexts.
Spectral analysis of nonlinear flows
We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an…
Analysis of Fluid Flows via Spectral Properties of the Koopman Operator
This article reviews theory and applications of Koopman modes in fluid mechanics. Koopman mode decomposition is based on the surprising fact, discovered in Mezic (2005), that normal modes of linear…
Koopman-mode decomposition of the cylinder wake
  • S. Bagheri
  • Engineering
    Journal of Fluid Mechanics
  • 2013
Abstract The Koopman operator provides a powerful way of analysing nonlinear flow dynamics using linear techniques. The operator defines how observables evolve in time along a nonlinear flow…
Spectral Properties of Dynamical Systems, Model Reduction and Decompositions
In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of…
Reduced-order models for control of fluids using the eigensystem realization algorithm
TLDR
This article shows that using the Eigensystem Realization Algorithm (ERA) one can theoretically obtain exactly the same reduced-order models as by balanced POD, and demonstrates them on an example of the flow past an inclined flat plate at a low Reynolds number.
...
...