Generalizing Koopman Theory to allow for inputs and control

Joshua L. Proctor; Steven L. Brunton; J. Nathan Kutz

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arXiv · 1602.07647

Generalizing Koopman Theory to allow for inputs and control

Joshua L. Proctor·Steven L. Brunton·J. Nathan Kutz

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Abstract

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 complex systems. In actuated systems, standard Koopman analysis and DMD are incapable of producing input-output models; moreover, the dynamics and the modes will be corrupted by external forcing. Our new theoretical developments extend Koopman operator theory to allow for systems with nonlinear input-output characteristics. We show how this generalization is rigorously connected and generalizes a recent development called Dynamic Mode Decomposition with control (DMDc). We demonstrate this new theory on nonlinear dynamical systems, including a standard Susceptible-Infectious-Recovered model with relevance to the analysis of infectious disease data with mass vaccination (actuation).

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Joshua L. Proctor, Steven L. Brunton, J. Nathan Kutz. 2016-02-24. Generalizing Koopman Theory to allow for inputs and control. https://arxiv.org/abs/1602.07647

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