Abstract
We develop a new generalization of Koopman operator theory that incorporates the e ects 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 coherent, spatio-temporal modes from data, connects local-linear analysis to nonlinear operator theory, and importantly creates an equation-free architecture for the study of complex systems. For 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 to a recent development called dynamic mode decomposition with control. 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).
Highlights
We introduce a new method called Koopman with inputs and control (KIC) that generalizes Koopman spectral theory to allow for the analysis of complex, inputoutput systems
We demonstrate how Koopman is fundamentally connected to dynamic mode decomposition with control (DMDc), a recently developed extension of DMD for input-output systems [36] which has already been successfully applied to simulation data of a rapidly pitching airfoil [10]
Koopman operator theory and DMD offer a data-driven method to characterizing complex systems [31, 42]
Summary
We introduce a new method called Koopman with inputs and control (KIC) that generalizes Koopman spectral theory to allow for the analysis of complex, inputoutput systems. We demonstrate how Koopman is fundamentally connected to dynamic mode decomposition with control (DMDc), a recently developed extension of DMD for input-output systems [36] which has already been successfully applied to simulation data of a rapidly pitching airfoil [10]. DMD has been shown to be intimately connected to ERA and OKID as well as other subspace identification methods such as the numerical algorithms for subspace state space system identification (N4SID) [38, 48, 36] In this manuscript, we will establish the connection between KIC and DMDc for linear input-output systems.
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