Extending Data-Driven Koopman Analysis to Actuated Systems

In recent years, methods for data-driven Koopman spectral analysis, such as Dynamic Mode Decomposition (DMD), have become increasingly popular approaches for extracting dynamically relevant features from data sets. However to establish the connection between techniques like DMD or Extended DMD (EDMD) and the Koopman operator, assumptions are made about the nature of the supplied data. In particular, both methods assume the data were generated by an autonomous dynamical system, which can be limiting in certain experimental or computational settings, such as when system actuation is present. We present a modification of EDMD that overcomes this limitation by compensating for the effects of actuation, and is capable of recovering the leading Koopman eigenvalues, eigenfunctions, and modes of the unforced system. To highlight the efficacy of this approach, we apply it to two examples with (quasi)-periodic forcing: the first is the Duffing oscillator, which demonstrates eigenfunction approximation, and the second is a lattice Boltzmann code that approximates the FitzHugh-Nagumo partial differential equation and shows Koopman mode and eigenvalue computation.