Principal component trajectories for modeling spectrally continuous dynamics as forced linear systems

Delay embeddings of time series data have emerged as a promising coordinate basis for data-driven estimation of the Koopman operator, which seeks a linear representation for observed nonlinear dynamics. Recent work has demonstrated the efficacy of dynamic mode decomposition (DMD) for obtaining finite-dimensional Koopman approximations in delay coordinates. In this paper we demonstrate how nonlinear dynamics with sparse Fourier spectra can be (i) represented by a superposition of principal component trajectories and (ii) modeled by DMD in this coordinate space. For continuous or mixed (discrete and continuous) spectra, DMD can be augmented with an external forcing term. We present a method for learning linear control models in delay coordinates while simultaneously discovering the corresponding exogenous forcing signal in a fully unsupervised manner. This extends the existing DMD with control (DMDc) algorithm to cases where a control signal is not known a priori. We provide examples to validate the learned forcing against a known ground truth and illustrate their statistical similarity. Finally, we offer a demonstration of this method applied to real-world power grid load data to show its utility for diagnostics and interpretation on systems in which somewhat periodic behavior is strongly forced by unknown and unmeasurable environmental variables.