Identification of noisy Koopman models

This work proposes a data-driven Koopman modeling approach for nonlinear systems that takes into account the effect of noise, where the underlying noise distribution, autocorrelations, and governing differential equations are unknown. The method uses block coordinate descent to solve a regularized least squares problem by utilizing the extended dynamic mode decomposition (eDMD) framework for linearization and the log-likelihood characterization of the noise. Specifically, it learns a mapping to lift the nonlinear system to a higher-dimensional approximated linear system while also identifying the noise components in the lifted space. Furthermore, we prove theoretically that the identified model, when given new noisy data, can be used to predict the succeeding states with a guaranteed upper bound on the prediction error. Case studies are performed on a Van der Pol oscillator and a four-stage binary distillation column where unknown noise is present.