The Effect of Sensor Fusion on Data-Driven Learning of Koopman Operators

Dictionary methods for system identification typically rely on one set of measurements to learn governing dynamics of a system. In this paper, we investigate how fusion of output measurements with state measurements affects the dictionary selection process in Koopman operator learning problems. While prior methods use dynamical conjugacy to show a direct link between Koopman eigenfunctions in two distinct data spaces (measurement channels), we explore the specific case where output measurements are nonlinear, non-invertible functions of the system state. This setup reflects the measurement constraints of many classes of physical systems, e.g., biological measurement data, where one type of measurement does not directly transform to another. We propose output constrained Koopman operators (OC-KOs) as a new framework to fuse two measurement sets. We show that OC-KOs are effective for sensor fusion by proving that when learning a Koopman operator, output measurement functions serve to constrain the space of potential Koopman observables and their eigenfunctions. Further, low-dimensional output measurements can be embedded to inform selection of Koopman dictionary functions for high-dimensional models. We propose two algorithms to identify OC-KO representations directly from data: a direct optimization method that uses state and output data simultaneously and a sequential optimization method. We prove a theorem to show that the solution spaces of the two optimization problems are equivalent. We illustrate these findings with a theoretical example and two numerical simulations.