A Sampling Theorem for Exact Identification of Continuous-Time Nonlinear Dynamical Systems

Low sampling frequency challenges the exact identification of continuous-time (CT) dynamical systems from sampled data, even when their models are identifiable. A necessary and sufficient condition is proposed, built from the Koopman operator, for the exact identification of CT systems from sampled data. This condition provides a Nyquist-Shannon-like critical frequency for the exact identification of CT nonlinear dynamical system which has Koopman invariant spaces. Firstly, it establishes a sufficient condition for a sampling frequency that permits a discretized sequence of samples to discover the underlying system. Secondly, it establishes a necessary condition for the sampling frequency to avoid system aliasing, which would render the underlying system distinguishable. Lastly, unlike the requirement in the Nyquist-Shannon sampling theorem, the CT signal associated with the system need not be band-limited. The theoretical criterion has been demonstrated on a number of simulated examples, including linear systems, nonlinear systems with equilibria, and limit cycles.