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 NyquistShannon-like critical frequency for the exact identification of the CT nonlinear dynamical system, which has Koopman invariant spaces. First, it establishes a sufficient condition for a sampling frequency that permits a discretized sequence of samples to discover the underlying system. Second, it establishes a necessary condition for the sampling frequency to avoid system aliasing, which would render the underlying system distinguishable. Finally, unlike the requirement in the NyquistShannon 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.