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

TL;DR

This paper introduces a sampling theorem based on Koopman operator theory that precisely determines the conditions for exact identification of continuous-time nonlinear dynamical systems from sampled data, relaxing traditional band-limited assumptions.

Contribution

It provides a necessary and sufficient sampling frequency condition for exact system identification using Koopman invariant subspaces, extending classical sampling theory to nonlinear systems.

Findings

Established a Nyquist-Shannon-like critical frequency for nonlinear systems

Demonstrated the criterion on linear and nonlinear simulated examples

Showed that the original signals need not be band-limited for exact identification

Abstract

Low sampling frequency challenges the exact identification of the continuous-time (CT) dynamical system from sampled data, even when its model is identifiable. The necessary and sufficient condition is proposed -- which is built from Koopman operator -- to the exact identification of the CT system from sampled data. The condition gives a Nyquist-Shannon-like critical frequency for exact identification of CT nonlinear dynamical systems with Koopman invariant subspaces: 1) it establishes a sufficient condition for a sampling frequency that permits a discretized sequence of samples to discover the underlying system and 2) it also establishes a necessary condition for a sampling frequency that leads to system aliasing that the underlying system is indistinguishable; and 3) the original CT signal does not have to be band-limited as required in the Nyquist-Shannon Theorem. The theoretical criterion has been demonstrated on a number of simulated examples, including linear systems, nonlinear systems with equilibria, and limit cycles.