Observation of nonlinear systems via finite capacity channels: Constructive data rate limits

The paper deals with observation of nonlinear and deterministic, though maybe chaotic, discrete-time systems via finite capacity communication channels. We introduce several minimum data-rate limits associated with various types of observability, and offer new tractable analytical techniques for their both upper and lower estimation. Whereas the lower estimate is obtained by following the lines of the Lyapunov's linearization method, the proposed upper estimation technique is along the lines of the second Lyapunov approach. As an illustrative example, the potential of the presented results is demonstrated for the system which describes a ball vertically bouncing on a sinusoidally vibrating table. For this system, we provide an analytical computation of a closed-form expression for the threshold that separates the channel data rates for which reliable observation is and is not possible, respectively. Another illustration is concerned with the celebrated Hénon system. The offered sufficient data rate bound is accompanied with a constructive observer that works whenever the channel capacity fits this bound.