Exponential state estimation, entropy and Lyapunov exponents

TL;DR

This paper explores the concept of estimation entropy in dynamical systems, relating it to Lyapunov exponents and entropy measures, with implications for state estimation under communication constraints.

Contribution

It establishes a connection between estimation entropy and $eta$-entropy, providing lower bounds based on Lyapunov exponents for measure-preserving systems.

Findings

Estimation entropy is related to $eta$-entropy.

Lower bounds are derived using Lyapunov exponents.

Results apply to Hamiltonian and symplectic systems.

Abstract

In this paper we study the notion of estimation entropy recently established by Liberzon and Mitra. This quantity measures the smallest rate of information about the state of a dynamical system above which an exponential state estimation with a given exponent is possible. We show that this concept is closely related to the $\alpha$-entropy introduced by Thieullen and we give a lower estimate in terms of Lyapunov exponents assuming that the system preserves an absolutely continuous measure with a bounded density, which includes in particular Hamiltonian and symplectic systems. Although in its current form mainly interesting from a theoretical point of view, our result could be a first step towards a more practical analysis of state estimation under communication constraints.