Limitations for nonlinear observation over erasure channel

TL;DR

This paper investigates the fundamental limitations of observing nonlinear systems over erasure channels, revealing how system dynamics and entropy constrain stability and observability.

Contribution

It establishes a new limitation criterion based on Lyapunov exponents and entropy, extending linear system results to nonlinear systems.

Findings

Limitations depend on erasure probability and Lyapunov exponents.

Positive Lyapunov exponents indicate inherent instability constraints.

Results unify nonlinear and linear system observation limitations.

Abstract

In this paper, we study the problem of state observation of nonlinear systems over an erasure channel. The notion of mean square exponential stability is used to analyze the stability property of observer error dynamics. The main results of this paper prove, fundamental limitation arises for mean square exponential stabilization of the observer error dynamics, expressed in terms of probability of erasure, and positive Lyapunov exponents of the system. Positive Lyapunov exponents are a measure of average expansion of nearby trajectories on an attractor set for nonlinear systems. Hence, the dependence of limitation results on the Lyapunov exponents highlights the important role played by non-equilibrium dynamics in observation over an erasure channel. The limitation on observation is also related to measure-theoretic entropy of the system, which is another measure of dynamical complexity. The limitation result for the observation of linear systems is obtained as a special case, where Lyapunov exponents are shown to emerge as the natural generalization of eigenvalues from linear systems to nonlinear systems.