Optimal estimation in spatially distributed systems: how far to share measurements from?

TL;DR

This paper investigates how measurement sharing in spatially distributed systems affects optimal estimation, revealing conditions for decentralization and introducing a new analytical technique for decay rate analysis.

Contribution

It provides detailed analysis of spatial decay rates in optimal estimators for invariant systems and introduces the Branch Point Locus method for quantifying these decay rates.

Findings

Decay rates depend on system dynamics and noise correlations.

Matching conditions lead to fully decentralized estimators.

The Branch Point Locus technique quantifies spatial decay analytically.

Abstract

We consider the centralized optimal estimation problem in spatially distributed systems. We use the setting of spatially invariant systems as an idealization for which concrete and detailed results are given. Such estimators are known to have a degree of spatial localization in the sense that the estimator gains decay in space, with the spatial decay rates serving as a proxy for how far measurements need to be shared in an optimal distributed estimator. In particular, we examine the dependence of spatial decay rates on problem specifications such as system dynamics, measurement and process noise variances, as well as their spatial autocorrelations. We propose non-dimensional parameters that characterize the decay rates as a function of problem specifications. In particular, we find an interesting matching condition between the characteristic lengthscale of the dynamics and the measurement noise correlation lengthscale for which the optimal centralized estimator is completely decentralized. A new technique - termed the Branch Point Locus - is introduced to quantify spatial decay rates in terms of analyticity regions in the complex spatial frequency plane. Our results are illustrated through two case studies of systems with dynamics modeled by diffusion and the Swift-Hohenberg equation, respectively.