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

TL;DR

This article examines the dependence of spatial decay rates on problem specifications such as system dynamics, measurement, and process noise variances, as well as their spatial autocorrelations, and proposes nondimensional parameters that characterize the decay rates as a function of problem specifications.

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.

Figures (6)

• Figure 1: Notions of spatial locality. a) $f_1$ is a spatially distributed signal. b) $f_2$ is a Gaussian centered at the origin: spatially distributed, but localized (rapidly decaying). c) $f_3$ is a bump function, with compact support --- an example of decentralized or space-limited kernel. d) $f_4$ is a Dirac delta distribution, point-supported and thus, completely decentralized. Vertical axis is the same in all panels.
• Figure 2: Diffusion with scaled spatiotemporally white process and measurement noises. Color-code is consistent among panels. a) BPL: trajectories of the branch points \ref{['eq:diffusion_BP_whiteNoises']} as $l_*$ is varied (branch cuts omitted). Arrows indicate the direction of increasing $l_*$. Branch points $z_{1,2,3,4}$ and analyticity strip $\mathscr{S}$ for $l_*= 1$ in red. b) Normalized Fourier transform $\hat{L}_{\lambda}/\hat{L}_0$ against spatial frequency $\lambda$ for different values of $l_*$, as indicated. c) Normalized Kalman gain kernel $L(x)/L(0)$ against the spatial coordinate $x$, for different values of $l_*$, as given.
• Figure 3: Diffusion with spatially autocorrelated measurement noise. Colors according to different values of $\Pi_*$, as indicated. a) Branch points, branch cuts, and analyticity region of the extension of \ref{['eq:diffusion_KalmanGain_FourierSymbol_dimensionless']} in the complex plane. Kalman gain operator: b) $\hat{\mathsf{L}}_{\Lambda}$ as a function of spatial frequency $\Lambda$, and c) $\mathsf{L}$ as a function of the spatial coordinate $\mathsf{x}$. All variables are dimensionless. Vertical axis is preserved in each row.
• Figure 4: Diffusion with spatially autocorrelated measurement noise. a) BPL of the extension of \ref{['eq:diffusion_KalmanGain_FourierSymbol_dimensionless']}, colored according to $\Pi_*$. Arrows indicate increasing $\Pi_*$. b) Dimensionless decay rate $\Theta$\ref{['eq:diffusion_dimensionless_decayRate']} of $\mathcal{L}$. The area shaded in blue (red) corresponds to $\Theta$ being dominated by $l_*$ ($l_v$). c) Steady-state dimensionless performance $\mathsf{var}(\mathsf{e})$ of the Kalman filter plotted as a function of $\Pi_*$.
• Figure 5: Swift-Hohenberg dynamics with scaled white noises. a) Fourier symbol $\hat{\mathcal{A}}_{\Lambda}$ for different values of $\Pi_*$ (darker color corresponds to higher $\Pi_*$ value) plotted as a function of $\Lambda$. b) Trajectories of some of the branch points (as indicated) as a function of $\Pi_*$. Arrows indicate direction of increasing $\Pi_*$. c) $\Omega$ and $\Theta$ as defined in \ref{['eq:SH_Omega_Theta']}. (d) Filter performance as a function of $\Pi_*$. Variables in all panels are dimensionless.

Theorems & Definitions (24)

• Definition II.1: Translation operator, $\mathcal{T}_z$
• Definition II.2: Translation invariant operator, from Bamieh:2002
• Definition II.3: Multiplication operator
• Definition II.4: spatial Fourier transform
• Definition II.5: Extension to the complex plane
• Theorem II.1: Paley-Wiener Theorem, from Bamieh:2002 originally adapted from Hormander:1990
• Proposition
• Theorem : informal
• Definition IV.1: Completely decentralized Kalman filter
• Proposition IV.1: Condition for complete decentralization