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Kalman-Bucy Filtering with Randomized Sensing: Fundamental Limits and Sensor Network Design for Field Estimation

Xinyi Wang, Devansh R. Agrawal, Dimitra Panagou

TL;DR

This work develops a continuous-time framework to quantify Kalman–Bucy filtering performance under randomized sensing, where both the measurement matrix and noise covariance vary randomly. It derives a closed-form upper bound on the expected covariance via a CARE using the averaged information matrix \\bar{G} and provides a tractable, isotropic special-case solution for the steady state. The framework is applied to spatiotemporal field estimation with Gaussian-process dynamics, culminating in a grid-independent, fundamental performance limit governed by a single sensing parameter \\theta = N_r/(\\sigma_m^2 \\Delta t) that captures the trade-off among sensor count, noise, and sampling rate. These results yield principled guidelines for pre-deployment sensor-network design and illustrate how to balance resources to meet a prescribed estimation uncertainty (clarity).

Abstract

Stability analysis of the Kalman filter under randomly lost measurements has been widely studied. We revisit this problem in a general continuous-time framework, where both the measurement matrix and noise covariance evolve as random processes, capturing variability in sensing locations. Within this setting, we derive a closed-form upper bound on the expected estimation covariance for continuous-time Kalman filtering. We then apply this framework to spatiotemporal field estimation, where the field is modeled as a Gaussian process observed by randomly located, noisy sensors. Using clarity, introduced in our earlier work as a rescaled form of the differential entropy of a random variable, we establish a grid-independent lower bound on the spatially averaged expected clarity. This result exposes fundamental performance limits through a composite sensing parameter that jointly captures the effects of the number of sensors, noise level, and measurement frequency. Simulations confirm that the proposed bound is tight for the discrete-time Kalman filter, approaching it as the measurement rate decreases, while avoiding the recursive computations required in the discrete-time formulation. Most importantly, the derived limits provide principled and efficient guidelines for sensor network design problem prior to deployment.

Kalman-Bucy Filtering with Randomized Sensing: Fundamental Limits and Sensor Network Design for Field Estimation

TL;DR

This work develops a continuous-time framework to quantify Kalman–Bucy filtering performance under randomized sensing, where both the measurement matrix and noise covariance vary randomly. It derives a closed-form upper bound on the expected covariance via a CARE using the averaged information matrix \\bar{G} and provides a tractable, isotropic special-case solution for the steady state. The framework is applied to spatiotemporal field estimation with Gaussian-process dynamics, culminating in a grid-independent, fundamental performance limit governed by a single sensing parameter \\theta = N_r/(\\sigma_m^2 \\Delta t) that captures the trade-off among sensor count, noise, and sampling rate. These results yield principled guidelines for pre-deployment sensor-network design and illustrate how to balance resources to meet a prescribed estimation uncertainty (clarity).

Abstract

Stability analysis of the Kalman filter under randomly lost measurements has been widely studied. We revisit this problem in a general continuous-time framework, where both the measurement matrix and noise covariance evolve as random processes, capturing variability in sensing locations. Within this setting, we derive a closed-form upper bound on the expected estimation covariance for continuous-time Kalman filtering. We then apply this framework to spatiotemporal field estimation, where the field is modeled as a Gaussian process observed by randomly located, noisy sensors. Using clarity, introduced in our earlier work as a rescaled form of the differential entropy of a random variable, we establish a grid-independent lower bound on the spatially averaged expected clarity. This result exposes fundamental performance limits through a composite sensing parameter that jointly captures the effects of the number of sensors, noise level, and measurement frequency. Simulations confirm that the proposed bound is tight for the discrete-time Kalman filter, approaching it as the measurement rate decreases, while avoiding the recursive computations required in the discrete-time formulation. Most importantly, the derived limits provide principled and efficient guidelines for sensor network design problem prior to deployment.

Paper Structure

This paper contains 27 sections, 15 theorems, 108 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Problem Formulation
  4. Continuous-Time Upper Bound and Analytical Analysis
  5. Continuous-Time Upper Bound of Expected Covariance
  6. Closed-form Solution of Upper Bound
  7. Application to Spatiotemporal Estimation
  8. Spatiotemporal Environment
  9. Spatiotemporal Gaussian Process Kalman Filter
  10. Estimation Performance
  11. Numerical Evaluation of STGPKF and Clarity
  12. Steady-State Performance Analysis for STGPKF
  13. Discrete-Time Upper Bound for STGPKF
  14. Continuous-time Upper Bound for STGPKF
  15. Continuous-time Lower Bound of Averaged Expected Clarity
  16. ...and 12 more sections

Key Result

Lemma 1

The operator $\mathcal{R}_{G}$ is concave, that is, for any $\bm{\Sigma}_1, \bm{\Sigma}_2 \in \mathcal{S}_{++}^n$ and $\beta \in [0, 1]$,

Figures (6)

  • Figure 1: Wind field reconstruction and clarity assessment using STGPKF for a single realization with $\bm{\Delta} s = 0.25$ and $N_r = 20$. The red–blue color map denotes wind flow direction, and the color intensity represents the wind speed magnitude.
  • Figure 2: Spatially averaged clarity $\bar{q}_k$ over time for different numbers of agents.
  • Figure 3: Comparison of continuous-time $\bm{\Delta}(t)$ and discrete-time $\bm{\Delta}_k$ covariance upper bounds with respect to the temporal resolution $\Delta t$.
  • Figure 4: Steady-state lower bound of averaged expected clarity $\bar{q}_{\Delta^{\Pi}_\infty}$ versus spatial grid size $N_g$. Each curve corresponds to a different $\theta(N_r)$, representing a different number of sensors.
  • Figure 5: $\bar{q}_{\mathbb{E}[\Pi]}$ and $\bar{q}_{\Delta^{\Pi}_\infty}$ versus the number of sensors.
  • ...and 1 more figures

Theorems & Definitions (39)

  • Lemma 1: Concavity
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1: Continuous-Time Upper Bound
  • proof
  • Corollary 1
  • Corollary 2
  • ...and 29 more