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

2025-10-29

AI Summary

This work investigates the stability of continuous-time Kalman–Bucy filtering under stochastic sensing, where both the measurement matrix and noise covariance are random processes—inducing measurement dropouts and dynamic uncertainty. To address this, we propose a differential-entropy-based “clarity” metric and derive, for the first time, a closed-form upper bound on the expected estimation error covariance and a mesh-independent lower bound on the spatially averaged clarity. Our key innovation is the introduction of a composite sensing parameter that jointly characterizes sensor count, noise intensity, and sampling frequency—thereby revealing their fundamental trade-offs in estimation performance. The theoretical bounds are tight and empirically validated to approximate well even in discrete-time settings. Crucially, they avoid recursive computation, enabling efficient, interpretable pre-deployment design of sensor networks.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Establishes fundamental limits for Kalman filtering with random sensing variability
• Derives performance bounds for spatiotemporal field estimation using sensor networks
• Provides design guidelines for sensor networks prior to deployment

Innovation

• Continuous-time Kalman filtering with random sensing
• Closed-form upper bound on estimation covariance
• Grid-independent clarity bound for sensor networks