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

Xinyi Wang; Devansh R. Agrawal; Dimitra Panagou

eess.SYOct 29, 2025

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

Authors: Xinyi Wang, Devansh R. Agrawal, Dimitra Panagou

Organizations: Department of Robotics, University of Michigan, Ann Arbor, MI 48109, USA · Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA · Departments of Robotics and Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA

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 lower bound on the expected spatially averaged clarity and show that its steady-state value converges to a grid-independent limit as the grid is refined. 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 show close agreement between the continuous-time analytical solution and the discrete-time covariance, with the agreement improving as the sampling interval decreases. The analytical solution also avoids the recursive computations required in the discrete-time formulation. Most importantly, the derived limits provide principled and efficient guidelines for sensor network design prior to deployment.

Figures & tables

Symbol	Definition
Covariance-related symbols
$\bm{\Sigma}$	Covariance of the latent-state estimation error $\bm{s}-\hat{\bm{s}}$ .
$\bar{\bm{\Sigma}}=\mathbb{E}[\bm{\Sigma}]$	Expected covariance of the latent-state estimation error under stochastic sensing.
$\bm{\Delta}$	Upper bound of $\bar{\bm{\Sigma}}$ .
$\bm{\Pi}$	Covariance of the field-state estimation error $\bm{f}-\hat{\bm{f}}$ .
$\bar{\bm{\Pi}}=\mathbb{E}[\bm{\Pi}]$	Expected covariance of the field-state estimation error under stochastic sensing.

Table 1: Summary of covariance- and clarity-related symbols.

Panel of Figure 1 — Figure 1: STGPKF field-estimation results. (a) Wind field reconstruction and clarity assessment for a single realization with spatial grid spacing $\delta=0.25\,\mathrm{km}$ and $N_{r}=20$ . The red–blue color map denotes wind flow direction, and the color intensity represents the wind speed magnitude. (b) Spatially averaged clarity $\bar{q}_{k}$ over time for different numbers of agents, with one representative randomized sensor-deployment realization shown for each agent count.

Parameter	Symbol / Value	Unit
Temporal kernel std.	$\sigma_{t}=2.0$	km/min
Temporal length scale	$l_{t}=60$	min
Spatial kernel std.	$\sigma_{s}=1.0$	km/min
Spatial length scale	$l_{s}=2.0$	km
Spatial resolution	$\delta=0.5$	km
Spatial domain	$[0,5]\times[0,5]$	km 2

Table 2: Simulation parameters.

Figure 2: Covariance comparison ( $T=120\,\mathrm{min}$ ): the spatially averaged covariance metric $\mathcal{L}(\cdot)$ evaluated for the Monte Carlo covariance estimate and the discrete- and continuous-time bounds at $\Delta t=0.02\,\mathrm{min}$ (left), and maximum relative errors in Eqs. ( 48 )–( 49 ) versus $\Delta t$ (right).

Figure 3: Steady-state lower bound on expected spatially averaged 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 agents.

Panel of Figure 4 — Figure 4: Sensor-network design results. (a) The estimate $\widehat{\mathbb{E}}[\bar{q}]$ and the steady-state lower bound $\bar{q}_{\Delta^{\Pi}_{\infty}}$ versus the number of sensors. (b) Steady-state continuous-time lower bound on expected spatially averaged clarity $\bar{q}_{\Delta^{\Pi}_{\infty}}$ as a function of the measurement noise variance $\sigma_{m}^{2}$ and normalized sampling interval $\Delta t/N_{r}$ , with $N_{r}=1$ . Dashed curves indicate constant $\theta$ , equivalently constant $\sigma_{c}^{2}=\sigma_{m}^{2}\Delta t$ .

$q_{\text{target}}$	$N_{r}$	$\widehat{\mathbb{E}}[\bar{q}]$	$\bar{q}_{\Delta^{\Pi}_{\infty}}$
0.50	1	0.522	0.528
0.60	3	0.628	0.632
0.70	7	0.709	0.712
0.80	21	0.801	0.803
0.90	112	0.899	0.900

Table 3: The estimate

\widehat{\mathbb{E}}[\bar{q}]

and

\bar{q}_{\Delta^{\Pi}_{\infty}}

versus the number of sensors.

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Neural Aided Kalman Filtering for UAV State Estimation in Degraded Sensing Environments

Akhil Gupta, Erhan Guven

Whiting School of Engineering EP Program, Johns Hopkins University, Baltimore, Maryland, USA

Bayesian Neural Networks Kalman Filtering Uncertainty Quantification Robot State Estimation

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Accurate state estimation of nonlinear dynamical systems is fundamental to modern aerospace operations across air, sea, and space domains. Online tracking of adversarial unmanned aerial vehicles (UAVs) is especially challenging due to agile nonlinear motion, noisy and sparse sensor measurements, and unknown control inputs; conditions that violate key assumptions of classical Kalman filter variants and degrade estimation performance. Neural networks (NNs) can learn complex nonlinear relationships from data, but lack principled uncertainty quantification, which is critical for state estimation tasks where confidence bounds drive downstream decisions. We address this with Bayesian Neural Networks (BNNs), which model uncertainty through distributions over network weights and produce predictive means and uncertainties via Monte Carlo sampling. Building on this, we propose the Bayesian Neural Kalman Filter (BNKF): a hybrid framework coupling a trained BNN with a Kalman correction step for robust online UAV state estimation. Unlike related neural Kalman approaches, BNKF produces full state predictions and incorporates Bayesian uncertainty directly into covariance propagation, improving robustness under high noise conditions. We evaluate BNKF under varying radar noise levels and sampling rates using synthetic nonlinear UAV flight data. Five fold cross validation demonstrates that BNKF outperforms Extended and Unscented Kalman Filters in accuracy, precision, and truth containment under degraded sensing. An ensemble variant (BNKFe) further improves precision in high-noise edge cases at a slight accuracy tradeoff. Runtime analysis confirms minimal inference overhead, supporting real-time deployment feasibility.

Abstract

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