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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Bayesian Neural Networks Kalman Filtering Uncertainty Quantification Robot State Estimation

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State estimations and noise identifications with intermittent corrupted observations via Bayesian variational inference

This paper focuses on the state estimation problem in distributed sensor networks, where intermittent packet dropouts, corrupted observations, and unknown noise covariances coexist. To tackle this challenge, we formulate the joint estimation of system states, noise parameters, and network reliability as a Bayesian variational inference problem, and propose a novel variational Bayesian adaptive Kalman filter (VB-AKF) to approximate the joint posterior probability densities of the latent parameters. Unlike existing AKF that separately handle missing data and measurement outliers, the proposed VB-AKF adopts a dual-mask generative model with two independent Bernoulli random variables, explicitly characterizing both observable communication losses and latent data authenticity. Additionally, the VB-AKF integrates multiple concurrent multiple observations into the adaptive filtering framework, which significantly enhances statistical identifiability. Comprehensive numerical experiments verify the effectiveness and asymptotic optimality of the proposed method, showing that both parameter identification and state estimation asymptotically converge to the theoretical optimal lower bound with the increase in the number of sensors.

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Robust Estimation Covariance Estimation Variational Inference

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Unscented KalmanNet: Structure-Preserving Deep Learning with Calibrated Posterior Uncertainty under Incomplete Physics and Unknown Noise

Nonlinear state estimation requires sequentially fusing model-based predictions with noisy measurements. Under imperfect dynamics and unknown, time-varying noise statistics, this fusion can degrade in both accuracy and statistical consistency. Existing learning-aided filters largely treat accuracy and uncertainty estimation separately, limiting their ability to correct model-mismatch-induced bias while retaining an explicit, calibrated posterior covariance. This paper introduces Unscented KalmanNet (UKN), a model-based deep learning architecture that extends the Unscented Kalman Filter (UKF) with learned mechanisms for these two sources of filtering error while preserving explicit posterior covariance propagation. NoiseNet learns time-varying process and measurement covariances as bounded multiplicative corrections to baseline covariances, guaranteeing positive definiteness, while GainNet learns a bounded residual correction to the analytical UKF gain to compensate for model-mismatch-induced bias. A calibration-aware training objective couples state error with posterior covariance and innovation consistency terms through adaptive weighting, jointly optimizing accuracy and calibration. UKN is benchmarked against UKF, KalmanNet, and Bayesian KalmanNet on three synthetic systems and real-flight UZH-FPV data. It achieves the lowest state-estimation error in all four examples and reduces RMSE by 26.4-49.7% compared with UKF in the synthetic cases. Leave-one-sequence-out cross-validation over 11 flights shows 22.4% and 34.3% reductions in mean position and velocity RMSE, respectively. UKN also yields the lowest fold-to-fold variability, with dimension-normalized NEES and empirical coverage closest to nominal values among covariance-reporting filters. These results show that structured learned adaptation improves estimation accuracy while retaining calibrated uncertainty.

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Department of Civil and Construction Engineering and Management, The University of Texas at Tyler, Tyler, TX, 75799, USA · Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, Columbus, OH, 43210, USA

Kalman Filtering Unscented Kalman Filtering Covariance Estimation Robot State Estimation Uncertainty Calibration

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