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Efficient Sensor Fusion Through Covariance-Constrained Observation Decimation (CCOD)
Andres Enriquez Fernandez, Erik Blasch, Angel Flores-Abad, John J. Bird
TL;DR
Observation decimation reduces sensing and assimilation requirements but makes steady-state covariance prediction difficult because standard DARE inputs omit covariance growth between updates. The paper introduces CCOD, which uses equivalent decimated system and process-noise matrices in a single DARE to predict covariance and select allowable decimation. Experiments on a high-dimensional linear system and space-object tracking demonstrate covariance-constrained decimation, including a maximum factor d = 39 in the tracking case.
Problem
Standard DARE evaluation does not directly predict steady-state covariance for decimated Kalman filters, while lifting and periodic Riccati formulations add computational complexity.
Method
CCOD replaces the DARE system and process-noise matrices with equivalent decimated matrices that capture covariance growth between measurement updates.
Results
The experiments maintain steady-state covariance below specified thresholds while maximizing measurement-update decimation; the space-object case permits d = 39 with a maximum P− value of 0.00049 km^2.
Takeaways & Limitations
A single DARE evaluation can predict DMU steady-state covariance and determine the minimum measurement assimilation frequency satisfying a prescribed covariance bound.
Abstract
from arXiv · showhide
Observation decimation is frequently employed in state estimation to reduce sensing, communication, and computational requirements, but decreasing the measurement assimilation frequency increases estimation uncertainty. Selecting an appropriate observation decimation factor therefore requires accurately predicting the resulting estimator performance. While the discrete algebraic Riccati equation (DARE) provides the steady-state estimation-error covariance for standard linear time-invariant Kalman filters, it is not directly applicable to estimators employing decimated measurement updates. Existing approaches address this limitation through lifted system representations or periodic Riccati equation formulations, both of which incur additional computational complexity. This paper presents a covariance-constrained observation decimation (CCOD) framework that reformulates the DARE inputs using equivalent decimated system and process-noise matrices that capture covariance growth between measurement updates. The proposed reformulation enables direct prediction of the steady-state estimation-error covariance through a single DARE evaluation without increasing the system dimension or solving coupled periodic Riccati equations. The resulting covariance prediction is used to determine the maximum observation decimation factor that satisfies a prescribed estimation uncertainty bound. Validation using a high dimensional linear time-invariant system and a space object tracking application demonstrates that the proposed approach accurately predicts steady-state estimator performance while reducing the measurement assimilation frequency required to satisfy specified covariance constraints.
I. Introduction
Decimated measurement updates reduce assimilation frequency but complicate steady-state covariance prediction because skipped updates allow uncertainty to grow. CCOD reformulates DARE inputs with equivalent decimated matrices that preserve system dimension and support direct covariance prediction.
- Motivation: DMU estimators perform consecutive state propagations before assimilating observations, addressing resource, environmental, and communication constraints.Observations are assimilated only when k mod d = 0, with d denoting the decimation interval.
- Motivation: Monte Carlo simulation can assess dropped-observation effects but is computationally expensive and unsuitable for runtime decimation selection.Steady-state covariance provides a more efficient measure of long-term estimator performance for linear time-invariant systems.
- Prior limitations: The standard DARE is not directly applicable to DMU estimators, while lifting and periodic Riccati approaches increase dimensionality or computational complexity.These limitations motivate a direct reformulation for decimated systems.
- CCOD approach: CCOD replaces A and Q with equivalent A_dec and Q_dec matrices that capture state evolution and covariance growth across skipped measurement updates.H and R remain unchanged because they are not involved in the time update.
- CCOD approach: Substituting A_dec and Q_dec into one standard DARE yields the steady-state a priori covariance P∞dec without increasing system dimensionality or solving coupled periodic Riccati equations.The matrices are constructed from covariance propagation over d consecutive prediction steps without intermediate measurement updates.
- CCOD approach: The decimated covariance formulation builds on multi-step state and covariance propagation to represent deterministic trajectories and process-noise-driven uncertainty growth between updates.Setting the arbitrary propagation interval i equal to d produces the decimated matrices.
IV. Experiment and Case Study
The experiments use steady-state covariance prediction to select the largest permissible observation-decimation factor while enforcing a user-specified covariance threshold.
- Selection criterion: The maximum permissible decimation factor is selected by iteratively evaluating P∞dec for increasing d until any covariance diagonal element exceeds the threshold.The constraint is governed by the maximum variance among individual states.
A. Arbitrary System Experiment
The arbitrary-system experiment evaluates CCOD on a scalable 20-state linear system designed with complex, near-marginally stable dynamics and controlled structural properties.
- System design: The synthetic system is scalable and designed to isolate controllability, observability, and stability while testing complex oscillatory and marginally stable dynamics.This construction enables controlled evaluation of decimated-filter boundaries.
- System design: The system matrix has n = 20 states with complex eigenvalues whose magnitudes satisfy 0.84 < ρ < 0.99.The selected eigenvalue parameters keep the dynamics close to the discrete-time stability limit.
- Structural properties: Only half of the states receive direct input, while a full-rank controllability matrix confirms full controllability.The input matrix assigns unity gain to the selected directly actuated states.
- Structural properties: Only half of the states are directly observed, and a full-rank observability matrix confirms full observability.Full controllability and observability imply the stabilizability and detectability conditions needed for steady-state covariance convergence.
- Noise model: Measurement noise is Gaussian with variance 0.1, while process noise on directly actuated states is Gaussian with variance 1 and yields Q = BΣωBᵀ.These settings define the synthetic experiment's noise covariances.
B. Space Object Tracking Case Study
The space-object case study evaluates CCOD on a linearized relative-tracking model derived from orbital propagation and expressed in the local RTN frame.
- Tracking model: The case study models proximity operations in which a chaser estimates a target's relative state using independently propagated SGP4 trajectories.The target is initialized with a stable in-track phase separation relative to the ISS.
- Tracking model: Transforming the nonlinear ECI trajectories into the local Radial, Along-Track, Cross-Track frame produces a continuous LTI model.The model is governed by linearized Clohessy-Wiltshire equations.
- Tracking model: The estimator uses a first-order discrete Euler integration of a constant 6 × 6 system matrix.The state is defined by relative kinematics.
- Noise model: The measurement-noise covariance is set to 10 m^2 as a conservative baseline for vision-based position measurements.The process-noise covariance represents omitted nonlinearities, orbital perturbations, and Earth-oblateness J2 effects.
C. Finding the Maximum Measurement Decimation Factor
The maximum permissible decimation factor is found by iteratively evaluating the decimated DARE covariance as the factor increases, then implementing a filter with the selected factor.
- The decimated DARE is evaluated for increasing d until any diagonal element of P∞dec exceeds the user-specified threshold.
- The selected decimation factor satisfying the covariance condition is then used in a discrete-time Kalman filter with DMU to evaluate P− evolution.
V. Results
The results evaluate CCOD using a generalized arbitrary linear system and a relative space-object tracking case, tracking covariance convergence under decimation.
- CCOD performance is demonstrated with a generalized arbitrary linear system and a relative space-object tracking case.Covariance behavior is tracked to verify convergence to a steady-state value under decimation factor d.
A. Arbitrary System
The arbitrary 20-state system permits decimation factor 21 under a variance threshold of 7.0, with simulated covariance converging just below that limit.
- Arbitrary System: d=21 is permissible for the arbitrary 20-state system under a maximum allowable variance threshold of 7.0.Vertical dotted red lines indicate the time steps when measurement updates occur.
- Arbitrary System: State variances bifurcate within the initial 0.025 seconds because directly measured states are reduced at t=0 while covariances propagate through single-step dynamics.
- Arbitrary System: 6.9995 is the maximum steady-state variance after 5 seconds, remaining below the specified threshold of 7.0 and validating the modified DARE prediction.
- Arbitrary System: Dynamics coupling causes the two initial variance groups to intermix during subsequent propagation steps.
- Space Object Tracking: The space-object tracking setup uses a 1 Hz non-decimated sensor measurement frequency as its standard sampling rate for vision-based position measurements.
- Space Object Tracking: A maximum allowable variance threshold of 0.0005 is specified for the tracking case.The units are km2 for relative position states or km2 s2 for relative velocity states.
- Space Object Tracking: d=39 is the resulting maximum decimation factor for the tracking case.
- Space Object Tracking: The along-track P− variance at the end of the run is 0.00049 km2, confirming that CCOD satisfies the covariance bound.
VI. Conclusion
The conclusion frames DMU as a trade-off between reduced data assimilation and increased covariance, and presents CCOD as a direct covariance-prediction approach validated on linear systems and tracking.
- DMU reduces assimilated data but increases estimator covariance, requiring DARE input reformulation to account for growth between measurement updates.
- CCOD develops decimated system and process-noise matrices that account for covariance growth without observations and proves their use with the DARE predicts DMU steady-state covariance.
- Experiments show DMU estimators maintain steady-state covariance below a desired threshold while maximizing decimation and reducing sensor-data assimilation in linear systems and space-object tracking.Extensions to nonlinear systems are left for future work.
Appendix: Structurally controlled System Matrix Design
The appendix constructs a scalable real-valued system matrix by combining controlled modal blocks with an orthogonal eigenvector transformation. Stability and oscillatory behavior are specified through eigenvalue placement relative to the discrete-time unit circle.
- System-matrix construction: An eigen-decomposition framework constructs A with controlled stability, damping, and oscillatory-mode profiles.The matrix is formed from an orthogonal eigenvector basis and a block-diagonal modal matrix.
- System-matrix construction: QR decomposition of a randomly initialized V0 produces an orthogonal eigenvector matrix with condition number 1.This improves numerical robustness and permits replacing V^-1 with V^T.
- Modal structure: The modal matrix Λ combines Nr scalar real eigenvalues and Nc 2 × 2 blocks for complex conjugate pairs, with n = Nr + 2Nc.Its block-diagonal structure is Λ = blockdiag(λ1, . . . , λNr, Λ1, . . . , ΛNc).
- Stability specification: The geometric placement of modal entries relative to the unit circle governs stability limits and modal frequencies.This provides direct control over oscillatory and non-oscillatory dynamics.
- Stability specification: Real-mode stability is set by eigenvalue magnitude: |λi| < 1 is asymptotically stable, |λi| = 1 marginal, and |λi| > 1 unstable.These scalar eigenvalues are assigned directly to the diagonal of Λ.
- Stability specification: Complex conjugate pairs are represented as real 2 × 2 rotation-scaling blocks whose spectral radius determines damping, marginal oscillation, or instability.The corresponding boundaries are ρ < 1, ρ = 1, and ρ > 1, respectively.
Compute the Real System Matrix
The final real-valued system matrix is computed through the matrix triple product, using real modal blocks and coordinate transformations. The work was partially funded by the Air Force Office of Scientific Research.
- Matrix computation: The final system matrix A is evaluated directly through the matrix triple product in Equation A1.Because the modal blocks are real-valued, the resulting system matrix is fully real-valued.
- Matrix computation: Real-valued coordinate transformations ensure that A remains fully real-valued when Λk ∈ R^2×2.This follows from constructing the modal matrix with real-valued rotation-scaling blocks.
- Funding: Portions of the work were funded by the Air Force Office of Scientific Research under award FA9550-24-1-0176.