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Overlapping Covariance Intersection: Fusion with Partial Structural Knowledge of Correlation from Multiple Sources

Leonardo Pedroso, Pedro Batista, W. P. M. H. Heemels

Abstract

Emerging large-scale engineering systems rely on distributed fusion for situational awareness, where agents combine noisy local sensor measurements with exchanged information to obtain fused estimates. However, at the sheer scale of these systems, tracking cross-correlations becomes infeasible, preventing the use of optimal filters. Covariance intersection (CI) methods address fusion problems with unknown correlations by minimizing worst-case uncertainty based on available information. Existing CI extensions exploit limited correlation knowledge but cannot incorporate structural knowledge of correlation from multiple sources, which naturally arises in distributed fusion problems. This paper introduces Overlapping Covariance Intersection (OCI), a generalized CI framework that accommodates this novel information structure. We formalize the OCI problem and establish necessary and sufficient conditions for feasibility. We show that a family-optimal solution can be computed efficiently via semidefinite programming, enabling real-time implementation. The proposed tools enable improved fusion performance for large-scale systems while retaining robustness to unknown correlations.

Overlapping Covariance Intersection: Fusion with Partial Structural Knowledge of Correlation from Multiple Sources

Abstract

Emerging large-scale engineering systems rely on distributed fusion for situational awareness, where agents combine noisy local sensor measurements with exchanged information to obtain fused estimates. However, at the sheer scale of these systems, tracking cross-correlations becomes infeasible, preventing the use of optimal filters. Covariance intersection (CI) methods address fusion problems with unknown correlations by minimizing worst-case uncertainty based on available information. Existing CI extensions exploit limited correlation knowledge but cannot incorporate structural knowledge of correlation from multiple sources, which naturally arises in distributed fusion problems. This paper introduces Overlapping Covariance Intersection (OCI), a generalized CI framework that accommodates this novel information structure. We formalize the OCI problem and establish necessary and sufficient conditions for feasibility. We show that a family-optimal solution can be computed efficiently via semidefinite programming, enabling real-time implementation. The proposed tools enable improved fusion performance for large-scale systems while retaining robustness to unknown correlations.
Paper Structure (17 sections, 11 theorems, 47 equations, 6 figures)

This paper contains 17 sections, 11 theorems, 47 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivating Example
  3. State-of-the-art
  4. Contributions
  5. Notation
  6. Problem Formulation
  7. Overlapping Covariance Intersection
  8. Analysis of Partial Knowledge
  9. A State-of-the-art Approach
  10. Efficient OCI Solution
  11. Conclusion
  12. Proofs
  13. Proof of Lemma \ref{['lem:inverse_bounds']}
  14. Proof of Lemma \ref{['lem:boundedness']}
  15. Proof of Theorem \ref{['th:equiv_OCI_prob']}
  16. ...and 2 more sections

Key Result

Lemma 1

The set $\mathcal{P}$ of admissible matrices $\mathbf{P}$ can be expressed as $\mathcal{P} := \{\mathbf{P} \in\! S_{++}^m \!:\mathbf{P}^{-1} \!\succeq \mathbf{Y}_b \;\forall b \in \{1,2,\ldots,M\} \}$, where $\mathbf{Y}_b := \mathbf{W}_b^\top\mathbf{X}_b^{-1} \mathbf{W}_b$ for $b = 1,2,\ldots, M$.

Figures (6)

  • Figure 1: Scheme of the cooperative localization toy problem.
  • Figure 2: Comparison of the estimation error covariance matrix with three estimates for distinct partial information structures. Dashed contours represent knowledge of bounds and solid contours represent exact knowledge.
  • Figure 3: Illustrative OCI bounds and admissible set $\mathcal{P}\subset S_{++}^2$.
  • Figure 4: Illustrative OCI bounds and admissible set $\mathcal{P}\subset S_{++}^3$.
  • Figure 5: Illustrative trace-minimization OCI solution for two-dimensional scenario in Example \ref{['eg:bounds']}.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Example II.1
  • Remark II.1
  • Remark II.2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Remark III.1
  • Example III.1
  • ...and 20 more