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A Unified Family-optimal Solution to Covariance Intersection Problems with Semidefinite Programming

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

Abstract

Covariance intersection (CI) methods provide a principled approach to fusing estimates with unknown cross-correlations by minimizing a worst-case measure of uncertainty that is consistent with the available information. This paper introduces a generalized CI framework, called overlapping covariance intersection (OCI), which unifies several existing CI formulations within a single optimization-based framework. This unification enables the characterization of family-optimal solutions for multiple CI variants, including standard CI and split covariance intersection (SCI), as solutions to a semidefinite program, for which efficient off-the-shelf solvers are available. When specialized to the corresponding settings, the proposed family-optimal solutions recover the state-of-the-art family-optimal solutions previously reported for CI and SCI. The resulting formulation facilitates the systematic design and real-time implementation of CI-based fusion methods in large-scale distributed estimation problems, such as cooperative localization.

A Unified Family-optimal Solution to Covariance Intersection Problems with Semidefinite Programming

Abstract

Covariance intersection (CI) methods provide a principled approach to fusing estimates with unknown cross-correlations by minimizing a worst-case measure of uncertainty that is consistent with the available information. This paper introduces a generalized CI framework, called overlapping covariance intersection (OCI), which unifies several existing CI formulations within a single optimization-based framework. This unification enables the characterization of family-optimal solutions for multiple CI variants, including standard CI and split covariance intersection (SCI), as solutions to a semidefinite program, for which efficient off-the-shelf solvers are available. When specialized to the corresponding settings, the proposed family-optimal solutions recover the state-of-the-art family-optimal solutions previously reported for CI and SCI. The resulting formulation facilitates the systematic design and real-time implementation of CI-based fusion methods in large-scale distributed estimation problems, such as cooperative localization.
Paper Structure (11 sections, 8 theorems, 19 equations)

This paper contains 11 sections, 8 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. Unifying Problem
  5. Overlapping Covariance Intersection Problem
  6. Efficient Family-optimal Solution
  7. Covariance Intersection
  8. Split Covariance Intersection
  9. Conclusion
  10. Proof of Lemma \ref{['lem:kahan_conservative']}
  11. Proof of Theorem \ref{['th:OCI_Kahan_optimal_Rz']}

Key Result

Lemma 1

The Kahan family defined in eq:kahan_family parameterizes bounds on $\mathcal{P}$ defined in eq:Pcal_def, in the sense that $\mathcal{P} \!\subseteq \!\mathcal{P}_{\mathrm{KF}}(\boldsymbol{\omega})$ for all $\boldsymbol{\omega} \!\in\! \Delta^M\!$.

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Theorem 1: $\!$PedrosoBatistaEtAl2025OCI
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • Remark 2
  • Theorem 4
  • Proposition 1: $\!\!$PedrosoBatistaEtAl2025OCI
  • ...and 4 more