Conservative fusion of unbiased partial state estimates: CI is optimal

TL;DR

This paper proves that Covariance Intersection (CI) is the optimal conservative unbiased linear fusion rule for combining partial state estimates, extending its known optimality from full to partial estimates and establishing equivalence with certain optimization problems.

Contribution

It generalizes the optimality of CI to the fusion of partial state estimates and shows the equivalence of multiple optimization formulations for this problem.

Findings

CI is optimal among conservative unbiased linear fusion rules for partial estimates.

Three optimization problems are shown to be equivalent, including a semi-definite program.

Conditions for solvability and optimal solutions are provided for specific cost functions.

Abstract

We show that Covariance Intersection (CI) is optimal amongst all conservative unbiased linear fusion rules also in the general case of information fusion of two unbiased partial state estimates, significantly generalizing the known optimality result for fusion of full state estimates. In fact, we prove the much stronger result that three different optimization problems are equivalent, namely the abstract optimal conservative unbiased linear information fusion problem with respect to a strictly isotone cost function, the scalar Covariance Intersection (CI) problem, and a simple semi-definite program (SDP). We provide a general solvability condition for these problems as well as equations characterizing the optimal solutions for the matrix determinant and matrix trace cost functions.