Approximating the Lower Bound on the Arithmetic Average-Based Distributed State Estimation

In multisensor state estimation, existing posterior Cramér–Rao lower bounds (PCRLBs) are generally derived under the measurement independence assumption. When correlations among measurements are unknown, the true PCRLB cannot be computed due to the inability to compute the likelihood function corresponding to the local measurements. Motivated by the random-variables-based arithmetic average (RV-AA) fusion rule, which combines the local variables with unknown correlations as a linear mixture variable (LMV), an LMV-based approximation to the performance lower bound (LMV-A-LB) on RV-AA is derived in this article to provide a more refined performance indicator for it. This approximation uses the posterior obtained from the LMV along with the Bayes' rule to estimate the true but unavailable likelihood function. We show that the LMV-A-LB generally lacks a closed-form solution, so it is approximated using sequential Monte Carlo approaches. Furthermore, by limiting the system model to the commonly used additive Gaussian noise case, we compute the exact expression of the derived bound with the aid of the Kalman filters. Numerical simulation experiments demonstrate that the LMV-A-LB obtains a tighter performance lower bound for RV-AA fusion by comparing with existing bounds.