Performance Analysis of Distributed Filtering under Mismatched Noise Covariances

TL;DR

The paper addresses distributed filtering with mismatched noise covariances in a sensor network by introducing three performance indices: the standard ${\Sigma}_{i,k|k}$, the nominal ${\Sigma}^f_{i,k|k}$, and the estimation-error ${\Sigma}^t_{i,k|k}$. It develops one-step and recursive relations among these indices, explicitly characterizing the influence of consensus fusion (via the fusion step $L$) on their interrelations. Under collective observability, it proves convergence of the nominal index to a discrete-time Riccati equation solution and of the estimation-error covariance to a Lyapunov equation solution, and provides a Frobenius-norm-based bound on the degradation caused by covariance deviations $\Delta Q$ and $\Delta R_i$. Simulations on a five-node network corroborate the theory and illustrate how increasing the fusion step attenuates consensus-induced discrepancies, guiding practical design of robust distributed filters. Overall, the work offers a rigorous framework to quantify and bound performance loss due to noise-covariance mismatches in consensus-based distributed filtering.

Abstract

This paper systematically investigates the performance of consensus-based distributed filtering under mismatched noise covariances. First, we introduce three performance evaluation indices for such filtering problems,namely the standard performance evaluation index, the nominal performance evaluation index, and the estimation error covariance. We derive difference expressions among these indices and establish one-step relations among them under various mismatched noise covariance scenarios. We particularly reveal the effect of the consensus fusion on these relations. Furthermore, the recursive relations are introduced by extending the results of the one-step relations. Subsequently, we demonstrate the convergence of these indices under the collective observability condition, and show this convergence condition of the nominal performance evaluation index can guarantee the convergence of the estimation error covariance. Additionally, we prove that the estimation error covariance of the consensus-based distributed filter under mismatched noise covariances can be bounded by the Frobenius norms of the noise covariance deviations and the trace of the nominal performance evaluation index. Finally, the effectiveness of the theoretical results is verified by numerical simulations.

Figures (8)

• Figure 1: The diagram of the communication topology in Example \ref{['example 1']}.
• Figure 2: The diagram of the communication topology.
• Figure 3: Illustration figure for the relations among different performance evaluation indices under $\Delta Q=0$ with the increasing fusion step $L$ in Case 1.
• Figure 4: Illustration figure for the relations among different performance evaluation indices under $\Delta Q=0$ and $\Delta R_i\geq 0$ with the increasing fusion step $L$ in Case 2.
• Figure 5: Illustration figure for the relations among different performance evaluation indices under $\Delta Q>0$ and $\Delta R_i= 0$ with the increasing fusion step $L$ in Case 3.

Theorems & Definitions (35)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Proposition 1
• Proposition 2
• Proposition 3