In this paper, we focus on the distributed set-membership filtering (SMFing) problem for a multi-agent system with absolute (taken from agents themselves) and relative (taken from neighbors) measurements. In the literature, the relative measurements are difficult to deal with, and the SMFs highly rely on specific set descriptions. As a result, establishing the general distributed SMFing framework having relative measurements is still an open problem. To solve this problem, first, we provide the set description based on uncertain variables determined by the relative measurements between two agents as the foundation. Surprisingly, the accurate description requires only a single calculation step rather than multiple iterations, which can effectively reduce computational complexity. Based on the derived set description, called the uncertain range, we propose two distributed SMFing frameworks: one calculates the joint uncertain range of the agent itself and its neighbors, while the other only computes the marginal uncertain range of each local system. Furthermore, we compare the performance of our proposed two distributed SMFing frameworks and the benchmark -- centralized SMFing framework. A rigorous set analysis reveals that the distributed SMF can be essentially considered as the process of computing the marginal uncertain range to outer bound the projection of the uncertain range obtained by the centralized SMF in the corresponding subspace. Simulation results corroborate the effectiveness of our proposed distributed frameworks and verify our theoretical analysis.

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