  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.
