  In this paper, we propose a distributed state-and-fault estimation scheme for
multi-agent systems. The proposed estimator is based on an $\ell_1$-norm
optimization problem, which is inspired by sparse signal recovery in the field
of compressive sampling. Two theoretical results are given to analyze the
correctness of the proposed approach. First, we provide a necessary and
sufficient condition such that state and fault signals are correctly estimated.
The result presents a fundamental limitation of the algorithm, which shows how
many faulty nodes are allowed to ensure a correct estimation. Second, we
provide a sufficient condition for the estimation error of fault signals when
numerical errors of solving the optimization problem are present. An
illustrative example is given to validate the effectiveness of the proposed
approach.
