  This paper considers the multi-dimensional consensus in networked systems,
where some of the agents might be misbehaving (or faulty). Despite the
influence of these misbehaviors, the benign agents aim to reach an agreement
while avoiding being seriously influenced by the faulty ones. To this end, this
paper first considers a general class of consensus algorithms, where each
benign agent computes an "auxiliary point" based on the received values and
moves its state toward this point. Concerning this generic form, we present
conditions for achieving resilient consensus and obtain a lower bound on the
exponential convergence rate. Assuming that the number of malicious agents is
upper bounded, two specific resilient consensus algorithms are further
developed based on the obtained conditions. Particularly, the first solution,
based on Helly's Theorem, achieves the consensus within the convex hull formed
by the benign agents' initial states, where the auxiliary point can be
efficiently computed through linear programming. On the other hand, the second
algorithm serves as a "built-in" security guarantee for standard average
consensus algorithms, in the sense that its performance coincides exactly with
that of the standard ones in the absence of faulty nodes while also resisting
the serious influence of the misbehaving ones in adversarial environment. Some
numerical examples are provided in the end to verify the theoretical results.
