Distributed Subspace Consensus in the Presence of Dynamic In-Network Disturbance

This paper studies distributed strategies for average-consensus of arbitrary vectors in multiagent systems, when the interagent information exchange is corrupted by the agents states within the same network. In particular, each neighboring state received by an agent has an additive component that consists of projections of the states at other agents; the agents corrupting this exchange are unknown to the receiving agent and may also change over time. We model such in-network disturbance with a dynamic disturbance graph over the agents, in addition to the static graph over which consensus is implemented. The problem in its full generality is quite challenging and in an attempt to simplify, we assume two particular disturbance cases: 1) sender based and 2) receiver based. In the former case, we assume that the (null spaces of the) projection subspaces are only known at the senders; while in the latter case, we assume this knowledge only at the receivers. We provide a concrete example of static, flat-fading multiple-input multiple-output channels to support this disturbance model. In the aforementioned context, we cast an algebraic structure over the disturbance subspaces and show that the average is reachable in a subspace whose dimension is complementary to the maximal dimension of the disturbance subspaces. To develop the results, we introduce the notion of information alignment to align the intended message to the null space of the unintended disturbance. We derive the conditions under which this alignment is invertible, that is, the intended message can be recovered. A major contribution of this work is to show that local protocols exist for (subspace) consensus even when the disturbance over the network spans the entire vector space.