Resilient Leader-Follower Consensus to Arbitrary Reference Values

Abstract

The problem of consensus in the presence of misbehaving agents has increasingly attracted attention in the literature. Prior results have established algorithms and graph structures for multi-agent networks which guarantee the consensus of normally behaving agents in the presence of a bounded number of misbehaving agents. The final consensus value is guaranteed to fall within the convex hull of initial agent states. However, the problem of consensus tracking considers consensus to arbitrary reference values which may not lie within such bounds. Conditions for consensus tracking in the presence of misbehaving agents has not been fully studied. This paper presents conditions for a network of agents using the W-MSR algorithm to achieve this objective.

A New Approach for Distributed Hypothesis Testing with Extensions to Byzantine-Resilience

“…Pick any agent i ∈ L m (θ , θ), and notice that it has a neighbor j (say) from the set (m−1) q=0 L q (θ , θ) at some time-step τ ∈ [(m − 1)T, mT ). The induction hypothesis coupled with (35) implies that c j,τ (θ) ≤ τ , and hence c i,τ +1 (θ) ≤ c j,τ (θ) + 1 ≤ τ + 1 based on (31). Appealing to (35) then reveals that c i,mT (θ) ≤ mT , thus completing the induction step.…”

“…The induction hypothesis coupled with (35) implies that c j,τ (θ) ≤ τ , and hence c i,τ +1 (θ) ≤ c j,τ (θ) + 1 ≤ τ + 1 based on (31). Appealing to (35) then reveals that c i,mT (θ) ≤ mT , thus completing the induction step. Finally, noting that (n−1) q=0 L q (θ , θ) = V completes our proof of the claim (33).…”

“…One can then easily prove via induction that c i,(n−1+r)T (θ) ≤ rT, ∀i ∈ C r (θ , θ), where 1 ≤ r ≤ (n − 1). The rest then follows from (35). We can repeat the above argument to establish that c i,m(n−1)T (θ) ≤ (n − 1)T, ∀i ∈ V, ∀m ∈ N + .…”

“…We can repeat the above argument to establish that c i,m(n−1)T (θ) ≤ (n − 1)T, ∀i ∈ V, ∀m ∈ N + . Finally, based on the above bound and (35), it follows that for each agent i ∈ V, c i,t (θ) is upperbounded by 2(n − 1)T at any time-step t ∈ (m(n − 1)T, (m + 1)(n − 1)T ), where m ∈ N + . This establishes (32) and completes the proof.…”

“…Finally, we emphasize that the non-adversarial agents are unaware of the identities of the adversaries in their neighborhood. As is fairly standard in the distributed fault-tolerant literature [29][30][31][32][33][34][35][36], we only assume that non-adversarial agents know the upper bound f on the number of adversaries in their neighborhood. The adversarial set will be denoted by A ⊂ V, and the remaining agents R = V \ A will be called the regular agents.…”