A Mean Square Consensus Protocol for Linear Multi-Agent Systems With Communication Noises and Fixed Topologies

The mean square consensus of linear multi-agent systems with communication noises is studied in this note. Each agent is modeled by a continuous-time linear time-invariant dynamics and the fixed communication topology is described by a digraph. The proposed consensus protocol is composed of two parts: the agent's own state feedback and the relative states between agent and its neighbor agents. Due to the existence of communication noises, the relative states cannot be obtained accurately. To attenuate the noise effect, a time-varying gain vector a(t)K is applied to the inaccurate relative states. It is proved that: 1) if the communication topology has a spanning tree and every node has at least one parent node, then the proposed protocol can solve the mean square consensus problem if and only if a(t) satisfies ∫
₀
^∞
a(s)ds = ∞ and ∫
₀
^∞
a
²
(s)ds <; ∞; and all roots of the polynomial whose coefficients are the elements of vector K are in the left half complex plane; 2) if the communication topology has a spanning tree and there exists one node without any parent node, then the condition ∫
₀
^∞
a
²
(s)ds <; ∞ is only sufficient but not necessary; and 3) if the communication topology has no spanning tree, then the proposed protocol cannot solve the mean square consensus problem.