Mean square average-consensus under measurement noises and fixed topologies: Necessary and sufficient conditions

In this paper, average-consensus control is considered for networks of continuous-time integrator agents under fixed and directed topologies. The control input of each agent can only use its local state and the states of its neighbors corrupted by white noises. To attenuate the measurement noises, time-varying consensus gains are introduced in the consensus protocol. By combining the tools of algebraic graph theory and stochastic analysis, the convergence of these kinds of protocols is analyzed. Firstly, for noise-free cases, necessary and sufficient conditions are given on the network topology and consensus gains to achieve average-consensus. Secondly, for the cases with measurement noises, necessary and sufficient conditions are given on the consensus gains to achieve asymptotic unbiased mean square average-consensus. It is shown that under the protocol designed, all agents’ states converge to a common Gaussian random variable, whose mathematical expectation is just the average of the initial states.