Necessary and Sufficient Conditions for Consensus of Double-Integrator Multi-Agent Systems With Measurement Noises

An average consensus protocol is proposed for continuous-time double-integrator multi-agent systems with measurement noises under fixed topologies. The time-varying control gain is employed to attenuate noises. The closed-loop system is therefore a time-varying linear stochastic differential equation. By determining the state transition matrix of this closed-loop system, the dynamic characteristics of the multi-agent system can be fully described. It is proved that in the noisy communication environment the average consensus can be achieved if and only if the communication topology is a balanced and strongly connected graph, and the time-varying control gain satisfies the stochastic approximation-type conditions. Under the proposed protocol, the position of each agent is convergent in mean square to a common random variable whose mathematical expectation is the average of initial positions and initial velocities of all agents in the system, while each agent's velocity is convergent in mean square to a common random variable whose mathematical expectation and variance are both zero.