Containment control of multi-agent systems in a noisy communication environment

The containment problem of first-order and second-order integral multi-agent systems with communication noises is investigated in this paper. Motivated by the previous work, a time-varying gain a(t)a(t) is introduced to attenuate the noise effect. It is proved that the proposed protocol can solve the mean square containment problem if the following conditions hold: (1) the communication topology graph has a spanning forest whose roots are leaders of the multi-agent system; (2) ∫0∞a(t)dt=∞; (3) limt→∞a(t)=0limt→∞a(t)=0. Moreover, if a(t)a(t) is uniformly continuous, these conditions are necessary for ensuring the mean square containment. In addition, the proposed protocol is also extended to allow the case where each agent has its own gain ai(t)ai(t). It is proved that the mean square containment problem can still be solved if all agent-dependent gains are infinitesimals of the same order as time goes to infinity. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed protocols.