Necessary and sufficient conditions for solving leader-following problem of multi-agent systems with communication noises

The leader-following problem of first-order integral multi-agent systems with communication noises is investigated in this paper. To attenuate the noise´s effect, a positive time-varying gain a(t) is employed in the protocol. It is proved that the proposed protocol can solve the mean square leader-following problem if the following conditions hold: 1) the communication topology graph has a spanning tree; 2) ∫₀^∞ a(t)dt = ∞; 3) lim_t→∞ a(t) = 0. The requirements on a(t) are different from most existing papers, where a(t) is required to satisfy that ∫₀^∞ a(t)dt = ∞ and ∫₀^∞ a²(t)dt <; ∞. It turns out that ∫₀^∞ a²(t)dt <; ∞ implies lim_t→∞ a(t) = 0, if a(t) is uniformly continuous. Therefore this paper relaxes the requirements on a(t) to some extent. In addition, under the mild condition (a(t) is uniformly continuous) these three conditions are necessary as well. Furthermore, if ∫₀^∞ a²(t)dt <; ∞, the employed protocol is proved to be able to solve the almost sure leader-following problem of first-order integral multi-agent system. Finally, a simulation example is provided to verify the effectiveness of the employed protocols.