Robust Consensus of Multi-agent Systems with Noise

The consensus problem of multi-agent systems has attracted wide attention from researchers in recent years, following the initial work of Jadbabaie et al [1] on the analysis of a simplified Vicsek model. While the original Vicsek model contains noise effects, almost all the existing theoretical results on consensus problem, however, do not take the noise effects into account. The purpose of this paper is to initiate a study of the consensus problems under noise disturbances. First, the class of multi-agent systems under study is transformed into the following general time-varying systems with noises: x(t+1) = P(t)x(t)+w(t+1), where {P(t)} is a sequence of nonnegative stochastic matrices. Then, for such a general time-varying systems, the equivalent relationships are established among (i) robust consensus, (ii) the positivity of the second smallest eigenvalue of a weighted Laplacian matrix, and (iii) the joint connectivity of the associated dynamical neighbor graphs. Finally, this basic equivalence result is shown to be applicable to several class of concrete multi-agent models with noises.