The consensus problem of multi-agent systems has attracted wide attention from researchers in recent years, following the initial work of Jadbabaie et al. 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 a general time-varying system with noise. Then, for such a system, 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 classes of concrete multi-agent models with noise.
The input uk and output yk of the multivariate ARMAX system A（x）yk = B（z）uk ＋ C（z）wk are observed with noises： uk^ob△=uk ＋ εk^u and yk^ob △=yk＋ εk^y, where εk^u and εk^y denote the observation noises. Such kind of systems are called errors-in-variables （EIV） systems. In the paper, recursive algorithms based on observations are proposed for estimating coefficients of A（z）, B（z）, C（z）, and the covariance matrix Rw of wk without requiring higher than the second order statistics. The algorithms are convenient for computation and are proved to converge to the system coefficients under reasonable conditions. An illustrative example is provided, and the simulation results are shown to be consistent with the theoretical analysis.
In this paper, sampled-data based average-consensus control is considered for networks consisting of continuous-time first-order integrator agents in a noisy distributed communication environment. The impact of the sampling size and the number of network nodes on the system performances is analyzed. The control input of each agent can only use information measured at the sampling instants from its neighborhood rather than the complete continuous process, and the measurements of its neighbors＇ states are corrupted by random noises. By probability limit theory and the property of graph Laplacian matrix, it is shown that for a connected network, the static mean square error between the individual state and the average of the initial states of all agents can be made arbitrarily small, provided the sampling size is sufficiently small. Furthermore, by properly choosing the consensus gains, almost sure consensus can be achieved. It is worth pointing out that an uncertainty principle of Gaussian networks is obtained, which implies that in the case of white Gaussian noises, no matter what the sampling size is, the product of the steady-state and transient performance indices is always equal to or larger than a constant depending on the noise intensity, network topology and the number of network nodes.
