Critical Sampling Rate for Sampled-Data Consensus over Random Networks

In this paper, we consider the consensus problem for a network of nodes with random interactions and sampled-data control actions. Each node independently samples its neighbors in a random manner over a directed graph underlying the information exchange of different nodes. The relationship between the sampling rate and the achievement of consensus is studied. We first establish a sufficient condition, in terms of the inter-sampling interval, such that consensus in expectation, in mean square, and in almost sure sense are simultaneously achieved provided a mild connectivity assumption for the underlying graph. Necessary and sufficient conditions for mean-square consensus are derived in terms of the spectral radius of the corresponding state transition matrix. These conditions are then interpreted as the existence of a critical value on the inter-sampling interval, below which global mean-square consensus is achieved and above which the system diverges in mean-square sense for some initial states. Finally, we establish an upper bound of the inter-sampling interval, below which almost sure consensus is reached, and a lower bound, above which almost sure divergence is reached. An numerical example is given to validate the theoretical results.