Interval consensus over random networks

This paper considers the interval consensus problems of discrete-time multi-agent systems over random interaction networks, where each agent can impose a lower and an upper bound, i.e., a local constraint interval, on the achievable consensus value. We show that if the intersection of the intervals is nonempty, it holds as a sure event that the states of all the agents converge to a common value inside that intersection, i.e., the interval consensus can be achieved almost surely. Convergence analysis is performed through developing a robust consensus analysis of random networks in view of a martingale convergence lemma. Numerical examples are also exhibited to verify the validity of the theoretical results.