Almost sure averagingwith relative-state-dependent measurement noises and linear noise intensity functions

In this paper, we consider the distributed averaging of high-dimensional first-order agents with relative-state-dependent measurement noises. Each agent can measure or receive its neighbors' state information with random noises, whose intensity is a linear vector-valued function of agents' relative states. Differently from the case with non-state-dependent measurement noises, we show that a negative control gain, though can not ensure mean square consensus, may ensure almost sure consensus. This tells us that the relative-state-dependent measurement noises will sometimes be helpful for the almost sure consensus of the network. For symmetric measurement models, the almost sure convergence rate is estimated by the Iterated Logarithm Law of Brownian motions.