Stochastic Lyapunov Analysis for Consensus Algorithms with Noisy Measurements

This paper studies the coordination and consensus of networked agents in an uncertain environment. We consider a group of agents on an undirected graph with fixed topology, but differing from most existing work, each agent has only noisy measurements of its neighbors´ states. Traditional consensus algorithms in general cannot deal with such a scenario. For consensus seeking, we introduce stochastic approximation type algorithms with a decreasing step size. We present a stochastic Lyapunov analysis based upon the total mean potential associated with the agents. Subsequently, the so-called direction of invariance is introduced, which combined with the decay property of the stochastic Lyapunov function leads to mean square convergence of the consensus algorithm.