Coordination and Consensus of Networked Agents with Noisy Measurements: Stochastic Algorithms and Asymptotic Behavior

This paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors' states. For consensus seeking, we propose stochastic approximation-type algorithms with a decreasing step size, and introduce the notions of mean square and strong consensus. Although the decreasing step size reduces the detrimental effect of the noise, it also reduces the ability of the algorithm to drive the individual states towards each other. The key technique is to ensure a trade-off for the decreasing rate of the step size. By following this strategy, we first develop a stochastic double array analysis in a two-agent model, which leads to both mean square and strong consensus, and extend the analysis to a class of well-studied symmetric models. Subsequently, we consider a general network topology, and introduce stochastic Lyapunov functions together with the so-called direction of invariance to establish mean square consensus. Finally, we apply the stochastic Lyapunov analysis to a leader following scenario.