Continuous-time stochastic consensus: Stochastic approximation and Kalman–Bucy filtering based protocols

This paper investigates the continuous-time multi-agent consensus with stochastic communication noises. Each agent can only use its own and neighbors’ information corrupted by random noises to design its control input. To attenuate the communication noises, we consider the stochastic approximation type and the Kalman–Bucy filtering based protocols. By using the tools of stochastic analysis and algebraic theory, the asymptotic properties of these two kinds of protocols are analyzed. Firstly, for the stochastic approximation type protocol, we clarify the relationship between the convergence rate of the consensus error and a representative class of consensus gains in both mean square and probability one. Secondly, we propose Kalman–Bucy filtering based consensus protocols. Each agent uses Kalman–Bucy filters to get asymptotically unbiased estimates of neighbors’ states and the control input is designed based on the protocol with precise communication and the certainty equivalence principle. The iterated logarithm law of estimation errors is developed. It is shown that if the communication graph has a spanning tree, then the consensus error is bounded above by O(t−1)O(t−1) in mean square and by O(t−1/2(loglogt)1/2)O(t−1/2(loglogt)1/2) almost surely. Finally, the superiority of the Kalman–Bucy filtering based protocol over the stochastic approximate type protocol is studied both theoretically and numerically.