Scaled consensus for second-order multi-agent systems subject to communication noise with stochastic approximation-type protocols

This work is dedicated to the leaderless/leader-following stochastic scaled consensus issue of second-order stochastic multi-agent systems (SMASs) in a noisy environment. Scaled consensus represents that the ratios among agents asymptotically tend to designated constants rather than the common convergence value. To lessen the influence of communication noise, some stochastic approximation protocols with time-varying gain are designed for our underlying system, where the time-varying gain remove the restriction of nonnegative value. Compared with the existing consensus results with communication noise, the major challenge is that the introduction of time-varying gain results in the inapplicability of Lyapunov-based technique. To cope with it, a state decomposition method is utilized, and a series of sufficient necessary conditions are set up for interacting agents with constant velocity and zero velocity if the topology includes a spanning tree. Furthermore, it is conducted that the consensus and bipartite consensus can be seen as two special cases of our work. Finally, the validity of our results is demonstrated by a simulation example.