Mean square performance analysis for torus-networked distributed consensus under stochastic disturbances

In this paper we mainly address the mean square performance problem for torus-networked distributed consensus under stochastic disturbances. Specifically, we answer the question of how a torus-networked distributed consensus system will behave when all communication links are unreliable and each agent is subject to the perturbation of additive white noise. We derive a necessary and sufficient condition to characterize the mean square performance. Due to the spatially invariant structure of torus lattices, such a condition can be computed in a closed form. Computation results show that the mean square performance depends on the network dimension. Therefore, it suggests a natural trade-off between mean square stability and performance for such systems.