White noise disturbance and topological heterogeneity analysis for the hybrid consensus protocol

In this note, we extend the robustness analysis for the hybrid consensus protocol to a more general case: the white noise disturbance effect on the hybrid system, which is a significant measurement when considering the robustness of control protocols. The expectation and covariance of the states for the system are formulated, and the Lyapunov stability for the hybrid system is developed. Furthermore, considering attacks to the graph of the multi-agent system, the topology of the network might be unconnected, therefore, the study of the disconnectedness effect on the system is necessary. Hence, the unconnected topologies are investigated for the hybrid system, and the necessary and sufficient conditions for weak formation and consensus are provided, respectively. Moreover, simulations are provided to verify our theoretical analysis.