Influence of algebraic connectivity and out-degree of leader on distributed observer

In this paper, the connection between the communication network and the consensus speed of the multi-agent cooperative control based on distributed observers is explored. The dynamic of the error is established based on the formulation of the observer. Consequently, two key factors of the graph topology are analyzed in connection to the error convergence. The concept of out-degree is considered along with the Laplacian matrix of the network topology to characterize the convergence features. By analyzing the Lyapunov function under different networks, it has been proven that the consensus speed positively correlates with the algebraic connectivity which is the smallest nonzero eigenvalue of the Laplacian matrix and the leader's out-degree. Finally, the theoretical results are validated through simulation on numerical examples and the application of a unicycle formation, in which the back-stepping control law is adopted.