Robustness analysis of leader-follower consensus

In this paper, robustness properties of the leader-follower consensus are considered. For simplicity of presentation, the attention is focused on a group of continuous-time first-order dynamic agents with a time-invariant communication topology in the presence of communication errors. In order to evaluate the robustness of leader-follower consensus, two robustness measures are proposed: the L2 gain of the error vector to the state of the network and the worst case L2 gain at a node. Although the L2 gain of the error vector to the state of the network is widely used in robust control design and analysis, the worst case L2 gain at a node is less conservative with respect to the number of nodes in the network. It is thus suggested that the worst case L2 gain at a node is used when the robustness of consensus is considered. Theoretical analysis and simulation results show that these two measures are sensitive to the communication topology. In general, the “optimal” communication topology that can achieve most robust performance with respect to either of the proposed robustness measures is difficult to characterize and/or obtain. When the in-degree of each follower is one, it is shown that both measures reach a minimum when the leader can communicate to each node in the network.