Lag-Bipartite Consensus of Linear Multi-agent Systems over Signed Directed Graph

This work focusses on studying lag-bipartite consensus for linear multi-agent systems (MAS) with a leader where the agents interact via a signed directed graph. In this consensus problem over a signed graph, one group of followers achieve the exact leader state value after a certain delay while another group achieves this value in magnitude, but opposite in sign after the same delay. The consensus is achieved by designing both distributed state feedback and distributed observer-based output feedback control protocols with appropriate design parameters. It is established in this work that if the underlying communication graph has a rooted directed spanning tree whose root is the leader, then the objective of realizing lag-bipartite consensus is achieved in this setting. The numerical examples verify the theoretical results of the work.