Formation control of directed multi-agent networks based on complex Laplacian

Real graph Laplacians are of great importance in consensus of multi-agent systems. This paper introduces complex graph Laplacians as a new tool to study the formation control problem in the plane. It is shown that complex graph Laplacians are of equally great importance for planar formation control like real Laplacians for consensus. First, complex graph Laplacians are used to characterize planar formations under given topology of networked agents. Second, complex graph Laplacians are used to derive local and distributed control strategies for asymptotically achieving formations. This paper explores the relations between graph topology, complex Laplacians, and planar formations, and obtains several graphical and algebraic conditions for realizability of spatial formations. Both simulation and experiment results are provided to illustrate our results.