Leader–follower formation via complex Laplacian

The paper introduces complex-valued Laplacians for graphs whose edges are attributed with complex weights and studies the leader–follower formation problem based on complex Laplacians. The main goal is to control the shape of a planar formation of point agents in the plane using simple and linear interaction rules related to complex Laplacians. We present a characterization of complex Laplacians that preserve a specific planar formation as an equilibrium solution for both single integrator kinematics and double integrator dynamics. Planar formations under study are subject to translation, rotation, and scaling in the plane, but can be determined by two co-leaders in leader–follower networks. Furthermore, when a complex Laplacian does not result in an asymptotically stable behavior of the multi-agent system, we show that a stabilizing matrix, which updates the complex weights, exists to asymptotically stabilize the system while preserving the equilibrium formation. Also, algorithms are provided to find stabilizing matrices. Finally, simulations are presented to illustrate our results.