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 leaderfollower 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. Published as: Z. Lin, W. Ding, G. Yan, C. Yu, A. Giua, Leader-Follower Formation via Complex Laplacian, Automatica, Vol. 49, No. 6, pp. 1900-1906, June 2013. DOI: 10.1016/j.automatica.2013.02.055.1111. ∗The work is in part supported by National Natural Science Foundation of China under Grant 61273113, Shandong Academy of Science Development Fund for Science and Technology, the Pilot Project for Science and Technology in Shandong Academy of Science, and Australian Research Council’s DP130103610. C. Yu is also supported by an ARC Queen Elizabeth II Fellowship under DP110100583 and also by the Overseas Expert Program of Shandong Province. †Zhiyun Lin (corresponding author, email: [email protected], Phone: +86-571-87951637; Fax: +86-571-87952152), Wei Ding (email: [email protected]) and Gangfeng Yan (email: [email protected]) are with the State Key Laboratory of Industrial Control Technology, College of Electrical Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027 P.R. China. Changbin Yu (email: [email protected]) is with Shandong Provincial Key Laboratory of Computer Network, Shandong Computer Science Center, Jinan 25001 P.R. China and with Research School of Engineering, Australian National University, Canberra ACT 2600, Australia. Alessandro Giua (email: [email protected]) is with DIEE, University of Cagliari, Piazza d’Armi, 09123 Cagliari, Italy and with LSIS, University of Aix-Marseille, France.