Consensus In Multi-Agent Complex-Weighted Networks And Its Applications

Consensus problems for multi-agent systems in the literature have been studied under real weighted networks. Complex-valued systems can be used to model many phenomena in applications including complex-valued signals and the motion in a plane.A natural question that arises is whether we can establish consensus under complex-weighted networks in some sense. In this paper, we provide a positive answer to this question. More precisely, we give that in a complex-weighted network all agents can achieve modulus consensus in which the states of all agents reach the same modulus.The second chapter gives some general notations used in this paper, as well as definition of the complex-weighted digraph. We give the properties of eigenvalues of the Laplacian matrix in terms of its associated digraph. Then, we give the relationship between the properties of eigenvalues of the perturbed Laplacian matrix and the connectivity of its associated digraph. Finally, his chapter gives the necessary and sufficient condition for Hurwitz stability of diagonally equiptent matrices in complex weighted networks.The third chapter gives the necessary and sufficient conditions for modulus consensus in both continuous-time and discrete-time cases, which explicitly reveal how the connectedness of networks and structural properties of complex weights jointly affect modulus consensus. This chapter, to achieve arbitrary shaped formation, gives scaled consensus under complex weighted networks. As a special case, the necessary and sufficient conditions for bipartite consensus on signed networks are obtained.The fourth chapter mainly uses the modulus consensus results to achieve circular formation in a plane. We first give the definition of circular formation with relative positions that requires all the agents converge to a common circle centered at a given point and are distributed along the circle in a desired pattern, expressed by the prespecified angle separations and ordering among agents. The main result is the circular formation with relative positions can be achieved if and only if the communication digraph has a spanning tree. However, it has the unspecified radius and absolute phases. To completely determine the circular formation, we give the control problem of circular formation with absolute positions. By using the pinning control strategy, we find that the circular formation with absolute positions can be achieved via a single local controller if and only if the communication digraph has a spanning tree.