Networked Multi-agent Systems: from Real Laplacian to Complex Laplacian

EXTENDED ABSTRACT. In recent years, networked multi-agent systems have attracted much research attention from various disciplines of engineering and science due to their interdisciplinary nature and broad potential applications. Collective behaviors arise from simple local interaction rules in complex networked systems. Graph Laplacian plays a very important role in analysis of networked multi-agent systems, in particular, for the consensus problem. A very simple local interaction law using relative state information for networked multiple agents gives rise to an overall system whose system matrix is related to the real-valued Laplacian corresponding to the control graph that defines the interaction topology. It is shown in [1], [2] and [3] that a collective behavior of consensus regarding the state of interest emerges if and only if the real-valued Laplacian has a simple eigenvalue, or equivalently the control graph is rooted, meaning that the graph has a root from which every node is reachable. Thus, the null space (kernel) of the Laplacian is exactly the consensus subspace, and is asymptotically stable. However, complex-valued Laplacians corresponding to graphs with edges attributed with complex weights have not come into notice in the study of networked multi-agent systems. This work introduces complex-valued Laplacians for networked multi-agent systems and shows that complex Laplacians are equally important in coordination and cooperation of multi-agent systems, particularly for planar formations. First, it is shown that a complex-valued Laplacian can be used to characterize a planar formation subject to four degrees of freedom (translation, rotation and scaling). That is, its kernel defines a planar formation of four degrees of freedom if and only if it has zero eigenvalues with both algebraic and geometric multiplicity two. For a given interaction topology called control graph and a formation basis with respect to a common reference frame, a complex-valued Laplacian can always be found, whose kernel defines the planar formations generated from the formation basis via rotation, translation and/or scaling if and only if the control graph is 2-rooted (an analogue to the notion of rooted, that is, the graph has two roots from which every other node is 2-reachable, i.e., every other node is still reachable after removing any node). Second, it is shown that a collective behavior of formations can be achieved via a lightweight distributed control law using complex-weighted relative state information related to the complex-valued Laplacian. But unlike real-valued Laplacians, complexvalued Laplacians distribute their eigenvalues everywhere in the complex plane and