Stability Conditions for Structured Multi-agent Linear Systems

In this work, multi-agent systems which consist of a finite set of agents featuring linear dynamics and influencing each other in a linear way are considered. On the assumption that the topology of the communication network that connects the various agents is known, while the gains of the communication channels, which can assume any real value, are unknown, the multi-agent system is modeled as a structured system whose dynamics is defined by a set of mutually independent real parameters. In this context, structural properties are defined as properties which hold for all the values that the parameters can take and, in particular, this work is focused on the study of the structural stability of the overall multi-agent system. In the case of interest, where all the agents are assumed to have asymptotically stable dynamics, it is shown that a necessary condition for the structural asymptotic stability of the multi-agent system is that the graph describing the relations between the state variables of the agents does not contain cycles of a special kind, defined herein as simple outer cycles. Namely, if the graph contains simple outer cycles, then the overall multi-agent system is not structurally asymptotically stable.