A Tool for Analysis and Synthesis of Heterogeneous Multi-agent Systems under Rank-deficient Coupling

The behavior of heterogeneous multi-agent systems is studied when the coupling matrices are possibly all different and/or singular (that is, its rank is less than the system dimension). Rank-deficient coupling allows exchange of limited state information, which is suitable for study of output coupling in multi-agent systems. We present a coordinate change that transforms the heterogeneous multi-agent system into a singularly perturbed form. The slow dynamics is still a reduced-order multi-agent system consisting of a weighted average of the vector fields of all agents, and some sub-dynamics of agents. The weighted average is an emergent dynamics, which we call a blended dynamics. By analyzing or synthesizing the blended dynamics, one can predict or design the behavior of heterogeneous multi-agent system when the coupling gain is sufficiently large. For this result, stability of the blended dynamics is required. Since stability of individual agent is not asked, stability of the blended dynamics is the outcome of trading stability among the agents. It can be seen that, under stability of the blended dynamics, the initial conditions of individual agents are forgotten as time goes on, and thus, the behavior of the synthesized multi-agent system are initialization-free and suitable for plug-and-play operation. As a showcase, we apply the proposed tool to two application problems; distributed state estimation for linear systems, and practical synchronization of heterogeneous Van der Pol oscillators (for which phase cohesiveness is achieved). We also present underlying intuition for two more applications; estimation of the number of nodes in a network, and a problem of distributed optimization.