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The behavior of heterogeneous multi-agent systems is studied when thecoupling matrices are possibly all different and/or singular, that is, its rankis less than the system dimension. Rank-deficient coupling allows exchange oflimited state information, which is suitable for the study of multi-agentsystems under output coupling. We present a coordinate change that transformsthe heterogeneous multi-agent system into a singularly perturbed form. The slowdynamics is still a reduced-order multi-agent system consisting of a weightedaverage 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 designthe behavior of a heterogeneous multi-agent system when the coupling gain issufficiently large. For this result, stability of the blended dynamics isrequired. Since stability of the individual agent is not asked, the stabilityof the blended dynamics is the outcome of trading off the stability among theagents. It can be seen that, under the stability of the blended dynamics, theinitial conditions of the individual agents are forgotten as time goes on, andthus, the behavior of the synthesized multi-agent system is initialization-freeand is suitable for plug-and-play operation. As a showcase, we apply theproposed tool to four application problems; distributed state estimation forlinear systems, practical synchronization of heterogeneous Van der Poloscillators, estimation of the number of nodes in a network, and a problem ofdistributed optimization.

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