Consensus in hierarchical multi-agent dynamical systems with low-rank interconnections: Analysis of stability and convergence rates

In this paper, we propose a fairly general model for hierarchical multi-agent dynamical systems (HMADSs) with fractal structure and investigate their stability or convergence condition for consensus. We first generalize a model introduced by Smith et al. to represent the weak cross-layer interconnection properly and explain the significance of focusing on the low-rank property instead of sparseness or small gain property. We then derive the analytical expression of eigenvalue distribution of the system matrix with rank 1 interconnection for cyclic pursuit. This provides us the stability or consensus condition for the whole system where each agent has a certain dynamics. We also clarify the relation between the property of interconnection and stability degree of multi-agent systems, which is confirmed by numerical examples. Further, we investigate the rank 2 case of the interconnection structure.