Eigenvector-based characterization for hierarchical multi-agent dynamical systems with low rank interconnection

In this paper, we investigate hierarchical multi-agent dynamical systems with low rank interconnection based on a new concept of eigen-interconnection. The eigenvector plays an important role in our framework. We first define the eigen-interconnection matrix for the rank one case. Then, we show eigenvalue distribution of a hierarchical interconnection with rank one eigen-interconnection matrix. The obtained eigenvalue distribution has explicit parameter dependency and the result allows us to place the eigenvalue selectively. Thus it gives useful knowledge to design hierarchical interconnection structure. This concept is naturally extended to the rank two case and a similar result for eigenvalue distribution is obtained. Those results completely include the previous researches as special cases. Furthermore, our characterization has a potential of straightforward extension to the higher rank cases. Finally, we show a numerical example where a hierarchical interconnection structure is designed based on our result.