Cartesian product-based hierarchical scheme for multi-agent systems

In this paper, we solve the average-consensus problem using a hierarchical scheme based on Cartesian product of strongly connected balanced graphs — an algebraic approach to design complex networks. We show that the Cartesian product based hierarchical scheme for multi-agent systems outperforms the single-layer control strategies for average-consensus problem in terms of convergence rate, and also the system matrix produced by Cartesian product-based hierarchical (CPH) scheme do not necessarily exhibit block circulant symmetry. We analyze that if the factors graphs in Cartesian product are cyclic pursuit graphs, then the CPH scheme provides the same convergence rate while requiring the same communication links as in the hierarchical cyclic pursuit (HCP). We provide simulation results to demonstrate the key theoretical results.