Discrete-Time Positive Edge-Consensus for Undirected and Directed Nodal Networks

This brief addresses the positive consensus of the edges with multi-input and multi-output positive dynamics for undirected and directed networks. The line graph is derived from the given nodal graph to represent the interaction of the edges, and the discrete-time positive systems are introduced to describe the edge dynamics. Based on the positive system theory and consensus analysis, a necessary and sufficient condition for positive edge-consensus is established, which reveals how the edge dynamics and the eigenvalues of the Perron matrix of line graph affect the consensus. Moreover, with further analysis, sufficient conditions for positive edge-consensus are derived without using the global interaction topology. It shows that the positive consensus can be achieved with the combined effect of edge dynamics, the number of edges, and the minimum diagonal element of the Perron matrix of line graph. The feedback matrix can be computed by solving linear programming problem. Finally, the simulations explicitly verify the proposed theoretical results.