Ring plant network

We employ our customized algorithm using graphsp_IP_res.m to grow a spatially-invariant resistive ring network with nodes. We set R = I and choose the state weight that penalizes the mean-square deviation from the network average, $Q = I - (1/n) mathbf{1} mathbf{1}^T$ .

As shown below, for gamma=0 , we obtain a centralized controller that requires information exchange between all nodes. As gamma increases, the number of added edges gradually decreases. Finally, for $gamma = 0.94 , gamma_{max}$ , each node establishes a link to the node that is farthest away in the network. This is in agreement with recent theoretical developments of Fardad, Zhang, Lin, and Jovanovic ’14 where perturbation analysis was used to identify optimal week links in edge-transitive consensus networks. This reference demonstrates that the efficiency of reaching consensus is maximally enhanced by the addition of long-range links.

Blue lines identify edges in the plant graph; red lines identify edges in the controller graph; $gamma_{max}$ identifies the value of the regularization parameter gamma for which all edge weights in the controller graph are equal to zero.

gamma , = , 0

$gamma , = , 0.11 ,gamma_{max}$

$gamma , = , 0.24 ,gamma_{max}$

$gamma , = , 0.94 ,gamma_{max}$