ℋ2-clustering of closed-loop consensus networks under generalized LQR designs

In this paper we present a Linear Quadratic Regulator (LQR) control design for large-scale consensus networks. When such networks have tens of thousands of nodes spread over a wide geographical span, the design and implementation of conventional LQR controllers become very challenging. Consider an n-node consensus network with both node and edge weights. Given any positive integer r, our objective is to develop a strategy for grouping the states of this network into r distinct non-overlapping groups. The criterion for this partitioning is defined as follows. First, an LQR state-feedback controller is defined over the n-node network for any given Q ≥ 0. Then, an r-dimensional reduced-order network is created by imposing a projection matrix P on the open-loop network, and a reduced-order r-dimensional LQR controller is constructed. The resulting controller is, thereafter, projected back to its original coordinates, and implemented in the n-node network. The problem, therefore, is to find a grouping strategy or P that will minimize the difference between the closed-loop transfer matrix of the original network with the full-order controller and that with the projected controller in the sense of ℋ
₂
norm. We derive an upper bound on this difference in terms of P, and, thereby propose a design for P using weighted k-means that tightens the bound. The weighting of k-means arises due to the node weights in the network, and the resulting asymmetry in its Laplacian matrix.