ℋ2-clustering of closed-loop consensus networks under a class of LQR design

Given any positive integer r, our objective is to develop a strategy for grouping the states of a n-node network into r ≤ n distinct non-overlapping groups. The criterion for this partitioning is defined as follows. First, a LQR controller is defined for the original n-node network. Then, a r-dimensional reduced-order network is created by imposing a projection matrix P on the n-node 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 H2 norm. We derive an upper bound on this difference in terms of P , and, thereby propose a design for P using K-means that tightens the bound while guaranteeing numerical feasibility.