H2-Clustering of Closed-loop Consensus Networks under a Class of LQR Design

In this paper we address the problem of clustering closed-loop consensus networks where the closed-loop controller is designed using a class of Linear Quadratic Regulator (LQR). Given any positive integer r, our objective is to develop a strategy for grouping the states of the 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 reducedorder r-dimensional LQR controller is constructed for this reduced-order system. 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. We discuss the computational benefits of the method, and illustrate the trade-off between r and H2performance using two network simulations.