  In this paper we present a set of projection-based designs for constructing
simplified linear quadratic regulator (LQR) controllers for large-scale network
systems. When such systems have tens of thousands of states, the design of
conventional LQR controllers becomes numerically challenging, and their
implementation requires a large number of communication links. Our proposed
algorithms bypass these difficulties by clustering the system states using
structural properties of its closed-loop transfer matrix. The assignment of
clusters is defined through a structured projection matrix P, which leads to a
significantly lower-dimensional LQR design. The reduced-order controller is
finally projected back to the original coordinates via an inverse projection.
The problem is, therefore, posed as a model matching problem of finding the
optimal set of clusters or P that minimizes the H2-norm of the error between
the transfer matrix of the full-order network with the full-order LQR and that
with the projected LQR. We derive a tractable relaxation for this model
matching problem, and design a P that solves the relaxation. The design is
shown to be implementable by a convenient, hierarchical two-layer control
architecture, requiring far less number of communication links than full-order
LQR.
