A Comparison of LQR Optimal Performance in the Decentralized and Centralized Settings

Here we consider the problem of LQR optimal control in the context of decentralized systems. It has been proven that if the system is centrally controllable and observable and the graph associated with the system is strongly connected, then there exists a linear periodic controller which provide LQR performance as close as desired to the optimal centralized performance. Here we demonstrate, via an example, that if the strongly connected assumption is violated then the optimal decentralized performance can be arbitrarily bad in comparison to the optimal centralized performance.