Discrete-Time Dynamical Networks with Diagonal Controllability Gramian

The controllability Gramian of a dynamical network carries rich information of the fundamental properties of the network. How to identify the connections from these fundamental properties to the network topology and weights is of great interest. It is, however, very challenging to do that because the Gramian is an extremely complicated function of the network topology and weights. In this paper, we consider the simplest case where the Gramian is diagonal. One of the main contributions of this paper is to prove the necessary and sufficient graphical conditions for a discrete-time dynamical network to feature a diagonal Gramian. The explicit relations between the values of the diagonal entries of the Gramian and the network weights are also established. The proposed results may be used to design networks with desired control energy and robustness performance.