Fundamental limits on robustness measures in networks of interconnected systems

We investigate robustness of interconnected dynamical networks with respect to external distributed stochastic disturbances. In this paper, we consider networks with linear time-invariant dynamics. The ℋ
₂
norm of the underlying system is considered as a robustness index to measure the expected steady-state dispersion of the state of the entire network. We present new tight bounds for the robustness measure for general linear dynamical networks. We, then, focus on two specific classes of networks: first- and second-order consensus in dynamical networks. A weighted version of the ℋ
₂
norm of the system, so called L
_Q
-energy of the network, is introduced as a robustness measure. It turns out that when L
_Q
is the Laplacian matrix of a complete graph, L
_Q
-energy reduces to the expected steady-state dispersion of the state of the entire network. We quantify several graph-dependent and graph-independent fundamental limits on the L
_Q
-energy of the networks. Our theoretical results have been applied to two application areas. First, we show that in power networks the concept of L
_Q
-energy can be interpreted as the total resistive losses in the network and that it does not depend on specific structure of the underlying graph of the network. Second, we consider formation control with second-order dynamics and show that the L
_Q
-energy of the network is graph-dependent and corresponds to the energy of the flock.