Title: Feedback Capacity over Networks,
Abstract: In this paper, we investigate the fundamental limitations of feedback
mechanism in dealing with uncertainties for network systems. The study of
maximum capability of feedback control was pioneered in Xie and Guo (2000) for
scalar systems with nonparametric nonlinear uncertainty. In a network setting,
nodes with unknown and nonlinear dynamics are interconnected through a directed
interaction graph. Nodes can design feedback controls based on all available
information, where the objective is to stabilize the network state. Using
information structure and decision pattern as criteria, we specify three
categories of network feedback laws, namely the
global-knowledge/global-decision, network-flow/local-decision, and
local-flow/local-decision feedback. We establish a series of network capacity
characterizations for these three fundamental types of network control laws.
First of all, we prove that for global-knowledge/global-decision and
network-flow/local-decision control where nodes know the information flow
across the entire network, there exists a critical number
$\big(3/2+\sqrt{2}\big)/\|A_{\mathrm{G}}\|_\infty$, where $3/2+\sqrt{2}$ is as
known as the Xie-Guo constant and $A_{\mathrm{G}}$ is the network adjacency
matrix, defining exactly how much uncertainty in the node dynamics can be
overcome by feedback. Interestingly enough, the same feedback capacity can be
achieved under max-consensus enhanced local flows where nodes only observe
information flows from neighbors as well as extreme (max and min) states in the
network. Next, for local-flow/local-decision control, we prove that there
exists a structure-determined value being a lower bound of the network feedback
capacity. These results reveal the important connection between network
structure and fundamental capabilities of in-network feedback control.