Feedback Capacity over Networks

TL;DR

This paper characterizes the fundamental limits of feedback control in network systems with uncertain and nonlinear node dynamics, revealing how network structure influences feedback capacity and stability.

Contribution

It introduces a novel classification of network feedback laws and establishes capacity bounds based on network structure and information flow, extending previous scalar system results.

Findings

Identifies a critical feedback capacity related to network adjacency and uncertainty.

Shows that max-consensus information flow achieves the same capacity as full knowledge.

Provides a lower bound on feedback capacity for local control based on network structure.

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.