A Bounded Complementary Sensitivity Function Ensures Topology-Independent Stability of Homogeneous Dynamical Networks

This paper investigates the topology-independent stability of homogeneous dynamical networks, composed of interconnected equal systems. Precisely, dynamical systems with identical nominal transfer function F(s) are associated with the nodes of a directed graph, whose arcs account for their dynamic interactions, described by a common nominal transfer function G(s). It is shown that topology-independent stability is guaranteed for all possible interconnections with interaction degree (defined as the maximum number of arcs leaving a node) equal at most to N if the infinity-norm of the complementary sensitivity function NF(s)G(s)[1 + NF(s)G(s)](-1) is less than 1. This bound is nonconservative in that there exist graphs with interaction degree N that are unstable for an -norm greater than 1. When nodes and arcs transferences are affected by uncertainties with norm bound K > 0, topology-independent stability is robustly ensured if the infinity-norm is less than 1/(1 + 2 N K ). For symmetric systems, stability is guaranteed for all topologies with interaction degree at most N if the Nyquist plot of NF(s)G(s) does not intersect the real axis to the left of - 1/2. The proposed results are applied to fluid networks and platoon formation.