A bounded complementary sensitivity function ensures topology-independent stability of homogeneous dynamical networks

Abstract

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 ∞-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 ∞-norm is less than 1/(1+2NK). 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.