On output consensus of heterogeneous dynamical networks

TL;DR

The paper tackles output consensus for networks of non-identical subsystems subject to external disturbances or references. It introduces a heterogeneity index $ extgamma_{ij}$ between adjacent subsystems, based on differences in input-output trajectories, and derives a connectivity-dependent condition for consensus. The key finding is that if $ extγ_m + extα oldsymbol{ extlambda}_2 > 0$, where $ extγ_m= obreak ext{min}_{(i,j) ext{ edge}} extγ_{ij}$, $ extα= obreak ext{min}_{(i,j) ext{ edge}} extα_{ij}$, and $oldsymbol{ extlambda}_2$ is the second-smallest eigenvalue of the Laplacian $L$, then there exist $ ho>0$ and $oldsymbol{ extvarepsilon} obreak \ge 0$ such that $ orm{D^{T}Y}_T obreak \le obreak ho orm{D^{T}W}_T + oldsymbol{ extvarepsilon}$ for all $T\,> 0$. This provides a rigorous bound linking heterogeneity and graph connectivity to output consensus under diffusive coupling.

Abstract

This work is concerned with interconnected networks with non-identical subsystems. We investigate the output consensus of the network where the dynamics are subject to external disturbance and/or reference input. For a network of output-feedback passive subsystems, we first introduce an index that characterises the gap between a pair of adjacent subsystems by the difference of their input-output trajectories. The set of these indices quantifies the level of heterogeneity of the networks. We then provide a condition in terms of the level of heterogeneity and the connectivity of the networks for ensuring the output consensus of the interconnected network.