Stability Analysis of Networked Systems Containing Damped and Undamped Nodes

TL;DR

This paper investigates the long-term behavior of heterogeneous networked systems with damped and undamped nodes, providing a comprehensive stability analysis based on eigenvector methods and graph reduction techniques.

Contribution

It introduces a novel eigenvector-based stability analysis for heterogeneous networks with damping, considering both original and reduced graph structures.

Findings

Damped nodes always reach a steady state.

Undamped nodes' convergence depends on network parameters and topology.

Eigenvector analysis links stability to graph properties.

Abstract

This paper answers the question if a qualitatively heterogeneous passive networked system containing damped and undamped nodes shows consensus in the output of the nodes in the long run. While a standard Lyapunov analysis shows that the damped nodes will always converge to a steady-state value, the convergence of the undamped nodes is much more delicate and depends on the parameter values of the network as well as on the topology of the graph. A complete stability analysis is presented based on an eigenvector analysis involving the mass values and the topology of both the original graph and the reduced graph obtained by a Kron reduction that eliminates the damped nodes.