Topological and Graph-coloring Conditions on the Parameter-independent Stability of Second-order Networked Systems

TL;DR

This paper introduces a graph coloring framework called richly balanced coloring to analyze parameter-independent stability in heterogeneous passive networked systems, linking it with zero forcing sets and providing efficient algorithms for complex graphs.

Contribution

It formulates the RBC problem as a new graph coloring condition for stability, extending zero forcing set concepts, and proposes an efficient algorithm for general graphs.

Findings

Parameter-independent stability is characterized by the absence of richly balanced colorings.

Richly balanced coloring generalizes zero forcing sets, guaranteeing stability in certain graphs.

An efficient chord node coloring algorithm outperforms brute-force methods for NP-complete RBC problem.

Abstract

In this paper, we study parameter-independent stability in qualitatively heterogeneous passive networked systems containing damped and undamped nodes. Given the graph topology and a set of damped nodes, we ask if output consensus is achieved for all system parameter values. For given parameter values, an eigenspace analysis is used to determine output consensus. The extension to parameter-independent stability is characterized by a coloring problem, named the richly balanced coloring (RBC) problem. The RBC problem asks if all nodes of the graph can be colored red, blue and black in such a way that (i) every damped node is black, (ii) every black node has blue neighbors if and only if it has red neighbors, and (iii) not all nodes in the graph are black. Such a colored graph is referred to as a richly balanced colored graph. Parameter-independent stability is guaranteed if there does not exist a richly balanced coloring. The RBC problem is shown to cover another well-known graph coloring scheme known as zero forcing sets. That is, if the damped nodes form a zero forcing set in the graph, then a richly balanced coloring does not exist and thus, parameter-independent stability is guaranteed. However, the full equivalence of zero forcing sets and parameter-independent stability holds only true for tree graphs. For more general graphs with few fundamental cycles an algorithm, named chord node coloring, is proposed that significantly outperforms a brute-force search for solving the NP-complete RBC problem.