The Controllability and Structural Controllability of Laplacian Dynamics

TL;DR

This paper analyzes the controllability and structural controllability of Laplacian dynamics in networks, revealing how eigenvalue multiplicity, topology, and weight configurations influence controllability properties.

Contribution

It provides new theoretical insights into eigenvalue multiplicity, controllable subspace invariance, and conditions for strong structural controllability in Laplacian-based systems.

Findings

Eigenvalue zero multiplicity depends on network cycles and node pairs.

Controllable subspace remains invariant under certain weight variations.

Connectivity without inaccessible nodes is necessary and sufficient for structural controllability.

Abstract

In this paper, classic controllability and structural controllability under two protocols are investigated. For classic controllability, the multiplicity of eigenvalue zero of general Laplacian matrix $L^*$ is shown to be determined by the sum of the numbers of zero circles, identical nodes and opposite pairs, while it is always simple for the Laplacian $L$ with diagonal entries in absolute form. For a fixed structurally balanced topology, the controllable subspace is proved to be invariant even if the antagonistic weights are selected differently under the corresponding protocol with $L$. For a graph expanded from a star graph rooted from a single leader, the dimension of controllable subspace is two under the protocol associated with $L^*$. In addition, the system is structurally controllable under both protocols if and only if the topology without unaccessible nodes is connected. As a reinforcing case of structural controllability, strong structural controllability requires the system to be controllable for any choice of weights. The connection between father nodes and child nodes affects strong structural controllability because it determines the linear relationship of the control information from father nodes. This discovery is a major factor in establishing the sufficient conditions on strong structural controllability for multi-agent systems under both protocols, rather than for complex networks, about latter results are already abundant.