Leader selection and weight adjustment problems for multi-agent systems

TL;DR

This paper investigates how to improve controllability in multi-agent systems by selecting leaders and adjusting edge weights, providing algebraic conditions, algorithms, and simulations for optimal control strategies.

Contribution

It introduces necessary and sufficient conditions for controllability with minimal leaders and develops an algorithm for weight adjustment to enhance controllability.

Findings

Controllability with fewest leaders characterized algebraically.

Structural controllability linked to the presence of a spanning tree.

Number of edges to adjust equals the controllability matrix's rank deficiency.

Abstract

For an uncontrollable system, adding leaders and adjusting edge weights are two methods to improve controllability. In this paper, controllability of multi-agent systems under directed topologies is studied, especially on leader selection problem and weight adjustment problem. For a given system, necessary and sufficient algebraic conditions for controllability with fewest leaders are proposed. From another perspective, when leaders are fixed, controllability could be improved by adjusting edge weights, and therefore the system is supposed to be structurally controllable, which holds if and only if the communication topology contains a spanning tree. It is also proved that the number of fewest edges needed to be assigned on new weights equals the rank deficiency of controllability matrix. An algorithm on how to perform weight adjustment is presented. Simulation examples are provided to illustrate the theoretical results.