[Paper Review] Distributed Surrounding Design of Target Region with Complex Adjacency Matrices

Youcheng Lou, Yiguang Hong|arXiv (Cornell University)|Oct 20, 2015

Distributed Control Multi-Agent Systems14 references3 citations

TL;DR

This paper proposes a distributed control protocol for multi-agent systems to surround a convex target region in the plane using complex adjacency matrices to specify desired relative angles and distances. Under mild connectivity assumptions, it establishes sufficient conditions for consistent and inconsistent surrounding behaviors, with a necessary and sufficient condition for the special case when the target is the origin, extending prior consensus and bipartite consensus results.

ABSTRACT

This is a complete version of the 6-page IEEE TAC technical note [1]. In this paper, we consider the distributed surrounding of a convex target set by a group of agents with switching communication graphs. We propose a distributed controller to surround a given set with the same distance and desired projection angles specified by a complex-value adjacency matrix. Under mild connectivity assumptions, we give results in both consistent and inconsistent cases for the set surrounding in a plane. Also, we provide sufficient conditions for the multi-agent coordination when the convex set contains only the origin.

Motivation & Objective

• To design a distributed controller enabling agents to surround a convex target set with uniform distance and user-defined angular projections.
• To analyze the system dynamics under switching communication graphs with complex-weighted configuration graphs.
• To characterize the role of cycle consistency in the configuration graph on the emergence of consistent or inconsistent surrounding behavior.
• To extend existing consensus and bipartite consensus results to the case of set surrounding, particularly when the target set reduces to a single point.

Proposed method

• The method employs a distributed control protocol based on complex-value adjacency matrices to encode desired relative angles and distances between agents and the target set.
• The configuration graph is defined using complex weights to represent desired relative orientations, with consistency of directed cycles determining system behavior.
• A projection-based dynamic is used where agents adjust their positions based on the difference between their current position and the projection onto the target set.
• The analysis leverages graph-theoretic concepts such as uniformly jointly strongly connected (UJSC) communication graphs and weak cycles to establish convergence properties.
• Theoretical results are derived using complex matrix analysis and Lyapunov-type arguments, particularly leveraging Babalat’s lemma for switching systems.
• Numerical simulations validate the theoretical findings under both consistent and inconsistent cycle configurations.

Experimental results

Research questions

• RQ1 Under what conditions can a group of agents achieve consistent surrounding of a convex target set using a distributed controller with complex adjacency matrices?
• RQ2 How does the consistency of directed cycles in the configuration graph affect the long-term behavior of the multi-agent system?
• RQ3 What are the necessary and sufficient conditions for multi-agent coordination when the target set reduces to a single point (the origin)?
• RQ4 How does the switching nature of the communication graph impact the convergence and stability of the surrounding protocol?
• RQ5 Can the proposed method extend existing consensus and bipartite consensus results to the more general setting of set surrounding?

Key findings

• For uniformly jointly strongly connected (UJSC) undirected communication graphs and fixed strongly connected graphs, the system achieves consistent surrounding if and only if all directed cycles in the configuration graph are consistent.
• In the inconsistent case, agents converge to the target set but fail to maintain consistent angular projections, resulting in a non-uniform surrounding pattern.
• When the target set is the origin, a necessary and sufficient condition for convergence is derived for the fixed strongly connected graph case.
• The proposed controller ensures that agents maintain a constant distance from the target set while achieving desired relative angles specified by the complex adjacency matrix.
• The results generalize classical consensus and bipartite consensus by extending them to the multi-agent set surrounding problem.
• Numerical examples confirm that consistent cycle configurations lead to stable, symmetric surrounding formations, while inconsistent configurations result in convergence to the target without angular consistency.