[Paper Review] Controllability of Ensemble Formation System over Digraph.

Xudong Chen|arXiv (Cornell University)|May 28, 2018

Markov Chains and Monte Carlo Methods3 citations

TL;DR

This paper proposes a framework for controlling an infinite ensemble of multi-agent formation systems using a common control input by co-designing information flow and network dynamics. It establishes that if the underlying digraph is strongly connected and each system has more than (n + 1) agents, the ensemble is approximately path-controllable over a dense, open, path-connected subset of the state space.

ABSTRACT

We propose in the paper a novel framework for using a common control input to simultaneously steer an infinite ensemble of networked control systems. We address the problem of co-designing information flow and network dynamics of every individual networked system so that a continuum ensemble of such systems is controllable. To keep the problem tractable, we focus in the paper on a special class of ensembles systems, namely ensembles of multi-agent formation systems. Specifically, we consider an ensemble of formation systems indexed by a parameter in a compact, analytic manifold. Every individual formation system in the ensemble is comprised of $N$ agents. These agents evolve in $\mathbb{R}^n$ and can access relative positions of their neighbors. The information flow within every individual formation system is, by convention, described by a directed graph where the vertices correspond to the $N$ agents and the directed edges indicate the information flow. For simplicity, we assume in the paper that all the individual formation systems share the same information flow digraph $G$. Amongst other things, we establish a sufficient condition for approximate path-controllability of the continuum ensemble of formation systems. We show that if the digraph $G$ is strongly connected and the number $N$ of agents in each individual system is great than $(n + 1)$, then every such system in the ensemble is simultaneously approximately path-controllable over a path-connected, open dense subset.

Motivation & Objective

• To develop a unified control framework for steering an infinite ensemble of networked formation systems using a single control input.
• To co-design the information flow (via a digraph) and the dynamics of each system to ensure ensemble controllability.
• To identify sufficient conditions under which the entire ensemble of formation systems is approximately path-controllable.
• To focus on ensembles indexed by a compact, analytic manifold, enabling tractable analysis of continuum dynamics.
• To address the challenge of controlling a continuum of systems with heterogeneous but structured dynamics.

Proposed method

• Model the ensemble as a continuum of multi-agent formation systems, each with N agents evolving in R^n and communicating via relative positions.
• Represent the information flow within each system using a fixed directed graph G, where vertices are agents and edges denote information access.
• Assume all systems in the ensemble share the same digraph G to ensure structural consistency across the ensemble.
• Use the concept of approximate path-controllability over a path-connected, open, dense subset of the state space to define controllability for the continuum.
• Apply graph-theoretic and differential geometric tools to analyze the controllability of the ensemble under the assumption of strong connectivity of G.
• Derive a sufficient condition based on the number of agents N > (n + 1) and strong connectivity of G for approximate path-controllability.

Experimental results

Research questions

• RQ1 Under what conditions can a common control input simultaneously steer an infinite ensemble of formation systems to desired trajectories?
• RQ2 How does the structure of the information flow digraph G affect the controllability of the entire ensemble?
• RQ3 What is the minimal number of agents N required for approximate path-controllability when the digraph G is strongly connected?
• RQ4 Can the ensemble be approximately path-controllable over a dense, open, path-connected subset of the state space?
• RQ5 How does the manifold structure of the ensemble parameter influence the controllability analysis?

Key findings

• If the digraph G is strongly connected and the number of agents N exceeds (n + 1), the ensemble of formation systems is approximately path-controllable.
• The set of reachable states for the ensemble is dense and open in a path-connected subset of the state space.
• The common control input enables simultaneous control of all systems in the ensemble, despite their parameterization over a compact, analytic manifold.
• The sufficient condition for controllability depends on both the graph topology (strong connectivity) and the agent count (N > n + 1).
• The analysis establishes that the ensemble's controllability is preserved under the specified structural constraints, enabling scalable control design.