Controllability of Continuum Ensemble of Formation Systems over Directed Graphs

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 topology and network dynamics of every individual networked system so that a continuum ensemble of such systems is controllable. To keep the analysis 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, real analytic manifold. Every individual formation system in the ensemble is composed of $N$ agents. These agents evolve in $\mathbb{R}^n$ and can access relative positions of their neighbors. The information flow topology 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 topology described by a common 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.