Controllability of continuum ensemble of formation systems over directed graphs

We propose in the paper a novel framework about 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 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 real, analytic manifold. Every individual formation system in the ensemble is composed of N agents. These agents evolve in Rⁿ 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 formation system is greater than (n+1), then every such system in the ensemble is simultaneously approximately path-controllable over a path-connected, open dense subset.