Non-fragility of multi-agent controllability

Controllability of multi-agent systems is determined by the interconnection topologies. In practice, losing agents can change the topologies of multi-agent systems, which may affect the controllability. In order to preserve controllability, this paper first introduces the concept of non-fragility of controllability. In virtue of the notion of cutsets, necessary and sufficient conditions are established from a graphic perspective, for almost surely strongly/weakly preserving controllability, respectively. Then, the problem of leader selection to preserve controllability is proposed. The tight bounds of the fewest leaders to achieve strongly preserving controllability are estimated in terms of the diameter of the interconnection topology, and the cardinality of the node set. Correspondingly, the tight bounds of the fewest leaders to achieve weakly preserving controllability are estimated in terms of the cutsets of the interconnection topology. Furthermore, two algorithms are established for selecting the fewest leaders to strongly/weakly preserve the controllability. In addition, the algorithm for leaders' locations to maximize non-fragility is also designed. Simulation examples are provided to illuminate the theoretical results and exhibits how the algorithms proceed.