Brief Paper - Consensus of swarm systems with time delays and topology uncertainties

Consensus problems for first-order continuous-time swarm systems in directed networks with time delays and topology uncertainties are dealt with. The state space of a swarm system is decomposed into a null space and a complement null space. By projecting the state of the system onto the two subspaces, a necessary and sufficient condition for consensus is proposed, and an explicit expression of the consensus function is shown based on the impacts of time delays and topology uncertainties. Sufficient conditions in terms of linear matrix inequalities are given for swarm systems with time-varying delays and topology uncertainties to achieve consensus. Numerical examples are shown to demonstrate theoretical results.