Distributed optimisation of second‐order multi‐agent systems by control algorithm using position‐only interaction with time‐varying delay

In this study, the distributed optimisation problem is investigated by using the framework of second‐order multi‐agent systems (MASs) over a directed communication topology. A distributed algorithm is proposed for the MASs to seek the solution of the optimisation problem. The proposed algorithm contains two layers, the cooperative compensator and the local optimiser, where the former layer is to compensate the cooperative errors and the latter will lead the agents to the optimal solution. In the algorithm, each agent only uses the position information from its neighbours in the presence of delay. By utilising the Lyapunov–Krasovskii functional, delay‐dependent conditions are derived for the slowly‐varying and fast‐varying delays, respectively, to guarantee the algorithm converges to the optimal solution. Furthermore, a criterion on solving the optimisation problem through the MASs with discrete‐time communication is also proposed directly from the result of the fast‐varying delay. Finally, a numerical simulation example is presented to illustrate the persuasive effectiveness of the theoretical results.