Optimal second order consensus protocol with time delay

In recent years, stability of consensus protocol with time delay has attracted a lot of interests. On the basis of local information exchange with limited time delay, the states of all the agents reach consensus with a second order consensus protocol under certain conditions, where the convergence speed is determined by eigenvalue spectrum of network Laplacian and system parameters. In this paper, the problem of fast consensus seeking in networked multi-agent systems is considered, which aims to optimize the convergence speed of consensus protocol with time delay. By using the frequency domain method, the optimization problem is decomposed into a series of convex optimization problems that can be solved by SeduMi toolbox. Furthermore, a consensus protocol with multi-hop relay scheme is investigated, where each agent can receive delayed information data transmitted by multi-hop agents. The results show that this scheme can improve the convergence speed without physically changing network topology. Finally, some examples are supplied to demonstrate theoretical results.