Consensus of double integrator multi-agents under communication delay

This paper deals with the consensus problem under network induced communication delays. It is well-known that introducing a delay generally leads to a reduce of the performance or to instability. Thus, investigating the impact of time-delays in the consensus problem is an important issue. Another important issue is to obtain an estimate of the convergence rate, which is not straightforward when delays appear in the network. In this paper, the agents are modelled as double integrator systems. It is assumed that each agent receives instantaneously its own output information but receives the information from its neighbors after a constant delay. A stability criterion is provided based on Lyapunov-Krasovskii techniques and is expressed in terms of LMI. An expression of the consensus equilibrium which depends on the delay and on the initial conditions taken in an interval is derived. The results are supported through several simulations for different network symmetric communication schemes.