Synchronization of multiple linear systems with communication delay

In this paper the problem of state and output synchronization of multi-agent systems with individual identical linear dynamics has been considered in the presence of constant and homogeneous communication delay. A linear synchronization protocol is introduced to perceive distributive control strategy which uses delayed information of states from the neighbouring agents. For marginally unstable dynamics of agents, maximum possible delay margin has been found. The delay margin depends both on the system matrix of the agent dynamics and network topology for directed graph whereas for undirected graph it depends only on the system matrix of the agents. However, for single-integrator and double-integrator agents, proposed protocol ensures synchronization with arbitrarily large bounded delay. A condition on coupling strength is derived for the proposed control methodology to tolerate delay less than the delay margin. It has been shown that the eigenvalues of the Laplacian matrix representing the communication topology are crucial in finding the condition for reaching synchronization. The idea of state synchronization is extended to output synchronization by the use of Luenberger observer. Finally, numerical simulations are provided to illustrate the results.