Analysis of the effects of communication delays for consensus of networked multi-agent systems

Achieving cooperation and coordination in a network of multi-agent systems is key to solving the consensus problem. Synchronization of such systems requires consistent communication between agents to reach a consensus, which is not feasible in the presence of delays, data loss, disturbances, and other unpredictable factors. Communication delays combined with environmental uncertainties can cause adverse effects and negatively change the behavior of the networked system preventing synchronization. In this paper, the stability of the delayed networked systems is analyzed using delay differential equations. Solving these equations has not been feasible, because of the infinite number of characteristic roots. The approach based on the LambertWfunction has the capability of analytically solving delay differential equations. The approach is used to quantify and analyze the stability of the delayed networked systems. The communication between the agents is modeled using the graph theory and the Laplacian matrix. The stability is analyzed by incorporating the Laplacian matrix into the Lambert W function based approach which provides the locus of the eigenvalues of the system as delay changes. Sensitivities and convergence speed with respect to delay for various topologies of the network are presented for comparison. The numerical results and implementation using MATLAB/Simulink are presented for illustration.