Minimal‐time complex consensus for multi‐agent systems with time delay

Abstract

This paper studies the minimum time consensus problem for discrete-time multi-agent systems with complex Laplacian delay networks such that each agent can find its complex consensus value in a minimum number of steps using its local observations. The stability analysis is first provided and the convergence condition is derived for complex weighted networks with time delays. Specifically, the delayed multi-agent system is modeled by employing the augmented graph representation. Via adding virtual agents in the augmented systems, the complex consensus is obtained in the networks with bounded time delay if the communication topology digraph of the system has a spanning tree. A decentralized algorithm is proposed for the minimal-time computation of complex consensus based on the information from the robot itself without relying on the external environment. The algorithm hinges on the minimal polynomial of the matrix concerning the augmented graph. Furthermore, the rearrangement of the virtual agents in the augmented system provides an upper bound for the number of agents required to compute the consensus value. Simulation examples demonstrate the effectiveness of our results. The advantage of this approach is that it can be easily deployed on a group of agents to rapidly achieve a complex consensus setting within any delayed networks.