Delay-margin design in consensus for high-order integrator agents

This paper analyzes the consensus problem for agents modeled with any number of integrators when these are subject to constant time-delays and interact over a directed network. Concretely, it provides convergence criteria that are used to devise a systematic methodology for designing the coupling gains of the consensus protocols. Such design is performed in a way that the multi-agent system attains a prescribed delay-margin (the maximum delay such that consensus is achieved), which can be made arbitrarily large at the cost of reducing the convergence rate. To that end, a modal decomposition is used to split the dynamics into fundamental modes and reduce the consensus problem to a stability problem. Frequency domain tools are then exploited to analyze stability and provide a unified approach to the consensus problem with constant time-delays. Initially, the proposed methodology is applied to design coupling gains for high-order consensus protocols. Then, the analysis focuses on first-, second- and third-order consensus protocols due to their high practical applicability, and explicit bounds on the coupling gains are derived. Finally, the proposed methodology is illustrated with examples.