First- and Second-Order Consensus with Constant Uniform Delays

This paper analyzes first- and second-order consensus protocols subject to constant uniform delays, when these are applied to single and double integrator agents interacting over a directed network. The goal is to derive explicit bounds on the coupling gains of these protocols such that consensus is achieved in the presence of delays, and consequently enable designing these gains in a structured manner. To that end, the consensus protocols are first analyzed using frequency domain tools, and a necessary and sufficient bound on the coupling gain of the first-order protocol is derived by inverting a bound on the delay. However, that approach does not readily provide bounds on the coupling gains such that consensus is achieved for the second-order case. Instead, the Pade approximation of the delay is used to derive sufficient bounds on the coupling gains for that case, which is the main contribution of the paper.