In this study, we investigate the consensus problem of high-order multi-agent systems (MASs) under directed topology. First, a distributed control scheme based on a reduced-order protocol, which only receives the relevant output measurement values of its neighbors through the network, is designed for MASs when communication delays are present. Second, with the help of the Sylvester equation, Riccati equation, and algebraic graph theory, a multi-step algorithm is proposed to construct the parameters involved in the designed protocol. Third, the convergence with respect to communication delays is analyzed, with the maximum estimated time delay obtained. Finally, an example is provided to illustrate the feasibility and effectiveness of the proposed theoretical results.