This paper brings forward the notion of "H-infinity almost output synchronization" for heterogeneous multi-agent systems under directed interconnection structures. Agents are assumed to be linear, right-invertible and introspective (i.e. agents have partial knowledge of their own states). The objective is to suppress the impact of disturbances on the synchronization error dynamics in terms of the H-infinity norm of the corresponding closed-loop transfer function. In addition, the problem of regulating the consensus trajectories to a reference signal is addressed. The application of the proposed method to the problem of formation is furthermore elucidated. Simulation results are provided to illustrate the method. (C) 2014 Elsevier Ltd. All rights reserved.