This paper is concerned with the model reduction of positive systems. For a given stable positive system, our attention is focused on the construction of a reduced-order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed H-infinity performance. Based upon a system augmentation approach, a novel characterization on the stability with H-infinity performance of the error system is first obtained in terms of linear matrix inequality (LMI). Then, a necessary and sufficient condition for the existence of a desired reduced-order model is derived accordingly. Furthermore, iterative LMI approaches with primal and dual forms are developed to solve the positivity-preserving H-infinity model reduction problem. Finally, a compartmental network is provided to show the effectiveness of the proposed techniques. (C) 2011 Elsevier Ltd. All rights reserved.