This paper investigates the positive edge-consensus problem via output feedback protocols for undirected networks and directed networks whose edge dynamics are defined on nonnegative spaces. First, the definition and topology of the line graph are presented based on a given nodal graph with positive edge dynamics. Two observer-type positive edge-consensus protocols are introduced, and sufficient conditions of positive consensus for uncertain positive systems are proposed without utilizing the information of algebraic connectivity. Furthermore, necessary and/or sufficient conditions of positive consensus for the nominal systems are derived. The analysis illustrates that the error variables defined in this paper are nonnegative in both protocols. Finally, the illustrative simulation examples are presented to verify the effectiveness of the proposed observer-type positive edge-consensus protocols.