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• The PD controller design problem of positive discrete-time systems with constant delay is tackled in this work. • A systematic formulation is proposed to design non-fragile PD controllers for such kind of systems. • Based on the necessary and sufficient conditions derived, two tractable linear-programming-based and semi-definite-programming-based algorithms are developed for calculating the solution. This paper addresses the issue of proportional-derivative (PD) controllers design for positive linear systems in the discrete-time domain, which still remains a challenging problem in positive systems theory. The specific aim is to design a PD controller for a system with constant time delay, which simultaneously ensures closed-loop system stability and preserves positivity. Moreover, additive gain variation of the controller is considered in the synthesis process. Systematic formulation and tractable algorithms are developed to find the PD controller gains for positive stabilization. The performance of such methods is validated by numerical examples. [ABSTRACT FROM AUTHOR]