We revisit the classical pole-placement controller design problem with the objective of ensuring non-overshooting and error-free reference tracking. To this end, we present a number of novel conditions for external positivity of discrete-time transfer functions which characterise this non-overshooting property. These conditions are both necessary and sufficient for first- and second-order systems, but only sufficient for higher-order systems. In addition, they have simple geometric interpretations in terms of allowed locations of the closed-loop poles. Based on these conditions, we propose a control design procedure that guarantees a stable closed-loop system with non-overshooting, error-free tracking of step changes in the reference.