An explicit relation between the open- and closed-loop system responses is established directly in the time domain by means of Bell polynomials. This relation is used to derive necessary and sufficient conditions on the open-loop impulse response that guarantees that its closed-loop impulse response under unity feedback is non-negative in a given time range. Based on these conditions, a new design technique for PID controllers is presented that guarantees a monotonic closed-loop step response over a desired time interval. The results hold for any (possibly infinite-dimensional) discrete-time LTI system with a known impulse response. Numerical examples are provided to assess the contribution.