This paper is concerned with torsional vibration control of drill-string systems. The objective is to develop a delay-dependent control scheme such that the downhole vibrations can be minimized by using ground measurement output with time-varying measurement delays. By regarding a drill-string as a series of lumped masses, a state space model is derived from a generic multi-degree-of-freedom model of the drill-string through a variable transformation. This provides the foundation of an observer-based output feedback control system, in which an internal model is inserted to represent the drill rig for improving the tracking performance, and a state observer is combined with a low-pass filter to estimate an equivalent effect of the downhole bit-rock interaction in the control input channel. To calculate the parameters of this control system, some sufficient conditions are derived in terms of linear-matrix-inequalities by taking into account a refined allowable delay set. It is shown through a numerical example that (i) the measurement of rotary table’s angular displacement helps to produce less conservative results and (ii) a small measurement delay is beneficial for designing a controller with a smaller gain in the sense of Euclidean norm, however it may also result in a larger control torque by enhancing the bit-rock interaction.