We develop an output feedback H-x control design method for linear time-delay systems based on the linear-matrix-inequality (LMI) approach. A sufficient condition for the existence of H-x controllers of any order is given in terms of three LMIs. The condition is valid for both singular and regular problems. We also give a characterization of H-x controllers in state space. The LMI existence condition can be easily determined whether or not, it is feasible and if it is then one can easily obtain H-x controller gain matrices via convex optimization.