This paper presents robust and optimal control methods to suppress vibrations of flexible payloads carried by robotic systems. A new improved estimator in discrete-time H2 optimal control design based on the Kalman Filter predictor form is developed here. Two control design methods using state-space models, LQR and H2 Optimal Design, in discrete-time domain are applied and compared. The manipulator joint encoders and the wrist-mounted six-degree-of-freedom force/torque sensor provide the control feedback. A complete dynamic model of the robot/payload system is taken into account to synthesize the controllers. Experimental verifications of both methods are performed using a Mitsubishi five-degree-of-freedom robot manipulator that carries a flexible beam. It is shown that both methods damp out the vibrations of the payload very effectively.