In this work, we address the problem of reducing the vibrations in the case of a thin flexible beam, with an Euler-Bernoulli configuration thanks to a new type of feedback controller based on a delayed proportional action. The flexible structure is equipped with two piezoelectric patches: one of them works as a sensor and the other as an actuator. Each one of them is bonded on one side of the beam and both are collocated. The model of this system is obtained from a finite element analysis, leading to a linear state space model. The purpose of this work is to investigate the properties of the proposed output feedback controller in terms of vibration damping and of robustness. This controller is based on a rightmost-characteristic root assignment approach. Simulation results show the effectiveness of this method of controller design.