Since the 80’s, the use of active bearings to influence the dynamic behaviour of flexible rotors has been intensively researched. Although theoretical knowledge and practical experience from laboratory models and industrial prototypes is available for various configurations of actuators, sensors and control strategies, active bearings are not yet broadly applied in industry. Possible reasons are the absence of an "ideal" active bearing as well as an automatic control concept, which provides sufficient performance under the constraints of practical system. Moreover there is lack of well-priced actuators which satisfy the construction and task requirements. In order to apply such active vibration controls one needs to solve various more or less complex problems. Researching methods and components to overcome such difficulties and to facilitate the industrial usage of these controls led to the present work. Therefore the main subject of this study is the development of a control concept for an active bearing aiming at the bending vibration damping of elastic rotors. This concept contains the proper selection, positioning and dimensioning of actuators and sensors including the appropriate control strategy. The first section of this work focuses on the mathematical description of flexible rotors which is achieved by defining an abstract rotor system to study typical vibration problems. Modelling elastic rotors is based on the RITZ-theory. Thereby many natural bending frequencies can be described with a low system order. The equations of motion are derived according to the LAGRANGE principle. The second section discusses the active support. This term describes here a bearing not directly placed on the machine structure but supported by an actuator. The control forces act as relative forces between the fundament and the bearing. In contrast to passive bearings no significant changes on the rotor system are necessary. It is thus easier to integrate the bearing in an already existing machine concept. A piezo-actuator is used in this study. Such actuators can generate forces or displacements and provide a wide application range. Because of the high actuator stiffness only minor changes of the passive system behaviour are expected, which is an advantage in case the actuator breaks down. Dimensioning criteria such as the maximal actuator stroke and diameter are determined for the required control forces and performance of piezo-actuators.New terms describing the active support are introduced to the equations of motion of the uncontrolled rotor, increasing the system’s degree of freedom and leading to a higher model order. The next section of this work is devoted to the control design. In the beginning the controller design purpose of active damping is determined by means of plant analysis. Developing an appropriate control strategy depends significantly on the kind and location of the applied sensors. Therefore sensors and their positioning are addressed subsequent to treating the actuators. This discussion proceeds in parallel to the study of control algorithms and shows the controller complexity and the achievable performance. In this framework it is dealt with linear, time invariant controls. A practical, systematical and robust concept is sought, which guarantees the stability of the closed loop system while preserving performance. Two control principles are suggested: collocation and robust synthesis. By using a collocated actor-sensor-pair one can benefit from simpler controllers which guarantee a good performance even with high uncertainty. If the actuator cannot be placed at the optimal position because of technical reasons, it is possible to increase the performance by positioning the sensor at a relevant place. In this case the collocation drops out and simple control algorithms cannot be applied any more since the stability of the closed loop would be limited. Such instabilities can also arise in the collocation case if the loss of amplitude and phase is large.With the formalistic robust control synthesis, which is based on a physical model, a second variant is presented. It is based on the well-known methods of the Hinfin- and the µ-synthesis. These methods offer a good performance and a high robustness of the closed loop control regarding uncertainties, thus allowing a practical implementation of the controller with a high damping ratio. Since the feasibility of the suggested concepts is a basic requirement for their industrial usage, the last section reports the measurement results on the basis of a laboratory test rig using robust control synthesis. It is shown that time invariant robust control synthesis by means of piezo actuator puts very well the theoretically computed damping ratio into effect and can significantly reduce the resonance amplitude with regard to active damping. The formalistic design steps of the robust synthesis shows a systematic realization of the automatic controller, whose first and second part, i.e. the modelling and the specification of the performance, must be mastered by the technician. The expenditure of the last step of the control design for robust synthesis is reflected as unfavourable but this complexity is cancelled using the commercial toolboxes.