Optimal Repetitive Control with Preview for Active Vibration Control

In this paper a new repetitive control scheme is presented for active vibration control of linear systems subject to periodic disturbances. The motion of the structure is controlled by momentum exchange with essentially collocated masses. Due to inherent structural limitations, performance trade-offs have to be made and perfect regulation cannot be achieved even when the disturbance is known. By combining ideas from optimal preview control, simultaneous state and input estimation, and repetitive control, a methodology is developed that allows for minimization of system vibration as defined by a quadratic performance index. The system disturbance is estimated based on the information from the most recent period and, assuming the disturbance to be repetitive, this estimate is used to compute a preview input for the consecutive cycle. Since the optimal preview control law requires a full state feedback and feedforward of a variable dependent on future inputs, an observer is required. A design procedure for a simultaneous state and disturbance observer is therefore developed. Necessary and sufficient conditions for the existence of a full order observer are formulated using straightforward matrix calculus. In principle, the system can achieve optimal performance after one cycle of operation. The proposed approach is illustrated on an example of a 2-DOF model representing a machine with an active vibration absorber.