On the finite settling time and residual vibration control of flexible structures

Rapid end-point positioning of a structure with minimal residual vibration under a single bounded control input is considered. The structure between the actuator and the end point is assumed to be flexible, and is the main cause of positioning error from residual vibration. A standard solution to this problem is offered by the well known bang-bang control. Implementation of such a bang-bang control strategy leads to decreased response time but introduces unwanted vibrations from repeated overshooting of the switching contour, caused by errors in the implementation of the discontinuous control law. An appropriately shaped input function based on minimum energy and bounded control for such end-point positioning systems totally eliminates the residual vibration while making the response time small. The proposed forcing input has two discontinuities at most and therefore does not suffer from the undesirable intermediate discontinuities present in the bang-bang control. This novel control force compares favorably with the bang-bang solution with respect to response time. It also compares favorably with the ramped sinusoidal input force proposed by Meckl and Seering 1] with respect to residual vibration. They shaped the input function by eliminating the resonant frequencies where as the resonant frequencies are retained in the function proposed. As an illustration, the newly shaped forcing input is applied to a rotating thin flexible beam to suppress residual vibration.