Nonreflective boundary control of a vibrating string

One of the important applications in the developing Large Space Systems Technology will be the electrostatically controlled membrane mirror antenna. A high level of surface quality is achievable using electrostatically tensioned membranes in which surface accuracy is obtained through active control. Electrostatic actuators behind the membrane surface would provide a means of obtaining the prescribed surface shape and also be utilized to suppress the structural vibrations in the system. The surface quality, in this case, would be limited by the size, force field shape, and the number of the electrostatic actuators. An additional control capability is to introduce boundary control at the membrane perimeter. Using this additional control mechanism, structural vibrations can be absorbed at the boundary without being reflected back into the interior regions of the membrane antenna. In this paper, boundary control of a vibrating string is studied. For this system, a nonreflective boundary control is developed in which waves reaching the boundary are absorbed by the appropriate control movement of the boundary. The control is closed-loop and utilizes a single measurement close to the boundary. The closed-loop control is a delay of the measurement. The delay is determined by the velocity of wave propagation in the string and the location of the sensor.