Control of a Vibrating String using Impedance Matching

The problem of adding damping to a vibrating string considered. An approach novel to the control community is used to define the time optimal control for this system. This approach treats the string as a waveguide and oscillations as a result of reflections at discontinuities -- namely the end points. Through the appropriate choice of the controller, the energy in the string can be completely absorbed by the control, eliminating reflections and likewise oscillations. In this paper, the characteristic impedance of a string is defined. It is shown that by choosing the control to be a damper of like impedance, all oscillations decay as quickly as possible. A finite element model of a string is used to verify the results.