A negative imaginary approach to control the vibrations of a string

A guitar string that is rigidly clamped at both the ends is considered. This string is actuated by a supplying current, through it, in the presence of magnetic field. A sensor measuring the string displacement is positioned such that the input, which is the supplied current, and the output, the string displacement, appear collocated. An experiment is performed by exciting the string using sine-sweep and measuring the corresponding output. A model is fit to the input-output data, and is shown to satisfy the negative imaginary property. An integral resonant controller is designed by exploiting negative imaginary property such that it damps the first five resonant modes of the string. This is verified experimentally.