Exponentially stabilizing boundary control of string-mass systems

Exponentially stabilizing controllers are derived for the transverse vibration of a string-mass system modeled by the one-dimensional wave equation with a pinned and a controlled boundary condition. Lyapunov's theory for distributed parameter systems, the Meyer-Kalman-Yakubovitch Lemma, and integral inequalities prove that a class of boundary controllers provide strong exponential stability. These controllers are designed so that the transfer function between boundary slope and velocity satisfies a restricted strictly positive real condition. An example controller, consisting of boundary position, velocity, slope, slope rate, and integrated slope feedback, is implemented on a laboratory test stand. In experimental impulse response tests, the controlled response decays six times faster than the open-loop response and has half the response amplitude.