Adaptive error feedback regulation problem for 1D wave equation

By using the adaptive control approach, we solve the error feedback regulator problem for the one-dimensional wave equation with a general harmonic disturbance anticollocated with control and with two types of disturbed measurements, ie, one collocated with control and the other anti-collocated with control. Different from the classical error feedback regulator design, which is based on the internal mode principle, we give the adaptive servomechanism design for the system by making use of the measured tracking error (and its time derivative) and the estimation mechanism for the parameters of the disturbance and of the unknown reference. Constructing auxiliary systems and observer and applying the backstepping method for infinite-dimensional system play important roles in the design. The control objective, which is to regulate the tracking error to zero and to keep the states bounded, is achieved.