Backstepping-based adaptive error feedback regulator design for one-dimensional reaction-diffusion equation

In this paper, we consider the error feedback regulator problem (EFRP) for one-dimensional reaction-diffusion equation with unknown harmonic boundary disturbance, where the regulated output is anti-collocated with the control. Some auxiliary systems are constructed in order to make the control and the disturbance become collocated, or make the measured tracking error become the output. Then the error feedback adaptive servomechanism is presented, in which the parameters of the disturbance and the reference signals are estimated. Our design is based on the motion planning and the backstepping approaches. In addition, we give a brief description that our approach is applicable to the heat equation with distributed disturbance. It is shown that the proposed adaptive control law regulates the tracking error to zero and keeps the states of all the internal loops bounded.