Adaptive output regulation for one-dimensional parabolic equation with nonlocal term

In this article, we develop the adaptive error feedback regulator design for one-dimensional heat equation with a nonlocal term and disturbances which are located all the channels. First, we construct a transformation to transform the original system into the first auxiliary system, and in this system the measurable tracking error becomes output, both the disturbances located at the in-domain and the control end are transformed to the uncontrolled end. Next, we design an observer for the first auxiliary system and an estimation mechanism for unknown parameters based on measurable tracking error. Then, by another transformation, we obtain the second auxiliary system, in which the disturbances and the reference signal are collocated with the control. Finally, the adaptive error feedback regulator is designed by using the backstepping method. The simulation results show that the regulator makes the tracking error asymptotically tend to zero and the state of the closed-loop system is bounded.