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.

中文翻译：

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具有非局部项的一维抛物线方程的自适应输出调节

在本文中，我们为具有非局部项和位于所有通道的扰动的一维热方程开发了自适应误差反馈调节器设计。首先，我们构造了一个变换，将原系统转化为第一个辅助系统，在这个系统中，可测量的跟踪误差成为输出，位于域内和控制端的扰动都转化为非受控端。接下来，我们设计了第一辅助系统的观测器和基于可测量跟踪误差的未知参数估计机制。然后，通过另一个变换，我们得到第二个辅助系统，其中扰动和参考信号与控制搭配。最后，采用反步法设计了自适应误差反馈调节器。