Output Regulation for a One-Dimensional Heat Equation with Input Delay and Unknown Exosystem

In this work, we investigate the performance output tracking for a one-dimensional heat equation with input delay and disturbances that enter all possible channels. Besides, both the frequencies and coefficients of disturbances are unknown. By treating the input delay as dynamics dominated by a first-order hyperbolic equation, we transform the heat equation to a PDE-PDE cascaded system. For this cascaded system, we first use the internal model principle to simplify the robust output tracking into a dynamic tracking error-based control. Then, a full state feedback control law is proposed to stabilize the system by virtue of the backstepping method. Using adaptive observer approach, we estimate states and disturbances of the system based on known tracking error, respectively. Finally, based on the observers, we develop an output feedback control law which makes the system to be robust to input delay and disturbances.