Robust tracking of fractional-order diffusion equation with uncertain parameter using second-order sliding mode boundary control

In the present paper, we are concerned with the output tracking problem for a one-dimensional fractional-order diffusion equation with disturbance at the boundary and an uncertain parameter in the system. First, when the system is free from disturbance, an appropriate reference system is designed, followed by the design of a controller to enable the system's output to track a given reference signal. Subsequently, when the system is considered with disturbance, a classic second-order sliding mode controller is ingeniously constructed to counteract this disturbance. Meanwhile, the Lyapunov method is employed to prove that the system's output tracks the given reference signal. Finally, numerical simulation examples are provided to demonstrate the rationality of the controller.