Adaptive Stabilization for Cascaded PDE–ODE Systems With a Wide Class of Input Disturbances

This paper focuses on the adaptive stabilization for a class of cascaded partial differential equation–ordinary differential equation systems with input disturbance. Remarkably, the disturbance includes harmonic disturbance and periodic disturbance as special cases and may not be smooth (but must be in the related literature). This substantially broadens the scope of input disturbance and highlights the main contribution. To solve the stabilization problem, a novel adaptive learning control scheme is successfully proposed. First, inspired by learning control idea and infinite-dimensional backstepping method, a state-feedback controller is constructed, in which some pivotal to-be-updated parameters are involved to compensate for the disturbance. Then, a crucial switching mechanism is proposed to online update the parameters in the designed controller. It is shown that the resulting closed-loop system has a unique solution, and all the closed-loop system states are bounded and converge to zero ultimately. Finally, a simulation example is provided to illustrate the effectiveness of the obtained theoretical results.