Boundary stabilization of a cascade of ODE-wave systems subject to boundary control matched disturbance

In this paper, we are concerned with a cascade of ODE-wave systems with the control actuator-matched disturbance at the boundary of the wave equation. We use the sliding mode control (SMC) technique and the active disturbance rejection control method to overcome the disturbance, respectively. By the SMC approach, the disturbance is supposed to be bounded only. The existence and uniqueness of solution for the closed-loop via SMC are proved, and the monotonicity of the ‘reaching condition’ is presented without the differentiation of the sliding mode function, for which it may not always exist for the weak solution of the closed-loop system. Considering that the SMC usually requires the large control gain and may exhibit chattering behavior, we then develop an active disturbance rejection control to attenuate the disturbance. The disturbance is canceled in the feedback loop. The closed-loop systems with constant high gain and time-varying high gain are shown respectively to be practically stable and asymptotically stable. Then we continue to consider output feedback stabilization for this coupled ODE-wave system, and we design a variable structure unknown input-type state observer that is shown to be exponentially convergent. The disturbance is estimated through the extended state observer and then canceled in the feedback loop by its approximated value. These enable us to design an observer-based output feedback stabilizing control to this uncertain coupled system.