Sliding mode control to stabilization of cascaded heat PDE–ODE systems subject to boundary control matched disturbance

In this paper, we are concerned with the boundary feedback stabilization of a cascade of heat PDE–ODE system with Dirichlet/Neumann interconnection and with the external disturbance flowing the control end. In order to deal with the disturbance, the sliding model control (SMC) is integrated with the backstepping approach, where the disturbance is supposed to be bounded only. The existence and uniqueness of the solution for the closed-loop system are proved, and the monotonicity of the “reaching condition” is presented without differentiation of the sliding mode function, for which it may not always exist for the weak solution of the closed-loop system. Finally the numerical simulations validate the effectiveness of this method for the system with periodic and normal random disturbances respectively.