A Lyapunov approach to second-order sliding-mode boundary control of an unstable heat system with spatiotemporal-varying parameters under boundary disturbances

This paper considers a boundary stabilization problem of an unstable heat system incorporated with spatial and temporal varying coefficients subjected to boundary uncertainties. The system model is governed by a second-order parabolic partial differential equation (PDE). By taking the Volterra integral transformation, we can obtain a target PDE with asymptotic stability characteristics in the new coordinates when an appropriate backstepping boundary control input is applied. The implicated backsteeping control law can be further integrated into the matched boundary disturbance. The associated Lyapunov function can then be used for designing an in nite-dimensional sliding surface, on which the system exhibits exponential stability, invariant of the bounded matched disturbance. Based on the Lyapunov method, a second-order sliding-mode boundary control, constructed by the integration of discontinuous signal, is employed to maintain the robustness to matched boundary disturbance. The closed-loop stability of the controlled system is also verified. Simulation results are provided to demonstrate the feasibility of this proposed control scheme.