Output-feedback boundary control of an uncertain heat equation with noncollocated observation: A sliding-mode approach

The boundary stabilization problem of a one-dimensional unstable heat conduction system with boundary disturbance is investigated using a sliding-mode approach. This infinite-dimensional system, mathematical modeled by a parabolic partial differential equation (PDE), is powered with a Dirichlet type boundary actuator and only sensing at opposite end. By applying the Volterra integral transformation, a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties. The associated Lyapunov function is used for designing an infinite-dimensional sliding manifold, on which the system exhibits the same type of stability and robustness against of bounded exogenous boundary disturbance. By utilizing the similar transformation, an infinite-dimensional sliding-mode observer is proposed to reconstruct the system´ states, which is with robustness to boundary disturbance. Moreover, the relative degree of the chosen sliding function with respect to the output-feedback boundary control input is zero. A continuous control law satisfying the reaching condition is obtained by passing a discontinuous (signum) signal through an integrator.