Stabilization of a class of infinite-dimensional systems subjected to unknown matched disturbances

In this paper, we consider the problem of constructing an output-feedback controller for stabilizing an un-stable linear PDE system with bounded control and observation operators driven by an unknown disturbance. The disturbance is matched with the input of the PDE system and its derivative is assumed to be bounded. Under a smoothness assumption on the observation operator and an observability matching condition, we first develop a linear observer with disturbance decoupling for state estimation and then we develop a sliding mode observer for estimating the disturbance. Using the estimated state and disturbance we implement a state-feedback control law which guarantees the exponential decay of the state of the PDE system to zero for all initial states. Our approach assumes certain prior knowledge regarding the stabilization and estimation of the PDE system in the absence of disturbances. Our contribution lies in performing stabilization (convergence of state to zero) and estimation in the presence of unknown disturbances. We illustrate our controller design approach in simulations using an unstable 1D heat equation.