Backstepping Boundary Control of a Wave PDE With Spatially Distributed Time Invariant Unknown Disturbances

An observer based boundary control scheme is suggested for an infinite dimensional system modeled as a wave partial differential equation (PDE) with in-domain and boundary disturbances. We consider a collocated case where the sensors and the actuators are located at the same boundary. The aim of the controller is to regulate the rate of change of the opposite boundary around a given reference. The unknown in-domain disturbance is spatially distributed but it is constant in time same as the unknown boundary disturbance. The problem is reformulated as a stabilization of an linear time invariant (LTI) system with simultaneous delays in input and output channels. The exponential stability of the equilibrium of the closed-loop system is proven. The performance of the controller is shown using numerical simulations.