Rejecting Unknown Harmonic Disturbances in $ 2 \times 2 $ Linear Hyperbolic PDEs

Abstract

In this paper, we solve the problem of rejecting a harmonic disturbance containing unknown frequencies and bias entering a linear hyperbolic system of 1-D partial differential equations at one boundary, using boundary sensing and actuation anticollocated with the disturbance. By combining a full state feedback controller with an adaptive observer that estimates system states and the disturbance’s bias, frequencies, amplitudes, and phases with exponential convergence, the effect of the disturbance is rejected exponentially fast. The theoretical results are applied to a relevant problem from the oil and gas industry, and the performance is demonstrated in a computer simulation.