Dead-Time Compensation for Wave/String PDEs

Smith Predictor-like designs for compensation of arbitrarily long input delays are commonly available only for finite-dimensional systems. Only very few examples exist where such compensation has been achieved for PDE systems, including our recent result for a parabolic (reaction-diffusion) PDE. In this paper we address a more challenging wave PDE problem, where the difficulty is amplified by allowing all of this PDE's eigenvalues to be a distance to the right of the imaginary axis. Anti-damping (positive feedback) on the uncontrolled boundary induces this dramatic form of instability. We develop a design which compensates an arbitrarily long delay at the input of the boundary control system and achieve exponential stability in closed loop. We derive explicit formulae for our controller's gain kernel functions. They are related to the open-loop solutions of the anti-stable wave equation system over the time period of input delay (this simple relationship is the result of the design approach).