Compensation of Spatially Varying Input Delay in Distributed Control of Reaction-Diffusion PDEs
Jie Qi, Miroslav Krstić
Abstract
We develop a predictor-based distributed feedback controller that guarantees exponential stability for a class of reaction-diffusion partial differential equations (PDEs) subject to spatially varying input delay. The delay depends on the location of the input in the spatial domain. In order to design a distributed controller to compensate the spatially varying delay, we first introduce an implicit backstepping transformation, which contains the state of the target system on both sides of the definition and then derive an additional backstepping transformation by a successive integration approach to arrive at a target system that is a distributed cascade of a 2-D transport PDE into a 1-D reaction-diffusion PDE. The resulting delay-compensated controller includes spatially weighted state feedback and feedback of the earlier inputs in four differential spatial regions, which avoids the need for feedback of future inputs, whereas the future input feedback would result if the conventional backstepping transformation was applied. The inverse transformation is also derived, to prove L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> exponential stability. The applicability and performance of the controller is evaluated in simulation studies.