Stabilization of a 2 × 2 system of hyperbolic PDEs with recirculation in the unactuated channel

A result by Guo and Jin (2010) considered boundary control of a string with point damping at the exact midpoint. An interpretation of the model admits a system of first-order hyperbolic (transport) PDEs with actuation at the boundary of one PDE. The unactuated second PDE exhibits recirculation phenomena. The recirculation coupling of these two PDEs gives rise to a stabilization problem with nonlocal terms that prior has not been considered. In this paper, we consider a general class of 2 × 2 hyperbolic PDEs where both the unactuated PDE and the actuated PDE have strict-feedback recirculation (from the outlet in the “upstream direction”). In addition, the state of the unactuated PDE feeds, in a non-local fashion, into the domain of the actuated PDE. We introduce a novel set of transformations through which we arrive at a simple target system with desirable exponential stability characteristics. A backstepping observer and output feedback controller design are also given for this 2 × 2 hyperbolic PDE system. We then apply our control design to the string-and-midway-antidamper application found in Guo and Jin and illustrate the result with simulations.