Output regulation of a reaction-diffusion PDE with long time delay using backstepping approach

In this paper, we are concerned with the output regulation of an unstable reaction-diffusion PDE with regulator delay and unmatched disturbances, which are generated by an exosystem. A backstepping regulator design procedure is presented by mapping the reaction-diffusion PDE cascaded with a transport equation into an error system, which is shown to be exponentially stable with a prescribed rate in a suitable Hilbert space. The output regulation is then verified by solving cascaded regulator equations. The solvability condition of the cascaded regulator equations is characterized by a transfer function and eigenvalues of the exosystem. Finally, the numerical simulations are provided to illustrate the effect of the regulator.