A Koopman-backstepping approach to data-driven robust output regulation for linear parabolic systems

In this paper a solution of the data-driven robust output regulation problem for linear parabolic systems is presented. Both the system as well as the ODE, i.e., the disturbance model, describing the disturbances are unknown, but finite-time sequential data obtained from measurements of the output to be controlled and additional boundary outputs are available. The data-driven controller is designed in the Koopman operator framework for PDEs, where the Koopman modes and eigenvalues are obtained from data using Hankel-DMD. It is shown that all system parameters and the eigenvalues of the disturbance model can be recovered from the available measurements by solving an inverse Sturm-Liouville problem. This allows to directly apply backstepping methods for the robust regulator design. For this, closed-loop stability in the presence of small errors in the Hankel-DMD is verified in the nominal case. Robust output regulation is shown for non-destabilizing model uncertainties. A numerical example demonstrates the results of the paper.