Hybrid external models, and the rejection of harmonic disturbances at the controller input

In this paper, we address the problem of rejecting harmonic disturbances occurring at the input of a linear controller in a stable loop. The task is to recover regulation of the plant output, using only information corrupted by harmonic noise of unknown frequency. The problem departs from the standard output regulation framework, as the assumption of the explicit availability of an error signal to be regulated to zero is not met. We take a novel approach of observing and rejecting the disturbance, reconstructing asymptotically the steady state behavior of the closed-loop system. The solution is provided by an adaptive external model (that is a device placed outside the stabilizing loop) that due to the specific structure of the associated certainty-equivalence, the solution has acquired hybrid system structure. The important issue of convergence in absence of persistence of excitation is addressed in full.