On the existence and uniqueness of solutions to the output regulation equations for periodic exosystems

In this paper we prove that, for a general class of control-affine systems, the output regulation equations are uniquely solvable whenever the exosystem is periodic and the linearized zero-dynamics of the plant does not contain periodic solutions of the same period as those of the exosystem. Our main result can therefore be applied to cases when the linearized zero-dynamics are non-hyperbolic. As an application, we consider the important case of when the exosystem is composed of k-uncoupled harmonic oscillators.