Robustness properties of controllers with reduced order internal models

In this paper we study robust output regulation for distributed parameter systems. In particular we are interested in the internal model principle, which can be used in characterizing controllers that achieve robust output tracking and disturbance rejection for a linear system. We show that if we do not require robustness with respect to arbitrary perturbations, then there may exist robust controllers that do not contain a full-sized internal model of the exosystem´s dynamics. The existence of such controllers depends on the class of admissible perturbations. Our approach also establishes a convenient way of testing the robustness of a controller with respect to given perturbations. The theoretic results are applied to analyzing the robustness properties of controllers for a system of two shock absorber models, and for a one-dimensional heat equation.