In this paper we shall extend and amplify the recent theory of controllers employing the so-called internal model structure. For exponentially stable closed loop control systems this structure has been shown in another paper to be necessary and sufficient for robust output regulation, also in infinite-dimensional spaces. Here we shall derive conditions under which two controller types occurring frequently in applications have the internal model structure. Under these conditions robust output regulation is achieved-also in infinite-dimensional spaces-if the closed loop system is sufficiently stable. In the case that the closed loop system is only strongly (but not exponentially) stable, it is sometimes possible to obtain conditional robustness. This means that asymptotic tracking/disturbance rejection is not destroyed by small perturbations so long as closed loop stability also persists. Our results allow for infinite-dimensional plants, controllers, and exogenous systems, and as an application of such a general setting we shall consider generalized repetitive control for exponentially stable infinite-dimensional SISO systems.