Recent research has proposed the use of feedback of the observed climate to adjust the radiative forcing of solar radiation management (SRM)-based geoengineering schemes, which involve deliberate and large-scale intervention in the planetary environment to counteract anthropogenic climate change. Feedback in an SRM scheme has the potential to compensate for uncertainty in both the forcing and the climate response, and for unexpected changes in the climate system. The long-term warming effects arising from anthropogenic emission of greenhouse gases are thereby (at least partially) offset, in such a way that neither natural climate variability nor measurement noise is unduly amplified. In this paper SRM is framed as a feedback control problem for disturbance rejection, drawing on $H_\infty$-synthesis as a formal framework in which the effect of anthropogenic climate disturbances can be minimized. The effectiveness of both an $H_\infty$-suboptimal SRM controller and a simple proportional-integral (PI) controller is demonstrated on the reduced-complexity climate model MAGICC. The extent and speed with which negative radiative forcing could feasibly be implemented and sustained impose tight constraints on the effectiveness of the control actuation authority in an SRM climate control loop. This in turn suggests caution in relying heavily on feedback to counterbalance uncertainty in the climate system when implementing SRM.