The design of optimal disturbance accommodation and servomechanism controllers with limited plant model information is considered in this paper. Their closed-loop performance are compared using a performance metric called competitive ratio which is the worst-case ratio of the cost of a given control design strategy to the cost of the optimal control design with full model information. It was recently shown that when it comes to designing optimal centralized or partially structured decentralized state-feedback controllers with limited model information, the best control design strategy in terms of competitive ratio is a static one. This is true even though the optimal structured decentralized state-feedback controller with full model information is dynamic. In this paper, we show that, in contrast, the best limited model information control design strategy for the disturbance accommodation problem gives a dynamic controller. We find an explicit minimizer of the competitive ratio and we show that it is undominated, that is, there is no other control design strategy that performs better for all possible plants while having the same worst-case ratio. This optimal controller can be separated into a static feedback law and a dynamic disturbance observer. For constant disturbances, it is shown that this structure corresponds to proportional-integral control.