This paper addresses the design of robust dynamic output feedback control for highly uncertain systems in which the unknown disturbance might be excited by the derivative of the control input. This context appears in many industrial problems such as the speed control of the hydraulic turbines and the frequency stabilization in micro grids to cite but few examples. A key feature that has to be carefully addressed in this context is that too agressive feedback might lead to the loss of controllability and/or a significant drop in the closed-loop performance. The paper formulates the problem, underlines its relevance and gives a rigorous solution in which a dynamic output feedback is given together with a realistic set of sufficient conditions on the controller's parameters that enable to assess the behavior of the closed-loop under different circumstances. A numerical example is given to illustrate the relevance of the proposed successful design characterization.