This note addresses the derivation of state-space realizations for the feedback control of linear, high-index differential-algebraic-equation systems that are not controllable at infinity. In particular, a class of systems is considered for which the underlying algebraic constraints involve the control inputs, and thus a state-space realization cannot be derived independently of the feedback controller. The proposed methodology involves the design of a dynamic state feedback compensator such that the underlying algebraic constraints in the resulting modified system are independent of the new inputs. A state-space realization of the feedback-modified system is then derived that can be used as the basis for controller synthesis.