The classical study of controllability of linear systems assumes unconstrained control inputs. The "distance to uncontrollability" measures the size of the smallest perturbation to the matrix description of the system rendering it uncontrollable and is a key measure of system robustness. We extend the standard theory of this measure of controllability to the case where the control input must satisfy given linear inequalities. Specifically, we consider the control of differential inclusions, concentrating on the particular case where the control input takes values in a given convex cone.