In this paper, the robust controllability for linear systems with constrained controls (x)over dot = Ax+Bu, u is an element of Omega, is studied under the assumption that Omega is an arbitrary subset satisfying the only condition 0 is an element of cl Omega and the system matrices are subjected to structured perturbations. The notion of the structured local and global controllability radii are defined. Based on properties of convex processes, some general formulas for the complex and the real controllability radii, for both local and global controllability concepts, are established with respect to structured affine perturbations and multiperturbations of matrices A, B. As particular cases, the general results are applied to linear systems with bounded controls and single- input linear systems, yielding some explicit and computable formulas of controllability radii. Examples are given to illustrate the obtained results.