Linear discrete-time dynamical systems xk + I = Axk + kwith constrained inputs ck ∈ ω, for which the matrix A possesses the property of leaving a proper cone AK + positively invariant, i.e. AK + ⊂ K + . Necessary and sufficient conditions guarantee that a non-empty set 𝒟(K; a, b) ⊂ Rn, obtained from the intersection of translated proper cones, is positively invariant for motions of the system. Both the homogeneous and inhomogeous cases are considered. In the latter case, the external behaviour of motions, i.e. for trajectories originating from x0 ⊂ Rn/𝒟(K; a, b) (respectively,xo ⊂ Rn) is studied in terms of attractive and contractivity of the set 𝒟(K; a, b). The global attractivity conditions of 𝒟(K; a, b) are also given. It is shown how the results presented can be used to solve the saturated state feedback regulator problem.