In this paper, we investigate the reachability analysis and safety controller synthesis problem for linear discrete-Time systems under additive bounded disturbances. We consider the following problem; design a state feedback controller such that any state trajectories starting from an initial set can be robustly controlled towards a target one in finite time, while at the same time avoiding any prohibited regions. One of the potential disadvantages of existing reachability algorithms when external disturbances are taken into account, may be that the solution to guarantee reachability becomes conservative. Motivated by this, this paper provides a new controller synthesis framework based on the notion of tube-based control strategy, in which a suitable sequence of polytopes is generated according to a convex feasibility problem. An illustrative simulation validates the effectiveness of our proposed method.