In this paper, we propose a control law for a discrete-time linear system with actuator saturation to track time-varying reference signals. The proposed control law consists of a feedback controller and a target recalculation mechanism. The feedback controller includes an integrator to achieve zero steady-state error in the case where the reference signal is constant. The feedback gains of the controller are parameterized by a single scheduling parameter. In the proposed control algorithm, when the tracking error is large, a modified reference signal is computed by the target recalculation mechanism so that feasibility of the algorithm and stability of the control system are guaranteed at all times. At this stage, the controller state is modified online so that the tracking control performance is improved. Further, when the tracking error becomes small, the scheduling parameter and the controller state are updated simultaneously so that the tracking control performance is improved. The problems of determining the scheduling parameter, the controller state, and the modified reference signal are reduced to convex optimization problems with linear matrix inequality constraints. The effectiveness of the proposed control method is shown through an experiment.