This paper provides a novel method for designing step tracking controllers for systems with saturating actuators. The approach is based on the theory of Quasi-linear Control (QLC), which offers methods for designing random reference tracking controllers for systems with nonlinear actuators and sensors. In the current paper, a QLC approach to designing step tracking controllers is presented. The development is based on two ideas: introducing a precompensator, which observes given step tracking specifications, and recasting the output of the precompensator into a random reference bandwidth requirement, with subsequent utilization of QLC. Unlike other techniques, the method developed here takes the saturation into account directly at the initial stage of the design and does not require subsequent augmentations (e.g., anti-windup). Nevertheless, for the sake of completeness, a comparison with the anti-windup approach is provided.