The limited output of various drives means a challenge in controller design whenever the acceleration need of the nominal trajectory to be tracked temporarily exceeds the abilities of the saturated control system. The prevailing control design methods can tackle this problem either in a single theoretical step or in two consecutive steps. In this latter case in the first step the design happens without taking into account the actuator constraints, then apply a saturation compensator if the phenomenon of windup is observed. In the Fixed Point Iteration- based Adaptive Control (FPIAC) that has been developed as an alternative of the Lyapunov function-based approach the actuator saturation causes problems in its both elementary levels: in the kinematic/kinetic level where the desired acceleration is calculated, and in the iterative process that compensates the effects of modeling errors of the dynamic system under control and that of the external disturbances. The here presented approach tackles this problem in both levels by relatively simple considerations. To illustrate the method's efficiency simulation investigations were done in the FPIAC control of a modification of the van der Pol oscillator to which an additional strongly nonlinear term was added.