The control action signals are always bounded as far as real technological plants are concerned. Due to the nonlinear influence of such limits, the control quality does not always coincide with the required one, even when the ideal controller parameters are used. In the case of adaptive control systems with a reference model, this can lead to a drift of the controller parameters, which is unacceptable. Considering a torque control loop of a separately excited DC drive, there are two restrictions (saturations), which are imposed on the control actions: 1) the maximum supply voltage of the motor to limit the output of the armature current controller, 2) the maximum armature current to limit the output of the speed controller. In this study, considering an adaptive system to adjust the DC drive speed controller parameters, a method is proposed to minimize the drift of such parameters values. This aim is achieved through minimization of the error between the outputs of the plant and the reference model. For this purpose, it is proposed to adjust the reference model proportionally to the difference between the values of the control action before and after the saturation block. In contrast to the classical approach of dead zones, the proposed one is proved mathematically to provide the stability of the control loop. Oppose to the existing approaches of the same type, it also takes into consideration a saturation of the integral part of the speed controller. The effectiveness of the proposed method is shown through mathematical modelling.