The backstepping and related block control principle is a widely used method of robust control. When external and parametric disturbances act on the plant, the block approach can be combined with active disturbance rejection control (ADRC). The ADRC estimates the matched uncertainties in the transformed model via a disturbance observer. Control consists of two parts: stabilizing and compensating for uncertainties. It is standard to use linear feedback in both the controller and observer, which may contribute to overshooting. The aim of this paper is to develop an approach that is free of this problem to control the Euler angles and altitude of a quadcopter in the presence of uncertainties in the plant model. To achieve this goal, we propose a block control principle with smooth and bounded nonlinear feedback in the form of a hyperbolic tangent. In this control law, we introduce acceleration-related parts to reduce the oscillations of the state variables. In practice, these oscillations are particularly strong in the roll and pitch channels because of the technical features of the plant and the uncertain dynamics of the motors. In addition, we propose a tracking differentiator to smooth out the stepped reference signals and to avoid sudden jerks during motion. A mathematical proof of the stability of the closed-loop system is presented. On the basis of this proof, we obtain the conditions for selecting the controller coefficients, which ensure the tracking of the smoothed reference signals for the Euler angles and altitude with a given accuracy. We present the results of the proposed algorithms on a quadcopter with a standard frame F450. In an experiment with a 10-degree yaw reference signal jump, the proposed approach allowed: reducing the magnitude of the control jump by a factor of 3.86, reducing the maximum altitude stabilization error by a factor of 5.43, and avoiding oscillations of the state variables compared with the standard approach with linear controls and without a tracking differentiator.