## 2 Materials and Methods
### Design of Tracking Differentiators
#### 2.3.2 Designing a Three-Block Tracking Differentiator
Let us first formulate sufficient conditions of the boundedness of solutions of the system (22) without considering (19). To simplify the setting, let us establish the following initial conditions in the tracking differentiator (18): \(z_{1}(0)=\chi_{1}(0)\), \(z_{i}(0)=\overset{\rightarrow}{0},i=2,3\), which by virtue of \(\sigma(0)=0\), will ensure zero initial conditions for the state variables in the virtual system (22):  
\[e_{i}(0)=\overset{\rightarrow}{0},i=1,2,3. \tag{24}\]  
As we see, the structure of Equation (22) is similar to the structure of the closed-loop system (9). We can treat all terms on the right side of each equation, except for the sigmoidal ones, as external bounded disturbances. While analyzing the system (22), let us take as a base the proof of lemma 1. By virtue of (24), the variables of the system (22) are in domains similar to (10) from the beginning at \(t=0\):  
\[ \begin{array}{l}\left\|e_{1}(t)\right\|=\left\|z_{1}(t)-\chi_{1}(t)\right\| \leq 2.2/l_{1},\\ \left|e_{i}(t)\right|=\left\|z_{i}(t)+p_{i-1}\sigma(l_{i-1}e_{i-1}(t))\right\| \leq 2.2/l_{i},i=2,3.\end{array}  \tag{25}\]  
Sufficient conditions under which inequalities (25) will be correct at \(t\in[0;T]\), similarly to (14), have the form  
\[ \begin{array}{l}\left\|e_{2}\right\|+\left\|\dot{\chi}_{1}\right\|<0.8p_{1},\\ \left\|e_{3}\right\|+\left\|\Lambda_{1}\right\|<0.8p_{2},\\ \left\|\Lambda_{2}\right\|<0.8p_{3}.\end{array}  \tag{26}\]  
Then, for the derivatives of \(\dot{e}_{1},\dot{e}_{2}\), and, consequently, for Equation (23), similarly to (12), we obtain the estimations  
\[\left\|\dot{e}_{i}\right\|\leq 2p_{i^{\prime}}\left\|\Lambda_{i}\right\|\leq p _{i}^{2}l_{i^{\prime}},i=1,2. \tag{27}\]  
Considering (17), (25), (27), the inequalities (26) take the form  
\[ \begin{array}{l}1.25(2.2/l_{2}+X_{1})<p_{1},\\ 1.25(2.2/l_{3}+p_{1}^{2}l_{1})<p_{2},\\ 1.25p_{2}^{2}l_{2}<p_{3}.\end{array}  \tag{28}\]  
In Equation (28) are sufficient conditions under which inequalities (25) will be fulfilled at \(t\in[0;T]\). Now, let us introduce constraints on the state variables of the tracking differentiator (18), its corrective influences, and the rate of their change. Considering (25) and the inverse Equation (20), namely \(z_{2}=e_{2}-p_{1}\sigma(l_{1}e_{1})\), \(z_{3}=e_{3}-p_{2}\sigma(l_{2}e_{2})\), we obtain a system of dual inequalities. \[\left[ \begin{array}{l}2.75/l_{2}+1.25X_{1}<p_{1}\leq Y_{1}-2.2/l_{2},\\ 2.75/l_{3}+1.25p_{1}^{2}l_{1}<p_{2}\leq Y_{2}-2.2/l_{3},\\ 1.25p_{2}^{2}l_{2}<p_{3}\leq Y_{3};\\ p_{3}^{2}l_{3}\leq Y_{4}.\end{array} \right. \tag{29}\]  
By virtue of a priori assumptions, namely the lower inequalities (4), considering (14), we assume the system (29) to be joint. It has infinitely many solutions. Under the assumption of non-smoothness of the external signal \(\chi_{1}(t)\), the accuracy of stabilization of the variables \(e_{2}(t),e_{3}(t)\) (25) is not important. The tracking error at special points is bounded \(\left\|e_{1}(t)\right\|\leq 2.2/l_{1}\). In the steady-state mode, we have an estimate similar to (11):  
\[\left\|e_{1}(t)\right\|\leq\frac{2.2}{l_{1}}\cdot\frac{2.75/l_{2}+1.25X_{1}}{p_ {1}}<\frac{2.2}{l_{1}}. \tag{30}\]To minimize the steady-state tracking error (30), an iterative procedure for setting the gains of sigmoid functions based on the system of inequalities (29) is required. For example, the following variant is proposed:  
\[ \begin{array}{l}p_{3}=Y_{3},l_{3}=Y_{4}/p_{3}^{2}=Y_{4}/Y_{3}^{2},\\ p_{2}=Y_{2}-2.2/l_{3}=Y_{2}-2.2Y_{3}^{2}/Y_{4},\\ l_{2}=(0.8p_{3}-\beta_{2})/p_{2}^{2}=(0.8Y_{3}-\beta_{2})/(Y_{2}-2.2Y_{3}^{2}/Y_ {4})^{2},\\ p_{1}=Y_{1}-2.2/l_{2}=Y_{1}-2.2(Y_{2}-2.2Y_{3}^{2}/Y_{4})^{2}/(0.8Y_{3}-\beta_{2 }),\\ l_{1}=(0.8p_{2}-\beta_{1}-2.2/l_{3})/p_{1}^{2}=\\ =(0.8(Y_{2}-2.2Y_{3}^{2}/Y_{4})-\beta_{1}-2.2Y_{3}^{2}/Y_{4})/[Y_{1}-2.2(Y_{2}-2.2Y_{3}^{2}/Y_{4})^{2}/(0.8Y_{3}-\beta_{2})]^{2},\end{array}  \tag{31}\]  
where \(\beta_{1,2}\) are very small positive constants that are introduced so that the left inequalities in the system (29) remain strict.