## 2 Materials and Methods
### Properties of the Sigma Function and Sigmoid Feedback
If \(\overline{p}>F\Leftrightarrow F/\overline{p}<1\), then there exists an acceptable value of \(c^{*}:F/\overline{p}<\sigma(c^{*})<1\), at which there is a non-empty interval for choosing a gain \(p^{*}:F/\sigma(c^{*})<p^{*}\leq\overline{p}\), that ensures (10). Thus, the sufficiency of the fulfillment of condition \(F<\overline{p}\) is proved. For fixed values of \(c^{*},p^{*}\) chosen from the above intervals, consider different variants of the initial conditions of System (9). If \(|x(0)|\leq\Delta\), then inequality (10) is correct at \(t\geq 0\). If \(|x(0)|>\Delta\), the state variable reaches the specified neighborhood in a finite time \(t^{*}>0\), and inequality (10) is correct at \(t\geq t^{*}\), where  
\[\frac{|x(0)|-\Delta}{p^{*}\sigma(c^{*})+F}\leq t^{*}\leq\frac{|x(0)|-\Delta}{p^ {*}\sigma(c^{*})-F}.\]  
Thus, if \(F<\overline{p}\), then for any initial conditions, the solution of System (8) is bounded. \(\square\)  
Note that in the neighborhood of \(|x|\leq\Delta\) (10), the sigma function is bounded from below by the oblique line (7), and for the derivative of the Lyapunov function, considering (8), let us take the following estimate:  
\[\dot{V}=x\dot{x}=x(f(t)-p\sigma(lx))\leq|x|(F-p\sigma(c)x/\Delta).\]  
At \(F<p\sigma(c)\), the inequality \(\dot{V}<0\) is correct outside the neighborhood  
\[|x|\leq\Delta\frac{F}{p\sigma(c)}=\frac{c}{l}\frac{F}{p\sigma(c)}<\Delta=\frac {c}{l}, \tag{11}\]  
to which the state variable converges. The radius of the domain (11) is smaller than the primary estimate (10). It follows from (11) that the radius of the convergence domain (11) is directly proportional to the parameter \(c>0\) and inversely proportional to the gains \(p,l>0\). The control \(u\) in System (9) is bounded by the amplitude \(p:|u(t)|<p,t\geq 0\), and the control rate \(\dot{u}\) also depends on the second gain \(l\). The maximum values of \(p,l\) in practical problems are usually constrained for physical reasons. As a criterion for choosing the parameter \(c>0\), we take the minimum of the basic estimate of the modulus of the control rate [24]. Considering (6), the following estimates  
\[ \begin{array}{l}\dot{u}(t)=-0.5 pl(1-\sigma^{2}(lx))\dot{x},\\ F<p\sigma(c)\Rightarrow\left|\dot{x}(t)\right|<F+p<2p,t\geq 0,\\ \left|\dot{u}(t)\right|<p^{2}l,t\geq 0,p^{2}l\geq\frac{F^{2}}{\sigma^{2}(c)} \frac{\dot{\Delta}}{\dot{\Delta}}\end{array}  \tag{12}\]  
are correct. If \(\Delta,F\) are fixed, the reference condition becomes \(c/\sigma^{2}(c)\to\min,c>0\) and is fulfilled at \(c\approx 2.2\). In the following research, this value, as well as \(l\Delta=2.2\)\(\sigma(2.2)\approx 0.8;1/\sigma(2.2)\approx 1.25\), will be used. At that, in the indicated intervals of (7) the estimations \[ \begin{array}{l}0.8p\leq p|\sigma(lx)|<p,|x|>2.2/l;\\ 0.8p\frac{l}{22}|x|\leq p|\sigma(lx)|\leq 0.8p,|x|\leq 2.2/l\end{array} \] (13) occur. Let us impose the following requirement on systems of type (9): \[F<0.8\overline{p}\Leftrightarrow 1.25F<\overline{p}.\] (14)  
In the next subsection, the discussed sigmoidal feedback will be used to design the synthesis of the tracking differentiator.