## 2 Materials and Methods
### Properties of the Sigma Function and Sigmoid Feedback  
In further constructions, we will use as corrective actions in the tracking differentiator a modification of the hyperbolic tangent \(\mathrm{th}(x)=1-2/(1+\exp(2x))\), namely the sigma function \(\sigma(x)=-\mathrm{th}(-x/2)\), with two gains \(p,l=\mathrm{const}>0\) in the form  
\[p\sigma(lx)=p\bigg{(}\frac{2}{1+\exp(-lx)}-1\bigg{)},|\sigma(lx)|<1. \tag{5}\]  
The sigma function (5) belongs to the class of logistic functions with the following features: they are defined on the entire numerical axis, smooth, odd, bounded, and have an S-shape. Unlike other logistic functions, such as rational and root sigmoid and arctangent, etc., the sigma function is easy to compute since the corresponding Maclaurin series converges on the entire numerical axis, and its derivative has a recursive form:  
\[p\sigma^{\prime}(lx)=0.5 pl(1-\sigma^{2}(lx)),0<p\sigma^{\prime}(lx)\leq 0.5 pl,x\in R. \tag{6}\]  
In a small neighborhood of zero, the sigma function is equivalent to the linear function \(p\sigma(lx)\underset{x\to 0}{\sim}0.5 plx\), and as the gain \(l\) increases, it is closer to the sign function \(p\sigma(lx)\underset{l\to+\infty}{\sim}p\mathrm{sign}(x)\). The above features allow us to constrain the sigma function from below to a piecewise linear function  
\[p|\sigma(lx)|\geq p|\mathrm{sat}(lx)|,p\mathrm{sat}(lx)=\left[ \begin{array}[ ]{c}p\sigma(l\Lambda)\mathrm{sign}(x),|x|>\Delta>0,\\ p\sigma(l\Lambda)x/\Delta,|x|\leq\Delta,\end{array} \right. \tag{7}\]  
where the points \(\sigma(\pm l\Delta)\) separate the sigma function into a conditionally constant and a conditionally linear function. Let us denote  
\[l\Delta=c>0,\underset{c\to+0}{\lim}\sigma(c)=+0,\underset{c\to+\infty}{\lim} \sigma(c)=1-0. \tag{8}\]  
Let us consider the features of sigmoidal feedback in the problem of stabilizing a state variable under the influence of an external disturbance. We will use the example of a first-order system  
\[\dot{x}=f(t)+u,u=-p\sigma(lx),p,l=\mathrm{const}>0, \tag{9}\]  
where \(x\in R\) is the state variable, \(u\in R\) is the sigmoid control, and \(f(t)\in R\) is an unknown function of time, which is bounded modulo a known constant \(|f(t)|\leq F=\mathrm{const}>0,t\geq 0\). If \(f(t)\) does not decay with time, then in the closed-loop system (9), it is possible to ensure stabilization of the state variable with some accuracy if certain conditions are fulfilled. **Lemma 1**. : _In System (9), let the choice interval gain \(p>0\) be bounded from above: \(p\leq\overline{p}\). Then, for any initial condition \(x(0)\), he solution of the system \(x(t),t\geq 0\) will be bounded if and only if \(F<\overline{p}\)._  
Proof of Lemma 1.: To analyze the steady state of System (9), let us use the second Lyapunov method and estimate the derivative of the Lyapunov function \(V=x^{2}/2\). Outside the neighborhood of \(|x|\leq\Delta=c/l\), the sigma function from below is bounded by the constant (7), so considering (8), we have  
\[\dot{V}=x\dot{x}=x\big{(}f(t)-p\sigma(lx)\big{)}\leq|x|(F-p\sigma(c)).\]If \(F>p\sigma(c)\), then \(\dot{V}>0\), and for all \(p<F/\sigma(c)\), System (9) is not stable, and its solution is unbounded \(\lim\limits_{t\to+\infty}x(t)=\infty\). If \(F=p\sigma(c)\), then \(\dot{V}=0\), and System (9) is neutral, is on the stability boundary, and its solution can be considered constant \(x(t)\approx x(0)\). If \(p\leq\overline{p}\leq F\Leftrightarrow F/\overline{p}\geq 1\), then by virtue of \(0<\sigma(c)<1\) (5), (8) in both cases, we have a strict inequality \(\overline{p}<F/\sigma(c)\), i.e., the stability of System (9) cannot be ensured. Thus, it has been proved that it is necessary to fulfill the condition \(F<\overline{p}\). If \(F<p\sigma(c)\), then \(\dot{V}<0\) and for all \(p>F/\sigma(c)\), by virtue of (7), the state variable converges to the following neighborhood:  
\[|x(t)|\leq\Delta=c/l, \tag{10}\]  
i.e., the solution of System (9) is bounded.