## 3 Model-Free Control
### Tracking Differentiator Design Based on Hyperbolic Tangent Function  
In MFC, both the reference signal and its derivative are essential components of the control law. However, the direct calculation of the derivative can lead to significant challenges, particularly when the reference changes abruptly, such as transition phases in a PKGS. In these cases, the derivative may yield excessively large values, potentially leading to the instability of the system. While the derivative of the trajectory is often set to zero when dealing with step responses, this approach is not suitable for systems with varying speed references, such as a PKGS during the traction phase. To address these issues and ensure robust control, the integration of a tracking differentiator into the MFC framework is proposed in this work. The TD developed in this work is based on the improved hyperbolic tangent function proposed in [39]. The hyperbolic tangent function is derived from the ratio of the hyperbolic sine (sinh) and the hyperbolic cosine (cosh) functions. It produces a smooth, continuous curve with distinct saturated regions. Near the origin, the hyperbolic tangent can be approximated as a linear function, despite its overall nonlinear nature. It is expressed as  
\[tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \tag{35}\]  
Equation (35) can be formed as  
\[tanh(x)=1-\frac{2}{e^{2x}+1} \tag{36}\]  
Figure 3: Block diagram of the estimator of \(F_{est}\). As can be deduced from (36), the curve of the tanh is a strictly monotone rising function with upper and lower bounds of \(1\) and \(-1\) for the saturation value. To modify the approximate linear range close to the origin, the exponential component \(\lambda\) is introduced in Formula (36), with the resulting function shown in Figure 4, and the new formula is obtained as follows:  
\[tanh(x,\lambda)=1-\frac{2}{e^{2\lambda x}+1} \tag{37}\]  
Consider the following second-order system,  
\[\left\{ \begin{array}{l}\dot{z}_{1}=z_{2}\\ \dot{z}_{2}=f(z_{1},z_{2},v)\end{array} \right. \tag{38}\]  
where\(f\) is the nonlinear hyperbolic tangent function given by  
\[f(z_{1},z_{2})=-\rho^{2}\bigg{(}c_{1}tanh(z_{1}(t)-v(t))+c_{2}tanh\bigg{(} \frac{z_{2}(t)}{\rho}\bigg{)}\bigg{)} \tag{39}\]  
where \(v(t)\) represents the input, \(\rho\) and \(c_{1}\) are utilized to adjust the tracking speed, \(c_{2}\) is associated with the differential effect. Figure 5 shows the TD output for different parameter values. It can be seen that for low values of \(\rho\) and \(c_{1}\), and high values of \(c_{2}\), there is a noticeable delay between the TD output and the input signal. As \(\rho\) and \(c_{1}\) increase and \(c_{2}\) decreases, this delay gradually diminishes, resulting in a faster system response. However, this improvement in delay reduction comes with trade-offs. When the parameters reach values that effectively eliminate the delay, the TD response speed can exceed the tracking capabilities of the real system, which can have a negative impact on system stability, significantly increase the required control effort, and potentially cause overshoots in the system response. To address this compromise, a nonlinear function is introduced for adjusting the approximate linear range of the hyperbolic tangent function. Consequently, the system becomes  
\[f(z_{1},z_{2})=-\rho^{2}\bigg{(}c_{1}tanh(\lambda e(t))+c_{2}tanh\bigg{(} \frac{z_{2}(t)}{\rho}\bigg{)}\bigg{)} \tag{40}\]  
where \(e\) is the normalized error between \(z_{1}\) and \(v(t)\), and \(\lambda\) is a nonlinear function defined as follows:  
\[\lambda(e,a,b,\xi)=\left\{ \begin{array}{l}\frac{|e|^{b}}{\xi^{b-a}}\text{ sign}(e)\quad|e|\leq\xi\\ |e|^{a}\text{sign}(e)\quad|e|>\xi\end{array} \right. \tag{41}\]  
Figure 4: tanh(x, \(\lambda\)) curves with the variation in \(\lambda\). \(a\), \(b\), and \(\xi\) are parameters which can be adjusted to meet the required performance levels.