## 3 Model-Free Control
### Tracking Differentiator Design Based on Hyperbolic Tangent Function
Figure 6 illustrates the diagram of the proposed TD based on Equations (38), (40) and (41), where \(\omega_{\textit{ref}_{n}}\) represents the normalized rotational speed. The traditional hyperbolic tangent function commonly used in control systems provides a smooth, symmetric curve with a linear region near the origin and saturation at the extremes. Despite being efficient in many situations, it might not be the best option due to the previously mentioned trade-off, and to enhance tracking accuracy during periodic phases (where errors are typically small) by increasing the gain parameter \(\lambda\), the system must compromise on the stability of the transition response, potenti  
Figure 5: TD output for different parameter values: (**a**) with \(c_{1}=7\) and \(c_{2}=7\); (**b**) with \(\rho=2\) and \(c_{2}=7\); (**c**) with \(\rho=2\) and \(c_{1}=7\). Figure 6: Block diagram of the proposed TD. less controlled transitions. The proposed function, described in (40), introduces a nonlinear modification to the error before passing it through the hyperbolic tangent function. To ensure good tracking accuracy in periodic phase, this function has to amplify the tracking error when it is near the equilibrium point. Conversely, it scales down the error when it is far from the equilibrium in order to have a stable transition response that respects the physical limit of the system. Figures 7 and 8 illustrate the influence of parameters \(a\) and \(b\) on the nonlinear gain function \(\lambda\). As depicted, \(a\) and \(b\) primarily affect the slope of the function in regions of small errors. Smaller values of \(a\) and larger values of \(b\) result in a low slope near the origin, while larger values of \(a\) and smaller values of \(b\) increase the sensitivity to small errors. In our case, we are looking for a higher slope in order to enhance accuracy in steady-state phases without noise amplification. The parameter \(\xi\) defines the threshold separating small and large error regions, marking the boundary where the gain function's behavior transitions. By appropriately adjusting these parameters, we can fine-tune the system's accuracy during steady-state operation when errors are small (traction phases) and control the aggressiveness of the system in correcting large deviations and guarantee a smooth transition between the two phases, all without compromising noise sensitivity. Figure 9 illustrates the performance advantages offered by the proposed TD incorporating a nonlinear \(\lambda\). This approach demonstrates the highest tracking accuracy compared to TDs with constant \(\lambda\), even with high values. Notably, this enhanced precision does not come at the cost of dynamic response quality. The figure reveals that despite achieving the best tracking accuracy, the proposed TD neither exhibits the fastest response nor produces  
Figure 8: \(\tanh(e,\lambda)\) function curves for nonlinear \(\lambda\) with different values of \(b\). Figure 7: \(\tanh(e,\lambda)\) function curves for linear and nonlinear variation in \(\lambda\) with different values of \(a\). overshoots. This balanced performance underscores a key benefit of the proposed TD: it simultaneously delivers a stable transition response that respects the system's physical limitations and maintains excellent tracking accuracy during periodic phases. Such dual optimization represents a significant advancement in control system design, offering a solution that does not compromise between transition stability and steady-state precision. The proposed MFC (PMFC) scheme is shown in Figure 10.