## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)
For the final reference signal variation, from 150 to 10, the behavior aligns with the patterns identified in Section 4.3.1. The PFDL algorithm demonstrates a slower response compared to CFDL, with a settling time difference of 0.2 s favoring the compact form (CFDL). Meanwhile, the MFC algorithms maintain their consistent performance, mirroring their behavior during the previous reference variations. Figure 18 provides a detailed view of the output signals presented in Figure 17, offering further insight into the dynamic responses of the control algorithms. The objective comparison based on the results of applying Equation (39) is shown in the Table 8 and highlights the previously mentioned information. Throughout the experiment, the MFC differences show superior results to the MFAC algorithms. Moreover, the PFDL law performed worse than the CFDL law for this type of system. Figure 17: Output signals of nonminimum phase nonlinear system. Figure 18: Output signals of nonminimum phase nonlinear system—detailed view. #### 4.3.3 Control of Nonlinear System with Time-Varying Parameter

The nonlinear system with time-varying parameter control is performed in the context of the time-varying parameter \(a_{k}\), which takes the following values:

\[a_{k}=\left\{ \begin{array}{c}1,t<2\,\mathrm{s}\\ 2,t\in(2,7]\,\mathrm{s}\\ 3,t\in(7,12]\,\mathrm{s}\\ 4,t>12\,\mathrm{s}\end{array} \right. \tag{40}\]

The results of the comparative analysis for the four control laws applied to the nonlinear system with time-varying parameters are depicted in Figure 20. The parameters for the intelligent proportional (iP) controller were set to \(K_{p}=45,\ \alpha=530\), while for the intelligent proportional-derivative (iPD) controller, the values \(K_{p}=17*10^{3}\), \(K_{d}=180,\ \alpha=2*10^{5}\) were used, with the sampling period consistent with that of the MFAC algorithms. During the first reference signal variation, all four control laws exhibited an overshoot of 100%, with the primary distinction lying in the settling times: 0.09 s for both iP and iPD, compared to 0.55 s for CFDL and 0.35 s for PFDL. The response to the change in the time-varying parameter \(a_{k}\) at 2 s was relatively similar across all algorithms. For the second reference signal variation, from 0.5 to 1, none of the control systems displayed overshoot, but differences in settling times were evident. The MFC algorithms (iP and iPD) achieved a settling time of 1.32 s, while CFDL required 1.87 s and PFDL 2.15 s. At the 7-s mark, corresponding to another change in \(a_{k}\), the control systems responded

Figure 19: Control signals of nonminimum phase nonlinear system. \begin{table}
\begin{tabular}{c c c c c} \hline  & **CFDL** & **PFDL** & **iP** & **iPD** \\ \hline \(|\overline{e}_{k}|\) & 4.3303 & 5.4945 & 1.9167 & 0.0965 \\ \hline \end{tabular}
\end{table} 
Table 8: Performance analysis for PFDL control of nonminimum phase nonlinear system. similarly, with a slight advantage for the MFC-based controllers, which demonstrated marginally faster behavior, as the detailed view of Figure 21 illustrates. During the final reference signal variation, the settling times were 0.44 s for CFDL, 0.45 s for PFDL, 0.33 s for iP, and 0.3 s for iPD. The last variation of the parameter \(a_{k}\) at 11 s had a comparable impact on all four control algorithms, with no significant differences in their responses. This analysis highlights the performance characteristics of each control law in managing a nonlinear system with time-varying parameters. Table 9 shows the results of the comparative analysis performed using Equation (39). For this type of system, the comparative results are the closest, but the superiority of MFC laws is maintained over MFAC. Figure 22 shows the values of the control signals applied by the four laws for the nonlinear system with time-varying parameter. The values presented in this picture are

Figure 21: Output signals for nonlinear system with time-varying parameter—detailed view. Figure 20: Output signals for nonlinear system with time-varying parameter.