## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)
The parameters for the intelligent proportional (iP) controller were set to \(K_{p}=100,\ \alpha=10000\), while for the intelligent proportional-derivative (iPD) controller, the values \(K_{p}=12*10^{5},\ K_{d}=5*10^{3},\ \alpha=12*10^{7}\) were used, with the sampling period consistent with that of the MFAC algorithms. During the initial reference signal variation from 0 to 700 rpm, the behavior of all four algorithms--MFAC-CFDL, MFAC-FFDL, iP, and iPD--was relatively similar, though all exhibited significant overshoot: 113% for CFDL, 45% for PFDL, 82% for iP, and 48% for iPD. The settling times were 0.1063 s for CFDL, 0.0404 s for PFDL, 0.043 s for iP, and 0.0225 s for iPD. For the second reference signal variation, from 700 rpm to 1500 rpm, the FFDL algorithm demonstrated distinct behavior compared to the other three, as highlighted in Figure 15, which provides a detailed view of Figure 14. The settling time for PFDL was 0.2 s, compared to 0.126 s for CFDL, 0.134 s for iP, and 0.11 s for iPD. Another notable observation from the detailed figure is the ability of each algorithm to reject disturbances introduced by the load torque. While all four control laws effectively handled the disturbance at the 0.2-s mark, the adaptive algorithms (CFDL and PFDL) struggled more with the disturbance at the 0.8-s mark, exhibiting higher-amplitude oscillations in their response. Figure 14: Speed of BLDC motor for CFDL, PFDL, iP, iPD comparison. Figure 13: Load torque applied to BLDC motor. Figure 15: Speed of BLDC motor for CFDL, PFDL, iP, iPD comparison—detailed view. During the final reference signal variation from 1500 rpm to 0, the previously observed trends persisted. CFDL showed an overshoot of 23%, iP of 3.8%, and iPD of 5.86%, with settling times of 0.188 s for CFDL, 1.47 s for PFDL, 0.014 s for iP, and 0.019 s for iPD. All four algorithms successfully rejected the disturbance occurring at the 1.2-s mark. To objectively evaluate the performance of the four algorithms in controlling the BLDC motor speed, the criterion defined in Equation (39) was applied, with the results summarized in Table 7.

Table 7 summarizes the results of the comparison between the presented Model Free algorithms. As in the Figure 14, a superior behavior may be distinguished for the MFC laws compared to the MFAC ones is shown. Moreover, the variant that should have provided a better response, PFDL, taking into consideration the results of the errors presented in Tables 2 and 5, has a slower dynamic and produces higher control errors. Figure 16 shows the input voltages for the inverter generated by the control signals. The maximum values for MFAC algorithms are around 600 V, while they tend towards 1000 V for MFC. #### 4.3.2 Control of Nonminimum Phase Nonlinear System

The comparative analysis of the Model-Free control laws applied to the non-minimum phase nonlinear system is illustrated in Figure 17. The parameters for the intelligent proportional (iP) controller were set to \(K_{p}=70,\ \alpha=1000\), while for the intelligent proportional-derivative (iPD) controller, the values \(K_{p}=10^{5},\ K_{d}=11*10^{2}\), \(\alpha=7*10^{5}\), were used, with the sampling period matching that of the MFAC algorithms. The results reveal a relatively consistent behavior among the control algorithms within each category. For the first step reference variation, the MFAC algorithms exhibit a settling time of approximately 0.35 s, whereas the MFC algorithms (iP and iPD) achieve a faster settling time of \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **CFDL** & **PFDL** & **iP** & **iPD** \\ \hline \(|\overline{\varepsilon}_{k}|\) & 39.1766 & 208.5407 & 13.3831 & 12.0203 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Performance analysis for PFDL control of BLDC motor. Figure 16: Voltage input signal of BLDC motor for CFDL, PFDL, iP, iPD comparison. 0.175 s. A similar performance trend is observed during the reference variation from 50 to 150.