## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)
#### 4.2.2 Control of Non-Minimum Phase Nonlinear System

The non-minimum phase system described by Equation (38) was used to analyze the applicability of Algorithm 2 in such a context. Starting from the set of initial parameters: \(\mu=1.2\), \(\eta=0.05\), \(\rho_{1}=0.15\), \(\rho_{2}=0.15\), \(\lambda=10\), \(\varepsilon=0.00001\), \(L=2\), \(\hat{\Phi}_{21}^{0}=[0.7\:0.5]\) with the same sample period as in the CFDL simulation. Following the application of \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **PFDL\({}_{0}\)** & **PFDL*\({}_{1}\)** & **PFDL*\({}_{2}\)** \\ \hline \(|\overline{\varepsilon}_{k}|\) & 12.21 & 9.81 & 6.88 \\ \(aEs\) & 18.9293 & 15.4857 & 13.2586 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance analysis for PFDL speed control of BLDC motor. Figure 8: BLDC motor speed for PFDL optimization algorithm. Figure 7: Average absolute error steps variation during Algorithm 2 for BLDC motor. signal, from 0.2 to 0.4, and by 0.1 s for the variation from 0.4 to 0.6. An improvement can also be observed in the settling time for the first variation of the reference, a fact that is also reflected in the results in Table 6. As Figure 11 suggests, the intermediate and final response, after applying Algorithm 2, provides relatively similar results. A similar conclusion to that of Algorithm 1 can be drawn regarding Algorithm 2, which presents the evolution of \(aEs\) during the optimization process, in Figures 7, 9 and 11, with the corresponding number of iterations performed for each value of \(\hat{\phi}\). The results presented in Figures 8, 10 and 12 depict the comparative responses of the system, starting from the initial version, showing that the application of Algorithm 2 brings improvements to the output of the controlled system. Tables 4-6 present this comparison between the initial, intermediate, and final version in an objective manner. The conclusions are the same, with the responses of the controlled system improved by the application of Algorithm 2 to the FFDL laws. ### Comparative Analysis of Model Free Control Laws

Comparative studies between the two proposed Model-Free algorithms are relatively scarce in the literature [43]. In this section, the performance of these algorithms is evaluated for the nonlinear systems described in Section 3. The focus is on the most widely used forms of each approach: for Model-Free Control (MFC), the intelligent proportional (iP) and intelligent proportional-derivative (iPD) controllers are examined, while for Model-Free Adaptive Control (MFAC), the compact form dynamic linearization (MFAC-CFDL) and partial form dynamic linearization (MFAC-CFDL) variants are analyzed. The parameters for the MFAC-CFDL and MFAC-FFDL algorithms were selected based on the optimization procedures detailed in Sections 4.1 and 4.2, respectively. For MFC, the parameters were chosen within the ranges defined by Equations (27) and (31), ensuring stability and optimal performance. This comparative analysis aims to highlight the strengths and limitations of each approach in controlling nonlinear systems. #### 4.3.1 Control Speed of BLDC Motor

BLDC motor speed control may represent a challenge for the Model Free algorithms presented in Section 2. In addition to the nonlinear nature of the control structure comprising the motor and inverter, another element that can influence the control result is the load torque. According to the theoretical details provided by [78], both methods should be able to reject such disturbances. This is due to the Model Free frame papers, which only acquire data from the control system and interpret it to create a way for the system's behavior to be understood in an online manner. In the comparative analysis carried out on the motor described in Section 3.1, the load torque presented in Figure 13 was applied. Figure 14 presents the results of the comparative analysis for speed control of the BLDC motor, with the previously defined load torque applied.