## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)
The comparative outputs of the systems controlled by the initial variant, CFDL\({}_{0}\), and the final CFDL* during 4.5 s of simulation are depicted in Figure 4, while the evaluation based on Equation (39) and _aEs_ is presented in Table 2.

The output signals depicted in Figure 4 show an improvement in response for the first step variation of the reference signal, from 0 to 20. For the other reference signal variations, the output signals of CFDL* and CFDL\({}_{0}\) are relatively similar. The numerical analysis is presented in \[T_{E}-T_{L}=J\frac{d\omega}{dt}+B\omega \tag{36}\]0 \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\mathbf{CFDL_{0}}\) & \(\mathbf{CFDL^{*}}\) \\ \hline \(|\vec{\epsilon}_{k}|\) & 1.9942 & 1.5959 \\ _aEs_ & 1.6809 & 1.6710 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance analysis for CFDL applied for non-minimum phase nonlinear system. Figure 3: Average absolute error for steps variation during Algorithm 1 for non-minimum phase nonlinear system. #### 4.1.3 Control of Nonlinear System with Time-Varying Parameter

The nonlinear system described by Equation (38) was optimized by means of Algorithm 1, using a constant value of \(a_{k}a_{k}\), so that its behavior does not greatly influence the optimization process. Thus, starting from the set of parameters \(\mu=1\), \(\eta=1\), \(\rho=1\), \(\lambda=25\), \(\epsilon=0.00001\), \(\hat{\theta}_{1}^{0}=2\) and the sampling period \(T_{s}=0.01\), following the application of the algorithm, an optimized value of \(\hat{\phi}_{1}^{*}=5.1199\) was obtained. The results of applying the algorithm are described by Figure 5, where the variation of \(aEs\) for the 15 iterations of the Algorithm 1 is presented. The comparative results between CFDL\({}_{0}\) and CFDL\({}^{*}\) for 10 s of simulation are shown in Figure 6. Table 3 presents the comparative values resulting from the application of Equation (38) as well as the \(aEs\) values. The output signals from Figure 6 show a significant improvement for the settling time for CFDL\({}^{*}\) compared to CFDL\({}_{0}\). For the step type variation from 0.2 to 0.4, the response time decreases by 0.3 s, while for the variation from 0.4 to 0.6, it decreases by 0.6 s. Thus, lower control error values result, which are also presented in Table 3.

Figure 4: Output signals for CFDL optimization algorithm applied for non-minimum phase nonlinear system. Figure 5: Average absolute error steps variation during Algorithm 1 for nonlinear system with time-varying parameter. As Figures 1, 3 and 5 show, Algorithm 1 achieves a minimization of the value of \(aEs\) for the CFDL control law, which leads to the improvement of the responses of the controlled systems. This aspect can be observed in Figures 2, 4 and 6, where the initial variants are presented compared to the final ones, and the results presents improvements. Also, Tables 1-3 show by objective evaluation that the application of Algorithm 1 on MFAC control laws produces improvements at the level of the controlled system. ### Applications of Using Algorithm 2 for Nonlinear Systems

#### 4.2.1 Speed Control of BLDC Motor Using FFDL Algorithm

The BLDC motor speed control application via MFAC-FFDL law started from the initial parameters \(\mu=1.2\), \(\eta=0.01\), \(\rho_{1}=0.1\), \(\rho_{2}=0.05\), \(\lambda=10\), \(\varepsilon=0.00001\), \(L=2\), \(\hat{\Phi}_{2



In Figure 8, progressive improvements from the point of view of settling time are shown. Both for the variation of the reference signal from 333 rpm to 666 rpm, and for that from 666 rpm to 999 rpm, using the resulting initial values for \(\phi_{1}^{*}\) and then \(\phi_{1}^{*}\), \(\phi_{2}^{*}\) provides faster system outputs, which also means values of lower error control. This advantage comes with the disadvantage of a higher overshoot for the first reference variation, 79% in comparison to 17%, but after reaching the steady state, there is no difference in overshoot level for subsequent step variations of the reference.