## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **CFDL** & **PFDL** & **iP** & **iPD** \\ \hline \(|\overline{e}_{k}|\) & 0.0390 & 0.0440 & 0.0299 & 0.0321 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Performance analysis for FFDL control of system with time-varying parameter. relatively similar, from the point of view of the range of values applied and the time points at which they are applied. The comparative analysis of the two MFAC-based control laws--CFDL and PFDL--reveals that the CFDL algorithm consistently delivers superior system responses across the three evaluated systems. In the case of the BLDC motor, the PFDL variant exhibited significant errors, particularly in the latter part of the simulation, and demonstrated weaker performance in rejecting disturbances introduced by the load torque. For nonlinear systems with non-minimum phase characteristics or time-varying parameters, the errors associated with PFDL were reduced, but CFDL maintained a clear advantage in terms of overall performance. Conversely, the comparison between the intelligent proportional (iP) and intelligent proportional-derivative (iPD) controllers within the MFC framework showed that their output signals were closely aligned. However, the iPD variant, which incorporates a derivative component, provided marginally better responses than the iP variant. Both MFC-based controllers exhibited similar capabilities in rejecting disturbances and adapting to the time-varying parameter nonlinear system. Nevertheless, the predictive nature of the iPD algorithm enabled it to achieve better performance metrics based on the objective analysis criterion applied. This suggests that while both MFC variants are effective, the iPD controller offers a slight edge in dynamic response and adaptability. ## 5 Conclusions

This paper conducted a comprehensive comparative analysis of two categories of Data-Driven Control (DDC) laws: gradient-based (MFAC) and non-gradient-based (MFC) approaches. For the gradient-based MFAC method, two algorithms were developed to determine the optimized initial values of the pseudo partial derivative (PPD) and pseudo gradient (PG) parameters. These algorithms, rooted in a data-driven philosophy, utilize system-acquired data to iteratively enhance system dynamics. When applied to three nonlinear systems, the algorithms demonstrated significant improvements in dynamic performance, validated through numerical evaluation criteria. A notable limitation, however, is their reliance on step-type reference signals, which are essential for capturing PPD and PG variations to derive optimized initial values. Figure 22: Control signals for nonlinear system with time-varying parameter. For the gradient-based approach, MFAC algorithms employing compact form dynamic linearization (CFDL) and partial form dynamic linearization (PFDL) were developed to optimize the initial values of time-varying parameters (PPD and PG), resulting in enhanced dynamic performance. On the other hand, the non-gradient-based MFC approach utilized intelligent proportional (iP) and intelligent proportional-derivative (iPD) controllers, which exhibit behavior analogous to classical PI and PID controllers. Both MFAC and MFC variants were tested and validated on nonlinear systems, enabling a direct performance comparison between the two methodologies. Regarding the comparative analysis between MFAC and MFC, the results clearly indicate that MFC algorithms outperform their MFAC counterparts. While MFAC algorithms, both in CFDL and PFDL forms, exhibited slower responses and higher control errors, MFC algorithms demonstrated faster settling times, reduced overshoot, and lowered control errors, as evidenced by the discrete-time integral absolute error criterion. Both approaches successfully achieved steady-state error, rejected disturbances, and adapted to time-varying nonlinear systems. However, MFC algorithms consistently achieved control objectives more efficiently and with greater consistency in settling times.