## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.1 Compact Form Dynamic Linearization (CFDL)  
The simplest form of the dynamic linearization model is CFDL, which uses the compact form, defined by:  
\[y_{k+1}=y_{k}+\phi_{k}\Delta u_{k} \tag{2}\]  
where \(\phi_{k}\in\mathbb{R}\) is a bounded time varying parameter called PPD. Obtaining the control algorithm is achieved by minimizing a cost function, based on the one-step-ahead prediction of the control error:  
\[J(u_{k})=|e_{k+1}|^{2}+\lambda|\Delta u_{k}|^{2}, \tag{3}\]  
where \(e_{k}\) is the control error, defined as \(e_{k}=\widetilde{y}_{k}-y_{k}\), and \(\lambda>0\) is a weighting factor to constrain the changing rate of the control signal. By using the model described by Equation (2) in the context of Equation (3), provided that \(\partial J(u_{k})/\partial u_{k}=0\), the control law provides the following results:  
\[u_{k}=u_{k-1}+\frac{\rho\phi_{k}}{\lambda+\left|\phi_{k}\right|^{2}}( \widetilde{y}_{k+1}-y_{k}), \tag{4}\]  
where \(\rho\in(0,1]\) is a step factor introduced to make the algorithm more general. For the calculation of the time-varying \(\phi_{k}\) parameter, an estimate can be made based on the cost function:  
\[J(\phi_{k})=|y_{k}-y_{k-1}-\phi_{k}\Delta u_{k-1}|^{2}+\mu|\phi_{k}-\phi_{k-1} |^{2}, \tag{5}\]  
where \(\mu>0\) is a weighting factor introduced to penalize the change rate of the PPD estimation, which is determining the learning speed of the algorithm. By minimizing Equation (5) according to \(\phi_{k}\), it can be estimated as:  
\[\hat{\phi}_{k}=\hat{\phi}_{k-1}+\frac{\eta\Delta u_{k-1}}{\mu+\Delta u_{k-1}^ {2}}\big{(}\Delta y_{k}-\hat{\phi}_{k-1}\Delta u_{k-1}\big{)}, \tag{6}\]  
where \(\eta\in(0,2]\) is a step factor introduced to make the algorithm more general and flexible, while \(\mu\) prevents the denominator to be equal to zero. A restart mechanism was defined for \(\phi_{k}\) estimation, and to obtain a better tracking of it:  
\[\hat{\phi}_{k}=\hat{\phi}_{1},\text{ if }\big{|}\hat{\phi}_{k}\big{|}< \varepsilon\text{ or }|\Delta u_{k-1}|<\varepsilon\text{ or }\text{sgn}(\hat{\phi}_{k})\
eq\text{sgn}(\hat{\phi}_{1}), \tag{7}\]  
where \(\hat{\phi}_{1}\) represents the initial value of the PPD and \(\varepsilon\) is a small scalar value. Based on the PPD estimate from Equation (6), the control law in Equation (4) of the MFAC-CFDL algorithm becomes:  
\[u_{k}=u_{k-1}+\frac{\rho\hat{\phi}_{k}}{\lambda+\left|\hat{\phi}_{k}\right|^{ 2}}(\widetilde{y}_{k+1}-y_{k}), \tag{8}\]  
The \(\rho\in(0,1]\), \(\eta\in(0,2]\), \(\lambda>0\) and \(\mu>0\) parameters are those which allow the user to adjust the performance and behavior of the control system. The MFAC-CFDL algorithm defined by Equation (8) uses the PPD estimate from Equation (6). The equation contains an important parameter for the behavior of the control system, as could be seen from [28]. In (6), \(\hat{\phi}_{k-1}\) represents the previous value of PPD, and the initial value \(\hat{\phi}_{1}\) is unknown for the first discrete time as it is chosen by the user as a control law parameter. An iterative algorithm is proposed to obtain the value of \(\hat{\phi}_{1}\), which offers the best dynamic performances of the control system in the context of the parameters mentioned in Equation (8). The Algorithm for Determining the Optimized Initial Value of PPD for the CFDL-MFAC Law  
The CFDL-MFAC variant described by Equation (2) is based on the concept of dynamic linearization. A critical aspect of this approach is the initial value required for the pseudo partial derivative (PPD) estimation in Equation (6), which, as demonstrated in [36], significantly influences the system's subsequent behavior. To address this, an iterative algorithm is proposed for selecting the optimized initial value \(\hat{\phi}_{1}\), leveraging data acquired from the control system. This algorithm was developed in response to the observation that systems controlled by MFAC-CFDL laws often exhibit suboptimal dynamic performance in the initial phase, with improvements occurring only after a reference signal variation is applied.