## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)  
The MFAC-PFDL algorithm is one that offers more degrees of freedom than the CFDL variant. It also uses a larger number of control signal values from the past. Thus, the system can be described based on the equation:  
\[y_{k+1}=y_{k}+\Phi_{Lk}\Delta U_{Lk}, \tag{9}\]  
where \(U_{Lk}= \begin{bmatrix}u_{k}&\ldots&u_{k-L+1}\end{bmatrix} ^{T}\) represents a vector containing the control signals within a moving time window \([k-L+1,k]\) with \(U_{Lk}=0_{L}\) for \(k\leq 0\), \(\Phi_{Lk}= \begin{bmatrix}\phi_{1k}&\ldots&\phi_{Lk}\end{bmatrix} \in\mathbb{R}^{L}\) is a time varying matrix with the bounded elements called pseudo gradient (PG), and \(L\) is the linearization length constant. The criterion that must be minimized to obtain the control law is the one described by Equation (3). When the PFDL data model from Equation (9) is entered as the cost function from Equation (3) by minimizing it, the control law results in the following. \[u_{k}=u_{k-1}+\frac{\rho_{1}\phi_{1k}(\widetilde{y}_{k+1}-y_{k})}{\lambda+\left| \phi_{1k}\right|^{2}}-\frac{\phi_{1k}\sum\limits_{i=2}^{L}\rho_{i}\phi_{ik} \Delta u_{k-i+1}}{\lambda+\left|\phi_{1k}\right|^{2}}, \tag{10}\]  
where \(\rho_{i}\in(0,1]\) are step factors introduced to make the controller algorithm more general. For the estimation of PG, analogously to what was presented in Equation (5), the criterion results are as follows. \[J(\Phi_{1k})=|y_{k}-y_{k-1}-\Phi_{1k}\Delta U_{k-1}|^{2}+\mu\|\Phi_{1k}-\Phi_{ 1k-1}\|^{2}, \tag{11}\]  
The Matrix Inversion Lemma from [7] explains the fact that, in the context in which it exists, there are four real matrices \(A,B,C,D\) with appropriate dimensions. If the inversions of matrices \(A,C\) and \(DA^{-1}B+C^{-1}\) exist, it follows that:  
\[[A+BCD]^{-1}=A^{-1}-A^{-1}B\Big{[}DA^{-1}B+C^{-1}\Big{]}^{-1}DA^{-1} \tag{12}\]  
Minimizing Equation (11) with respect to \(\Phi_{1k}\), according to optimality conditions and using the Matrix Inversion Lemma, results in the estimation of \(\Phi_{1k}\) as:  
\[\hat{\Phi}_{Lk}=\hat{\Phi}_{Lk-1}+\frac{\eta\big{(}\Delta y_{k}-\hat{\Phi}_{Lk -1}\Delta U_{Lk-1}\big{)}\Delta U_{Lk-1}^{T}}{\mu+\left\|\Delta U_{Lk-1}\right\| ^{2}}. \tag{13}\]  
Analogous to Equation (7), a restart mechanism is introduced, for better tracking:  
\[\hat{\Phi}_{Lk}=\hat{\Phi}_{L1},\text{ if }\big{\|}\hat{\Phi}_{Lk}\big{\|}<\varepsilon \text{ or }\|\Delta U_{Lk-1}\|<\varepsilon\text{ or }\text{sgn}(\hat{\Phi}_{Lk})\
eq\text{sgn}(\hat{\Phi}_{L1}), \tag{14}\]  
where \(\varepsilon\) represents a small scalar value. Using the PG estimate in Equation (10), the MFAC-PFDL control law results in:  
\[u_{k}=u_{k-1}+\frac{\rho_{1}\hat{\phi}_{1k}(\widetilde{y}_{k+1}-y_{k})}{ \lambda+\left|\hat{\phi}_{1k}\right|^{2}}-\frac{\hat{\phi}_{1k}\sum\limits_{i =2}^{L}\rho_{i}\hat{\phi}_{ik}\Delta u_{k-i+1}}{\lambda+\left|\hat{\phi}_{1k} \right|^{2}}, \tag{15}\]  
where \(\lambda>0\) and \(\mu>0\), \(\eta\in(0,2]\), \(\rho_{i}\in(0,1]\), \(i=1,2,\ldots,L\). The Algorithm for Determining the Optimized Initial Value of PG for the  
MFAC-PFDL Law  
Similar to the MFAC-CFDL algorithm, the PFDL variant requires prior values of \(\hat{\Phi}_{Lk}\), and for the initial iteration, \(\hat{\Phi}_{L1}\) was selected by the user as a control law parameter. Thus, an iterative algorithm was also developed for this type of control structure to obtain optimized variant \(\hat{\Phi}_{L1}^{*}\) of the PG initial value. The principles are preserved as in the case of CFDL, and only the algorithm is extended. Thus, a nested iterative algorithm results, starting from a set of control law parameters and \(\hat{\Phi}_{L1}\) values that provide stable dynamics to the control system, and error minimization can be achieved. The algorithm performs step-by-step optimization, first by obtaining a value that provides the best performance for \(\hat{\phi}_{11}^{*}\).