## 2 Model Free Algorithms
### Model Free Adaptive Control (MFAC)
#### 2.1.2 Partial Form Dynamic Linearization (PFDL)
\tag{18}\]

for a more compact form, and \(T_{s}\) is the sampling period. In this way, the discrete-time ultralocal model was obtained as:

\[\Delta_{E}^{v}y_{k}=F_{k}+\alpha u_{k}, \tag{19}\]

The unmodelled function \(F_{k}\) can be estimated by using:

\[\hat{F}_{k}=\Delta_{E}^{v}y_{k}-\alpha u_{k-1}. \tag{20}\]

The discrete-time form of the iPID control law may be obtained similarly to the discrete-time variant of the ultralocal model, resulting in:

\[u_{k}=\frac{1}{\alpha}\bigg{(}\Delta_{E}^{v}\tilde{y}_{k}-\hat{F}_{k}+K_{p}e _{k}+\frac{K_{i}}{\Delta_{E}}e_{k}+K_{d}\Delta_{E}e_{k}\bigg{)}, \tag{21}\]

Using the approximation form of \(\hat{F}_{k}\) from Equation (20) in the law described by Equation (21), the iPID control law can be rewritten as:

\[u_{k}=u_{k-1}+\frac{1}{\alpha}\bigg{(}\Delta_{E}^{v}e_{k}+K_{p}e_{k}+\frac{K _{i}}{\Delta_{E}}e_{k}+K_{d}\Delta_{E}e_{k}\bigg{)}. \tag{22}\]From Equation (22), it immediately follows that the parameters of the iPID controller depend on the sampling period due to the presence of \(\Delta_{E}\) in the expression of the control law, as Equation (18) suggests. Introducing the control law from Equation (22) in the ultralocal model described in Equation (19) results in the following control error equation:

\[\Delta_{E}^{v}\epsilon_{k}+K_{p}\Delta_{E}\epsilon_{k}+\frac{K_{i}}{\Delta_{E}} \epsilon_{k}+K_{d}\Delta_{E}\epsilon_{k}+\delta F_{k}=0, \tag{23}\]

where the estimation error \(\delta_{k}=F_{k}-\hat{F}_{k}\) and is usually neglected [32]. Taking into account the previously mentioned approximation of \(\delta_{k}\), the control error equation is obtained as:

\[\Delta_{E}^{v+1}\epsilon_{k}+K_{p}\Delta_{E}\epsilon_{k}+K_{i}\epsilon_{k}+K_{ d}\Delta_{E}^{2}\epsilon_{k}=0, \tag{24}\]

To obtain a null control error in the steady state, \(\underset{k\rightarrow\infty}{\lim}\epsilon_{k}=0\), the roots of the characteristic polynomial of the control error equation must be stable. For \(v\geq 2\), the conditions for roots to be located inside the circle of unit radius in the Z-plane can be determined using the Jury test from [39]. #### 2.2.1 iP Controller with \(v=1\)

Considering \(K_{i}\) and \(K_{d}\) equal to 0, in Equation (22), the iP control law will result as:

\[u_{k}=u_{k-1}+\frac{1}{\alpha}\big{(}\Delta_{E}\epsilon_{k}+K_{p}\Delta_{E} \epsilon_{k}\big{)}, \tag{25}\]

As [8] demonstrates and [27] shows, an iP type controller behaves similarly to a classic PI, except that it has the advantage of being based only on the use of data from the system and not on a known mathematical model of the process. The control error equation may be obtained from Equation (24), with \(K_{i}=K_{d}=0\), as:

\[\Delta_{E}\epsilon_{k}+K_{p}\Delta_{E}\epsilon_{k}=0, \tag{26}\]

Using the forward Euler approximation from Equation (18), the characteristic polynomial of the control error is obtained as:

\[P_{\alpha}(z)=z+K_{p}T_{s}-1=0, \tag{27}\]

whose root \(z_{r}\) is stable if \(\left|z_{r}\right|=\left|1-K_{p}T_{s}\right|\leq 1\), which leads to the conclusion that \(K_{p}\in[0;2/T_{s}]\). #### 2.2.2 iPD Controller with \(v=2\)

The iPD type law is obtained by using \(K_{i}=0\) in Equation (22):

\[u_{k}=u_{k-1}+\frac{1}{\alpha}\Big{(}\Delta_{E}^{2}\epsilon_{k}+K_{p}\epsilon _{k}+K_{d}\Delta_{E}\epsilon_{k}\Big{)}, \tag{28}\]

which has behavior similar to classical PID law, but is data driven. The control error equation may be obtained from Equation (24), with \(K_{i}=0\) as:

\[\hat{F}_{k}=\Delta_{E}^{v}y_{k}-\alpha u_{k-1}. \tag{20}\]0

while the characteristic polynomial of the control error results by using the forward Euler approximation, from Equation (18) as:\[\hat{F}_{k}=\Delta_{E}^{v}y_{k}-\alpha u_{k-1}. \tag{20}\]1

A variant to obtain a domain of the controller parameters that ensures the fact that the \(P_{ce}(z)\) roots are stable is represented by the Jury test [39]. Based on it, from Equation (30) the relationships are obtained:

\[\left\{