## 2 Study of a Robust Nonlinear Control Strategy
### Preliminaries
It is established that:  
\[Z_{2}(t)=y^{(1)}(t)=L_{\rm T}H(X)+\beta(X)D_{i}L_{\rm G}H(X)+L_{\lambda}H(X)D_{i} \tag{25}\]  
For \(C_{i}>1\), one has:  
\[Z_{3}(t)=y^{(2)}(t)=L_{\rm T}H(X)+L_{\lambda}L_{\rm F}H(X)D_{i}+\beta(X)D_{i}L_ {\rm G}L_{\rm F}H(X) \tag{26}\]  
For \(C_{i}>2\), the previous Equation (26) is rewritten as:  
\[Z_{3}(t)=y^{2}(t)=L_{\rm F}^{2}H(X)=\sum\limits_{l\geq 1}\Lambda_{l}^{2}X^{[l]}, \tag{27}\]  
Stepping forward to:  
\[y^{(C_{i})}(t)=L_{\rm T}L_{\rm F}^{C_{i}-1}H(X)+L_{\lambda}L_{\rm F}^{C_{i}-1} H(X)D_{i}+\beta(X)D_{i}L_{\rm G}L_{\rm F}^{C_{i}-1}H(X) \tag{28}\]  
and seeing that \(q>C_{i}\), one has:  
\[y^{(C_{i})}(t)=L_{\rm F}^{C_{i}}H(X)+L_{\lambda}L_{\rm F}^{C_{i}-1}H(X)D_{i}, \tag{29}\]  
Then,  
\[ \begin{array}{ll}y^{(C_{i}+1)}(t)&=L_{\rm T}L_{\rm F}^{C_{i}}H(X)+\beta(X)D_ {i}L_{\rm G}L_{\rm F}^{C_{i}}H(X)+L_{\lambda}L_{\rm F}^{C_{i}}H(X)D_{i}\\ &+\frac{d}{dt}\Big{(}L_{\lambda}L_{\rm F}^{C_{i}-1}H(X)D_{i}\Big{)}\end{array}  \tag{30}\]  
In the case where \(q>C_{i}+1\), the computation is repeated until the \(q^{th}\) derivative corresponding to the output is reached; the following expression results:  
\[ \begin{array}{ll}y^{(q)}(t)&=L_{\rm F}^{q}H(X)+a(X)L_{\rm G}L_{\rm F}^{q-1} H(X)\\ &+\beta(X)D_{i}L_{\rm G}L_{\rm F}^{q-1}H(X)+L_{\lambda}L_{\rm F}^{q-1}H(X)D_{i} \\ &+\frac{d}{dt}\Big{(}L_{\lambda}L_{\rm F}^{q-2}H(X)D_{i}\Big{)}+\ldots+\frac{ d^{q-C_{i}}}{dt^{q-C_{i}}}\Big{(}L_{\lambda}L_{\rm F}^{C_{i}-1}H(X)D_{i}\Big{)}\end{array}  \tag{31}\]  
Then,  
\[y^{(q)}(t)=L_{\rm T}L_{\rm F}^{q-1}H(X)+\beta(X)L_{\rm G}L_{\rm F}^{q-1}H(X)D _{i}+\sum\limits_{k=0}^{q-C_{i}}\frac{d^{k}}{dt^{k}}\Big{(}L_{\lambda}L_{\rm F }^{q-k-1}H(X)D_{i}\Big{)} \tag{32}\]  
which gives:  
\[y^{(q)}(t)=\sum\limits_{k\geq 0}O_{k}^{s}X^{[k]}+\beta(X)D_{i}\sum\limits_{k=0 }^{q-1}\Pi_{k}^{l}X^{[k]}+\sum\limits_{k=0}^{q-c_{i}}\frac{d^{k}}{dt^{k}} \Bigg{(}D_{i}\sum\limits_{p\geq 0}\Delta_{p}^{i}X^{[p]}\Bigg{)} \tag{33}\]  
where \(O_{s}^{k}=\sum\limits_{l=0}^{k}\Lambda_{k-l+1}^{q-1}V^{(k-l+1)}\big{(}T_{s^{i} }\otimes I_{n^{(k-l)}}\big{)}\), and \(\Delta_{p}^{i}=\sum\limits_{l=0}^{p}\Delta_{p-l+1}^{q-p-l}V^{(p-l+1)}\big{(} \Delta_{\bar{\mu}}\otimes I_{n^{(p-l)}}\big{)}\). The decoupling of the output and class \(\zeta\) disturbance \(D_{i}\) is ensured using:  
\[\beta(X)=-\frac{\sum\limits_{k=0}^{q-c_{i}}\sum\limits_{l\geq 0}\Psi_{l}^{i}}{ \sum\limits_{k=0}^{q-1}\Pi_{k}^{l}X^{[k]}} \tag{34}\]where \(\left\{ \begin{array}{l}\bar{\Psi}_{l}^{0}=\Delta_{l}\\ \bar{\Psi}_{l}^{i}=\sum\limits_{j=1}^{l}\bar{\Psi}_{l-j+1}^{i-1}V^{(l-j+1)} \big{(}F_{j}\otimes I_{n^{(l-j)}}\big{)}\end{array} \right..\)  
The flowchart given in Figure 1 illustrates the different cases related to both values of local relative degree \(q\) and disturbance relative degree \(C_{i}\). The requirement is that the output \(y(t)\) corresponds to a specific reference trajectory. Using the state variable transformation [6], the manipulated variable \(U\) can be described as:  
\[U=\frac{-L_{F}^{q}H(X)+v}{L_{G}L_{F}^{q-1}H(X)}+\beta(X) \tag{35}\]  
This last equation is defined in the polynomial form by:  
\[\left\{ \begin{array}{l}U\ \ =-\left[\begin{array}{l}\sum\limits_{i=1}^{ \frac{C}{2}\Lambda_{i}^{i}X^{[i]}+K\sum\limits_{k=1}^{q}\Gamma_{k}X^{[i]}\\ \sum\limits_{k=0}^{l}\Gamma_{k}^{i}X^{[i]}\end{array}\right]}+\beta(X)\\ \ \ =\sum\limits_{i=1}^{l}a_{i}X^{[i]}+\sum\limits_{j=1}^{l}\mu_{i}X^{[i]}+\beta( X)\\ v=-K^{\mathrm{T}}\Gamma(X)=-K\sum\limits_{k=1}^{q}\Gamma_{k}X^{[i]},\ \ \Gamma_{q}=1\end{array} \right. \tag{36}\]  
Figure 1: Flowchart for the determination of the function \(\beta(X)\). \[\text{where}\ \ \left\{ \begin{array}{l}\mu_{1}=\left(\Pi_{0}^{-1}\right)KT_{1}\\ \vdots\\ \mu_{j}=\Pi_{0}^{-1}\Bigg{[}KT_{j}-\sum\limits_{p=1}^{j-1}\left(\Pi_{j-p}\otimes \mu_{p}\right)\Bigg{]}\end{array} \right..\]  
where \(v\) denotes a new reference varying set point that maybe computed to provide a designed-equivalent linear model with specific eigenvalues sets. One challenge concerning this control technique is that only local robustness can be ensured.