## 2 Study of a Robust Nonlinear Control Strategy
### Preliminaries
\begin{array}{l}\Lambda_{l}^{0}=H_{l}\\ \Lambda_{l}^{q}=\sum\limits_{j=1}^{l}\Lambda_{l-j+1}^{q-1}V^{(l-j+1)}\big{(}T _{j}\otimes I_{n^{(l-j)}}\big{)}\end{array} \right..\)  
Overall, there is no term in the control law \(U\) to discard case 1 disturbances because the state vector contains all pertinent information. **Lemma 3**. : _If \(C_{i}=q\) the control input is written as [38]:_  
\[U=a(X)+b(X)v+\beta(X)D_{i} \tag{18}\]  
_where \(\left\{ \begin{array}{l}a(X)=\sum\limits_{i=1}a_{i}X^{[i]},\\ b(X)=\sum\limits_{j=1}b_{j}X^{[i]},\\ \beta(X)=-\frac{\sum\limits_{j=1}^{l-1}g^{\kappa}X^{[i]}}{\sum\limits_{k=0}^{l }\Gamma_{l}^{k}X^{[k]}}\end{array} \right.\)._  
**Proof. ** If \(C_{i}=q\), the system output is similarly impacted by the control input and disturbance, there is a need for feed-forward steps to facilitate decoupling. The control law specified by Equation (18) will be considered, whereby there is feed-forward action specific to \(D_{i}\). The original system after such control can be represented as:  
\[\left\{ \begin{array}{l}\dot{X}=F(X)+G(X)b(X)v+G(X)a(X)+(G(X)\beta(X)+\lambda( X))D_{i}\\ y=H(X)\end{array} \right. \tag{19}\]  
Consequently, the initial \((q-1)\) state equations under the Byrnes-Isidori form appear similarly to the equations derived for an undisturbed model. However:  
\[y^{(q)}(t)=L_{T1}^{q}H(X)+L_{T_{2}}L_{F}^{q-1}H(X)D_{i} \tag{20}\]  
where \(T_{1}=F+G(a+bv)\) and \(T_{2}=G\beta+\lambda\)  
\[ \begin{array}{ll}L_{T_{2}}L_{F}^{q-1}H(X)&=L_{G}L_{F}^{q-1}H(X)\beta(X)+L_{ \lambda}L_{F}^{q-1}H(X)\\ &=\sum\limits_{k=0}^{q-1}\Pi_{k}^{i}X^{[k]}\beta(X)+\sum\limits_{s=0}^{q-1}I_ {s}^{p}X^{[s]}=0\end{array}  \tag{21}\]  
where \(\Pi_{k}^{l}=\sum\limits_{i=0}^{k}\Pi_{k-i+l}^{q-1}V^{(k-i+1)}\big{(}G_{li} \otimes I_{n^{(k-l)}}\big{)}\) and \(J_{s}^{p}=\sum\limits_{k=0}^{s}\Lambda_{s-k+1}^{q-1}V^{(s-k+1)}\big{(}\lambda _{p^{k}}\otimes I_{n^{(p-k)}}\big{)}\). Next, if \(y(t)\) is not dependent on \(D_{i}\), and the input \(U\) should be defined so that:  
\[y^{(q)}(t)=\dot{Z_{q}}=v, \tag{22}\]  
The function of decoupling \(\beta(X)\) is chosen as:  
\[\beta(X)=-\frac{\sum\limits_{s=0}^{q-1}J_{s}^{p}X^{[s]}}{\sum\limits_{k=0}^{ q-1}\Pi_{k}^{l}X^{[k]}} \tag{23}\]  
wherein matrixes \(J_{i}^{p}\) and \(\Pi_{k}^{l}\) are represented in Equation (21). **Lemma 4. **_If \(C_{i}<q\) the control input is obtained by [38]:_  
\[U=a(X)+b(X)v+\beta(X)D_{i} \tag{24}\]  
_where \(\left\{ \begin{array}{ll}a(X)=\sum\limits_{i=1}a_{i}X^{[i]},&a_{i}=\Pi_{0}^ {-1}\Bigg{[}\Lambda_{i}-\sum\limits_{p=1}^{i-1}\[PERSON]_{i-p}\otimes a_{p}\Bigg{]} \\ b(X)=\sum\limits_{j=1}b_{j}X^{[j]},&b_{j}=-\[PERSON]_{0}^{-1}\sum\limits_{p=1}^{i-1 }\[PERSON]_{i-p}\otimes b_{p}\\ \beta(X)=-\frac{\sum\limits_{k=0}^{q-1}\sum\limits_{j\geq 0}^{q-1}\sum\limits_{k=0}^{ q}\Pi_{k}^{l}X^{[k]}}{\sum\limits_{k=0}^{q-1}\Pi_{k}^{l}X^{[k]}}\end{array} \right.\) where: \(\Pi_{k}^{l}=\sum\limits_{i=0}^{k}\Pi_{k-i+1}^{q-1}V^{(k-i+1)}\big{(}G_{li} \otimes I_{n^{(k-l)}}\big{)}\) and \(\left\{ \begin{array}{l}\Psi_{l}^{0}=\lambda_{l}\\ \Psi_{l}^{i}=\sum\limits_{j=1}^{l}\Psi_{l-j+1}^{i-1}V^{(l-j+1)}\big{(}F_{j} \otimes I_{n^{(l-j)}}\big{)}\end{array} \right..\)  
**Proof. ** In the case where \(C_{i}<q\), the disturbance has a higher impact on the output compared to the control input. It is necessary to take \"anticipative action\" to fulfill disturbance-rejection. The control rule specified in Equation (18) is considered since the order of disturbance appearance is the \(C_{i}\) derivative corresponding to the output.