## 2 Study of a Robust Nonlinear Control Strategy
### Preliminaries
\right]\) is the compensator. It is straight obvious that if \(q<n\) the original nonlinear model will come into a nonlinear \(\mathfrak{R}^{n-q}\) subsystem whose behavior does not assign the desired system output and a linear \(\mathfrak{R}^{q}\) one, which evolves with linear dynamics. Such a nonlinear subsystem is designated as the \(ZD\). A nonlinear model where \(q<n\) is I/O linearizable only in case of minimum phase characteristic, which signifies that the \(ZD\) is locally asymptotically stable [39]. In order to overcome the output decoupling problem from process disturbances, [39] have addressed measurable disturbances as external inputs. In this respect, they have defined the new concept of disturbances relative degree. Definition 2 (38): Let the disturbance \(D\in\mathfrak{R}^{m}\) affects the dynamics of the nonlinear affine system (1). The system output \(y(t)\) is said to have a relative degree related to the disturbance \(D_{i}(0\leq i\leq m)\) if there exists an integer \(C_{i}\) satisfying:  
\[L_{\lambda_{i}}L_{\mathrm{F}}^{C_{i}-1}H(X)=\sum\limits_{k=0}^{C_{i}-1}P_{k}^ {i}X^{[k]}\
eq 0, \tag{9}\]  
where \(P_{k}^{i}=\sum\limits_{l=0}^{k}\Lambda_{k-l+1}^{C_{i}-1}V^{(k-l+1)}\big{(} \lambda_{l}\otimes I_{n^{(k-l)}}\big{)}\). The following lemma is stated based on the concept of disturbance relative degree stated on Definition 2. Lemma 2: If \(C_{i}>q\) then the output of the system \(y(t)\) is decoupled from the disturbance \(D_{i}\) and the control input can be obtained by [38]:  
\[U=a(X)+b(X)v \tag{10}\]  
where \(\left\{ \begin{array}{cc}a(X)=\sum\limits_{i=1}a_{i}X^{[i]},&a_{i}=\Pi_{0}^{ -1}\Bigg{[}\Lambda_{i}-\sum\limits_{p=1}^{i-1}\Pi_{i-p}\otimes a_{p}\Bigg{]}& \mbox{and $v$ is the control input of the $\left.\right.$}\\ b(X)=\sum\limits_{j=1}b_{j}X^{[j]},&b_{j}=-\Pi_{0}^{-1}\sum\limits_{p=1}^{i-1} \Pi_{i-p}\otimes b_{p}&\end{array} \right.\)  
Proof: The disturbance affects to a lesser degree compared to the control input in case \(C_{i}>q\). The information corresponding to disturbance \(D_{i}\) can be obtained using the system states; therefore, decoupling \(y(t)\) and \(D_{i}\) needs no additional effort. Here, it can be easily confirmed that for \(i=0,1,\ldots,q-1\) one has \(L_{\lambda}L_{\mathrm{F}}^{i}H(X)=0\). Next, the disturbance vector components \(D_{i}\) are not expected to affect the initial \(q\) equations of the observability Byrnes-Isidori form, which is similar to the controllability canonical one. The other equations concerning the ZD would depend on the disturbance components vector \(D\). Presume that the subsequent feedback control input is selected:  
\[U=-\frac{B(Z)}{A(Z)}+\frac{v}{A(Z)} \tag{11}\]  
Then, the feedback control is defined by Equation (8) and:  
\[\dot{\vartheta}=Q(Z)+K(Z)D_{i} \tag{12}\]  
where it may be observed that the output (\(Z\)) is entirely disconnected from the disturbance term \(D_{i}\). Here, the condition \(C_{i}>q\) must be fulfilled if the output is to not influence \(D_{i}\), while \(U\) is the linearizing control input. Consider \(U=a(X)+b(X)v\) so such extent that the system output \(y(t)\) has no influence of \(D_{i}\). The closed-loop system can be expressed as:  
\[\left\{ \begin{array}{l}\dot{X}=F(X)+G(X)b(X)v+G(X)a(X)+\lambda(X)D_{i}\\ y=H(X)\end{array} \right. \tag{13}\]  
Next,  
\[y^{(1)}(t)=Z_{2}(t)=L_{F+G\kappa}H(X)+L_{\lambda}H(X)D_{i} \tag{14}\]  
becomes autonomous from the disturbance \(D\) if and only if \(L_{\lambda}H(X)=0\). Assuming this is obtained,  
\[y^{(2)}(t)=Z_{3}(t)=L_{F+G\kappa}^{2}H(X)+L_{\lambda}L_{F+G\kappa}H(X)D_{i} \tag{15}\]  
where \(L_{\lambda}L_{T}^{q-1}H(X)\) should be different from zero. Calculate iteratively:  
\[y^{(q)}(t)=L_{T}^{q}H(X)+L_{\lambda}L_{T}^{q-1}H(X)D_{i} \tag{16}\]  
where \(L_{\lambda}L_{T}^{q-1}H(X)\) is equal to zero. Consequently, it can be concluded that \(L_{\lambda}L_{T}^{q-1}H(X)=0\) so far as \(i=0,1,\ldots,q-1\), \(C_{i}>q\). Then, Equation (16) is written  
\[y^{(q)}(t)=\sum\limits_{l\geq 1}\Lambda_{l}^{q}X^{[l]}, \tag{17}\]  
where \(\left\{