## 2 Study of a Robust Nonlinear Control Strategy  
### Preliminaries  
There is a short description of the primary factors comprising the I/O FBL technique in order to touch upon the fundamental topics. If the reader wishes to have more information concerning the background and comprehensive aspects of the framework, reading the works of [31; 37] is suggested. The following expression concerns the general form of a nonlinear affine system  
\[\left\{ \begin{array}{l}\dot{X}=F(X)+G(X)U+\lambda(X)D\\ y=H(X)\end{array} \right. \tag{1}\]  
where \(X\in\Re^{n},U\in\Re,y\in\Re,D\in\Re^{m};F(.):\Re^{n}\rightarrow\Re^{n};G(. ): \Re^{n}\rightarrow\Re\) and \(\lambda(. ):\Re^{n}\rightarrow\Re^{m}\) are \(C^{\infty}\) vector fields, and \(H(. ):\Re^{n}\rightarrow\Re\) is a \(C^{\infty}\) function. It is straightforward to transform the model into the Kronecker product and Taylor expansions so that it can be converted to a polynomial expression specified below:  
\[\left\{ \begin{array}{l}\dot{X}=\sum\limits_{i=1}^{q}\left(F_{i}X^{[i]}\right)+ \sum\limits_{j=0}^{q}\left(G_{j}X^{[i]}\right)U+\sum\limits_{k=0}^{q}\lambda_{k }X^{[k]}D\\ y=\sum\limits_{s=1}^{q}\left(H_{s}X^{[s]}\right)\end{array} \right., \tag{2}\]  
**Lemma 1**[37]. : _Let the real function \(H(X)\) expressed under the polynomial form by: \(H(X)=\sum\limits_{i\geq 1}H_{i}X^{[i]}\) and the vector field:_  
\[F(X)=\sum\limits_{i\geq 0}F_{i}X^{[i]}, \tag{3}\]  
_The Lie derivative of \(H(X)\) with respect to\(F(X)\) denoted by \(L_{F}H(X)\) is given by:_  
\[L_{F}H(X)=\sum\limits_{k\geq 0}\Lambda_{k}X^{[k]}, \tag{4}\]  
_where_  
\[\Lambda_{k}=\sum\limits_{j=1}^{k}\Lambda_{k-j+1}V^{(k-j+1)}\left(F_{i}\otimes I _{n^{(k-j)}}\right), \tag{5}\]  
_By exploiting this lemma, the quantities \(L_{F}^{i}H(X)\) are then written:_  
\[L_{F}^{i}H(X)=\sum\limits_{l\geq 1}\Lambda_{l}^{i}X^{[l]}\text{ for }1\leq i \leq n, \tag{6}\]  
_where_  
\[\left\{ \begin{array}{l}\Lambda_{l}^{0}=H_{l}\\ \Lambda_{l}^{i}=\sum\limits_{j=1}^{l}\Lambda_{l-j+1}^{i-1}V^{(l-j+1)}\left(F_ {j}\otimes I_{n^{(l-j)}}\right)\end{array} \right.\]  
_where \(V^{[k]}=\sum\limits_{j=0}^{i-1}\left(U_{n^{j}\times n}\otimes I_{n^{(i-j-1)}}\right)\) and \(U_{p\times q}\) denotes the Kronecker permutation matrix. Moreover, we express in a similar way the quantities_  
\[L_{\mathbb{G}}L_{F}^{q-1}H(X)=\sum\limits_{k\geq 0}J_{k}^{j}X^{[k]}, \tag{7}\]  
_where_  
\[J_{k}^{i}=\sum\limits_{l=0}^{k}\Lambda_{k-l+1}^{q-1}V^{(k-l+1)}\left(G_{j} \otimes I_{n^{(k-l)}}\right)\]  
**Definition 1**[38]. : _System (1) is characterized by a local relative degree \(q\) at an operating point \(X_{0}\) if_  
* \(L_{\mathbb{G}}L_{F}^{k}H(X)=\sum\limits_{j\geq 0}N_{j}^{j}X^{[j]}=0\) _for all_ \(X\) _close to_ \(X_{0}\) _and for all_ \(k<q-1\)_;_ _where_ \(N_{j}^{i}=\sum\limits_{l=0}^{j}\Lambda_{j-l+1}^{k}V^{(j-l+1)}\left(G_{j} \otimes I_{n^{(l-1)}}\right)\)__
* \(L_{\mathbb{G}}L_{F}^{q-1}H(X)=\sum\limits_{k\geq 0}J_{k}^{j}X^{[k]}\
eq 0\)_;_  
_Considering that the system has a measurable relative degree, it is feasible to determine a coordinate function \(Z=\Gamma(X)\) that is obtained by repeatedly differentiating the output. This system is now specified as a function of the new coordinates having the normal form. Consideringthe situation where there is a lack of disturbance, the output of the normal form is the canonical controllability form:_  
\[\left\{ \begin{array}{c}\dot{Z}_{1}=Z_{2}=L_{\mathrm{F}}H(X)=\sum\limits_{l\geq 1 }\Lambda_{l}^{1}X^{[l]}\\ \dot{Z}_{2}=Z_{3}=L_{\mathrm{F}}^{2}H(X)=\sum\limits_{l\geq 1}\Lambda_{l}^{2}X^{[l]}\\ \vdots\\ \dot{Z}_{q-1}=Z_{q}=L_{\mathrm{F}}^{q-1}H(X)=\sum\limits_{l\geq 1}\Lambda_{l}^{q- 1}X^{[l]}\\ \dot{Z}_{q}=v\\ \dot{Z}_{i}=\theta(Z),q+1\leq i\leq n\\ y=Z_{1}=H(X)\end{array} \right. \tag{8}\]  
\(Z=\left[ \begin{array}{cc}Y_{1}&Y_{2}\end{array} \right]^{T}\) with \(Y_{1}=\left[ \begin{array}{cc}H(X)&L_{\mathrm{F}}H(X)&L_{\mathrm{F}}^{2}H(X) &\ldots&L_{\mathrm{F}}^{q-1}H(X)\end{array} \right]\) and \(Y_{2}=\left[ \begin{array}{cc}\varphi_{1}(X)&\ldots&\varphi_{n-q}H(X)\end{array}