## 2 Preliminaries and Problem Statement  
A class non-integral chain form of an uncertain non-linear system with \(n^{th}\) order and mismatched disturbances is considered in [13] as:  
\[ \begin{cases}\dot{x}=Ax+B_{u}u+B_{f}d(x,w(t),t),\\ y=Cx,\end{cases}  \tag{1}\]  
where \(x=[x_{1}\cdots x_{n}]^{T}\), \(u\in R\), \(y\in R\), and \(w\in R\) are the state vector, input, controlled output, and external disturbances respectively. \(A\in R^{n\times n}\), \(B_{u}\in R^{n\times 1}\), \(B_{f}\in R^{n\times 1}\), \(d(\cdot)\in R\) may be mismatched uncertain disturbance function of \(x\) and \(w\) and \(C\in R^{(1\times n)}\). **Remark 1**. : _The generalized term for lumped disturbance \(d(x,w(t),t)\) in Equation (1), is assumed to be bounded. This uncertainty function \(d(x,w(t),t)\) includes unmodeled dynamics, external noise, and parametric uncertainty, which may be challenging for a simple feedback controller._  
**Remark 2**. : _The matching condition with respect to disturbance matrix \(B_{f}\) is given by \(B_{u}\)=\(\lambda B_{f}\), \(\lambda\in R\)[13]. Equation (1) describes a more specific class of systems because the system (1) is not only restricted to an integral chain form and it can also be subject to mismatched uncertainties [37]._  
The GESO for system (1) with state feedback controller and disturbance compensation gain law was proposed in [13], whereas the enhanced GESO based controller for the same class of system is given in [14]. The control law provided by [13] is stated as,  
\[u=K_{x}\hat{x}+K_{f}\hat{d} \tag{2}\]  
where  
\[K_{f}=-[C(A+B_{u}K_{x})^{-1}B_{u}]^{-1}C(A+B_{u}K_{x})^{-1}B_{f} \tag{3}\]  
disturbance compensation gain is denoted as \(K_{f}\), and state feedback gain is represented as \(K_{x}\). \(\hat{x}\) and \(\hat{d}\), are the estimations of \(x\), \(d\), respectively, and are obtained by following GESO structure:  
\[ \begin{bmatrix}\hat{\hat{x}}\\ \hat{x}_{(n+1)}\end{bmatrix} = \begin{bmatrix}A_{(n\times n)}&B_{f}\\ 0_{(1\times n)}&0\end{bmatrix}\begin{bmatrix}\hat{\hat{x}}\\ \hat{\hat{x}}_{(n+1)}\end{bmatrix} + \begin{bmatrix}B_{u}\\ 0\end{bmatrix} u+L_{0}(y-[C\;0_{1\times 1}]\hat{x}) \tag{4}\]  
where \(\hat{x}= \begin{bmatrix}\hat{x}_{1}&\hat{x}_{2}&\cdots&\hat{x}_{n}\end{bmatrix} ^ {T}\) are the estimates of the state variable \( \begin{bmatrix}x_{1}&x_{2}&\cdots&x_{n}\end{bmatrix} ^{T}\), \(L_{0}\) is observer gain matrix with dimension \((n+1)\times 1\), which has to be design, and \(\hat{x}_{(n+1)}\) is an estimation of lumped disturbance \(d\). The GESO simultaneously provides an estimate of the states and the uncertainties, as the uncertainties are part of the states of the extended order system [13]. **Remark 3**. : _The lumped disturbances assumed to be bounded and they must have constant steady state value as \(t\rightarrow\infty\) i.e., \(lim_{t\rightarrow\infty}\;d((x,w(t),t))=0\), means \(lim_{t\rightarrow\infty}\;d((x,w(t),t))=C_{d}\), where \(C_{d}\) is a constant value. Under this assumption and control structure given from (2) to (4), the GESO based control gives stable bounded response. Here Remark 3 is completed, whereas the enhanced generalized observer-based control law proposed by [14] stipulates as,_  
\[u=K_{x}\hat{x}_{g}+u_{g}\Big{(}\hat{\eta},\hat{d},\hat{\lambda},\cdots,\hat{d} ^{(k_{d})}\Big{)} \tag{5}\]  
_where \(K_{x}\) is selected so that \(\tilde{A}\cong A+B_{u}K_{x}\) will be Hurwitz, \(\hat{x}_{g}\) is estimated GESO states which consist of estimation of original states along with extended state and it is given as \([\hat{x}\;\hat{d}\;\hat{d}\;\cdots\;\hat{d}^{(k_{d})}]^{T}\), \(u_{g}\) is a function of combined term which consists of \(\hat{\eta}\), \(\hat{d}\), \(\hat{d}\), and \(\hat{d}^{(k_{d})}\) which are the dynamic component, estimation of lumped disturbance, estimation of disturbance derivative, and estimation of disturbance \(k_{d}^{th}\) order derivative respectively.