## 3 Proposed Control Scheme
### Design of Enhanced Generalized Extended State Observer  
This sub-section is intended to design mismatched disturbance and states estimation method. The lumped disturbances shown by matrix \(B_{f}\) are effectively estimated by (7). The EGESO is defined as  
\[\dot{\hat{x}}_{g}=A\hat{x}_{g}+B_{g}u+L_{g}(y-C_{g}\hat{x}_{g}) \tag{7}\]  
where \(L_{g}\in R^{(n+2)\times 1}\) is the observer gain matrix which needs to be designed, and  
Figure 1: Block diagram of the proposed control scheme.  
\[A_{g}=\begin{bmatrix}A&B_{f}&0_{n\times k_{d}}\\ 0_{k_{d}\times n}&0_{k_{d}\times 1}&I_{k_{d}\times k_{d}}\\ 0_{1\times n}&0_{1\times 1}&0_{1\times k_{d}}\end{bmatrix},B_{g}=\begin{bmatrix}B_{u}\\ 0_{(k_{d}\times 1)}\\ 0_{1\times 1}\end{bmatrix},C_{g}=\begin{bmatrix}C\\ 0_{1\times 1}\\ 0_{1\times k_{d}}\end{bmatrix}^{T}, \tag{8}\]  
\[x_{g}=\begin{bmatrix}x&d&\dot{d}&\cdots&d^{(k_{d})}\end{bmatrix}^{T}.\]  
are the generalized extended observer states, \(A_{g}\) is new GESO system matrix, \(B_{g}\) is GESO control matrix, \(C_{g}\) is new GESO output matrix, \(k_{d}=\max\{0,\,m_{f}^{tot}-m_{u}^{tot}\}\) and \(\hat{x}\), \(\hat{d}\), \(\hat{d}\), \(\cdots\), \(\hat{d}^{(k_{d})}\) are estimations of \(x,d,\hat{d},\cdots,d^{(k_{d})}\) respectively.  
**Assumption 1**.: _The matrix pair \((A,B_{u})\) is controllable and the pair \((\bar{A},C)\) is observable. Furthermore, the observability of matrix pair \((\bar{A},C)\) is a necessary condition for the observability of \((A_{g},C_{g})\), as given in [13]._  
**Assumption 2**.: _The matrix \((\bar{A},B_{u},C)\) does not have zeros at the imaginary axis._  
**Assumption 3**.: \(d^{(k_{d}+1)}\) _is bounded._  
Here we define estimation error of the ESO as \(e_{0}=x_{g}-\hat{x}_{g}\) with \(x_{g}\) defined in (8), by considering the observer dynamics as given in Equation (7). The following statement indicates throughout proof for boundedness of error \(e_{0}\),  
**Proposition 1**.: _Under Assumptions 1-3, the boundedness of error \(e_{0}\) is guaranteed if \(L_{g}\) in (7) is chosen such that \((A_{g}-L_{g}C_{g})\) is Hurwitz._  
Proof.: By considering the system in Equation (1), the boundation of \(e_{0}\) is given by the following proof.  
\[\begin{cases}\dot{x}=Ax+B_{u}u+B_{f}x_{d,0}\\ x_{d,0}=d\\ \dot{x}_{d,0}=x_{d,1}=\dot{d}\\ \vdots\\ \dot{x}_{d,k_{d}-1}=x_{d,k_{d}}=d^{k_{d}}\\ \dot{x}_{d,k_{d}}=d^{(k_{d}+1)}\end{cases} \tag{9}\]  
where \(x_{d,k_{d}}=d^{k_{d}}\), value of \(k_{d}\) varies from \(0\) to positive integer, and it is given as \(k_{d}=\max\{0,\,m_{f}^{tot}-m_{u}^{tot}\}\). Disturbance terms can be expressed as \([x_{d,0},\,\,x_{d,1},\,\,\cdots,\,\,x_{d,k_{d}}]^{T}=[d,\dot{d},\cdots,d^{k_{d }}]^{T}\). Using dynamics given in Equation (9), we are estimating disturbance and their derivatives from Equation (10). Expressing Equation (9) in matrix form leads to,  
\[\dot{x}_{g}=A_{g}x_{g}+B_{g}u+B_{g,f}d^{k_{d}+1} \tag{10}\]  
with \(x_{g}\), \(A_{g}\), \(B_{g}\) defined in (8) and \(B_{g,f}=[0_{1\times(n+k_{d})},1]^{T}\).  
By differentiating \(e_{0}\triangleq x_{g}-\hat{x}_{g}\) and inserting Equations (7) and (10) in it, we get  
\[\dot{e}_{0}\triangleq(A_{g}-L_{g}C_{g})(x_{g}-\hat{x}_{g})+B_{g,f}d^{k_{d}+1} \tag{11}\]  
which is bounded for any bounded \(f^{k_{d}+1}\), since \((A_{g}-L_{g}C_{g})\) is Hurwitz [38]. The ESO stability proof is finished here.