## 4 Integrated FE/FTC
### FE Design  
The augmented state \(\overline{x}(t)\) is estimated by a NUIO as follows:  
\[\left\{ \begin{array}{c}\dot{z}=Mz+Gu+Ly\\ \dot{\overline{x}}=z+Hy\end{array} \right. \tag{10}\]  
where \(z\in R^{n+q}\) is the observer system state and \(\dot{\overline{x}}\in R^{n+q}\) is the estimate of \(\overline{x}\). The matrices \(M\in R^{(n+q)\times(n+q)},G\in R^{(n+q)\times m},L\in R^{(n+q)\times p},\ H\in R ^{(n+q)\times q}\) are to be designed. Defining the estimation error as \(e=\overline{x}-\dot{\overline{x}}\), the error dynamics as follows:  
\[\dot{e}=\dot{\overline{x}}-\dot{\overline{x}}=\left(\overline{\Xi}\overline{A }-L_{1}\overline{C}\right)e+\left(\overline{\Xi}\overline{A}-L_{1}\overline{C }-M\right)z+\left(\overline{\Xi}\overline{B}-G\right)u+\left[\left(\overline{ \Xi}\overline{A}-L_{1}\overline{C}\right)H-L_{2}\right]y+\overline{\Xi} \overline{D}\overline{\epsilon} \tag{11}\]  
where \(\Xi=I_{n+q}-H\overline{C},\ L=L_{1}+L_{2}\), The matrices \(M,G,\) and \(L_{2}\) are defined as  
\[M=\overline{\Xi}\overline{A}-L_{1}\overline{C},\ G=\overline{\Xi}\overline{B },\ \ L_{2}=\left(\overline{\Xi}\overline{A}-L_{1}\overline{C}\right)H \tag{12}\]  
With the definition given in Equation (12), the error dynamics (11) become  
\[\dot{e}=\left(\overline{\Xi}\overline{A}-L_{1}\overline{C}\right)e+\overline{ \Xi}\overline{D}\overline{\epsilon} \tag{13}\]  
**Theorem 1**. : _There exists a robust NUIO (10) if the error System (13) is robustly asymptotically stable._  
Proof with Equation (12), the error System (13) is equivalent to the original error System (11). So, if Equation (13) is robustly and asymptotically stable, then Equation (11) is also robustly and asymptotically stable, indicating that \(lim_{t\rightarrow\infty}e(t)=0\) in the presence of disturbance. Note that the design matrices \(M,\ G\) and \(L\) can be derived after the matrices \(L_{1}\) and \(H\) are obtained. Therefore, the main task of obtaining the NUIO (10) as described in the sequel is to design \(L_{1}\) and \(H\) such that Equation (13) is robustly and asymptotically stable. **Remark 3**. : _Note from Equation (13) that the disturbance is completely decoupled from the state/FE if the following one condition holds: (\(Q\)) rank (\(CD\)) = rank (\(D\)) + \(q\). The disturbance can be completely decoupled by designing \(H\) such that \((I_{n+q}-H\overline{C})\overline{D}=0\) with \(rank\big{(}\overline{CD}\big{)}=rank\big{(}\overline{D}\big{)}\). Considering the case of partially decoupled disturbance, the method allows disturbance of the partial decoupling process in the observer design. An augmented state vector composed of current state, delay state and additive fault is constructed, and the augmented system is described inthe singular form. Then, an unknown input observer which can decouple some disturbances is obtained, and the uncoupled disturbances are reduced, but only one kind of integral measurement system is studied [14]. A robust NUIO (10) is proposed in this paper using \(H_{\infty}\) optimization. This optimization method has no requirement of rank, so it is suitable for more practical engineering systems._  
**Remark 4**. : _It is worth noting that unlike traditional FDI, the traditional FDI includes fault detection and isolation processes based on residual generators, and the residual threshold is used to design the proposed NUIO to directly estimate the fault shape when faults occur. The observer can estimate the faults naturally and regard them as new states in an enhanced system._  
In the proposed NUIO, the faults are assumed to be bounded and continuously smooth, and it is seen from Equation (13) that the FE performance is affected by the fault modelling error \(\tilde{f}_{a}\). The augmented perturbation \(\tilde{\epsilon}\) is not completely decoupled but attenuated by robust design. Thus, the robust FE performance depends on the robustness of the error System (11). **Remark 5**. : _Since the matrix \(C\) in this paper is not the identity matrix, it can be assumed that_  
\[C=C_{1}+C_{2}=