## 4 Integrated FE/FTC
### Integrated Synthesis of FE/FTC  
The augmented closed-loop system composed of (13) and (18) is  
\[\dot{x}=(\Theta_{1}A-BK_{x})x+BK_{e}+D_{1}\overline{\varepsilon}\dot{e}=\left( \Xi\overline{A}-L_{1}\overline{C}\right)e+\Xi\overline{D}\overline{ \varepsilon}\;z_{1}=C_{x}x+C_{e}\text{e} \tag{19}\]  
where \(z_{1}eR^{\gamma}\) is the measured output used to verify the closed-loop system performance with matrices \(C_{x}eR^{\gamma\times n}\), \(C_{e}eR^{\gamma\times(n+q)}\), \(D_{1}=\left[ \begin{array}{cc}\Theta D&0\end{array} \right]\). It can be seen from (19) that the system disturbance affects the state/FE, and the estimation error directly affects the control system. This shows that there is a bi-directional robustness interaction between the FE and FTC models, which breaks the so-called Separation Principle and produces an integrated FE/FTC design approach to achieve optimal and robustness overall performance of the FTC system. Now, the problem of integrated design can be expressed as follows: design the controller gain \(K_{x}\) and the observer gains \(H\) and \(L_{1}\) to ensure the robust stability of the augmented closed-loop System (19). This integrated design is essentially an observer-based robust control problem. To solve this problem, a one-step LMI formulation is proposed in Theorem 2. This one-step approach is modified from the method used in [23]. **Theorem 2. **_Given positive scalars \(\gamma_{1},\ \epsilon_{1},\ \epsilon_{2},\ \epsilon_{3},\ \delta_{1}\) and \(\delta_{2}\), the closed-loop System (19) is stable with \(H_{\infty}\) performance \(\|G_{z_{1}\overline{z}}\|_{\infty}<\gamma_{1}\), if there exist three symmetric positive definite matrices \(Z\in R^{n\times n},Q\in R^{n\times n},R\in R^{q\times q}\), and matrices \(M_{1}\in R^{m\times n},M_{2}\in R^{n\times p},M_{3}\in R^{n\times p},M_{4}\in R ^{q\times p}\), \(M_{5}\in R^{q\times p}\) such that_  
\[\left[ \begin{array}{cc}\Pi_{1}&\Pi_{2}\\ *&\Pi_{3}\end{array} \right]<0 \tag{20}\]  
_with_  
\[\Pi_{1}=\left[ \begin{array}{cc}\Xi_{1,1}&\Xi_{1,2}\\ *&J_{2,2}\end{array} \right],\ \Pi_{2}=\left[ \begin{array}{cc}\Xi_{1,3}&\Xi_{1,4}&0& \Xi_{1,6}&\Xi_{1,7}&0&0&\Xi_{1,10}\\ J_{2,3}&J_{2,4}&J_{2,5}&0&0&I&J_{2,9}&0\end{array} \right]\]  
\[\Pi_{3}=-diag\bigg{\{}\gamma_{1}{}^{2}I,I,\epsilon_{1}I,\left(\epsilon_{1}+ \epsilon_{2}\right)^{-1}\!,\epsilon_{3}{}^{-1}Z,\epsilon_{3}Z,\delta_{1}I, \left(\delta_{2}L_{f}^{2}\right)^{-1}\!I\bigg{\}}\]  
\[J_{2,2}=\left[ \begin{array}{cc}\[PERSON]_{2,2}&\[PERSON]_{2,3}\\ *&\[PERSON]_{3,3}\end{array} \right],\ J_{2,3}=\left[ \begin{array}{cc}QD-M_{2}CD&0 \\ -M_{4}CD&R\end{array} \right],\ J_{2,A}=\left[ \begin{array}{cc}C_{\rm e}^{ \top}\\ C_{\rm e/a}^{\top}\end{array} \right]\]  
\[J_{2,5}=\left[ \begin{array}{cc}QM_{0}-M_{2}CM_{0}\\ -M_{4}CM_{0}\end{array} \right],\ J_{2,9}=\left[ \begin{array}{cc}Q-M_{2}C&0 \\ -M_{4}C&R\end{array} \right],\]  
\[\Xi_{1,1}=He(\Theta AZ-BM_{1})+\epsilon_{2}{}^{-1}M_{0}M_{0}{}^{\top}+\delta_{ 2}{}^{-1}\Theta\Theta^{\top},\ \ \[PERSON]_{1,2}=\left[ \begin{array}{cc}0&F\end{array} \right]\[PERSON]_{1,3}=\left[ \begin{array}{cc}\Theta D&0\end{array} \right],\]  
\[\[PERSON]_{1,A}=ZC_{x}^{\top},\ \[PERSON]_{1,6}=ZN_{0}^{\top},\ \[PERSON]_{1,7}=BM_{1},\ \[PERSON]_{1,10}=Z,\ \[PERSON]_{2,2}= He(QA-M_{2}CA-M_{3}C)+\delta_{1}L_{f}^{2}I_{n}\]  
\[\[PERSON]_{2,3}=QF-M_{2}CF-A^{\top}C^{\top}M_{4}^{\top}-C^{\top}M_{5}^{\top},\ \[PERSON]_{3,3}= He(-M_{4}CF).\]  
_where \(He(W)=W+W^{\top}\), '\(*\)' denotes the symmetric part of a matrix. Then the gains are given by_  
\[K_{X}=M_{1}Z^{-1},H_{1}=Q^{-1}M_{2},H_{2}=R^{-1}M_{4},L_{11}=Q^{-1}M_{3},L_{12}= R^{-1}M_{5}\]  
Proof Consider a Lyapunov function \(V_{e}={\rm e}^{\top}P_{1}{\rm e}\) with a symmetric positive matrix \(P_{1}\).