## 4 Integrated FE/FTC
### Integrated Synthesis of FE/FTC
The time derivative of \(V_{e}\) along (13) is as follows:  
\[\dot{V}_{e}=\dot{e}^{\top}P_{1}e+e^{\top}P_{1}\dot{e}=e^{\top}He\big{[}P_{1} \big{(}\Xi\overline{A}-L_{1}\overline{C}\big{)}\big{]}e+He\Big{(}e^{\top}P_{1} \Xi\overline{D}\overline{e}\Big{)} \tag{21}\]  
Consider another Lyapunov function \(V_{x}=x^{\top}P_{1}x\). Then, the time derivative of \(V_{x}\) along (18) is  
\[\dot{V}_{x}=\dot{x}^{\top}P_{1}x+x^{\top}P_{1}\dot{x}=x^{\top}He[P(\Theta A- BK_{x})]x+He\Big{(}x^{\top}PBKe\Big{)}+He\Big{(}x^{\top}PD_{1}\overline{\epsilon} \Big{)} \tag{22}\]  
The \(H_{\infty}\) performance \(G_{z_{1}\overline{z}}<\gamma\) can be represented as  
\[J=\int_{0}^{\infty}\Big{(}z_{1}^{\top}z_{1}-\gamma_{1}^{2}\overline{z}^{\top} \overline{\epsilon}\Big{)}dt<0 \tag{23}\]  
Under zero initial conditions, it follows that  
\[J=\int_{0}^{\infty}\Big{(}z_{1}^{\top}z_{1}-\gamma_{1}^{2} \overline{\epsilon}^{\top}\overline{\epsilon}+\dot{V}_{x}+\dot{V}_{e}\Big{)} dt-\int_{0}^{\infty}\Big{(}\dot{V}_{x}+\dot{V}_{e}\Big{)}dt\] \[=\int_{0}^{\infty}\Big{(}z_{1}^{\top}z_{1}-\gamma_{1}^{2} \overline{\epsilon}^{\top}\overline{\epsilon}+\dot{V}_{x}+\dot{V}_{e}\Big{)} dt-(V_{x}(\infty)+V_{e}(\infty))+(V_{x}(0)+V_{e}(0))\] \[\leq\int_{0}^{\infty}\Big{(}z_{1}^{\top}z_{1}-\gamma_{1}^{2} \overline{\epsilon}^{\top}\overline{\epsilon}+\dot{V}_{x}+\dot{V}_{e}\Big{)}dt\]A sufficient condition for the satisfaction of (23) is  
\[J_{1}=z_{1}^{\top}z_{1}-\gamma_{1}^{2}\overline{\epsilon}^{\top}\bar{\epsilon}+ \dot{V}_{x}+\dot{V}_{\epsilon}<0 \tag{24}\]  
Substituting (21) and (22) into (24) yields  
\[J_{1}=\left[ \begin{array}{c}x\\ e\\ \bar{\epsilon}\end{array} \right]^{\top}\left[ \begin{array}{ccc}J_{11}&PBK&PD_{ 1}\\ *&J_{22}&P_{1}\Xi\overline{D}\\ *&*&-\gamma_{1}^{2}I\end{array} \right]\left[ \begin{array}{c}x\\ e\\ \bar{\epsilon}\end{array} \right]<0 \tag{25}\]  
where \(J_{11}=He(P(\Theta A-BK_{x}))+\delta_{2}^{-1}P\Theta\Theta^{\top}P+\delta_{2} L_{f}^{2}I_{n}+(\epsilon_{1}+\epsilon_{2})N_{0}^{\top}N_{0}+\epsilon_{2}^{-1}PM_{0}M_{0}^{ \top}P\)  
\(+C_{x}^{\top}C_{x}\), \(J_{22}=He\big{(}P_{1}\big{(}\Xi\overline{A}-L_{1}\overline{C}\big{)}\big{)}+ \delta_{1}^{-1}P_{1}\Xi\Xi^{\top}P_{1}+\delta_{1}L_{f}^{2}A_{0}^{\top}A_{0}I_{ n+q}+\epsilon_{1}^{-1}P_{1}\Xi\overline{M}_{0}\overline{M}_{0}^{\top}\Xi^{\top}P_{1}\)  
\(+C_{\epsilon}^{\top}C_{\epsilon}\). Define \(Z=P^{-1}\). Pre-multiplying and post-multiplying both sides of (25) with \(\text{diag}(Z,\ I,\ I)\) gives  
\[\left[ \begin{array}{ccc}J_{11}&BK&D_{1}\\ *&J_{22}&P_{1}\Xi\overline{D}\\ *&*&-\gamma_{1}^{2}I\end{array} \right]<0 \tag{26}\]  
where \(J_{11}=He((\Theta A-BK_{x})Z)+\delta_{2}^{-1}\Theta\Theta^{\top}+\delta_{2}L _{f}^{2}ZZ+(\epsilon_{1}+\epsilon_{2})ZN_{0}^{\top}N_{0}Z+\epsilon_{2}^{-1}M_{ 0}M_{0}^{\top}+ZC_{x}^{\top}C_{x}ZJ_{22}=He\big{(}P_{1}\big{(}\Xi\overline{A} -L_{1}\overline{C}\big{)}\big{)}+\delta_{1}^{-1}P_{1}\Xi\Xi^{\top}P_{1}+ \delta_{1}L_{f}^{2}A_{0}^{\top}A_{0}I_{n+q}+\epsilon_{1}^{-1}P_{1}\Xi\overline {M}_{0}\overline{M}_{0}^{\top}\Xi^{\top}P_{1}++C_{\epsilon}^{\top}C_{\epsilon}\). It follows from the [PERSON]'s relation that for some positive scalar \(\epsilon_{3}\),  
\[\text{He}\left\{\left[ \begin{array}{c}BK_{x}\\ 0\\ 0\end{array} \right]\left[ \begin{array}{c}0\\ I\\ 0\end{array} \right]^{\top}\right\}\leq\epsilon_{3}\left[ \begin{array}{c}BK_{x }Z\\ 0\\ 0\end{array} \right]Z^{-1}\left[ \begin{array}{c}BK_{x}Z\\ 0\\ 0\end{array} \right]^{\top}+\epsilon_{3}^{-1}\left[ \begin{array}{c}0\\ I\\ 0\end{array} \right]Z^{-1}\left[ \begin{array}{c}0\\ I\\ 0\end{array} \right]^{\top}\]  
Further define  
\[P_{1}=\left[ \begin{array}{cc}Q_{n\times n}&0\\ 0&R_{q\times q}\end{array} \right],L_{1}=\left[ \begin{array}{c}L_{11}\\ L_{12}\end{array} \right],H=\left[ \begin{array}{c}H_{1}\\ H_{2}\end{array} \right]\]  
\[M_{1}=K_{x}Z,M_{2}=QH_{1},M_{3}=QL_{11},M_{4}=RH_{2},M_{5}=RL_{12}.\]  
Using the Schur complement repeatedly, (26) can be finally reformulated into (20). **Remark 7**.