## 3 Controller Design
\begin{bmatrix}\Omega_{1}&\Omega_{3}{}^{\mathrm{T}}\\ \Omega_{3}&-\Omega_{2}\end{bmatrix} <0\]  
**Lemma 5  
Let \(\overline{P}=\mathrm{P}^{-1}\), combined with Lemma 3, we have the following:  
\[ \begin{bmatrix}-\overline{P}&\overline{P}G^{\mathrm{T}}&\overline{P}H_{1}{}^{ \mathrm{T}}\\ G\overline{P}&-\overline{P}&0\\ H_{1}\overline{P}&0&-\overline{P}\end{bmatrix} <0\]  
through Equation (12), the above equation becomes:  
\[ \begin{bmatrix}-\overline{P}&\overline{P}(A+\Delta A)^{\mathrm{T}}&\overline{ P}(B+\Delta B)^{\mathrm{T}}\\ (A+\Delta A)\overline{P}&-\overline{P}&0\\ (B+\Delta B)\overline{P}&0&-\overline{P}\end{bmatrix} <0\]  
take,  
\[Z= \begin{bmatrix}-\overline{P}&\overline{P}A^{\mathrm{T}}&\overline{P}B^{ \mathrm{T}}\\ A\overline{P}&-\overline{P}&0\\ B\overline{P}&0&-\overline{P}\end{bmatrix} \]  
then,  
\[Z+ \begin{bmatrix}0&\overline{P}\Delta A^{\mathrm{T}}&\overline{P}\Delta B^{ \mathrm{T}}\\ \Delta A\overline{P}&0&0\\ \Delta B\overline{P}&0&0\end{bmatrix} \]  
\[=Z+ \begin{bmatrix}0&\overline{P}(EMF_{a})^{\mathrm{T}}&\overline{P}(EMF_{b}) ^{\mathrm{T}}\\ (EMF_{a})\overline{P}&0&0\\ (EMF_{b})\overline{P}&0&0\end{bmatrix} \]  
\[=Z+\begin{bmatrix}EM& \begin{bmatrix}0&0&0\\ F_{a}\overline{P}&0&0\\ F_{b}\overline{P}&0&0\end{bmatrix} + \begin{bmatrix}0&\overline{P}F_{a}{}^{ \mathrm{T}}&\overline{P}F_{b}{}^{\mathrm{T}}\\ 0&0&0\\ 0&0&0\end{bmatrix}\begin{bmatrix}(EM)^{\mathrm{T}}&\\ (EM)^{\mathrm{T}}&\\ &(EM)^{\mathrm{T}}\end{bmatrix} <0 \tag{24}\]  
by Lemma 1, the above inequality applies to the following inequality,  
\[Z+\varpi \begin{bmatrix}E&\\ &E&\\ &&E\end{bmatrix}\begin{bmatrix}E^{\mathrm{T}}&\\ &&E^{\mathrm{T}}\end{bmatrix} +\varpi^{-1} \begin{bmatrix}0&\overline{P}F_{a}{}^ {\mathrm{T}}&\overline{P}F_{b}{}^{\mathrm{T}}\\ 0&0&0\\ 0&0&0\end{bmatrix}\begin{bmatrix}0&0&0\\ F_{a}\overline{P}&0&0\\ F_{b}\overline{P}&0&0\end{bmatrix} \]  
\[= \begin{bmatrix}\Omega&\overline{P}A^{\mathrm{T}}&\overline{P}B^{\mathrm{T}}\\ A\overline{P}&-\overline{P}+\varpi EE^{\mathrm{T}}&0\\ B\overline{P}&0&-\overline{P}+\varpi EE^{\mathrm{T}}\end{bmatrix} <0\]  
were, \(\Omega=-\overline{P}+\varpi EE^{\mathrm{T}}+\varpi^{-1}\overline{P}F_{a}{}^{ \mathrm{T}}F_{a}\overline{P}+\overline{P}F_{b}{}^{\mathrm{T}}F_{b}\overline{P}\). Let \(\overline{P}=\varpi X\), that is  
\[ \begin{bmatrix}\wedge&XA^{\mathrm{T}}&XB^{\mathrm{T}}\\ AX&-X+EE^{\mathrm{T}}&0\\ BX&0&-X+EE^{\mathrm{T}}\end{bmatrix} <0\]  
where,  
\[\wedge=-X+EE^{\mathrm{T}}+XF_{a}{}^{\mathrm{T}}F_{a}{}^{\mathrm{T}}+XF_{b}{}^{ \mathrm{T}}F_{b}{}^{\mathrm{T}}F_{b}{}^{\mathrm{X}}\]  
Using Theorem 2, replace A with \(A-BK\) and so on, respectively, and a new LMI similar to (20) appears. The proof of this theorem is completed. The proof is complete. \(\square\)  
**Remark 3**. : _The quadratic stabilizability of uncertain systems can be stabilized in the same system, but the opposite is not true._Similarly, given constant \(\epsilon_{1}>0\), if and only if there is an appropriate scalar function or value, and there are positive definite symmetric matrix \(\text{W}^{-1}\) and matrix \(\text{V}^{-1}\), then the guaranteed cost controller is stable to the system within a certain limit.