## 3 Controller Design  
For uncertain systems, the Riccati equation method for constructing the optimal quadratic guaranteed cost controller under the Lyapunov function is given. Symmetric matrix \(P\) is a stable Riccati differential equation solution of \(A^{\rm T}P+PA+PMP+Q=0\). **Lemma 3**. : _For any \(\epsilon>0\) and any \(M\) satisfying Equation (16), then combined with Lemma 1, there are \(x\in\mathbb{R}^{n}\), all of which have the following:_  
\[0\leq\epsilon x^{\rm T}PYY^{\rm T}Px+\epsilon^{-1}x^{\rm T}C^{\rm T}Cx-symm \Big{(}x^{\rm T}PYMCx\Big{)} \tag{20}\]  
**Theorem 2**. : _Given \(Q\in\mathbb{R}^{n\times n}\) and \(R\in\mathbb{R}^{n\times n}\), the symmetric matrix satisfies the Riccati equation and has a constant \(\epsilon>0\), in combination with references [5; 6], the Riccati equation is as follows:_  
\[\Big{(}A+B\big{(}\epsilon R+F_{b}{}^{\rm T}F_{b}\big{)}^{-1}F_{b}{}^{\rm T}F_{ a}\Big{)}^{\rm T}P+P\Big{(}A+B\big{(}\epsilon R+F_{b}{}^{\rm T}F_{b}\big{)}^{-1}F_{b}{}^{ \rm T}F_{a}\Big{)}+\epsilon PEE^{\rm T}P- \tag{21}\] \[\epsilon PB\big{(}\epsilon R+F_{b}{}^{\rm T}F_{b}\big{)}^{-1}B^{ \rm T}P+\frac{1}{\epsilon}F_{a}{}^{\rm T}\Big{(}I-F_{b}\big{(}\epsilon R+F_{b}{ }^{\rm T}F_{b}\big{)}^{-1}F_{b}{}^{\rm T}\Big{)}F_{a}+Q=0\]  
_With positive definite symmetric solution \(P\), the uncertain system (10) is quadratic stable. Then, the appropriate stable control rate of the state feedback system is given as:_\(K=-\left(\epsilon R+F_{b}{}^{\mathrm{T}}F_{b}\right)^{-1}\left(\epsilon B^{ \mathrm{T}}P-F_{b}{}^{\mathrm{T}}F_{a}\right)=-W^{-1}V\), which meets the assumed design criteria and makes the system control stable._  
**Proof. ** Combined with Lemma 1, for any \(\epsilon>0\) and any \(\mathrm{m}\) satisfy Equation (16), there are \(\mathrm{x}\in\mathbb{R}^{n}\), all of which have the following:  
\[0\leq\epsilon\mathrm{x}^{\mathrm{T}}PY\mathrm{Y}^{\mathrm{T}}P\mathrm{x}+ \epsilon^{-1}\mathrm{x}^{\mathrm{T}}C^{\mathrm{T}}C\mathrm{x}-symm\left( \mathrm{x}^{\mathrm{T}}PYMCx\right)\]  
For the Riccati equation, define \(\overline{A}=A-BK\), \(\widetilde{R}_{2}=\epsilon R+F_{b}{}^{\mathrm{T}}F_{b}\), \(\overline{E}=F_{a}+F_{b}K\), \(\overline{Q}=F_{a}+K^{\mathrm{T}}RK\), \(\tilde{P}=\epsilon P\) and when \(\widetilde{P}=(\epsilon/\epsilon_{1})P=(1/\epsilon_{1})\tilde{P}>P\), any constant \(\epsilon_{1}<\frac{1}{n\sigma[P(t)]}\in(0,\epsilon)\). According to the above expression, we have the following equation:  
\[\overline{A}^{\mathrm{T}}P+P\overline{A}+\epsilon PEE^{\mathrm{T}}P+ \epsilon PB\widetilde{R}_{2}{}^{-1}B^{\mathrm{T}}P+\frac{1}{\epsilon}F_{a}{} ^{\mathrm{T}}\left(I-F_{b}\widetilde{R}_{2}{}^{-1}F_{b}{}^{\mathrm{T}}\right) F_{a}+Q\] \[\qquad=\overline{A}^{\mathrm{T}}P+P\overline{A}+\epsilon PEE^{ \mathrm{T}}P+\epsilon PB\widetilde{R}_{2}{}^{-1}B^{\mathrm{T}}P+\frac{1}{ \epsilon}\overline{E}^{-1}\overline{E}+\overline{Q}\] \[\qquad=\overline{A}^{\mathrm{T}}\tilde{P}+\tilde{P}\overline{A}+ \tilde{P}EE^{\mathrm{T}}\tilde{P}+PB\widetilde{R}_{2}{}^{-1}B^{\mathrm{T}}P+ \overline{E}^{-1}\overline{E}+\epsilon\overline{Q}\] \[\qquad\qquad<\overline{A}^{\mathrm{T}}\tilde{P}+\tilde{P}\overline {A}+\tilde{P}EE^{\mathrm{T}}\tilde{P}+\overline{E}^{-1}\overline{E}+ \epsilon_{1}\overline{Q}\] \[\qquad\qquad<\overline{A}^{\mathrm{T}}\tilde{P}+\tilde{P} \overline{A}+\tilde{P}EE^{\mathrm{T}}\tilde{P}+\overline{E}^{-1}\overline{E}+ \epsilon_{1}\overline{Q}\] \[\qquad\qquad<\overline{A}^{\mathrm{T}}\tilde{P}+\tilde{P} \overline{A}+\epsilon_{1}\widetilde{P}EE^{\mathrm{T}}\tilde{P}+\frac{1}{ \epsilon_{1}}\overline{E}^{-1}\overline{E}+\overline{Q}<0\]  
There is a constant \(\epsilon_{1}\) such that the controller makes the system stable. The proof is completed. \(\square\)  
**Lemma 4** ([PERSON]'s lemma) ([PERSON], 1988). : _Given the appropriate dimensions, we have matrices \(\Omega_{1}\), \(\Omega_{2}\) and \(\Omega_{3}\) also \(\Omega_{1}=\Omega_{1}{}^{\mathrm{T}}\). There are inequalities \(\Omega_{1}+\Omega_{3}{}^{\mathrm{T}}\Omega_{2}{}^{\mathrm{T}}\Omega_{3}<0\) that are equivalent to_  
\[