## 2 Problem Description and Model Analysis
### Bounded Analysis  
In order to achieve dynamic stability and robustness for uncertainty, a class of systems with norm bounded uncertainties were introduced into the H\(\infty\) control theory to ensure that the closed-loop system was asymptotically stable at the specified H\(\infty\) disturbance attenuation level [20]. But it is difficult to achieve a stable and uncertain optimal cost. So, for the non-zero term uncertainty, we assume that there is a positive scalar constant and a certain value limit. Therefore, we define the following:  
\[S=\big{\{}A^{*}(t),\ \ B_{1}}^{*}(t),\ \ B_{2}}^{*}(t),\ \ C(t)\big{\}} \tag{10}\]  
Then, if \(\hat{\Theta}(t)\in S\), we have that \(\hat{\Theta}(t)=\hat{\Theta}+\Delta\hat{  
_where: \(E\), \(F_{a}\), \(F_{b}\) are units of a known real constant, and \(M\) is an uncertain perturbation matrix that can be measured._  
Therefore, the system model changes to the following:  
\[\dot{x}(t)=(A+BK+EM(F_{a}+F_{b}K))x(t) \tag{12}\]  
The cost functions related to uncertain systems are as follows:  
\[J=\int_{0}^{\infty}\Bigl{(}x^{\mathrm{T}}(t)Qx(t)+u^{\mathrm{T}}(t)Ru(t) \Bigr{)}dt \tag{13}\]  
\(Q=Q^{\mathrm{T}}\) is a positive semi-definite symmetric matrix, \(R=R^{\mathrm{T}}\) is a positive definite symmetric matrix and is controllable for (A, B). **Lemma 1** (2016). : _For any \(\epsilon>0\), given the matrices \(Y,C\) and \(D\) with appropriate dimensions, and \(Y\) is a symmetric matrix, then \(Y+CDM+DTMCT<0\). For any \(M\), the following equation is satisfied._  
\[symm(x^{\mathrm{T}}PYMCx)\leq ex^{\mathrm{T}}PYY^{\mathrm{T}}Px+\epsilon^{-1 }x^{\mathrm{T}}C^{\mathrm{T}}Cx\quad\forall x\in\mathbb{R}^{n} \tag{14}\]  
**Lemma 2** ([PERSON] and [PERSON], 2017). : _Considering the system uncertainty (without disturbance), given an admissible set \(D\) and cost function (13), it is assumed that there are some positive \(\epsilon_{1},\epsilon_{2}\). There is an invariant set if and only if \(\epsilon>0\), \(\epsilon\subset\mathbb{R}^{n}\), \(\epsilon\supset\mathcal{D}\), a symmetric positive definite matrix \(P\) and a matrix \(K\). We then set the following formula:_  
\[ \begin{split}& x^{\mathrm{T}}\{Q+K^{\mathrm{T}}RK+symm(P[A-BK]+[A-BK]^{ \mathrm{T}}P)\}x+\epsilon_{1}x^{\mathrm{T}}PDD^{\mathrm{T}}Px+\\ &\epsilon_{1}{}^{-1}x^{\mathrm{T}}(E_{1}-E_{2}K)^{\mathrm{T}}(E_ {1}-E_{2}K)x+\epsilon_{2}x^{\mathrm{T}}P[I_{n}\otimes(x^{\mathrm{T}}D)][I_{n} \otimes(D^{\mathrm{T}}x)]Px<0\end{split}  \tag{15}\]  
**Definition 1**. : _If the system has no other interference, the designed state-feedback controller is: \(u(t)=Kx(t)\) to keep the uncertain system stable. Given the cost function (14), an admissible set \(\mathcal{D}\) and a positive definite cost matrix \(P>0\), the following satisfied:_  
\[x^{\mathrm{T}}\Bigl{[}Q+K^{\mathrm{T}}RK\Bigr{]}+2x^{\mathrm{T}}[A+BK+EM(F_{a }+F_{b}K)]x<0 \tag{16}\]  
For all admissible uncertainties \(\Delta\hat{\Theta}(t)\) in system (9), if the following conditions are satisfied, the feedback controller is a quadratic guaranteed cost controller with a related cost matrix. The relationship between the above quadratic stability definition and quadratic stabilizability definition is extended to ensure performance. **Theorem 1**. : _Considering the system stability under the assumption, the control law is a quadratic guaranteed cost control with cost matrix \(P>0\). Then the closed-loop uncertain system (12) is quadratic stable at any time under constraints \(M^{\mathrm{T}}(t)M(t)\leq 1\). In addition, for all allowable uncertainties \(\Delta\hat{\Theta}(t)\), the corresponding value of the cost function (13) satisfies the following bound:_  
\[J\leq x_{0}^{\mathrm{T}}Px_{0} \tag{17}\]  
Proof. : The uncertain system performance index (13) satisfies the boundary (17) in the initial state, and it is proved by Lemmas 1 and 2.