## 4 Robust Controller Design Based on LMIs  
### \(\mathcal{D}\)-Stability  
The chosen \(\mathcal{D}\)-stability region is given by the intersection of a semi-plane sector (\(\rho\)) to guarantee a lower limit on settling time. Disc sector (\(\sigma\)) is used as an upper limit on settling time. It restricts the magnitude of the eigenvalues indirectly and imposes a limitation on the control effort. LMI regions [28] are defined as regions of the complex plane that can be described in terms of characteristic functions as follows:  
\[\mathcal{D}=\left\{s\in\mathbb{C}:f_{\mathcal{D}}(s)<0\right\},f_{\mathcal{D} _{\sigma,\rho}\beta}(s)=L+sM+sM^{\mathrm{T}} \tag{28}\]  
Lemma 1 below presents a condition that ensures the quadratic \(\mathcal{D}\)-stability of the closed loop system (26) in a given LMI region. **Lemma 1**. : _Quadratic stability of \(\mathcal{D}\)-Region [28]. Consider the LMI region with a characteristic function (28). Additionally, consider that \(M=M_{1}^{\mathrm{T}}M_{2}\), where both \(M_{1}\) and \(M_{2}\) are matrices with \(k\) lines and rank per complete line. The system (26) presents quadratic \(\mathcal{D}\)-stability with region (28) if there are real symmetric matrices \(\mathcal{P}>0\) and \(\mathcal{U}>0\) such that:_  
\[\left[ \begin{array}{ccc}L\otimes\mathcal{P}+He\left\{M\otimes\left( \mathcal{P}\left(\tilde{A}+B_{u}K\right)\right)\right\}&M_{1}^{\mathrm{T}} \otimes\left(\mathcal{P}B_{\Delta}\right)&\left(M_{2}^{\mathrm{T}}\mathcal{U }\right)\otimes C_{\Delta}^{\mathrm{T}}\\ M_{1}\otimes\left(B_{\Delta}^{\mathrm{T}}\mathcal{P}\right)&-\mathcal{U} \otimes I&0\\ \left(\mathcal{U}M_{2}\right)\otimes C_{\Delta}&0&-\mathcal{U}\otimes I\end{array}  \right]<0 \tag{29}\]  
When the LMI region of interest is formed by the intersection of other LMI regions, quadratic \(\mathcal{D}\)-stability is guaranteed when the same matrix \(\mathrm{P}\) meets (26) simultaneously with the characteristic functions of each region [29]. In particular, the region of our work can be described as \(\mathcal{D}=\mathcal{D}_{\sigma}\cap\mathcal{D}_{\rho}\), where  
\[\mathcal{D}_{\sigma}=\left\{\delta_{i}\in\mathbb{C}:\Re\left( \delta_{i}\right)<-\sigma,\sigma\geqslant 0,\sigma\in\mathbb{R}\right\},L_{ \sigma}=2\sigma;M_{\sigma}=1 \tag{30}\] \[\mathcal{D}_{\rho}=\left\{\delta_{i}\in\mathbb{C}:\left|\delta_{i }+q\right|<\rho,\rho>0,\rho\in\mathbb{R}\right\},L_{\rho}=\left[ \begin{array} []{cc}-\rho&q\\ q&-\rho\end{array} \right],M_{\rho}=\left[ \begin{array}{cc}0&1\\ 0&0\end{array} \right]\]  
The following theorem details the conditions in the form of LMIs for the positioning of the eigenvalues of \(\tilde{A}+B_{\Delta}\Delta(t)C_{\Delta}+\tilde{B}_{u}K\) in \(\mathcal{D}=\mathcal{D}_{\sigma}\cap\mathcal{D}_{\rho}\)  
**Remark**. : _One can notice that the quadratic \(\mathcal{D}\)-stability is wider than the \(\mathcal{H}_{\infty}\) condition. Indeed, the quadratic \(\mathcal{D}\)-stability allows to guarantee the stability of the uncertain system as well as the \(\mathcal{H}_{\infty}\) performance. We refer the interested reader to [28] and the references therein._  
**Theorem 1**. : _When real scalars \(\sigma\) and \(\rho\) are known, the closed-loop system \(\hat{\tilde{x}}(t)=\left(\tilde{A}\left(Y_{load}(t)\right)+\tilde{B}_{u}K\right) \tilde{x}(t)\) is asymptotically stable for all \(\left(Y_{load}\right)_{\min}\leq Y_{load}(t)\leq\left(Y_{load}\right)_{\max},\) and the poles are located in the region \(S(\sigma,\rho)\), if there exists a symmetric positive definite matrices \(Q\in R^{\left(n+2h\right)\times\left(n+2h\right)},\)\(W\in R^{1\times\left(n+2h\right)}\) such that the following LMI are feasible:_  
\[