## 3 LPV Control Design
\begin{array}{c|cc}A_{We}&B_{We}\\ \hline C_{We}&D_{We}\end{array} \right]\left[ \begin{array}{c}x_{We}\\ e\end{array} \right] \tag{21}\]  
The state-space realization for LPV controller synthesis is given as:  
\[\left[ \begin{array}{c}\dot{x}_{p}\\ \dot{x}_{We}\\ z_{1}\\ z_{2}\end{array} \right]=\left[ \begin{array}{cccc}A_{p}(\rho)&0&0&B_{w}(\rho )V_{d}&B_{p}(\rho)V_{u}\\ -B_{We}C_{p}(\rho)&A_{We}&B_{We}&0&0\\ \hline-D_{We}C_{p}(\rho)&C_{We}&D_{We}&0&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{c}x_{p}\\ x_{We}\\ w_{2}\\ \dot{u}\end{array} \right] \tag{22}\]  
The scaled control input \(\hat{u}\) is given by  
\[\hat{u}=V_{u}^{-1}u \tag{23}\]  
The LPV controller is designed using the following well-known lemmas, which describe the \(\mathcal{H}_{\infty}\)-norm and regional pole placement in the LMI region [32,33]. **Lemma 1**. : \(\mathcal{H}_{\infty}\)_-norm_  
_The \(\mathcal{H}_{\infty}\)-norm of closed-loop LPV system from exogenous inputs \(w\) to exogenous outputs \(z\) will be lower than a positive scalar \(\gamma\), if and only if there exists a positive-definite symmetric matrix \(X(\rho)\), and a matrix \(Y(\rho)\) for all the admissible values of time-varying parameter vector \(\rho\) satisfying  
Figure 6: The mixed sensitivity control problem. \[ \begin{cases}\begin{aligned} & min&\gamma\\ \text{s.t.}& X(\rho)>0\\ &\begin{bmatrix}\mathcal{A}(\rho)X(\rho)+X(\rho)\mathcal{A}(\rho)^{T}+\mathcal{ B}_{2}(\rho)Y(\rho)+Y(\rho)^{T}\mathcal{B}_{2}(\rho)^{T}&\mathcal{B}_{1}(\rho)&X(\rho)G_{z}( \rho)^{T}+Y(\rho)^{T}\mathcal{D}_{12}(\rho)^{T}\\ \mathcal{B}_{1}(\rho)^{T}&-\gamma I&\mathcal{D}_{11}(\rho)^{T}\\ \mathcal{C}_{z}(\rho)X(\rho)+\mathcal{D}_{12}(\rho)Y(\rho)&\mathcal{D}_{11}( \rho)&-\gamma I\end{bmatrix}<0\end{aligned}\end{cases}  \tag{24}\]  
**Lemma 2**. : _Regional pole constraints_  
_The closed-loop LPV system's eigenvalues will be located in the desirable LMI region \(D_{(\alpha,\beta,\varphi)}\), if and only if there exist \(X(\rho)\), and \(Y(\rho)\), such that_  
\[ \begin{cases}X(\rho)>0\\ 2\alpha X(\rho)+\mathcal{A}(\rho)X(\rho)+\mathcal{B}_{2}(\rho)Y(\rho)+X(\rho )\mathcal{A}(\rho)^{T}+Y(\rho)^{T}\mathcal{B}_{2}(\rho)^{T}<0\\ 2\beta X(\rho)+\mathcal{A}(\rho)X(\rho)+\mathcal{B}_{2}(\rho)Y(\rho)+X(\rho )\mathcal{A}(\rho)^{T}+Y(\rho)^{T}\mathcal{B}_{2}(\rho)^{T}>0\end{cases}  \tag{25}\]  
_and_  
\[ \begin{bmatrix}\begin{bmatrix}(X(\rho)\mathcal{A}(\rho)^{T}+\mathcal{A}(\rho) X(\rho)+Y(\rho)^{T}\mathcal{B}_{2}(\rho)^{T}+\mathcal{B}_{2}(\rho)Y(\rho) \end{bmatrix}sin\varphi&\begin{pmatrix}X(\rho)\mathcal{A}(\rho)^{T}-\mathcal{ A}(\rho)X(\rho)-\mathcal{B}_{2}(\rho)Y(\rho)+Y(\rho)^{T}\mathcal{B}_{2}(\rho)^{T} \end{pmatrix}cos\varphi\\ \begin{pmatrix}\mathcal{A}(\rho)X(\rho)-X(\rho)\mathcal{A}(\rho)^{T}+\mathcal{ B}_{2}(\rho)Y(\rho)-Y(\rho)^{T}\mathcal{B}_{2}(\rho)^{T}\end{pmatrix}cos\varphi& \begin{pmatrix}A(\rho)X(\rho)+X(\rho)\mathcal{A}(\rho)^{T}+\mathcal{B}_{2}(\rho )Y(\rho)+Y(\rho)^{T}\mathcal{B}_{2}(\rho)^{T}\end{pmatrix}sin\varphi\end{bmatrix} <0 \tag{26}\]  
_where the LMI region \(D_{(\alpha,\beta,\varphi)}=D_{(\alpha,\beta)}\cap D_{(\varphi)}\). The system will be \(D\)-stable if both conditions (25) and (26) are satisfied. The sub-LMI regions \(D_{(\alpha,\beta)}\) and \(D_{(\varphi)}\) are shown in Figures 7 and 8._The LPV controller gains \(K(\rho)\) can be determined as follows:  
\[K(\rho)=Y(\rho)X(\rho)^{-1} \tag{27}\]  
Equations (24)-(26) impose an unlimited number of LMIs that are difficult to solve. The total number of LMIs can be squeezed to a finite set using a polytopic LPV system. Using convexity, (24)-(26) will hold if it holds at the vertices of the polytope.