## 3 Design of SIRAC
### Design of RAC  
\(H_{\infty}\) control is applied to process the wind disturbance. The system equations belonging to \(P(s)\) in Figure 4 are given, as follows:  
\[ \begin{array}{l}\dot{x}(t)=Ax(t)+B_{1}(t)w(t)+B_{2}u(t)\\ z(t)=C_{1}x(t)+D_{11}w(t)+D_{12}u(t)\\ y(t)=C_{2}x(t)+D_{21}w(t)\end{array}  \tag{35}\]  
where \(u(t)\) is the control signal, \(w(t)\) is the exogenous input, \(z(t)\) is the exogenous output, and \(y(t)\) is the sensed output. Additionally, the RAC \(K(s)\) can be represented, as follows:  
\[\left[ \begin{array}{c}\dot{x}_{k}(t)\\ u(t)\end{array} \right]=\left[ \begin{array}{cc}A_{k}&B_{k}\\ C_{k}&D_{k}\end{array} \right]\left[ \begin{array}{c}x_{K}(t)\\ y(t)\end{array} \right] \tag{36}\]  
\[K(s)=\left[ \begin{array}{cc}A_{k}&B_{k}\\ C_{k}&D_{k}\end{array} \right] \tag{37}\]  
Substitute \(u(t)\) in Equation (36) into Equation (35) and \(y(t)\) in Equation (35) into Equation (36), the closed-loop system can be written as:  
Figure 4: Generalized P-K form. Figure 3: Flowchart of the SI algorithm. \[\left[ \begin{array}{c}\dot{x}(t)\\ \dot{x}_{k}(t)\\ z(t)\end{array} \right]=\left[ \begin{array}{ccc}A+B_{2}D_{k}C_{2}&B_{2}C_{k}&B_{ 1}+B_{2}D_{k}D_{21}\\ B_{k}C_{2}&A_{k}&B_{k}D_{21}\\ C_{1}+D_{12}D_{k}C_{2}&D_{12}C_{k}&D_{11}+D_{12}D_{k}D_{21}\end{array} \right] \left[ \begin{array}{c}x(t)\\ x_{k}(t)\\ w(t)\end{array} \right] \tag{38}\]  
Let  
\[ \begin{array}{c}A_{cl}=\left[\begin{array}{cc}A+B_{2}D_{k}C_{2}&B_{2}C_{k} \\ B_{k}C_{2}&A_{k}\end{array}\right],B_{cl}=\left[\begin{array}{c}B_{1}+B_{2}D _{k}D_{21}\\ B_{k}D_{21}\end{array}\right]\\ C_{cl}=\left[\begin{array}{cc}C_{1}+D_{12}D_{k}C_{2}&D_{12}C_{k}\end{array} \right],D_{cl}=\left[D_{11}+D_{12}D_{k}D_{21}\right]\end{array}  \tag{39}\]  
and  
\[\underline{S}(P,K)=\left[ \begin{array}{cc}A_{cl}&B_{cl}\\ C_{cl}&D_{cl}\end{array} \right] \tag{40}\]  
The desired performance criterion to be minimized can be taken as a \(H_{\infty}\) norm of the closed loop transfer function. Hence, the cost function takes the following form [30]  
\[J_{\infty}(K)=\left\|\underline{S}(P,K)\right\|_{\infty} \tag{41}\]  
where \(K(s)\) should make the closed-loop system stable and satisfy the following \(H_{\infty}\) norm constraints  
\[J_{\infty}(K)<\gamma,\gamma>0 \tag{42}\]  
Linear Matrix Inequalities (LMIs) are used to solve the \(H_{\infty}\) optimal control problem for a minimum \(\gamma\) without losing convexity in order to find the controller \(K(s)\). Next, we give three steps to find the controller \(K(s)\)[31]. **Step 1:** Find symmetric matrices \(R\) and \(S\). \[\left[ \begin{array}{cc}N_{R}&0\\ 0&I\end{array} \right]^{T}\left[ \begin{array}{cc}AR+RA^{T}&*&*\\ C_{1}R&-\gamma I&*\\ B_{1}^{T}&D_{11}^{T}&-\gamma I\end{array} \right]\left[ \begin{array}{cc}N_{ R}&0\\ 0&I\end{array} \right]<0 \tag{43}\]  
\[\left[ \begin{array}{cc}N_{S}&0\\ 0&I\end{array} \right]^{T}\left[ \begin{array}{cc}A^{T}S+SA&*&*\\ B_{1}^{T}S&-\gamma I&*\\ C_{1}&D_{11}&-\gamma I\end{array} \right]\left[ \begin{array}{cc}N_{S}&0\\ 0&I\end{array} \right]<0 \tag{44}\]  
\[\left[ \begin{array}{cc}R&I\\ I&S\end{array} \right]\geq 0 \tag{45}\]  
where \(N_{R}\) is the basis of the null space of \(\left[ \begin{array}{cc}B_{2}&D_{12}\end{array} \right]^{T}\), and \(N_{S}\) is the basis of the null space of \(\left[ \begin{array}{cc}C_{2}&D_{21}\end{array} \right]\). **Step 2:** Find symmetric positive definite matrix \(X\). When the \(R\) and \(S\) are found, we solve \(MN^{T}=I-RS\), and let  
\[XF_{1}=F_{2} \tag{46}\]  
where  
\[F_{1}=\left[ \begin{array}{cc}R&I\\ M^{T}&0\end{array} \right],F_{2}=\left[ \begin{array}{cc}I&S\\ 0&N^{T}\end{array} \right] \tag{47}\]  
**Step 3:** Find the controller \(K(s)\). When the \(X\) is found, according to the bounded real lemma, if \(\underline{S}(P,K)\) is stable and \(J_{\infty}(K)<\gamma\), then \(X\) should satisfy the LMI  
\[\left[ \begin{array}{ccc}A_{cl}^{T}X+XA_{cl}&*&*\\ B_{cl}^{T}X&-\gamma I&*\\ C_{cl}&D_{cl}&-\gamma I\end{array}