## 3 Design of a H\({}_{2}\)-SFC Controller  
The objective of this paper is to design a robust H\({}_{2}\)-SFC controller that generates a better behavior than a conventional PI/PID controller against the effect of complex dynamic behavior influenced by external disturbances. For this reason, this work raises the design of a robust H\({}_{2}\)-SFC controller. Figure 4 shows the block diagram of a control system with robust H\({}_{2}\) controller, conformed by the controller \(K\), the linear dynamic process \(P\), the output of the system \(y\), the control signal \(\mathbf{u}\), the output signal \(\mathbf{z}\), and the disturbance of input \(\mathbf{d}\) with which we have \(\left( \begin{array}{cc}z\\ y\end{array} \right)=\left( \begin{array}{cc}P_{11}&P_{12}\\ P_{21}&P_{22}\end{array} \right)\left( \begin{array}{c}d\\ u\end{array} \right)\) and \(u=Ky\). The realization of \(P\) is equal to \(\left[ \begin{array}{cc}A&B_{1}&B_{2}\\ C_{1}&D_{11}&D_{12}\\ C_{2}&D_{21}&D_{22}\end{array} \right]\). This controller is characterized by using a state space model to obtain the sequence of control signals \(u(k)\). The model of states spaces is found through Equation (6), thus we have:  
\[ \begin{array}{l}x(k+1)=Ax(k)+B_{1}d(k)+B_{2}u(k)\\ z_{2}(k)=C_{1}x(k)+D_{11}d(k)+D_{12}u(k)\\ y(k)=C_{2}x(k)+D_{21}d(k)+D_{22}u(k)\end{array}  \tag{7}\]  
Figure 4: Traditional block diagram of the robust H\({}_{2}\) controller. Figure 3: Validation results of the mathematical model obtained. \[A=\left[ \begin{array}{cccc}0&0&0&-0.4664\\ 0.5&0&0&+0.7370\\ 0&1&0&-1.2180\\ 0&0&2&+2.4280\end{array} \right]\]  
\[B_{1}=B_{2}=\left[ \begin{array}{cccc}-0.09152&-0.009942\\ +0.06893&0.01625\\ -0.003494&-0.0151\\ -0.06956&0.01072\end{array} \right]\]  
\[C_{1}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\end{array} \right]\]  
\[D_{11}=\left[ \begin{array}{cccc}0&0\\ 0&0\\ 0&0\\ 0&0\\ 0&0\\ 0&0\end{array} \right]\]  
\[D_{12}=\left[ \begin{array}{cccc}0&0&0\\ 0&0\\ 0&0\\ 1&0\\ 0&1\end{array} \right]\]  
\[C_{2}=\left[ \begin{array}{cccc}0&0&0&0.125\end{array} \right]\]  
\[D_{21}=\left[ \begin{array}{cccc}1&0\end{array} \right]\]  
\[D_{22}=\left[ \begin{array}{cccc}0&0&0\end{array} \right]\]  
\[u(k)=-Kx(k)+Fv(k) \tag{8}\]  
where:  
\[K\,\epsilon\,R^{2x1}\]  
--stabilizer controller  
\[K\,\epsilon\,R^{2x1}\]  
--tracking gain  
\[v\]  
--new system entry  
\[\left(v(k)=Kp\epsilon(k)+K_{l}\sum_{i=0}^{k}e(i)\right)\]  
The following assumptions were made:  
(i) \((A,B_{2})\) is stabilizable and \((C_{2},A)\) is detectable. (ii) \(D_{12}\) and \(D_{21}\) have full rank. (iii) \(\left[ \begin{array}{cc}A-jwl&B_{2}\\ C_{1}&D_{12}\end{array} \right]\) has full rank. (iv) \(\left[ \begin{array}{cc}A-jwl&B_{1}\\ C_{2}&D_{21}\end{array} \right]\) has full rank. Assumption (i) guarantees the stabilizability by the output feedback. Assumption (ii) guarantees that the optimal H\({}_{2}\) controller is linear and invariant in time and dimensional finite (nonsingular case). Assumptions (iii) and (iv) guarantee the solvability of algebraic Riccati equations. For \(D_{22}=0\) and \(D_{22}=0\). The assumption \(D_{22}=0\) is made to simplify algebra. With the assumption \(D_{11}\ =\ 0,P_{11}\) becomes strictly proper. The problem H\({}_{2}\) is to find a stabilizer controller K that minimizes the \(\|H\|_{2}\) norm of the states and the action of the controller given the realization of the state space [42; 43; 44]. The tracking gain \(F\) allows the system output to reach the \(R\) reference by solving the nonzero reference tracking problem of the system being the error \(e=R-y\). Here, it should be fulfilled:  
\[F=\left[C_{2}(I-(A-B_{2}K))^{-1}B_{2}\right]^{-1} \tag{9}\]  
After inserting Equation (8) into the dynamic system, Equation (7) turns out:  
\[x(k+1)=(A-B_{2}K)x(k)+B_{2}Fv(k)+B_{1}d(k) \tag{10}\]  
\[z_{2}(k)=(C_{1}-D_{12}K)x(k)+D_{12}Fv(k)+D_{11}d(k)\]  
We determine the matrices \(K\) and \(F\) as follows:  
\[K=\left[ \begin{array}{ccc}K_{11}&K_{12}&K_{13}&K_{14}\\ K_{21}&K_{22}&K_{23}&K_{24}\end{array}