## 2 Materials and Methods
### Design of a DTC-H\(\infty\)-SFC Controller  
The H\(\infty\) controller design is a current and active research area [49; 63; 64]. From its beginnings to the present there has been made substantial growth in this field [50; 51; 54]. It has been extensively proposed to solve diverse complex industrial and academic control problems [55; 56; 57]. In this paper Ho-based robust controller is proposed to solve the problem of operative temperature control in a precalciner. The control system is characterized agreed to configuration showed in Figure 4, where \(P\) is the generalized plant or interconnected system, \(w\) represents all the external inputs, that includes reference signal and disturbances, \(z\) denoted the output signals to be to minimized (consisting of all error signals), \(y\) is the vector of measurements available to the controller \(K\), and \(u\) is the vector of control signals [57; 59]. The measured variable \(y\) is used in \(K\) to calculate the control signals \(u\) (\(u=Ky\)). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Model** \\ **Structure** \\ \end{tabular} } & \multicolumn{5}{c}{**Model Order**} & \multicolumn{1}{c}{**Performance**} \\ \cline{2-7}  & **na** & **nb** & **nc** & **nd** & **nf** & **nk** & **Index(FIT)** \\ \hline ARMAX & 4 & 4 & 4 & - & - & 10 & 90.04\% \\ \hline BJ & - & 4 & 4 & 4 & 4 & 9 & 86.72\% \\ \hline ARX & 4 & 4 & - & - & - & 10 & 84.25\% \\ \hline OE & - & 2 & - & - & 2 & 11 & 85.45\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Validation results. The controller design trouble is to meet a stabilizing controller \(K\) that produces a signal \(u\) regarding the information from \(y\) to decrease the effect of \(w\) on \(z\), minimizing the closed loop norm from \(w\) to \(z\)[50]. It is correspondent to minimizing the Hoo-norm of the transfer function from \(w\) to \(z\)[51]. The generalized plant \(P\) is partitioned as:  
\[P=\left[ \begin{array}{cc}P_{11}&P_{12}\\ P_{21}&P_{22}\end{array} \right] \tag{7}\]  
Therefore, the control system can be represented as:  
\[\left[ \begin{array}{c}z\\ y\end{array} \right]=P\left[ \begin{array}{c}w\\ u\end{array} \right]=\left[ \begin{array}{cc}P_{11}&P_{12}\\ P_{21}&P_{22}\end{array} \right]\left[ \begin{array}{c}w\\ u\end{array} \right] \tag{8}\]  
Figure 4: Standard diagram of the robust Hoo configuration. Figure 3: Validation results. From (8) it is obtained:  
\[z=\Big{[}P_{11}+P_{12}K(I-P_{22}K)^{-1}P_{21}\Big{]}w \tag{9}\] \[z:=F_{I}(P,K)\]  
where \(F_{I}(P,K)\) is the lower linear fractional transformation of \(P\) and \(K\)[50]. Thus, the controller design objective becomes [51]:  
\[\min_{K\text{ }stabilizing}\lVert F_{I}(P,K)\rVert_{\infty} \tag{10}\]  
where the expression (10) is denoted as the H\(\infty\) optimization problem [55]. However, in industrial controller design it is enough to meet a stabilizing controller \(K\), such that the H\(\infty\)-norm of the closed-loop transfer function is minor than a given positive number \(\gamma\)[50; 57]:  
\[\lVert F_{I}(P,K)\rVert_{\infty}<\gamma \tag{11}\]  
where \(\gamma>\gamma_{0}:=\min_{K\text{ }stabilizing}\lVert F_{I}(P,K)\rVert_{\infty}\)  
The H\(\infty\) controller commonly uses a state-space model to obtain the control signals \(u(k)\)[65]. Therefore, from the derived model in z-domain of the precalciner temperature (6), the following process model was obtained in discrete-time states-space with time delay:  
\[ \begin{split} x(k+1)&=Ax(k)+B_{1}w(k)+B_{2}u(k- nk)\\ z(k)&=C_{1}x(k)+D_{11}w(k)+D_{12}u(k-nk)\\ y(k)&=C_{2}x(k)+D_{21}w(k)+D_{22}u(k-nk)\end{split}  \tag{12}\]  
where \(x(k)\in\Re^{n}\) is the state vector, \(w(k)\in\Re^{p}\) is the disturbance input, \(u(k)\in\Re^{m}\) is the control input, \(y(k)\in\Re^{r}\) is the measurement, \(z(k)\in\Re^{q}\) is the controlled output, \(nk\) is the discrete-time delay, \(A\), \(B_{1}\), \(B_{2}\), \(C_{1}\), \(C_{2}\), \(D_{11}\), \(D_{12}\), \(D_{21}\), \(D_{22}\) are matrices appropriately dimensioned,  
\[A=\left[