## 2 Materials and Methods
### Design of a DTC-H\(\infty\)-SFC Controller
\begin{array}{cccc}1.783&-0.0356&-1.279&0.5489\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ \end{array} \right],\;B_{1}=B_{2}=\left[ \begin{array}{c}8\\ 0\\ 0\\ 0\\ \end{array} \right],\;C_{1}=\left[ \begin{array}{cccc}0&0&0&1\\ \end{array} \right],\]  
\[C_{2}=\left[ \begin{array}{cccc}2.489&-5.768&4.138&-0.8552\\ \end{array} \right],\;D_{11}=\left[0\right],\;D_{12}=\left[1\right],\;D_{21}= \left[1\right],\;D_{22}=\left[0\right],\]  
\[nk=10T=600\;\text{s}\]  
Figure 5 shows the diagram of the precalciner temperature control system with the proposed DTC-H\(\infty\)-SFC controller, where \(Y_{m1}(k)\) is the model output without time delay and \(Y_{m2}(k)\) is the delayed model output. It is well-known that a null steady state error is achieved both in the tracking of a step command and in the rejection of a step disturbance applied to the process input if an integral action is added to an SFC in the direct path [57; 66]. Therefore, to guarantee a null steady state error of the error signal \(e(k)\), a PI controller was added, which provides said integral action and has two gains whose tuning allows improving the performance of the control system. Consequently, the discrete-time PI controller is represented by the expression [67]:  
\[v(k)=K_{\text{C}}e(k)+K_{I}e(k)+P_{I}(k-1) \tag{13}\]  
where  
\[P_{I}(k)=K_{I}\sum_{j=0}^{k}e(j)=K_{I}e(k)+P_{I}(k-1) \tag{14}\]  
\(K_{\text{C}}\) is the proportional gain, \(K_{I}=\frac{K_{\text{C}}T_{\text{s}}}{T_{I}}\) is the integral gain, \(T_{I}\) is the integral time constant. The state feedback control law is described by the expression [49]:  
\[u(k)=-Kx(k)+Fv(k) \tag{15}\]  
where \(K\in\Re^{1\times 4}\) is the stabilizing controller, and \(F\in\Re^{1\times 1}\) is the tracking gain. The next suppositions were considered:  
1. \((A,B_{1})\) is stabilizable and \((C_{1},A)\) is detectable;
2. \((A,B_{2})\) is stabilizable and \((C_{2},A)\) is detectable. Supposition (i) guarantees that internal stability of \(P\) is equivalent to BIBO (bounded-input, bounded-output) stability from \(w\) to \(z\). Supposition (ii) is necessary and hopeful that \(P\) can be internally stabilized by output feedback. The H\(\infty\) problem is to meet a stabilizing controller \(K\) that minimizes the H\(\infty\)-norm of the transfer function from \(w\) to \(z\)[50; 57]. The tracking gain \(F\) permits the system output \(y\) to achieve the reference \(R\) by settling the non-zero reference tracking problem of the system being the error \(e=R-y\). Here, it should be complied:  
\[F=\left[C_{2}(I-(A-B_{2}K))^{-1}B_{2}\right]^{-1} \tag{16}\]  
After inserting (15) into the discrete-time state space process model without time delay (12) it is obtained:  
\[ \begin{split} x(k+1)&{=(A-B_{2}K)x(k)+B_{2}Fv(k) +B_{1}w(k)}\\ z(k)&{=(C_{1}-D_{12}K)x(k)+D_{12}Fv(k)+D_{11}w(k)} \end{split}  \tag{17}\]  
Figure 5: Diagram of the designed precalciner temperature control system with the designed DTC-Ho-SFC controller. Thus, the controller \(K\) and the tracking gain \(F\) are determined as:  
\[K=\left[ \begin{array}{cccc}K_{11}&K_{12}&K_{13}&K_{14}\end{array} \right],\;F=[f_{ 11}] \tag{18}\]  
The transfer matrix from \(w(k)\) to \(z(k)\) is obtained by means of:  
\[G_{w,z}(z)=(C_{1}-D_{12}K)(zI-(A-B_{2}K))^{-1}B_{1}+D_{11} \tag{19}\]  
Considering the process dominant time delay (\(nk=10T_{s}=600\) s) the use of a dead-time compensator is proposed to compensate said delay in the feedback [32; 66]. Therefore, the Hoo control problem studied in this work is specified as: to meet a controller \(K\) such that \((A-B_{2}K)\) is Hurwitz and \(\left\|G_{w,z}\right\|_{\infty}<\gamma\) for the minimum value of \(\gamma>0\) that meets the following linear matrix inequality [50; 57]:  
\[\left[ \begin{array}{cccc}(A-B_{2}K)X+X(A-B_{2}K)^{T}&B_{1}&X(C_{1}-D_{12}K)^ {T}\\ B_{1}^{T}&-I&D_{11}^{T}\\ (C_{1}-D_{12}K)X&D_{11}&-\gamma^{2}I\end{array} \right]<0 \tag{20}\]  
\[X>0 \tag{21}\]  
where, \(X\) is a symmetric matrix and \(\gamma\) is a given positive number. From (20) it was obtained:  
\[K=\left[