## 3 Results and Discussion  
To assess the performance and accuracy of the DTC-Hoo-SFC controller designed for precalciner temperature control, some tests based on simulations of real industrial operating scenarios of the control system were developed. The first test consisted of verifying the capacity of the DTC-Hoo-SFC controller for tracking variations in the command signal \(R(k)\). Figure 6 exhibits the time response of the control system with the DTC-Hoo-SFC controller designed, as well as the control effort against a change in the command signal from 920 to 950 \({}^{\circ}\)C.  
From Figure 6, it can be seen that the temperature in the precalciner reaches the new reference value in a time period of approximately 1245 s, without overshoot and with a steady state error \(e(k)=0\). These results show that the designed DTC-Hoo-SFC controller has sufficient capacity to effectively follow the variation in the command signal in a time period less than 1250 s, and, therefore, it is within the established limits of nominal operating temperature in the precalciner [6; 11]. Considering that, currently, the PI and the DTC+PI are the most used controllers for precalciner temperature control of cement rotary kilns [30; 31], the second test consisted of comparatively analysing the tracking capacity of the DTC-Hoo-SFC vs. DTC+PI controllers to variations in the command signal, as well as the rejection of the negative effect of external disturbances in environments close to the nominal operating conditions of the precalciner under study. Therefore, a discrete-time DTC with PI (DTC+PI) controller was designed. For this reason, the model obtained from the temperature in the precalciner (6) was used as the internal model of the DTC. The parameters obtained from the tuning of this PI controller were: \(K_{C}=0.00575\) and \(K_{I}=0.0008225\). Figure 7 exhibits the time response of the control system with DTC-Ho-SFC vs. DTC+PI controllers against a change in the command signal from 910 to 950 \({}^{\circ}\)C. From said figure it can be observed that with the DTC-Ho-SFC controller the temperature in the precalciner reaches the reference value in a time period of approximately 1400 s, while with the PI controller the new reference is reached with overshoot and in a time period of 4000 s, that is, in a time interval 2.85 greater. From Figure 7 it is also observed that in time \(t=4400\) s an external step-type disturbance \(w(k)\) occurs due to a decrease in the opening magnitude of the control valve that regulates the fuel flow to the burner of the rotary kiln (see Figure 1), causing a decrease in said fuel flow of 5 m\({}^{3}\)/h. As a result, the hot gases that reach the precalciner (from the rotary kiln) decrease in temperature and cause a variation in the precalciner temperature. The DTC-Ho-SFC controller completely rejects this negative effect in a time period of approximately 800 s, while the PI refuses it in a time period of approximately 2300 s, that is in a time interval 2.87 larger. These results show that the DTC-Ho-SFC controller designed has sufficient capacity to effectively reject the negative effects of external disturbances within the established limits of nominal operation of the precalciner temperature. Figure 6: Time response of the control system with the DTC-Ho-SFC controller designed, as well as the control effort against a change in the command signal from 920 to 950 \({}^{\circ}\)C.  
To assess the performance of the control system with the controllers designed, the performance indexes of the integral squared error (ISE) and the integral absolute error (IAE) were used, which are represented by the following expressions [67]:  
\[J_{ISE}=\sum_{k=0}^{N}e{(k)}^{2} \tag{23}\]  
\[J_{IAE}=\sum_{k=0}^{N}|e{(k)}| \tag{24}\]  
Tables 2 and 3 show the comparative results of the ISE and IAE performance indices defined in (23), (24) of the temperature control system in the precalciner with the DTC-Hoo-SFC vs. DTC+PI controllers, considering the effect of an external disturbance.