# Hierarchical Coordinated Predictive Control of Multiagent Systems for Process Industries
## 5 Numerical Results and Analysis  
In this section, the temperature-control performances, power consumption and corresponding economic performances for the eight microwave heating scenarios are given and analyzed qualitatively and quantitatively. The IEPs of the six microwave sources and the corresponding temperature evolution of the eight scenarios are shown in Figures 6-13, while the variations in the total IEP of each of the eight scenarios is shown in Figure 14. The metrics of integral of absolute error (IAE), integral of time-weighted absolute error (ITAE), integral of squared error (ISE) and integral of time-weighted squared error (ITSE), which are used to evaluate the temperature-control performances for the eight scenarios, are given quantitatively in Table 4. The four metrics are defined as  
\[\text{IAE}=\int|d_{\text{y},n}(k)|\text{d}k \tag{32a}\] \[\text{ITAE}=\int k|d_{\text{y},n}(k)|\text{d}k\] (32b) \[\text{ISE}=\int(d_{\text{y},n}(k))^{2}\text{d}k\] (32c) \[\text{ITSE}=\int k(d_{\text{y},n}(k))^{2}\text{d}k \tag{32d}\]  
where \(d_{\text{y},n}(k)\) is the deviation of the \(n\)-th system output from the target at time step \(k\), i.e., the deviation of the temperature of the \(n\)-th temperature measurement point from the temperature target in our study. The total power consumption in kWh for the eight scenarios is also shown in Table 4. To further demonstrate the economic performance, by substituting the unit power normalized cost shown in Table 2 and the microwave source \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Constraint** & **Symbol** & **Value** & **Unit** \\ \hline Upper bound of the IEP & \(u_{\text{max}}\) & [1000, 1000, 1000, 1000, 1000] & W \\ Lower bound of the IEP & \(u_{\text{min}}\) & [0, 0, 0, 0,0,0] & W \\ Upper bound of the IEP increments & \(\Delta u_{\text{max}}\) & [100, 100, 100, 100, 100, 100] & W \\ Lower bound of the IEP increments & \(\Delta u_{\text{min}}\) & [\(-100\), \(-100\), \(-100\), \(-100\), \(-100\), \(-100\)] & W \\ Upper bound of the temperature & \(\mathbf{y}_{\text{max}}\) & [\(700,700,700,700\)] & \({}^{\circ}\)C \\ Lower bound of the temperature & \(\mathbf{y}_{\text{min}}\) & [\(20,20,20,20\)] & \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Constraint of each variable. IEP values for the time step \(k=100\) for all scenarios into Equation (33), the terminal normalized cost (TNC) for each scenario as shown in the last column of Table 4 can be obtained. Notably, the smaller value of the terminal normalized cost indicates the better economic performance of the scenario. \[\text{TNC}=-0.307u_{1}(k)+0.876u_{2}(k)-0.005u_{3}(k)+0.697u_{4}(k)+0.307u_{5}(k )+0.860u_{6}(k),k=100 \tag{33}\]  
As shown in each subfigure (b) of Figures 6-13, the material temperatures in all eight scenarios are able to enter the steady-state range (\(\pm 5\%\)) of the target at a time step of approximately \(k=18\) and subsequently remain steady. Table 4 shows that for scenario 8 (CMPC), the IAEs corresponding to scenarios 1-7 are 101.08%, 100.01%, 98.60%, 98.93%, 98.85%, 99.06% and 99.72%, respectively, of those of scenario 8. The ITAEs corresponding to scenarios 1-7 are 113.86%, 110.36%, 97.14%, 98.11%, 99.26%, 103.14% and 107.15%, respectively, of those in scenario 8. The ISEs corresponding to scenarios 1-7 are 100.07%, 99.97%, 99.94%, 99.96%, 99.95%, 99.94% and 99.97%, respectively, of those in scenario 8. The ITSEs corresponding to scenarios 1-7 are 100.29%, 99.87%, 99.62%, 99.71%, 99.68%, 99.71% and 99.83%, respectively, of those of scenario 8. From Figures 6-14 and Table 4, although the control performance of the HCPC strategy in scenario 1 is not as good as that of CMPC in all four performance metrics, it provides a 27.16% reduction in power consumption with a 13.86% loss in control performance even in terms of the worst metric of ITAE and it provides the lowest energy consumption among the eight scenarios (12.6512 kWh, from Table 4 and Figure 14).