# Hierarchical Coordinated Predictive Control of Multiagent Systems for Process Industries
## 4 Case Study: Multisource High-Power Microwave Heating Process
### Settings of the Algorithms  
We designed eight different microwave heating temperature-control scenarios. The proposed HCPC strategy is used in scenarios 1-7 but with different communication topologies. Scenario 8 is controlled by conventional centralized model predictive control (i.e., the CMPC mentioned previously) [35]. The purpose of setting up these eight scenarios is to demonstrate that the proposed HCPC can effectively reduce energy consumption while providing satisfactory control performance by comparing the differences in control performance and energy consumption between the proposed HCPC in this study and the conventional CMPC. In scenarios 1-7, the microwave heating device described in Section 4.1 is modeled as a microwave heating MAS, where each microwave source and its component parts are individual agent. The number of each agent is the same as the corresponding microwave source. The inputs are the IEP of each microwave source, and the outputs are the temperatures of the five measurement points. The structure of the information flow shown in Figure 4 is adopted for all seven scenarios, and their different communication topologies are shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Microwave Source** & **Energy-Conversion** & **Single Point Steady-State** & \
\begin{tabular}{c} **Average Steady-State** \\ **Temperature Rise** \\ **(\({}^{\circ}\)C/W)** \\ \end{tabular} &
\begin{tabular}{c} **Normalized Cost** \\ **(per W)** \\ \end{tabular} \\ \hline
1 & 60\% & [0.256, 0.262, 0.257, 0.263, 0.269] & 0.261 & \(-\)0.307 \\
2 & 70\% & [0.024, 0.025, 0.024, 0.025, 0.026] & 0.025 & 0.876 \\
3 & 64\% & [0.196, 0.200, 0.199, 0.202, 0.208] & 0.201 & \(-\)0.005 \\
4 & 61\% & [0.060, 0.060, 0.059, 0.061, 0.063] & 0.061 & 0.697 \\
5 & 55\% & [0.135, 0.139, 0.137, 0.140, 0.142] & 0.139 & 0.307 \\
6 & 65\% & [0.027, 0.028, 0.028, 0.028, 0.029] & 0.028 & 0.860 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Energy-conversion efficiency, heating capacity and normalized cost of each microwave source agent. The control horizon is \(N_{\rm C}=30\), the prediction horizon is \(N_{\rm P}=30\), the time step size is \(\Delta k=120\) s, the temperature target is 600 \({}^{\circ}\)C and the entire control process lasts 100 time steps. The constraints used for numerical calculations are shown in Table 3. The weights, the number of iterations for consistent open-loop prediction and the steps of prediction error filtering are set as follows:  
\[ \begin{split}&\mathbf{Q}^{\rm PC1}=\text{diag}\big{\{}\mathbf{Q}_{1}^{\rm PC1 }\quad\mathbf{Q}_{2}^{\rm PC1}\quad\cdots\quad\mathbf{Q}_{5}^{\rm PC1}\big{\}};\mathbf{Q}_{ n}^{\rm PC1}=\text{diag}\big{\{}30\quad 30\quad\cdots\quad 30\big{\}};\\ &\mathbf{Q}^{\rm DPC1}=\text{diag}\big{\{}\mathbf{Q}_{1}^{\rm DPC1}\quad \mathbf{Q}_{2}^{\rm DPC1}\quad\cdots\quad\mathbf{Q}_{5}^{\rm DPC1}\big{\}};\mathbf{Q}_{n}^ {\rm DPC1}=\text{diag}\big{\{}30\quad 30\quad\cdots\quad 30\big{\}};\\ &\mathbf{Q}_{i}^{\rm PC2}=\text{diag}\big{\{}20\quad 20\quad\cdots\quad 20 \big{\}};\mathbf{Q}_{i}^{\rm DPC2}=\text{diag}\big{\{}20\quad 20\quad\cdots\quad 20\big{\}};\\ &\mathbf{Q}_{i}^{\rm PC3}=\text{diag}\big{\{}10\quad 10\quad\cdots\quad 10 \big{\}};\mathbf{Q}_{i}^{\rm DPC3}=\text{diag}\big{\{}10\quad 10\quad\cdots\quad 10\big{\}};\\ & q_{n}=1,n=1,2,\cdots,N_{\rm O};q_{i}=1,n=1,2,\cdots,N_{\rm A} ;N_{\rm iter}=20;N_{\rm L}=2\end{split} \]  
In scenario 8, which was used for comparison with scenarios 1-7 where the HCPC strategy was used, the centralized model predictive control strategy (CMPC) mentioned in [35] is used to obtain the optimal control input increments at each time step \(k\) by solving the optimization problem (31). Figure 4: Structure of information flow. Figure 5: Communication topologies used in: (**a**) Scenario 1. (**b**) Scenario 2. (**c**) Scenario 3. (**d**) Scenario 4. (**e**) Scenario 5. (**f**) Scenario 6. (**g**) Scenario 7.