## 5 Results  
Results of the proposed PG-DMPC are presented in this section. The objective of the PG-DMPC algorithm is to provide consensus among local decisions from an FL-DMPC algorithm to achieve control objectives. The proposal PG-DMPC is compared with fuzzy  
Figure 5: Gradient estimation variance approach by exploration policy based on categorical distribution. DMPC (FL) [30], DMPC using a cooperative game [31], and the centralized MPC. FL and Coop. The game was also implemented in the upper layer. Furthermore, the influence of the baseline method is evaluated. Three validation cases (Table 2) were employed to demonstrate the proper performance of the proposal under the different initial, reference state vectors and disturbance states. The sampling time used in simulations is \(T_{s}=5\) s.  
**Case 1:** The reference vector is the same as the one used for training, and the state vector of the operating point was considered as the initial state vector, \(\widetilde{x}_{\mathcal{N}}=\widetilde{x}_{\mathcal{N}}^{0}\). **Case 2:** The reference vector and the initial state vector are the same as those used in Case 1, but for this case, the disturbances \(\widetilde{d}_{\mathcal{N}}=[0.02;0.02;0.02]\) were included. **Case 3:** The reference vector changes at \(t>200\) from the operation point as initial reference vector to \(\widetilde{T}_{\mathcal{N}}^{2}\) with \(h_{\mathcal{T}_{1}}=h_{1}^{0}+0.04\), \(h_{\mathcal{T}_{2}}=h_{2}^{0}+0.03\), \(h_{\mathcal{T}_{3}}=h_{5}^{0}+0.04\), \(h_{\mathcal{T}_{4}}=h_{6}^{0}+0.03\). The initial state vector was taken from \([h_{n}^{0}-0.03,h_{n}^{0}+0.03]\). Note that the initial state vector is taken out of the interval for initial state vectors during training. Tables 3-5 show the performance of the proposal and the techniques considered in Cases 1, 2, and 3, respectively. In the same order as the validation cases, Figures 6-8 portray the evolution of the state vectors until the reference vector is reached by Conf. 1 and Conf. 2. Tables 3-5 show that the centralized control has the best results because it has the availability of full plant information for prediction. For validation Case 1, Coop game offers better results in Pe, ISE, and e. Both configurations of the PG-DMPC show an advantage in \(P_{e}\) over the FL, while _ISE_ and \(e\) display very close values to FL. (Table 3). In validation Case 2, Coop. game shows better \(P_{e}\) than PG-DMPC and FL techniques. In this case, the worst \(P_{e}\) are given by PG-DMPC. On the other hand, these three techniques \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** \\ \hline Control problem & regulation & regulation & tracking \\ Disturbance & none & from \(t>245\) to \(t<305\) & none \\ Reference & steady state & steady state & step from one to another steady state \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Validation cases. Figure 6: Validation Case 1: Evolution of the levels of the controlled variables: (**a**) Conf. 1 performance; (**b**) Conf. 2 performance. show \(ISE\) and \(e\) are very close to each other. Although the measure disturbance was added, an acceptable response was raised by PG-DMPC (Table 4). Finally, in Case 3, the Coop game and FL show better \(P_{e}\) values. At the same time, the worst of \(ISE\) and \(e\) are given by FL. The two PG-DMPC configurations display similarly in the four indices. The PG-DMPC agent has an acceptable response and is comparable to the other techniques even though it was not trained for reference tracking (Table 5). Figure 8: Validation Case 3: Evolution of the levels of the controlled variables involving steps: (a) Conf. 1 performance; (b) Conf. 2 performance.