## 5 Experimental Results
### Examples
Firstly, the properties of deadlock and legal markings in matrix form are present in Tables 4 and 5, respectively._  
Figure 5: The real maximally permissive reachability graph of Figure 4.  
Figure 6: Another classical and complex Petri net model of FMS. Please notice that above all leading transitions can be viewed as the first control transition of this example. Besides, anyone of them can recover all eight deadlock markings. Under the proposed algorithm, the first one of all leading transition is chosen. Accordingly, \(t_{107,A}=[3p_{8}-p_{9}-p_{10}-p_{11}+p_{17}+p_{18}+p_{19}]\) is chosen as the first control transition since it is the first one identified from CGAM. After the controller is added into this system eight deadlock markings are still left (i.e., \(16-8=8\)). Therefore, we must run second iteration for recovering the other deadlock markings. In second reachability graph, eight deadlock markings and 246 legal markings are identified. Therefore, the new GCAM's size is: \(8\times 246\times 19\). In the following, accordingly, 25 sets of all leading transitions are identified since they all can recover the maximal five same deadlock markings, which are listed as follows (refer to Table 7):  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Leading Transition \(t_{i,j}\)**} & \multicolumn{2}{c}{**The Number of**} & \\  & **Deadlock Markings** & **The Property of Leading Transition** \\ \hline \(t_{107,A}\) & 8 & \(3p_{8}-p_{9}-p_{10}-p_{11}+p_{17}+p_{18}+p_{19}\) \\ \(t_{107,10}\) & 8 & \(2p_{8}-p_{10}-p_{11}+p_{17}+p_{18}\) \\ \(t_{107,20}\) & 8 & \(2p_{8}-p_{9}-p_{11}+p_{18}+p_{19}\) \\ \(t_{107,66}\) & 8 & \(-p_{1}+p_{7}+3p_{8}-p_{9}-p_{10}-p_{11}+p_{17}+p_{18}\) \\ \(t_{107,124}\) & 8 & \(-p_{1}+p_{7}+2p_{8}-p_{9}-p_{11}+p_{18}\) \\ \(t_{162,16}\) & 8 & \(3p_{8}-p_{9}-p_{10}-Again, under the proposed algorithm, the first leading transition is chosen. Accordingly, \(t_{89,8}=[p_{1}-p_{4}+2p_{8}-p_{9}-p_{10}+p_{14}+p_{17}+p_{19}]\) is chosen as the second control transition since it is the first one identified from CGAM. After the controller is added into this system, three deadlock markings are still left (i.e., \(8-5=3\)). Therefore, we must run third iteration for recovering the other deadlock markings. In the third iteration, three deadlock markings and 276 legal markings are identified. Therefore, the GCAM's scale is: \(3\times 276\times 19\). Further, there are 81 sets of leading transitions identified and they all can recover final three deadlock markings. Due to the space limitation of this paper, the author just shows the first leading transition and it is \([2p_{1}-p_{5}-p_{6}+2p_{8}-p_{9}-p_{10}+p_{16}+p_{17}+p_{18}+p_{19}]\). Further, it is chosen as the third control transition. When the third control transition is added into the system with previous two controllers the system is deadlock free. Most importantly, the total reachable legal markings are 282. In other words, the three controllers recover the system. Figure 7 presents the controlled system of Figure 6 when the above three controllers is added into Example 2. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Leading Transition \(\mathbf{t}_{i,j}\)**} & \multicolumn{2}{c}{**The Number of**} & \\  & **Deadlock Markings** & **The Property of Leading Transition** \\ \hline \(t_{89,8}\) & 5 & \(p_{1}-p_{4}+2p_{8}-p_{9}-p_{10}+p_{14}+p_{17}+p_{19}\) \\ \(t_{89,17}\) & 5 & \(p_{1}-p_{4}+p_{8}-p_{10}+