## 5 Comparison with Previous Approaches
### [PERSON] et al. Example 3  
The last PN example (see Figure 8) is taken from [10], which presents the forbidden state transition problem (FSTP) problem using the classical policy of the theory of regions to control a real flexible manufacturing system. The PN model considers the liveness requirement and uncontrollable transitions. Based on the experiment results in Table 8, 215 markings are needed to solve seven MTSI conditions if one uses the \(G\) approach in this example. In addition, the reduced technology is involved in \(R_{1}\) and \(R_{3}\). For both these control methods, we need to handle 120 or 0 markings which prove the big reduction of PN supervisor computation time decreasing from 29,150 ms to 14,450 ms and  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Control** & **No. of PN** & \begin{tabular}{c} **No. of** \\ **Contrilers** \\ \end{tabular} & \begin{tabular}{c} **Maximally** \\ **the RG** \\ \end{tabular} & \begin{tabular}{c} **No. of** \\ **Permissive** \\ \end{tabular} & \
\begin{tabular}{c} **No. of** \\ **MTSI** \\ \end{tabular} &
\begin{tabular}{c} **Computation** \\ **Time (ms)** \\ \end{tabular} & **Approach** \\ \hline \(R_{1}\) & 3 & 20 & Yes & 10 & 13,245 & Minimal Cuts \\ \hline \(R_{2}\) & 4 & 20 & Yes & 10 & 13,400 & Canonic markings \\ \hline \(HP\) & 3 & 27 & Yes & 4 & 15,500 & RG based \\ \hline \(R_{3}\) & 3 & 0 & Yes & 4 & 6700 & Without RG \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison results of example 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Additional Control Places** & **Initial Marking \(M_{0}(P_{c})\)** & **Incidence Vector \(\textbf{CV}_{cip}\)-** \\ \hline \multirow{3}{*}{\(R_{1}\)} & \(P_{c1}\) & 0 & (\(-2\), 0, 0, 0, 0, 0) \\  & \(P_{c2}\) & 0 & (\(0\), \(-2\), 0, 0, 0, 0) \\  & \(P_{c3}\) & 0 & (\(0\), \(0\), \(-2\), 0, 0, 0) \\ \hline \multirow{3}{*}{\(R_{2}\)} & \(P_{c1}\) & 1 & (\(0\), 0, 0, 0, \(-2\), 0) \\  & \(P_{c2}\) & 1 & (\(0\), 0, 0, \(-2\), 0, 0) \\  & \(P_{c3}\) & 1 & (\(0\), 0, 0, \(-2\), -2, 0) \\  & \(P_{c4}\) & 1 & (\(0\), \(-2\), \(-2\), 0, 0, 0) \\ \hline \multirow{3}{*}{\(HP\)} & \(P_{c1}\) & 2 & (\(0\),\(-1\), 1, 0, 1, \(-1\)) \\  & \(P_{c2}\) & 3 & (\(-1\), 1, \(-1\), 0, 1, 0) \\  & \(P_{c3}\) & 2 & (\(-1\), 1, 0, 1, \(-1\), 0) \\ \hline \multirow{3}{*}{\(R_{3}\)} & \(P_{c1}\) & 3 & (\(0\),\(-2\), 2, 0, 2, \(-2\)) \\  & \(P_{c2}\) & 4 & (\(-2\), 2, \(-2\), 0, 2, 0) \\ \cline{1-1}  & \(P_{c3}\) & 3 & (\(-2\), 2, 0, 2, \(-2\), 0) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Petri net controllers of example 2.  
Figure 7: Huang and Pan example 2.  
9960 ms for \(R_{1}\) and \(R_{3}\), respectively. Moreover, one can remark in this example that for the same number of control places in \(G\) and \(R_{3}\) methods, one can control the net system and satisfy the control specifications without needing to generate the graph/markings and with less calculation time for the controllers. The controlled system net of example 3 was obtained with three methods and is shown in Table 9. To summarize, our control policy \(R_{3}\) is still the best computationally control algorithm among existing methods.