# Theory of Regions for Control Synthesis without Computing Reachability Graph
## 5 Comparison with Previous Methods  
This section presents three tables for comparing our new method with previous works. For convenience, the policy of this work is called \(\mathbf{R}_{1}\), our prior work [23] is called \(\mathbf{R}_{3}\) and the related research of [PERSON] et al. [8] is called \(\mathbf{G}_{2}\).  
Table 13 lists the comparison results of the first case study with three approaches of supervisory control. Since the RG is generated and contains eight states, one can know that \(\mathbf{G}_{2}\) uses eight markings to design the PN monitor. Then, \(\mathbf{R}_{3}\) improves \(\mathbf{G}_{2}\) by introducing minimal cuts in RG and uses three markings to synthesize the PN controller. However, both of these efficient methods must pay for time consuming. Here, our proposed approach (i.e., \(\mathbf{R}_{1}\)) that does not generate RG reduced the technology and the theory of regions to obtain the same number of PN monitors. It enhances the computation efficiency of \(\mathbf{R}_{3}\). Based on the experimental results in Table 13, 3018 ms are needed if one uses \(\mathbf{R}_{1}\) in this example against 6015 ms and 3226 ms for \(\mathbf{G}_{2}\) and \(\mathbf{R}_{3}\), respectively. As a result, we can infer that \(\mathbf{R}_{1}\) is the most efficient among \(\mathbf{G}_{2}\) and \(\mathbf{R}_{3}\).  
Example 2 is a k-bounded PN model. Based on Table 14, 13 states and five forbidden state transitions are needed to be processed if \(\mathbf{G}_{2}\) or \(\mathbf{R}_{3}\) is used, respectively. Besides, \(\mathbf{R}_{3}\) still needs 7190 ms to design two PN monitors even though the computation time is reduced compared with \(\mathbf{G}_{2}\) which needs 10,593 ms to calculate the same number of control places. Nonetheless, 0 markings and three forbidden state transitions are needed in our policy. Furthermore, \(\mathbf{R}_{1}\) only takes 6216 ms to control the system, which shows that \(\mathbf{R}_{1}\) is faster than \(\mathbf{G}_{2}\) and \(\mathbf{R}_{3}\). This time difference can be significantly increased if bigger models are analyzed.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Approach** & \begin{tabular}{c} **No. of** \\ **Monitors** \\ \end{tabular} & \begin{tabular}{c} **No. of** \\ **Markings** \\ \end{tabular} & \begin{tabular}{c} **No. of** \\ **State Transitions \(\Omega\)** \\ \end{tabular} & \
\begin{tabular}{c} **Computation** \\ **Time (ms)** \\ \end{tabular} &
\begin{tabular}{c} **Method** \\ \end{tabular} \\ \hline \(R_{1}\) & 1 & 0 & 1 & 3018 & Without reachability graph \\ \(G_{2}\) & 1 & 8 & 1 & 6015 & Reachability graph based \\ \(R_{3}\) & 1 & 3 & 1 & 3226 & Minimal cuts based \\ \hline \hline \end{tabular}
\end{table}
Table 13: Comparison results of case study 1.  
Figure 9: The controlled PN 3.  
Based on the comparison results (i.e., Table 15), 27 markings and 12,120 ms are needed to obtain one PN monitor in \(\mathbf{G}_{2}\). However, under the minimal cuts policy (i.e., \(\mathbf{R}_{3}\)), only six markings are handled to calculate the supervisor. It does not seem to be efficient enough since it requires 6110 ms to supervise the quality control station of the FMS. Nevertheless, for the same number of forbidden state transitions and monitors, only 5930 ms are required in our new method that is able to control the FMS without computing any marking. To summarize, \(\mathbf{R}_{1}\) is the best computationally improved optimal control algorithm among existing literatures that use the theory of regions.