# Theory of Regions for Control Synthesis without Computing Reachability Graph
## 6 Conclusions and Future Works  
The proposed control synthesis method can be implemented for FMSs and generally for bounded PNs. The underlying notion of the previous work is that the TR must be solved by handling a number of markings which may increase according to the structural size of the system. For this purpose, one must generate the RG and all MTSI in the graph such that they have to pay for the high computation cost. However, PN properties and mathematical concepts are merged in our proposed method to reduce efficiently the computational burden of the TR.  
As mentioned above, three case studies are solved using the CPLEX software with a C program in order to compare our new method with the existing optimal policies; as input, one can give the PN model and the GMEC, then one can obtain all information concerning the controlled system in the program output (computing time, control places, incidence vector, etc.). In fact, our control method determines all equations of the linear system of the TR without generating the RG, thus one can avoid the combinatorial explosion of states if bigger models are analyzed. Additionally, PN controllers can be determined with a drastically reduced time. Consequently, this achievement will positively impact the performance of the production systems since the production constraints require immediate actions through rapid adjustments in order to maximize the production. Furthermore, any efficient supervision of any system has a high cost; if a constraint is not satisfied in a timely manner, the production or the security can be directly affected. There are numerous examples, such as the change over time with different constraints, the generation of a new flight plan to face with disruptions in air traffic networks, etc.  
Finally, this work provides scientific novelty to control bounded PNs (such as FMS, telecommunication systems, safety systems, air traffic management, rail traffic, etc.) since most previous studies on computing optimal supervisor depend on a complete state enumeration and mixed programming problem. In addition, since the RG-generation step is skipped, our program can handle any bounded PN model, even the biggest.  
Hence, in the near future, how to deal with deadlock prevention policy without generating RG is to become an important issue.  
The authors would like to thank the _Applied Sciences_ and the reviewers for possible evaluation of our manuscript entitled: Theory of Regions for Control Synthesis without Computing Reachability Graph. The authors would also like to thank the ICN Staff and the Lorraine University Senior Technicians for the use of the Flexible Manufacturing System on which the experiments and methods were implemented.  
The authors would like to thank everyone who helped complete this work with their continued efforts and support. [PERSON] and [PERSON] conceived and designed the methods and the experiments; [PERSON] developed the approaches and the algorithms; [PERSON] and [PERSON] analyzed and evaluated the data; [PERSON] supervised the results; and [PERSON] wrote the paper.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Approach** & **No. of Monitors** & **No. of Markings** & **No. of forbidden State Transitions \(\Omega\)** & **Computation Time (ms)** & **Method** \\ \hline \(R_{1}\) & 1 & 0 & 2 & 5930 & Without reachability graph \\ \(G_{2}\) & 1 & 27 & 2 & 12,120 & Reachability graph based \\ \(R_{3}\) & 1 & 6 & 2 & 6110 & Minimal cuts based \\ \hline \hline \end{tabular}
\end{table}
Table 15: Comparison results of case study 3.