# Theory of Regions for Control Synthesis without Computing Reachability Graph
## 4 Case Studies
### Case Study 1  
This first case study addresses a PN model with two machines sharing two resources \(\mathrm{R}_{1}\) and \(\mathrm{R}_{2}\) shown in Figure 4. The places \(\mathrm{P}_{1}\) and \(\mathrm{P}_{3}\) materialize two operations. To deal with the organizational and synchronization problem between two processes in parallel which is a current problem, we have to satisfy the following control specification: \(\mathrm{M}(\mathrm{P}_{1})+\;\mathrm{M}(\mathrm{P}_{3})\leq 1\). \(\bullet\) Basic cycle equations:  
The resolution of the equation: \(\mathrm{C}.\left[\mathrm{x}_{1}\;\mathrm{x}_{2}\ldots\mathrm{x}_{\mathrm{n}} \right]^{\mathrm{T}}=0\) gives the basic cycle Equation (10). \[\mathrm{C}.\mathrm{X}_{\mathrm{t}}=\left[ \begin{array}{cccccc}1&-1&0&0&0&0\\ 0&1&-1&0&0&0\\ 0&0&0&1&-1&0\\ 0&0&0&0&1&-1\\ 0&-1&1&-1&1&0\\ -1&1&0&0&-1&1\end{array} \right]\left[ \begin{array}{c}\mathrm{x}_{\mathrm{t}} (\mathrm{t}_{1})\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{2})\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{3})\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{4})\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{4})\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{5})\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{6})\end{array} \right]=\left[ \begin{array} []{c}0\\ 0\\ 0\\ 0\\ 0\end{array} \right]\]  
\[\left\{ \begin{array}{l}\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{1})-\mathrm{x}_{ \mathrm{t}}(\mathrm{t}_{2})=0\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{2})-\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{3} )=0\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{4})-\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{5} )=0\\ \mathrm{x}_{\mathrm{t}}(\mathrm{t}_{5})-\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{6})=0 \\ -\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{2})+\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{3} )-\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{4})+\mathrm{x}_{\mathrm{t}}(\mathrm{t}_ {5})=0\\ -\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{1})+\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{2} )-\mathrm{x}_{\mathrm{t}}(\mathrm{t}_{5})+\mathrm{x}_{\mathrm{t}}(\mathrm{t}_ {6})=0\\ \end{array} \right.\Rightarrow\quad\left(\mathrm{X}_{\mathrm{t}}^{1}\right)^{ \mathrm{T}}=\left[000111\right]\]  
The corresponding sequences of the basic cycle equations are: \(\sigma_{1}=\left(\mathrm{t}_{4}\mathrm{t}_{5}\mathrm{t}_{6}\right)\) and \(\sigma_{2}=\left(\mathrm{t}_{1}\mathrm{t}_{2}\mathrm{t}_{3}\right)\). Thus, the basic cycle Equation (10) can be listed as follows:  
\[\mathrm{C}(\mathrm{P}_{\mathrm{ci}},\mathrm{t}_{4})+\mathrm{C}(\mathrm{P}_{ \mathrm{ci}},\mathrm{t}_{5})+\mathrm{C}(\mathrm{P}_{\mathrm{ci}},\mathrm{t}_ {6})=0\]  
\[\mathrm{C}(\mathrm{P}_{\mathrm{ci}},\mathrm{t}_{1})+\mathrm{C}(\mathrm{P}_{ \mathrm{ci}},\mathrm{t}_{2})+\mathrm{C}(\mathrm{P}_{\mathrm{ci}},\mathrm{t}_ {3})=0\]  
\(\bullet\) Reachability conditions:  
According to the Definition 6, the set \(\tau_{\mathrm{cycle}}\) contains six transitions, i.e., \(\mathrm{card}\left(\tau_{\mathrm{cycle}}\right)=6\):  
\[\tau_{\mathrm{cycle}}=\left\{\mathrm{t}_{1},\;\mathrm{t}_{2},\mathrm{t}_{3}, \mathrm{t}_{4},\mathrm{t}_{5},\mathrm{t}_{6}\right\}\]  
The PN is safe (k = 1). Then, conforming to the Proposition 1 the set S of firing sequences of transitions is generated through combinations without repetition. \[\left|\mathrm{S}\right|=\sum_{i=0}^{\mathrm{card}\left(\tau_{\mathrm{cycle}} \right)}C_{\mathrm{card}\left(\tau_{\mathrm{cycle}}\right)}^{\mathrm{i}}=\sum_{ i=0}^{6}C_{6}^{\mathrm{i}}=C_{6}^{0}+C_{6}^{1}+C_{6}^{2}+C_{6}^{3}+C_{6}^{4}+C_{6}^{5}+C_{6}^{6}=64\]  
Figure 4: PN model with shared resources. Afterwards, the sequences S will be filtered conforming to the criteria mentioned in Section 3.4 in order to obtain the usable sequences listed in Table 1.  
Finally, by identifying the usable sequences \(\text{card}(\sigma_{\text{U}})=\)8, one can obtain the reachability conditions (11).