# Theory of Regions for Control Synthesis without Computing Reachability Graph
## 4 Case Studies
### Case Study 2  
A flexible manufacturing system in the Lorraine University contains two assembly stations in which engraved glass pieces are assembled with stands. The physical system and the PN model are presented in Figure 6a,b. The place P\({}_{2}\) models our production machine, while the place P\({}_{3}\) represents a subcontractor production. The capacities of three pieces of the machine and the subcontractor are modeled by C\({}_{M}\) and C\({}_{S}\), respectively. In addition, the place P\({}_{1}\) represents our stock production. Moreover, the entry and exit events of the pallets in annex conveyors are modeled by transitions \((t_{1},t_{2})\) and \((t_{4},t_{5})\), respectively. Alternatively, the pallet continues its way in the main conveyor by \(t_{3}\). Note that \(t_{3},\ t_{4}\) and \(t_{5}\) are uncontrollable. The following scenario is employed: If P\({}_{3}\) and P\({}_{2}\) represent the subcontractor and the machine, the subcontractor production should be minimized to increase our profits. This control specification can be expressed by the following GMEC: M\((\text{P}_{2})+2\)M\((\text{P}_{3})\leq 3\). \(\bullet\) Basic cycle equations: C.\(\left[\text{x}_{1}\text{ x}_{2}\dots\text{x}_{n}\right]^{T}=0\)  
Figure 5: The controlled PN 1. Figure 6: Flexible manufacturing system: (**a**) experimental environment; and (**b**) PN model. \[\text{C}.\text{X}_{\text{t}}=\left[ \begin{array}{ccccc}-1&-1&0&1&1\\ 1&0&0&-1&0\\ 0&1&0&0&-1\\ -1&0&0&1&0\\ 0&-1&0&0&1\end{array} \right]\left[ \begin{array}{c}\text{x}_{\text{t}}(1_{ \text{t}})\\ \text{x}_{\text{t}}(2_{\text{t}})\\ \text{x}_{\text{t}}(3_{\text{t}})\\ \text{x}_{\text{t}}(4_{\text{t}})\\ \text{x}_{\text{t}}(5_{\text{t}})\end{array} \right]=\left[ \begin{array}{c}0 \\ 0\\ 0\\ 0\\ 0\end{array} \right]\]  
\[\left\{ \begin{array}{l}-\text{x}_{\text{t}}(\text{t}_{1})-\text{x}_{\text{t} }(\text{t}_{2})+\text{x}_{\text{t}}(\text{t}_{4})+\text{x}_{\text{t}}(\text{t }_{5})=0\\ \text{x}_{\text{t}}(\text{t}_{1})-\text{x}_{\text{t}}(\text{t}_{4})=0\\ \text{x}_{\text{t}}(\text{t}_{2})-\text{x}_{\text{t}}(\text{t}_{5})=0\\ -\text{x}_{\text{t}}(\text{t}_{1})+\text{x}_{\text{t}}(\text{t}_{4})=0\\ -\text{x}_{\text{t}}(\text{t}_{2})+\text{x}_{\text{t}}(\text{t}_{5})=0\end{array}  \right.\Rightarrow\quad\left(\text{X}_{\text{t}}^{1}\right)^{\text{T}}=\left[ \begin{array}{ccccc}0&0&1&0&0\\ 0&0&1&0\\ \text{X}_{\text{t}}^{3}\right)^{\text{T}}=\left[ \begin{array}{ccccc}0&1&0&0& 1\\ 0&1&0&0&1\end{array} \right]\]  
Consequently, the basic cycle equations can be calculated as follows:  
\[\text{C}(\text{P}_{\text{ci}},\text{t}_{3})=0\]  
\[\text{C}(\text{P}_{\text{ci}},\text{t}_{1})+\text{C}(\text{P}_{\text{ci}}, \text{t}_{4})=0\]  
\[\text{C}(\text{P}_{\text{ci}},\text{t}_{2})+\text{C}(\text{P}_{\text{ci}}, \text{t}_{5})=0\]  
\(\bullet\) Reachability conditions: The set \(\tau_{cycle}\) contains four transitions, i.e., \(\tau_{cycle}=\{\text{t}_{1},\text{ t}_{2},\text{t}_{4},\text{t}_{5}\}\)  
The PN is k-bounded with k \(>\) 1. Thus, one can calculate the firing sequences of transitions using combinations with repetition (i.e., see Proposition 2):  
\[|\text{S}|=\sum_{\text{i}=0}^{\text{card}(\tau_{cycle})}\Gamma_{\text{card}( \tau_{cycle})}^{\text{i}}=\sum_{\text{i}=0}^{4}\Gamma_{4}^{\text{i}}=\Gamma_{4} ^{0}+\Gamma_{4}^{1}+\Gamma_{4}^{2}+\Gamma_{4}^{3}+\Gamma_{4}^{4}=70\]  
Next, the set S is filtered in order to calculate the usable sequences listed in Table 5, and then one can determine the reachability conditions. \(\bullet\) MTSI conditions:  
The MTSI conditions are calculated using the forbidden sequences \(\sigma_{\text{MI}}\). The sets of forbidden markings and sequences are listed in Tables 6 and 7. MTSI conditions are:  
\[M_{0}(P_{ci})+2C(P_{ci},t_{2})<0\]  
\[M_{0}(P_{ci})+C(P_{ci},t_{1})+2C(P_{ci},t_{2})<0\]  
\[M_{0}(P_{ci})+C(P_{ci},t_{2})+2C(P_{ci},t_{1})<0\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sequence No. ** & \(\sigma_{\text{i}}\) **= (t\({}_{1}\)t\({}_{2}\)t\({}_{3}\)t\({}_{4}\)t\({}_{5}\))** & **Sequence No.