# Theory of Regions for Control Synthesis without Computing Reachability Graph
## 3 Design of PN Controller  
### Problem Setting  
This work is focused on the minimization of the computational cost of the theory of regions by reducing the computation time for the synthesis of PN controllers. For this purpose, a new method based on PN tools and mathematical concepts, is proposed to design a PN supervisor without calculating the reachability graph.  
**Definition 1**.: (Combinations without repetition) _A \(k\)-combination without repetition of a set \(A\) with \(n\) elements is an arbitrary subset of \(A\) having \(k\) elements. If we take \(k\) objects among \(n\) without repetition and regardless of the appearance order, one can represent the \(k\) objects by a part of \(k\)-elements from a set with \(n\) elements. To determine the number of these provisions, one can determine the number of arrangements of \(k\) objects, and divide them by the number of provisions obtained from each other by a permutation: \(\left(\begin{array}{c}n\\ k\end{array}\right)=\frac{A_{k}^{n}}{k!}=C_{n}^{k}\)._  
**Definition 2**.: (Combination with repetition) _A \(k\)-combination with repetition of the set \(A\) is an arbitrary \(k\)-element multi-set of elements from \(A\). If we take \(k\) objects among \(n\) without repetition and regardless of the appearance order, these objects can appear several times and we cannot represent them either with a part of \(k\) elements or with a \(k\)-tuple as their placement order is not involved. However, it is possible to represent such provisions with applications called combinations with repetition. The number of \(k\)-combinations with repetition of a set with \(n\) elements is denoted by \(\Gamma_{n}^{k}\) and formally: \(\Gamma_{n}^{k}=\left(\begin{array}{c}n+k-1\\ k\end{array}\right)\)._  
**Definition 3**.: (Usable sequence) _A sequence is said to be a usable sequence if it leads to a reachable state that may exist in the presumed reachability graph. Let \(\sigma_{U}\) be the set of usable sequences_:  
\[\sigma_{U}=\left\{\sigma\in\sigma^{*}|\,M_{0}[\sigma>M,M\in G(N,M_{0})\right\} \tag{7}\]  
**Definition 4**.: (Unusable sequence) _A sequence is said to be an unusable sequence if it leads to a state that does not exist in the presumed reachability graph. Let \(\sigma_{UN}\) be the set of unusable sequences_:  
\[\sigma_{UN}=\left\{\sigma\in\sigma^{*}|\,M_{0}[\sigma>M,M\
otin G(N,M_{0})\right\} \tag{8}\]  
**Definition 5**.: (Prefix of a sequence) _Let \(M1\) be the set of forbidden markings that does not fulfill a given GMEC and let \(E\) be the set of events. A firing sequence of a given forbidden marking \(\sigma_{M1i}\) is a prefix of a sequence \(v\in E\), if there is another sequence \(w\) such that \(v=\sigma_{M1i}w\). That is, \(v\) is the concatenation of \(\sigma_{M1i}\) and \(w\) (see Figure 2). If \(v\) is a physically possible sequence in the system, then naturally any prefix of \(v\) is also a possible sequence in this system._The key idea of the proposed method is to focus on the determination of the linear system of the theory of regions. Once the three classes of conditions of the TR, i.e., Equations (4)-(6), are defined, one can obtain the PN supervisor by adopting the interpretation of the theory of regions.