## 3 Computationally Improved Optimal Control Policy
### Supervisory Control Problem  
The theory of regions is recognized as one of the powerful methods for the supervisory control problem. However, the analyzing steps of the RG to determine the linear system of the TR weigh down heavily on the calculation process of PN controllers. This may lead to a failure to respect the control constraints. Based on PN properties and mathematical concepts, one can obtain the linear system of TR without needing the RG. In this work, the non-generated RG will be appointed a virtual graph (VG). **Definition 1**. : _[_30_]_ _A GMEC (generalized mutual exclusion constraint) is a condition that limits a weighted sum of tokens contained in a subset of places \(P\). A control specification expressed by a GMEC is represented by a couple \((\vv{w},K)\), which defines a set of legal markings: \(\mathrm{ML}\ =\ \left\{M\in\mathrm{M}\middle|\vv{w^{t}}M\leq K\right\}\), where \(\vv{w}\) is a weight vector of nonnegative integers and \(K\) is a positive integer. Markings which fail to comply with a GMEC will be denoted forbidden state \(MF\)._  
**Definition 2**. : _[_27_]_ _Let a net \(N\ =\ (P,T,\Pre,Post)\) be a Petri net with an incidence matrix \(C\). \(X_{t}\) is a nonnegative vector of \(n\) integers, it is the solution of the equation: \(C.X_{t}\ =\ 0\). \(\sigma\) is the firing sequence for a legal marking \(M\) with \(\sigma\ =\ X_{t}\ (\card(\sigma)\ =\ n)\). The sequence \(\sigma\) is called transition invariant (t-invariant): \(C.[X_{t1}\ X_{t2}\ldots X_{tn}]^{t}\ =\ 0\) The resolution of this system generates the transition invariants that can identify the cycle Equation (3)._  
**Definition 3**. : _[_29_]_ _A \(p\)-invariant of a given PN is a nonnegative vector \(X\) of \(n\) integers \(\card(P)\ =\ n) verifying: \(X.C\ =\ 0\ \Rightarrow[X_{1}\ X_{2}\ldots\ X_{n}].C\ =\ 0\)._  
**Property 1**. : _An important property arises from Definition 3. Indeed, for any legal marking \(M\in\mathrm{ML}\), a \(p\)-invariant verifies: \(X^{t}.M\ =\ X^{t}.M_{0}\), which means that the sum of places markings weighted by the vector \(X\) is constant, i.e., \(\sum_{i\ =\ 1}^{n}X_{i}m_{i}\ =\ Ct,\ \forall Ct\geq 0\), where \(m_{i}\) is the marking of \(p_{i}\)._  
**Definition 4**. : _A primitive state \(M_{p}\) is a vector presented with the following coordinates: \(M_{p}^{t}\ =\ \left[p_{1}^{t}p_{2}^{t}\ldots p_{n}^{t}\right]^{t}\), where \(n\) is the number of places in the given PN and \(p_{x}^{t}\) is the corresponding marking of the place \(P_{x}\)._  
**Definition 5**. : _A reachable state \(M_{xi}\) is a marking that exists in the \(VG\) obtained by applying the PN state equation: \(M_{xi}\ =\ M_{0}+C.\vv{x_{xi}}\), \(M_{xi}\) can easily be determined if p-invariants exist._  
**Definition 6**. : _An event separation instance is represented by a couple \((M,t)\), where \(M\in\mathrm{M}_{(\vv{w},K)}\) is a legal state and \(t\) is a forbidden state transition \(\Omega\ =\ \{(M,t)|\ \exists M[t>M^{\prime},\ M\in\mathrm{M}_{(\vv{w},K)^{ \prime}},\ M^{\prime}\in M\}\) (see Figure 2)._  
Figure 2: Marking/transition separation instance (MTSI) example. **Definition 7**. : _A state is said to be dangerous if it leads to the forbidden markings MF by firing uncontrollable transitions. Let MD be the set of dangerous states: MD \(=\)\([M\in RG\big{|}\ \exists t\in T_{u}\), \(\exists\sigma\in T_{u}^{*}M[\sigma>M^{\prime}M^{\prime}[t>M^{\prime}(M^{\prime},t)\in\Omega]\)._  
**Definition 8**. : _Let \(M_{0}\) be the initial marking of a given PN with its incidence matrix such as \(M_{0}^{t}\ =\ (y_{0},y_{1}\ldots y_{n})\) where \(n\) is the number of places in the PN. The Linear System of Paths (LSP) is defined as follows:_  
\[C.\sigma\ =\ M_{p}-M_{0}\ =\ \left( \begin{array}{c}p_{1}^{\prime}-y_{1}\\ p_{2}^{\prime}-y_{2}\\.\\.\\ p_{n}^{\prime}-y_{n}\end{array} \right)\geq\left( \begin{array}{c}-y_{1}\\ -y_{2}\\.\\.\\ -y_{n}\end{array} \right) \tag{4}\]  
_The calculated paths will be denoted \(P_{s}\ =\ \sum_{i}\sigma_{i}\)._  
**Remark 1**.