# A Program Model of Fuzzy Interpreted Petri Net to Control Discrete Event Systems
## 3 The Formal Basis and the Conception of FIPN
: _The transition \(t\in T\) with marking M:P \(\rightarrow\) [0, 1] is enabled from the moment in which the degree to fulfil the condition \(\Theta(t)=\vartheta\), which is assigned to the transition, is greater than zero and the following conditions are satisfied [51]:_  
\[\forall p\in{}^{\bullet}t,M(p)\geq W(p,t)/K(p) \tag{1}\]  
_and_  
\[\forall p\in{}^{\bullet},M(p)\leq 1-W(t,p)/K(p), \tag{2}\]  
_to the moment at which:_  
\[\exists p^{\prime}\in{}^{\bullet}t,M(p^{\prime})=0 \tag{3}\]  
_or_  
\[\exists p^{\prime}\in{}^{\bullet},M(p^{\prime})=1, \tag{4}\]  
_where: \({}^{\bullet}t=\{p\in P|(p,t)\in R\}\) is the set of input places of the transition \(t\), and_  
\(t^{\bullet}=\{p\in P|(t,p)\in R\}\) _is the set of its output places._  
In FIPN, an analogue signal can be assigned to a transition and its value is normalized into the range [0, 1]. The transfer of a marker across the fired transition is a process whose duration is longer than one clock cycle that synchronizes the net operation. This duration depends on the increment of the sensor value. Such work of transitions permits more precise observation of changes in the controlled system. The change of the marking for places connected to the enabled transition depends on the increment of the degree to which the condition corresponding to the transition is satisfied. The method that calculates the new marking is described by Definition 3. The transition remains active until the markers are transferred from the input places to the output places of the transition. **Definition 3**. : _Let \(M\) be the marking for which the transition \(t\in T\) is enabled. The degree \(\Theta(t)=\vartheta\in[0,1]\) to which the condition corresponding to the enabled transition is satisfied will be changed by \(\Delta\vartheta\geq 0\) and there will be an event \(e\) which synchronizes the work of all transitions. The new marking of the net \(M^{\prime}\) is computed by the following rule [51]:_  
\[M^{\prime}(p)\!=\! \begin{cases}M(p)\!-\!\frac{\Delta\vartheta\!\cdot\!W(p,t)}{ K(p)}&\text{for}\quad p\in{}^{\bullet}t\!\setminus\!t^{\bullet},\\ M(p)\!+\!\frac{\Delta\vartheta\!\cdot\!W(t,p)}{K(p)}&\text{for}\quad p\in{}^{ \bullet}t\!\setminus\!t,\\ M(p)\!-\!\frac{\Delta\vartheta\!\cdot\![W(p,t)-W(t,p)]}{K(p)}&\text{for}\quad p \in{}^{\bullet}t\!\cap\!t^{\bullet},\\ M(p)&\text{for}\quad p\
ot\in{}^{\bullet}t\!\cup\!t^{\bullet}.\end{cases}  \tag{5}\]  
_The increment \(\Delta\vartheta<0\) does not introduce any changes in marking of the net._  
Figure 1: Two different input and output places of the transition \(t_{1}\). Figure 2 shows the exemplary change of the marking for places connected to the active transition \(t_{1}\). The increment of the degree to which the condition corresponding to the transition \(t_{1}\) is satisfied has the value of \(\Delta\theta=0.2\) at the time when the synchronization signal arrives. Such a change of the marking gives a more precise visualization in comparison to discrete PNs and enables the monitoring of progress in the movement of tokens between the input and output places across the transition. At the end of this section, a simple example is shown to clarify the main advantage of FIPN in comparison to classic Petri nets. In Figure 3a-d, the tank \(TK_{1}\) that can be filled using the valve \(V_{1}\) is presented, and two different graphs as control systems to fill \(TK_{1}\) are proposed. The first is created using SFC, and the second is based on FIPN. Different levels of liquid in \(TK_{1}\) and corresponding to the states of both systems are also shown. In general, the systems operate in a similar manner. First, the filling of \(TK_{1}\) is started by active step \(S_{1}/\) the place \(p_{1}\) which opens the valve \(V_{1}\) (variable \(V_{1}\) is set to _true_). Next, when the tank \(TK_{1}\) is completely filled, active step \(S_{2}/\)place \(p_{2}\) closes the valve (variable \(V_{1}\) is set to _false_). However, both systems differ in one important aspect.