## 2 Basic Concepts
* The firing of a transition \(t_{j}\), enabled at \(M_{k}\), leads to a new marking \(M_{k+1}\) (denoted as \(M_{k}[t_{j}>M_{k+1})\), caused by removing \(I(p_{i},t_{j})\) tokens to each place \(p_{i}\in{}^{\bullet}t_{j}\) and adding \(O(p_{i},t_{j})\) tokens to each place \(p_{i}\in t_{j}{}^{\bullet}\). The marking function \(M\) can be expressed as a column vector \(\mathbf{M}\in\mathbb{N}_{0}^{|P|}\), such that \(\mathbf{M}[i]=M(p_{i})\), \(\forall p_{i}\in P\). In this work, both \(M\) and \(\mathbf{M}\) will be called marking, since both represent the same information. Thus, a marking \(M_{k+1}\), reached from \(M_{k}\) after firing a transition \(t_{j}\), can be computed as \(\mathbf{M}_{k+1}=\mathbf{M}_{k}+\mathbf{C}\vec{t_{j}}\), where \(\vec{t_{j}}\) is the \(j\)-th column of the identity matrix of dimension \(|T|\). The function \(\varphi\) can be represented by a \(|\Sigma_{O}|\times|P|\) matrix \(\boldsymbol{\varphi}\), in which \(\boldsymbol{\varphi}[i,j]=1\), if \(p_{j}\) is labeled with the \(i\)-th output symbol and \(\boldsymbol{\varphi}[i,j]=0\) otherwise. The output vector of the \(1 PN\) at \(M_{k}\) is defined as \(\mathbf{y}_{k}=\boldsymbol{\varphi}\mathbf{M}_{k}\). A _firing sequence_ is a sequence of transitions \(\sigma=t_{i}t_{j}\ldots t_{k}\), such that \(M_{0}[t_{i}>M_{1}[t_{j}>\cdots M_{m}[t_{k}>M_{m+1}\). The marking \(\mathbf{M}^{\prime}\) reached after the firing of \(\sigma\) from a marking \(\mathbf{M}\) can be computed as \(\mathbf{M}^{\prime}=\mathbf{M}+\mathbf{C}\vec{\sigma}\), where \(\vec{\sigma}\in\mathbb{N}_{0}^{|T|}\) is a column vector, named the _Parikh vector_, defined such that \(\vec{\sigma}[j]=k\), if \(t_{j}\) is fired \(k\) times in \(\sigma\). This is denoted as \(M[\sigma>M^{\prime},\text{ and }\ M^{\prime}\) is said to be _reachable_ from \(M\). A \(PN\) system is said to be _bounded_ if \(\exists k\in\mathbb{N}\), such that \(\forall p\in P\) and for any \(M\) reachable from \(M_{0}\), it holds that \(M(p)\leq k\). A \(PN\) system is said to be _safe_ if it is bounded with \(k=1\). A \(PN\) system is said to be _live_ if \(\forall t_{j}\in T\) and for any \(M\) reachable from \(M_{0}\) there exists a fireable sequence \(\sigma\), such that \(M[\sigma>M^{\prime}\) and \(t_{j}\) is (marking) enabled at \(M^{\prime}\). \(IPN\) models for complex systems can be built from \(IPN\) component models, by using the synchronous composition. Given two safe \(IPN\)\(Q^{1}\) and \(Q^{2}\), the synchronous composition results in another safe \(IPN\)\(Q^{3}=Q^{1}\|Q^{2}\), defined as the union of \(Q^{1}\) and \(Q^{2}\) in which the transitions sharing the same identity label are merged, keeping all the transition and place labels. The synchronous composition is commutative and associative. Consider the double-acting electro-pneumatic actuator driven by a 5/2 valve and its \(IPN\) model shown in Figure 1. The incidence matrix \(\mathbf{C}\) and the output labeling matrix \(\mathbf{\varphi}\) is given by the following:  
\[\mathbf{C}= \begin{bmatrix}-1&1&0&0&0&0\\ 1&-1&0&0&0&0\\ 0&0&1&0&-1&0\\ 0&0&-1&1&1&-1\\ 0&0&0&-1&0&1\end{bmatrix} ,\ \ \mathbf{\varphi}= \begin{bmatrix}0&0&1&0&0\\ 0&0&0&0&1\end{bmatrix} .\]  
The input alphabet is defined as \(\Sigma_{I}=\{a,b\}\), where each symbol represents a command signal that allows the change of the valve position (i.e., energizing a coil), which leads to an actuator movement. The output alphabet is defined as \(\Sigma_{O}=\{A,B\}\), where each symbol represents the activation of a sensor witnessing the actuator position. Then, transitions \(\{t_{1},t_{2}\}\) are controllable, while transitions \(\{t_{3},t_{4},t_{5},t_{6}\}\) are uncontrollable. At the initial state (Figure 1a), the actuator is retracted, the initial marking is given by \(\mathbf{M}_{0}=[0\;1\;1\;0\;0]^{T}\) and the output vector is \(\mathbf{y}_{0}=[1\;0\;]^{T}\), meaning that sensor \(A\) is active.