## 3. Matherial and Methods
### Data Models and Equations  
#### 3.6.1 Digital Data Model  
The model of the system is based on the monitoring and comparison of the data obtained from the behavior of the physical system and the data obtained from the ideal behavior of the virtual system. The adjacency matrix is composed of rows of data representing information from the sensors and actuators; therefore, the same information can be represented with a set of vectors. For physical inputs, an array of physical binary inputs \(DI_{PS_{\text{S}}}\) is defined, where each place represents the value of different sensors \(Sp\) with corresponding physical PLC addresses. The virtual environment simulates the behavior of  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Input** & **Denomination** & **Output** & **Denomination** & **Input** & **Denomination** & **Output** & **Denomination** \\ \hline I10.0 & Start button L & Q10.0 & Start (Light) & I11.0 & Sensor mov X & Q11.0 & Forks right \\ \hline I10.1 & Stop Button L & Q10.1 & Start L (Light) & I11.1 & Sensor mov Z & Q11.1 & Forks left \\ \hline I10.2 & Reset button L & Q10.2 & Stop L (Light) & I11.2 & Emergency stop & Q11.2 & Forks Up \\ \hline I10.3 & Sensor conveyor 1L & Q10.3 & Stop (Light) & I11.3 & Start button D & Q11.3 & Start D (Light) \\ \hline I10.4 & Sensor conveyor 2L & Q10.4 & Reset L (Light) & I11.4 & Stop button D & Q11.4 & Stop D (Light) \\ \hline I10.5 & Sensor forks right & Q10.5 & Reset (Light) & I11.5 & Reset button D & Q11.5 & Reset D (Light) \\ \hline I10.6 & Sensor forks middle & Q10.6 & Conveyor 1L & I11.6 & Sensor conveyor 1D & Q11.6 & Conveyor 1D \\ \hline I10.7 & Sensor forks left & Q10.7 & Conveyor 2L & I11.7 & Sensor conveyor 2D & Q11.7 & Conveyor 2D \\ \hline  & & & I12.0 & Sensor current & QW30 & Target position \\ \hline \hline \end{tabular}
\end{table}
Table 2: Input and Output Denomination.  
Figure 8: Adjacency Matrix.  
Figure 7: Complex network of the physical system.  
the physical system, and there is a corresponding array \(DI_{VS_{k}}\) of the virtual binary with inputs for simulated sensors \(Sv\), as shown below:  
\[DI_{PS_{k}}= [Sp_{I0,0},Sp_{I0,1},Sp_{I0,2},Sp_{I0,3},Sp_{I0,4},Sp_{I0,5},Sp_{I0,6}, Sp_{I0,7}, \tag{1}\] \[Sp_{I1,0},Sp_{I1,1},Sp_{I1,2},Sp_{I1,3},Sp_{I1,4},Sp_{I1,5},Sp_{I 1,6},Sp_{I1,7}]\] \[DI_{VS_{k}}= [Sv_{I0,0},Sv_{I0,1},Sv_{I0,2},Sv_{I0,3},Sv_{I0,4},Sv_{I0,5},Sv_{I 0,6},Sv_{I0,7},\] (2) \[Sv_{I1,0},Sv_{I1,1},Sv_{I1,2},Sv_{I1,3},Sv_{I1,4},Sv_{I1,5},Sv_{I 1,6},Sv_{I1,7}]\]  
As can be observed in the adjacency matrix (Appendix A), only one column (strong blue) provides information on the actuators that have been activated by showing a value of \"1\" when the controller has activated them. The algorithms holds the information of activated outputs by means of a row arrangement (gray row). This arrangement of output elements can be represented as a vector. For the physical output, an array of physical binary outputs \(DO_{PS_{k}}\) is defined, where each place represents the value of different output \(Op\) with corresponding physical PLC addresses. For the virtual system, there is a corresponding array \(DO_{VS_{k}}\) of virtual binary outputs for simulated outputs \(Ov\), as follows:  
\[DO_{PS_{k}}= [Op_{Q0.0},Op_{Q0.1},Op_{Q0.2},Op_{Q0.3},Op_{Q0.4},Op_{Q0.5},Op_{ Q0.6},Op_{Q0.7}, \tag{3}\] \[Op_{Q1.0},Op_{Q1.1},Op_{Q1.2},Op_{Q1.3},Op_{Q1.4},Op_{Q1.5},Op_ {Q1.6},Op_{Q1.7},Op_{Qw0}]\] \[DO_{VS_{k}}= [Op_{Q0.0},Op_{Q0.1},Op_{Q0.2},Op_{Q0.3},Ov_{Q0.4},Op_{Q0.5},Op_ {Q0.6},Ov_{Q0.7},\] (4) \[Ov_{Q1.0},Op_{Q1.1},Op_{Q1.2},Ov_{Q1.3},Ov_{Q1.4},Op_{Q1.5},Ov_ {Q1.6},Ov_{Q1.7},Op_{Qw0}]\]