## 2 System Modeling
### Modeling of the IPA System  
As indicated in Figure 1, the IPA employed in this work consists of five comprehensive components which form an all-in-one actuator system. The five components are optically encoded data, laser stripe code, pressure sensor, programmable system on chip (PSoC) board, and valves. The main benefit of this actuator is its ability to determine the desired output from the feedback inputs through real-time interaction. This actuator features a 0.06 mm laser stripe pitch, a stroke of 200 mm, and a force of 100 N, which allows for providing great accuracy for position control.  
Using a pressure sensor to monitor the cylinder's chamber pressure is an important part of the control process. Driving the cylinder is accomplished by utilizing two on/off valves installed on the cylinder. The PSoC board, which is attached to the top of the actuator, is the central processing unit and it has an optical reflecting surface mount encoder chip. Three components make up this encoder chip: an LED light source, optical lenses, and a photodetector. Table 1 summarizes the cylinder stroke movements based on the valve's conditions.  
Figure 2 shows the experimental setup for this work, which includes an SHC68-68-EPM cable, a DAQ card PCI/PXI-6221 (68-Pin) board, and SCB-68 M series devices for connecting the IPA plant to a computer through MATLAB.  
\begin{table}
\begin{tabular}{c c c} \hline \multicolumn{3}{c}{**Valve Condition**} \\ \hline
**V1** & **V2** & **Cylinder Stroke Motions** \\ \hline OFF & OFF & Stop \\ OFF & ON & Retract \\ ON & OFF & Extend \\ ON & ON & Stop \\ \hline \end{tabular}
\end{table}
Table 1: Cylinder stroke movements.  
Figure 1: IPA components.  
The experimental input and output data are utilized to create the mathematical model of the IPA system using the SI approach. A total of 1600 input and output data were collected from a real-time experiment with a sampling time of 0.01 s; 800 samples were used for training, and 800 samples were used for validation. The input and output data from the experiment are plotted in Figure 3.  
To characterize the dynamic behavior and properties of the IPA system, a parametric model with autoregressive with exogenous input (ARX) was used as the model structure.  
Figure 3: The measured input and output data of the system.  
Figure 2: The experimental setup of the system.  
Equation (1) describes the discrete ARX transfer function employed in this study. Figure 4 depicts a plot of the output from the measured and simulated models.  
\[\text{A}=\left[\begin{array}{ccc}2.9900&-2.9810&0.9917\\ 1&0&0\\ 0&1&0\\ \end{array}\right]\text{B}=\left[\begin{array}{c}1\\ 0\\ 0\\ \end{array}\right] \tag{1}\]  
\[\text{C}=\left[\begin{array}{cc}0.1187&-0.2350&0.1169\\ \end{array}\right]\text{D}=0\]  
According to Equation (1) and Figure 5, the system is considered stable because all the poles of the system are within the unit circle. In addition, the best fitting criteria of the system is more than 90% after completing the model validation phase in the system identification process.  
Figure 4: The output of the model, both measured and simulated.  
Figure 5: The zero-pole plot for the model.