## 4 Experimental Setup and Methodology  
### Test Rig Setup  
In this section, the experimental setup and measurement methods for characterizing the piezoelectric actuator system are described. Both static and dynamic characteristics are presented in detail, which are crucial for switching technology and for open-loop as well as closed-loop position control. Two main parameters are measured and analyzed, (1) the stroke, or displacement, of the piezoelectric actuator system (\(x_{\text{pa}}\)) and (2) the electrical voltage applied to each piezo stack actuator (\(U_{\text{PE1}}\), \(U_{\text{PE2}}\), \(U_{\text{PE3}}\)). The experimental setup is illustrated in Figure 6, with a real picture of the test rig shown in Figure 7. A personal computer with NI LabVIEW programming and a graphical user interface was employed to conduct test cycles, manage open-loop position control, monitor measured variables, and save the results. Standard block diagrams and the NI LabVIEW library were used to create all functional diagrams. The BNC-2120 external terminal (DAQ--data acquisition device) from National Instruments was used to capture the measured variables, including the stroke of the piezoelectric actuator system (measured by an Eddy current sensor) and the electrical voltage at each PE. This device can acquire both analogue and digital signals and is compatible with NI LabVIEW, allowing for the setting of data types, sampling frequency, and triggers. It supports high-frequency data acquisition, ranging from 100 Hz to 1 MHz [21]. Pulse number modulation (PNM) was utilized for open-loop position control to determine the three discrete values of the piezoelectric actuator system's stroke (with one, two, or three PEs activated). Custom digital control electronics, including low-voltage high-response switching electronics and high-voltage control electronics, have been developed to achieve high response rates, high voltage signals, and stable control. The PNM method employs standard on/off pulses (low-voltage electronics with 5 V pulses to control high-voltage switches producing a 200 V output control pulse signal--\(U_{\text{S,PEi}}\)). Figure 5: Theoretical stroke of piezo actuator system considering the disc spring stiffness. Additionally, the pulse width modulation (PWM) method was combined with closed-loop position control using a PID controller and a position sensor in the feedback loop. The desired position of the piezoelectric actuator system was used as the reference signal [22]. For PWM, the parameters were set at the low-voltage electronics (5 V output signals) with a frequency of 100 Hz, resulting in 100 signals per second. The duty cycle varied from 10% to 100% depending on the required signal in the closed-loop control. A period of 10 microseconds was used, with a linear step response set to 0.1 milliseconds for rise and fall times. Based on the static characteristics of the piezoelectric actuator system, the control activated one, two, or three PEs for a specified time period to achieve the desired stroke. Only the proportional (P) gain was used in the high-voltage control electronics (amplifier) to achieve the desired step response and fast charging of the PEs. The P gain was set to 30, while the integral (I) and derivative (D) gains were set to zero or excluded from the closed-loop control. The overall displacement of the piezoelectric actuator system (stroke) was measured using the Micro-Epsilon U1 eddy current sensor, which has a measuring range of up to 1 mm and a stroke resolution of less than 0.1 microns. Calibration was performed using the Micro-Epsilon online tool. To accurately capture signals from the eddy current sensor, the  
Figure 6: Scheme of experimental setup and measured voltage at each PE. Figure 7: Real picture of experimental test rig. DT 3100 SM controller, equipped with an integrated amplifier and null shift function, was used. The extension of the piezoelectric stack actuator is indirectly influenced by the supply voltage (\(U_{\text{s,PEi}}\)).