## 4 Experimental Setup and Methodology
### Static and Dynamic Characteristics' Measurement  
The static characteristics include piezoelectric actuator system stroke \(x_{\text{pa}}\) for several different operating scenarios (one active PE, two active PEs, and three active PEs) at the maximum control voltage \(U_{\text{S,PEi}}\) = 200 V. The PNM method is used here to simplify control. We expected at least three discrete stroke values for the piezoelectric actuator system. Three scenarios were presented for activating a single PE (PE1, PE2, or PE3), as shown in Figure 8a. Similarly, three scenarios were planned for activating two PEs (PE1 + PE2, PE1 + PE3, and PE2 + PE3), as illustrated in Figure 8b. The final scenario involved activating all three PEs, as depicted in Figure 8c. For a better understanding, the voltage measured at each piezoelectric stack actuator was analyzed, which is directly measured at the positive and negative poles of the PEs (Figure 6).  
The static characteristics, particularly the potential for a higher stroke resolution of the piezoelectric actuator system, were analyzed using the PWM method. The width of the PWM control signal determines how long the PE remains activated. Since the PE requires a certain amount of time to reach full extension, low-width PWM signals can be used to achieve a partially extended PE, resulting in the partial displacement of the piezoelectric actuator system.  
The dynamic characteristics include the step response of the piezoelectric actuator system. Theoretically, the piezoelectric stack actuator (PE) can be modeled as an RC electrical circuit, as presented in [22] and shown schematically in Figure 6. In this case, the charging curve represents the response of a first-order system. Additionally, since the piezoelectric actuator system operates as a mechanical system (a mass-spring-damper system), the step response can be characterized as a second-order system, as defined by Equation (9) [23].  
\[m_{\text{s}}\cdot\ddot{x}_{\text{s}}(t)=F_{\text{s}}(t)-b_{\text{s}}\cdot\dot{x }_{\text{s}}(t)-k_{\text{s}}\cdot x_{\text{s}}(t) \tag{9}\]  
where \(m_{\text{s}}\) represents the effective mass of the moveable parts, \(F_{\text{s}}\) denotes the induced (or generated) force, \(b_{\text{s}}\) is the damping coefficient of the system, and \(k_{\text{s}}\) represents the stiffness of the system.  
The second-order system response is graphically shown in Figure 9. The \(x\)-axis represents time and the \(y\)-axis denotes the amplitude, corresponding to the piezoelectric actuator system stroke (response). The transient response of a practical system often displays damped oscillations before reaching a steady state. The response curve is characterized by several key parameters, namely the rise time \(t_{\text{r}}\) (time taken to rise from 10% to 90% of the  
Figure 8: Control scenarios. one active PE (**a**), two active PEs (**b**) and three active PEs (**c**).  
final value), the peak time \(t_{\text{p}}\) (time required for the response to reach the first peak of the overshoot), and the settling time \(t_{\text{s}}\) (time taken for the response to settle within its specified tolerance, which in our case is \(\pm 2\%\)). The response also exhibits a typical peak overshoot \(A\), along with a steady-state error \(e\)[23].  
In this paper, the step response time is defined by the rise time \(t_{\text{n}}\), which corresponds to the piezoelectric actuator system stroke increasing from 10% to 90% of its amplitude value [23]. Our aim is to achieve a step response for the piezoelectric actuator system without overshoot, stable conditions without oscillations, and a minimal steady-state error of less than 1 micron.