## 2 Theoretical Background of Piezolectric Actuators  
Piezoelectric materials can be utilized in both sensor and actuator technologies. When pressure is applied to piezoelectric materials such as quartz crystals, an electric potential is generated; this phenomenon is known as the piezoelectric effect and is employed in sensor technologies. Conversely, piezoelectric actuators operate based on the inverse piezoelectric effect. In this process, applying an electrical potential to the piezoelectric material causes a change in its shape, specifically an extension in the direction of the material polarization [11]. A detailed background on the practical use and functionality of piezoelectric actuators is provided in [12; 13], where authors focus on the potential applications of piezoelectric actuators and usage in various types of hydraulic valves, including conventional hydraulics and pneumatics, as well as in high-response continuously operated spool valves. In this paper, we focus only on the essential piezoelectric characteristics and operational principles to highlight the key parameters of piezoelectric actuators that significantly impact the static and dynamic performance of the new piezoelectric actuator system. Linear multilayer piezoelectric stack actuators, as shown in Figure 1a, are suitable for hydraulic valves, including DFCUs. The image depicts a 32.4 mm long piezo stack with a \(7\times 7\) mm cross-sectional area. It has two electrical connections; the blue one represents the negative pole, while the red one represents the positive pole [14]. In our application, we use an electrical voltage ranging from 0 V to 200 V. Figure 1b illustrates a schematic representation of a multilayer piezoelectric stack actuator. In this figure, 1 represents the initial height, h denotes the thickness of the PZT layer, P stands for the polarization of the piezo ceramic material, and \(\Delta\)l indicates the extension of the piezo stack when electrical voltage is applied (active state). F represents the generated force, while U refers to the applied electrical voltage, with positive and negative poles. The main properties for selecting a piezoelectric actuator (PZT) or piezo element (PE) for a particular application are as follows: (1) dimension (cross-section, length, or height _l_), (2) force generation (_F_), (3) extension (\(\Delta\)_l_), and (4) response time. Piezoelectric stack actuators consist of several piezoceramic plates with thickness \(h\), as shown schematically in Figure 1b. An advantage of these multilayer stack actuators is that they require a lower supply voltage (up to 200 V), which is important for applications in fluid power technology and assembly automation where a high response, energy efficiency, and a high stroke resolution are crucial. These actuators can generate high forces up to several kN (e.g., the piezo stack shown in Figure 1a generates 2 kN) but have small extensions up to 0.15% of the actuator's length or height. The fundamentals of piezoelectric actuators and the simplified mathematical formulations described in this paper are derived from several sources [15,16]. A piezoelectric actuator is characterized by its force-displacement curve and the electrical charge versus electrical voltage curve. Figure 2 illustrates the force-displacement characteristic curve for a piezoelectric stack actuator, which describes its mechanical performance. This performance can be expressed using a basic mathematical formula that defines the electro-mechanical relationship as presented in [15,16]. The characteristic curve defines the operating point of the piezoelectric stack actuator (OP) under specific loading conditions. Equation (1) represents the displacement \(x_{\text{pzt}}\) of the piezoelectric stack actuator. In this paper, we focus only on Equation (1) and its associated characteristic curve, while the electrical charge versus electrical voltage curve is discussed in detail in [15,16].