## 1 Introduction  
Hydraulic control systems have been developed and are frequently used in various industrial fields, including in several different aerial vehicles. Hydraulic systems are integrated with many types of mechanical systems. The performances of the mechanical components integrated with hydraulic systems usually depend on the performance of the hydraulic parts, comprising the pumps, cylinders, and valves. In particular, valves which regulate the flow rate and pressure of the gas or the liquid significantly affect the performance [1]. The hydraulic control valve is the most important part of the hydraulic and hydraulic-mechanical systems. Especially in the aerospace field, the hydraulic control valve is a crucial component or system for the attitude control or tracking control of a desired flight trajectory because the hydraulic system can generate a high-power density with high accuracy [2; 3; 4]. More specifically, the deflection angle of the aileron, elevator, and rudder are normally controlled by hydraulic actuators associated with the servo valve system [5]. Among many applications in aerospace engineering, the trajectory control of missiles requires certain special performances such as reliable resistance against high temperatures, high tracking accuracy and fast dynamic motion [6; 7; 8]. In order to meet these requirements, a special valve system needs to be developed. As is well known, the spool valve is the most widely used valve for hydraulic servo systems to control the flow rate and pressure drop, which are proportional to the spool displacement. The spool-valve system can be classified into single and two-stage valves. Figure 1 shows the schematic configuration of each valve system. In the two-stage valve, the air pressure is used to minimize the operating force [9]. As shown in Figure 1a, air pressures (\(P_{s\_air}\)) are always applied on the end of the spools and the flapper can control the air pressure by preventing air flow. When one air flow is prevented, the air pressure is increased. Finally, the spool can move according to the pressure change. The input pressure of the hydraulic oil (\(P_{s\_Fluid}\)) is applied and the area of the outlet port is changed by movement of the spool. This means that the small actuating force acting on the pilot is converted into a large actuating force for the spool [10]. A solenoid actuator is generally used for the pilot to generate the actuating force for the flapper. The flow rate can be controlled in a wide range using the two-stage spool valve. However, it has a low operating frequency (lower than 50 Hz) because of the inertia of the flapper [11]. The single-stage valve has been studied since the 1990s, as it can solve the problems of the two-stage valve [12; 13]. Because an actuator directly controls the spool, the single-stage valve requires a large actuating force but has a wide range of operating frequencies. It is natural that the actuator affects the performance of the valve. Hence, many types of actuators have been applied for single-stage valves. Currently, the solenoid-type actuator is the most widely used for valve control systems, but it has the disadvantage of a slow response [14; 15]. Furthermore, an advanced actuator capable of operating sufficiently quickly at high temperatures is needed in many industrial applications. One solution to these problems involves using piezoelectric actuators in the hydraulic servo valve. Piezoelectric materials are actively studied and applied in various fields because of their many advantages, such as their fast response, high power density, and infinite resolution [16]. The piezoelectric actuator is advantageous for driving a hydraulic servo valve system [17; 18]. Hence, single-stage valves using piezoelectric actuators have been studied [19; 20; 21; 22]. [PERSON] et al. designed a single-stage valve system actuated using a piezostack actuator [23; 24]. The performance of the valve system was evaluated via a simulation and an experiment conducted at room temperature.