## 3 Thermal Effect on Valve Performance  
To evaluate the control performances of the proposed valve system and controller under high-temperature conditions, an apparatus is used together with a large heat chamber. Figure 5a shows the schematic configuration of the experimental flow chart. The temperature is regulated by the heat chamber (BK-HTDO-8100, Bookwang Tech Co., Gyeonggi-do, Korea) to obtain the required experimental conditions, and the air temperature can be controlled between room temperature and 300 \({}^{\circ}\)C. First, an electric apparatus is constructed using a computer and sensors. The signals obtained from the sensors are saved, and the input signal is generated using a computer with the help of analog-to-digital converters (AD/C) and digital-to-analog converters (DA/C) (DS1104 R&D Controller Board, dSPACE, Paderborn, Germany). A voltage amplifier (EC750 SA, NF Corporation, Yokohama, Japan) is used to amplify the input signal obtained from the computer and is applied to the piezostack up to 150 V. A gap sensor (KD-2306 2 SMT, KAMAN, Bloomfield, CT, USA) is used to detect the spool displacement, and the position data are transferred to the computer. Secondly, the hydraulic part is set up to generate the pressure drop. The initial pressure is generated by a nitrogen pressure vessel, and the pressure is changed to the hydraulic pressure using the pneumatic-hydraulic cylinder. The cylinder has a hydraulic volume of 1 L, and three cylinders are used for the experiment. Two pressure sensors (210-40-011-04, Paine Electronics, LLC, East Wenatchee, WA, USA) are placed in the inlet and outlet ports of the valve system. A flow meter (FT-08 AEPJB-LEA-3, Flow Technology Inc., Tempe, AZ, USA) is used to detect the flow rate of the hydraulic fluid. The sensors and the piezostack valve system are located in the heat chamber. The piezostack is insulated from the heat using an aerogel. Figure 5b shows an image of the experimental apparatus inside the heat chamber. In this study, the piezostack actuator is made of lead-zirconate-titanate (PZT) ceramic. The maximum operating temperature of the piezostack used with the PZT materials is 80 \({}^{\circ}\)C, which is approximately half the Curie temperature, as the used PZT ceramic has a Curie temperature of 150 \({}^{\circ}\)C. The operating temperature in this study is 150 \({}^{\circ}\)C. Hence, the spool displacement of the valve system is first evaluated considering the effect of heat. The force generated by the PZT piezostack actuator decreases as the temperature increases. Figure 6 shows the input voltage required to obtain a spool displacement of 0.3 mm. To control the displacement of spool, a traditional PID controller is used. The P, I, and D gains are chosen via the trial-and-error method. To evaluate the heat influence, the temperature of the piezostack actuator is increased from 30 to 120 \({}^{\circ}\)C. As shown in Figure 6a, an input voltage of 90 V yields a spool displacement of 0.3 mm. However, the input voltage is increased by increasing the temperature, as shown in Figure 6b-d. At 120 \({}^{\circ}\)C, an input voltage of 147 V is needed to obtain a spool displacement of 0.3 mm. The spool cannot reach the desired position when the temperature of the piezostack actuator is above 120 \({}^{\circ}\)C. This implies that the actuating performance of the piezostack significantly decreases and the possibility of permanent damage to the piezostack increases.