## 4 Control Responses  
The piezostack is influenced by heat; thus, the properties of the piezostack such as the blocking force, displacement, and hysteresis change. This implies that the piezostack is a temperature-dependent, nonlinear system. Therefore, a fuzzy algorithm is used to control the system at various temperatures, because a fuzzy controller can control a nonlinear system. Furthermore, the fuzzy controller does not require a precise mathematical model. The fuzzy logic is far closer to human language than traditional logical systems and can process ambiguous input values such as warm and cold. The fuzzy logic can convert the linguistic strategy into an automatic control strategy on the basis of expert knowledge. The expert knowledge yields better results than conventional control logic. In particular, the methodology of the fuzzy logic controller (FLC) is very useful in cases of complex calculations for qualitative, inexact, or uncertain interpretations. The FLC comprises two essential parts: a set of linguistic control rules and a compositional rule of inference. In this study, a PID controller which can change gains automatically using a fuzzy algorithm is designed. Figure 8 shows a schematic of the PID controller together with the fuzzy algorithm. The PID controller has a feedback mechanism, which is commonly used in industrial fields. In the PID controller, an error (\(e(t)\)) between the desired and actual output values is used. This error is then applied for the correction based on the proportional, integral, and derivative terms. The normal PID controller is expressed as follows:  
\[u(t)=k_{p}e(t)+k_{i}\int e(t)+k_{d}\frac{d}{dt}e(t) \tag{7}\]  
Figure 7: Temperature variation and step responses of the valve system, (**a**) Temperature variation, (**b**) Step response at 30 \({}^{\circ}\)C, (**c**) Step response at 150 \({}^{\circ}\)C.  
However, the PADDV system is a temperature-dependent system because the properties of the piezostack actuator change based on the temperature, as previously mentioned. Therefore, the gains of the PID controller must be changed to control the valve system at various temperatures. The fuzzy algorithm has two input and two output variables. The inputs are the temperature and operating frequency, and the outputs are the \(k_{p}\) and \(k_{i}\) gains, which are the gains of the PID controller. Figure 9 shows the membership functions of the input and output variables. In this research, a triangular fuzzy number is used because the method has a low calculation amount compared with other method such as trapezoidal, bell-shaped, non-convex, etc. As shown in Figure 9a,b, the temperatures and frequencies are classified as \"cold\", \"warm\", \"hot\", and \"extreme hot\" and \"extreme low\", \"medium low\", \"low\", \"normal\", \"high\", \"medium high\", and \"extreme high\", respectively. The figures show the range of each value and perform the function of the fuzzification of the input temperature and frequency. This can provide necessary definitions to define linguistic control rules. Tables 3 and 4 list the fuzzy inference rule between the input and the output, and the rules are determined by many experimental results. The input membership functions and the fuzzy rules can provide necessary definitions to define the linguistic control rules. Figure 9c,d show the membership functions of the output variables; from those, the gains of the PID controller are determined. In other words, the figures perform the function of the defuzzification and provide control gains. Figure 8: Schematic of the fuzzy PID(proportional-integral-derivative) controller. Figure 9: Membership functions, (**a**) Temperature input, (**b**) Frequency input, (**c**) P gain output, (**d**) I gain output.