## 2 Materials and Methods
### Analytical Modelling of the Proposed IAC Actuator  
Mathematical modelling of the SMA spring is necessary to describe two operability states of the proposed actuator using scientific expressions. From the closing to opening state, the shape memory effect (SME) acts undergo the converted current and SMA models to fulfil the actuation of the IAC actuator, as shown in Figure 4. The function that defines the electric current (\(I\)) in the converter is dependent on the SMA spring's size (\(A\)) and holding temperature circulating into the spring (T\({}_{\text{w}}\)) [37]. This function is expressed as follows:  
\[I=\text{F}(\text{T}_{\text{w}},A) \tag{4}\]  
Figure 3: Operational mechanism of the actuator: (**a**) closing state; (**b**) opening state. The desired temperature of the actuator's pseudo-plasticity is the final austenite temperature of the SMA spring (\(A_{F}\)). The converter uses this temperature as the input of the current function to predict the required electric current to supply the SMA spring. The system uses a proportional integrated derivative (PID) controller to achieve fast smooth current output responses. However, the controller input is the output signal from the current function. The scientific behind of this controller is the tuning technique of its gains to the significant values. In this work, the parameterisation of the PID controller is performed with a genetic algorithm (GA) method to obtain the preference controller gains (\(K_{P,I,D}\)). The objective function to give the optimum gain is a function of the length of the integrated time absolute error (ITAE). The variables are three, the lower boundaries are [0 0 0], and the upper boundaries are [50 50 50]. The system output displays K and fitness values. After using the optimum gains, the output signals from PID pass through the current limiter. To prevent saturation output current signals, we opt to use a saturation limiter with upper and lower boundaries [38]. The output signals of the current converter are shown in Figure 5.  
Figure 5a shows the best fitness value of the PID controller's optimization process achieved within 25 generations. The best fitness value and the last minimum length value of the integrated time absolute error (ITAE) are the key parameters to confirm the optimum gains of the PID controller. Figure 5b shows the frequency of ITAE during controller optimization. The recorded optimum controller gains (K) are [1.03 0.64 0.31]. The controller gain values are important in the controller outputs, accuracy, and time response. Figure 5c shows the electric current output response. The electric current of 2 A is achieved after 700 ms from multiple simulation duty cycles. This current is enough to heat the SMA spring and becomes the input of the SMA models to activate the operation of the SMA-based actuator. The required SMA models to determine the appropriate responses of the actuator include thermal, constitutive, and kinetic models. The models are arranged consecutively to use controlled electric current as the principal input [7; 39]. The SME links the models to operate. However, SME is activated by heating the SMA spring at the final required temperature, as described by the thermal model. The thermal model is expressed by Equation (5). For the actuator to perform those two antagonistic states, the heat transfer of  
Figure 4: Shape memory effect (SME) analytical model to operate the actuator. Figure 5: Current converter’s characteristics. (**a**) PID optimization fitness values (Best: 504.366, Mean: 1.8951 \(\times\) 10\({}^{15}\)); (**b**) power spectrum of integrated time absolute error (ITAE); and (**c**) output current response. the SMA spring is followed by a heating cycle (\(T_{hc}\)) for opening and a cooling cycle (\(T_{cc}\)) for the actuator's closing process. The temperature response (\(T_{w}\)) from this model becomes the input of the phase transformation model.