## 2 BLDC Motor Dynamic Model  
The transfer function considers one of the most important techniques of control theory. Some control design and analysis methods, such as the root-locus method and the frequency-response method, are also developed based on the system transfer function. Usually, a 3-phase BLDC motor is connected by six electronic switches (power transistors) to produce a 3-phase voltage simultaneously to a full-bridge configuration power converter. The switching sequence can be adjusted by the position sensor [25]. The cases of rotor position are monitoring using three hall sensor devices. The hall sensors provide the information to the decoder block to create the sign of reference current signal vector to the back electromotive force (BEMF) [26]. To operate the motor in the opposite direction, the current is altered in the reverse direction or the switching order of the controller is changed. The controlled DC supply value depends on the PID controller output to can track a preselected speed profile [27]. The inverter used to transform the DC voltage to AC voltage likewise has a DC voltage converter to switch DC to this framework. Figure 1 demonstrates the main components of the BLDC motor drive system [28]. The state-space equation method is one of the most important analysis methods in modern control theory. The state-space method is becoming more and more popular in designing control systems with the fast development of computer techniques. The state-space model of the BLDC motor can be considered as Equations (1) and (2) [1]. \[\left[ \begin{array}{c}\dot{i_{a}}\\ \dot{i_{b}}\\ w_{m}\\ \theta_{m}\end{array} \right]=\left[ \begin{array}{cccc}-\frac{R}{L}&0&0&0\\ 0&-\frac{R}{L}&0&0\\ 0&0&-\frac{R}{L}&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{c}i_{a}\\ i_{b}\\ \frac{1}{L}&\frac{1}{L}&0\\ 0&0&\frac{1}{L}\\ 0&0&0\end{array} \right]\left[ \begin{array}{c}v_{ab}\_e_{ab}\\ v_{ab}\_e_{ab}\\ v_{e}\_T_{e}\_L\end{array} \right] \tag{1}\]  
\[\left[ \begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\\ w_{m}\\ \theta_{m}\end{array} \right]=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ -1&-1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{c}i_{a}\\ i_{b}\\ w_{m}\\ \theta_{m}\end{array} \right] \tag{2}\]  
The \(v_{AB}\) and \(v_{BC}\) consider line input voltage. The \(i_{a}\) and \(i_{b}\) are the phase current through BLDC stator winding. The \(\epsilon_{AB}\) and \(\epsilon_{BC}\) stand for the equivalent line back emf. The \(R\) is the phase resistance of the winding. The \(L\) is the phase inductance of winding. The rotor moment of inertia is \(J\) while the rotor speed is \(\omega\). The \(T_{L}\) is the load torque and \(T_{e}\) is the electrical torque. The \(K_{e}\) and \(K_{T}\) are the coefficient of lineback-EMF and the coefficient of line torque constant, respectively. Recently, the \(B_{v}\) stands for the viscous friction coefficient. The parameters of the BLDC motor are listed in Table 1.  
Figure 1: BLDC Motor Block Diagram. Figure 2 demonstrates the open-loop dynamic response of the BLDC motor drive system model. The no-load rotor speed stabilizes at 4000 RPM through 0.04 s. A fixed 50% of rated torque was applied on the motor shaft at time 0.1. Due to the sudden load, the corresponding rotor speed will be reduced to 3200 RPM. This behavior of the BLDC model indicates that the model was developed correctly. Figure 3 shows the corresponding phase current of the BLDC motor. Usually, the starting current of the BLDC motor is very high and decreases to zero in the case of no-load. it provides this fact where the starting current reaches 18 Ampere and decays to zero at stabilizes the rotor speed. Besides, it can noted that the phase current will be increased at time 0.1 s to \(\pm\)2.5 A due to the sudden load.