## 2 Boost Converter-Fed BLDC Machine Model
The boost converter is a kind of single pipe non-isolated DC conversion circuit, which is mainly composed of four parts: power device, diode, filter capacitance and energy storage inductance. Its operating modes can be divided into two categories: continuous current mode (CCM) and discontinuous current mode (DCM) [20]. Under the CCM condition, the boost converter-fed BLDC motor operates in two states and the state-space equations can be expressed as [21]  
\[\left\{ \begin{array}{ll}\dot{x}=A_{1}x+B_{1}u_{i}&0\leq t\leq DT\\ \dot{x}=A_{2}x+B_{2}u_{i}&DT\leq t\leq T\\ \end{array} \right., \tag{6}\]  
where \(D\) is the duty cycle, \(x\) is the state variable, \(x=(i_{L}\ u_{C})\), \(\dot{x}\) is the differential of the state variable, \(i_{L}\) is the inductance current, \(u_{C}\) is the capacity voltage, \(u_{i}\) is the input voltage, and \(A_{1}\), \(A_{2}\), \(B_{1}\) and \(B_{2}\) are the coefficients related to the circuit structure. The state-space equation can be obtained by  
\[\dot{x}=Ax+Bv\ \ \ \ \ 0\leq t\leq T\ \, \tag{7}\]  
where \(A=DA_{1}+(1-D)A_{2}\), \(B=DB_{1}+(1-D)B_{2}\). Figure 3 shows the equivalent circuits when the switch \(T9\) is on and off, respectively, regardless of the protection circuit. Using the state-space averaging method, the converter is transformed into an equivalent linear, time-varying, continuous circuit. When switch \(T9\) is ON, the state-space equations are shown, ignoring the resistance of diode \(D\) and switch \(T8\)[22; 23]:  
\[\left\{ \begin{array}{ll}\frac{d{i_{L}}}{dt}=\frac{1}{L}u_{in}\\ \frac{d{u_{out}}}{dt}=\frac{1}{C}(-\frac{u_{out}}{R})\\ \end{array} \right.. \tag{8}\]  
In case the switch is OFF, the state-space equations are given by  
\[\left\{ \begin{array}{ll}\frac{d{i_{L}}}{dt}=\frac{1}{L}(u_{in}-u_{out})\\ \frac{d{u_{out}}}{dt}=\frac{1}{C}(i_{L}-\frac{u_{out}}{R})\\ \end{array} \right.. \tag{9}\]  
Using the state-space averaging method these equations can be shown by \[\frac{d}{dt}\left[ \begin{array}{c}i_{L}\\ u_{out}\end{array} \right]=\left[ \begin{array}{cc}0&-\frac{1-D}{L}\\ \frac{1-D}{C}&-\frac{1}{RC}\end{array} \right]\left[ \begin{array}{c}i_{L}\\ u_{out}\end{array} \right]+\left[ \begin{array}{c}\frac{1}{L}\\ 0\end{array} \right]u_{in}, \tag{10}\]  
where \(u_{in}\) and \(u_{out}\) are the input and output voltages of the boost converter, \(L\) is the energy storage inductance, \(C\) is the filter capacitance, \(D\) is the duty cycle of switch \(T9\), \(i_{L}\) is the inductance current, \(R\) is the single-phase resistance, including stator resistance and transistor on-resistance. Ignoring the resistance of switch \(T8\), it can be rewritten as:  
\[u_{out}=u_{in}\frac{1}{1-D}-L\frac{di_{L}}{dt}. \tag{11}\]  
According to the design requirements of the system, the specifications of the front-end boost converter can be obtained by  
\[L=\frac{U_{in}D}{2\Delta i_{L}f_{s}}, \tag{12}\]  
\[C=\frac{U_{out}D}{2\Delta uR^{\prime}f_{s}}, \tag{13}\]  
where \(\Delta i_{L}\) and \(\Delta u\) are the current and voltage ripples, respectively, \(R^{\prime}\) is the load resistance, and \(f_{s}\) is the switch frequency of power device \(T9\). Here \(L\) takes 3.5 mH and \(C\) takes 200 uF according to the requirements of the system. These equations are utilized to set up the simulation model of the machine and the converter, offering the design parameter reference and supports for the prototype. To gain an optimized dynamic response, the bus-voltage should be kept at a high level by utilizing the boost converter. As mentioned above, the output voltage of the converter adopted only two levels. The DC-link voltage states \(u_{1}\) and \(u_{2}\) can be obtained according to the logical synthesis operation of control factors including \(S\), \(K_{1}\), \(K_{2}\) and \(K_{3}\), as shown in Table 1. The operating logic results are stored in a digital signal processing (DSP) chip and the reference LUT method is adopted according to the requirements under different situations during operation.