## 2 The Overall Structure of PMSM Sensorless Drive System  
### Equivalent of PMSM Mathematical Model  
It is well known that PMSM sensorless system is composed mainly of the motor ontology, the inverse model, the current controller and the speed controller. Accurate equivalence of each component model is the basis of stability control system.  
Generally speaking, the voltage equation of the PMSM in the \(d-q\) rotary transformer can be expressed as (1):  
\[\left\{\begin{array}{l}u_{d}=Ri_{d}+L_{d}\frac{di_{d}}{dt}-\omega_{e}L_{q}i_ {q}\\ u_{q}=Ri_{q}+L_{q}\frac{di_{d}}{dt}+\omega_{e}(L_{d}i_{d}+\psi_{f})\end{array}\right. \tag{1}\]  
Among them, \(u_{d}\) and \(u_{q}\) are the \(d-q\) axis component of the stator voltage. \(i_{d}\) and \(i_{q}\) are the \(d-q\) axis component of the stator current; \(R\) is the stator resistance. \(\omega_{e}\) is electric angular velocity. \(L_{d}\) and \(L_{q}\) are the inductance component of \(d-q\) axis. \(\psi_{f}\) stands for permanent magnet flux.  
When the \(i_{d}=0\) control strategy is used, the voltage equation can be reduced as shown in (2):  
\[\left\{\begin{array}{l}u_{d}=-\omega_{e}L_{q}i_{q}\\ u_{q}=Ri_{q}+L_{q}\frac{d_{d}}{dt}+\omega_{e}\psi_{f}\end{array}\right. \tag{2}\]  
When there is no external load disturbance, the torque equation and the motion equation of the motor will be written as [36]:  
\[\left\{\begin{array}{l}f\frac{d\omega_{m}}{dt}=T_{e}+T_{L}-B\omega_{m}\\ T_{e}=\frac{3}{2}P_{n}i_{q}\psi_{f}\end{array}\right. \tag{3}\]  
In (3), \(\omega_{m}\) is mechanical angular velocity. \(B\) is the damping coefficient. \(J\) is usually thought of as the moment of inertia. \(T_{e}\) and \(T_{L}\) are electromagnetic torque and load torque, respectively. \(P_{n}\) is polar logarithm of PMSM.  
After the laplace-transform, the transfer function of the PMSM electrical part \(G_{PMSMm}\) and mechanical part \(G_{PMSMm}\) can be equivalent to:  
\[\left\{\begin{array}{l}G_{PMSMm}(s)=\frac{1}{L_{q}s+R}\\ G_{PMSMm}(s)=\frac{1}{l_{q}s+B}\end{array}\right. \tag{4}\]  
It is obviously that PMSM are a second order system composed of two inertial processes.  
Meanwhile, the phase voltage of the inverter module output is not in response to the input vector pulse synchronization, and the inverter does not do the voltage amplitude scale transformation. Hence, the inverter gain \(K_{in}=1\), and it is the initial equivalent of the driving part:  
\[G_{SVPWM}(s)=\frac{K_{in}}{T_{s}s+1} \tag{5}\]  
where \(T_{s}\) is defined the switching frequency.  
Unfortunately, the inverter is generally used to switch dead areas and delay in action. Therefore, the small inertia link is compensated for (5). The transfer function for an inertia element can be written:  
\[G_{inte}(s)=\frac{1}{t_{d}s+1} \tag{6}\]In (6), \(t_{d}\) is the inertia time constant.  
If the current controller is equivalent to a differential link and an integral link, the transfer function block diagram of PMSM sensorless system can be expressed in Figure 1.  
It is worth noting that the speed regulator control object \(G_{obj}(s)\) is made up of five poles and a zero. Among them, \(ASC\) and \(ACR\) are the automatic speed regulator and the automatic current regulator, respectively.