## 2 Mathematical Model of PMSM with Friction Disturbance  
The structural diagram of vector control (VC) for PMSMs is shown in Figure 1. The principle of vector control is to measure the stator current vector of the motor, and control the excitation current and torque current, respectively. By taking the rotor coordinate as a reference coordinate, the dynamic model of a PMSM can be described as follows [29]:  
\[u_{d}=Ri_{d}-\omega_{e}L_{d}i_{q}+L_{d}\frac{di_{d}}{dt} \tag{1}\]  
\[u_{q}=Ri_{q}+\omega_{e}L_{d}i_{d}+\omega_{e}\psi+L_{q}\frac{di_{q}}{dt} \tag{2}\]  
\[T_{e}=\frac{3}{2}p\big{[}\psi i_{q}+(L_{d}-L_{q})i_{d}i_{q}\big{]} \tag{3}\]  
\[J\ddot{\theta}=T_{e}-T_{f}-\xi_{o} \tag{4}\]  
where \(i_{d}\) and \(u_{d}\) are the current and voltage of d-axis, respectively; \(i_{q}\) and \(u_{q}\) are the current and voltage of q-axis, respectively; \(R\) is the stator resistance; \(L_{d}\) and \(L_{q}\) are the inductance on AC and DC sides, respectively; \(\omega_{e}\) is the electric angular velocity; \(\psi\) is the rotor flux linkage; \(T_{e}\) is the electromagnetic torque; \(p\) is the number of pole pairs; \(J\) is rotational inertia of the motor and load; \(T_{f}\) is the friction torque; \(\xi_{o}\) is the sum of residual uncertain disturbances and \(\theta\) is the angle of the rotor.  
Two current loops are included in the VC of the PMSM. By controlling \(i_{d}=0\) and substituting Equation (3) into Equation (4), the system model with two current loops is changed as:  
\[J\ddot{\theta}=\frac{3p\psi}{2}i_{q}^{*}-T_{f}-\xi_{o} \tag{5}\]  
By setting \(b=\frac{3p\psi}{2}\), \(u=i_{q}^{*}\), Equation (5) is rewritten as:  
\[J\ddot{\theta}=bu-T_{f}-\xi_{o} \tag{6}\]  
Figure 1: Vector control for PMSMs.  
Figure 2 shows the simplified diagram of the dynamic characteristics of PMSMs, in which the friction torque \(T_{f}\) can be accurately described by LuGre model [30]. The functions of the LuGre model are as follows:  
\[T_{f}=\sigma_{0}z+\sigma_{1}z+\sigma_{2}\dot{\theta} \tag{7}\]  
\[\dot{z}=\dot{\theta}-\frac{\left|\dot{\theta}\right|}{g(\dot{\theta})}z \tag{8}\]  
\[g(\dot{\theta})=\left[f_{c}+(f_{s}-f_{c})e^{-(\dot{\theta}/\dot{\theta}_{s})^{ 2}}\right]\text{sgn}(\dot{\theta})+k_{v}\dot{\theta} \tag{9}\]  
where \(\sigma_{0}\), \(\sigma_{1}\) and \(\sigma_{2}\) are the stiffness of the bristles, damping and viscous coefficient, respectively; \(z\) is the immeasurable internal state; \(g(\cdot)\) is nonlinear function; \(f_{c}\) is Coulomb friction; \(f_{s}\) is maximum static friction; \(k_{v}\) is viscous friction coefficient and \(\dot{\theta}_{s}\) is the Stribeck velocity.  
The mathematical model of PMSMs with friction disturbance is shown in Figure 3. By setting \([x_{1},x_{2},x_{3}]^{T}=\left[\theta,\dot{\theta},z\right]^{T}\) and \(f_{g}(\dot{\theta})=\frac{\left|\dot{\theta}\right|}{g(\dot{\theta})}\), the model of PMSMs with friction disturbance can be described by a third-order state equation as follows:  
\[\left\{\begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=\frac{1}{T}\big{[}(\sigma_{1}+\sigma_{2})x_{2}+(\sigma_{0}-\sigma _{1}f_{g}(x_{2})x_{3}+bu+\xi_{o}\big{]}\\ \dot{x}_{3}=x_{2}-f_{g}(x_{2})x_{3}\\ y=x_{1}\end{array}\right. \tag{10}\]