## 3 An Integral Sliding Mode Compensation Method with Weakened Integrals
### Weakened Integral Sliding Mode Compensation Method  
Equation (2) can be rewritten as follows when considering only the model parameter errors:  
\[\left\{\begin{array}{c}\frac{d_{ij}}{dt}=\frac{1}{L_{0}}u_{d}-\frac{R_{0}}{L_{0} }i_{d}+\omega_{e}i_{q}+f_{d}\\ \frac{d_{iq}}{dt}=\frac{1}{L_{0}}u_{q}-\frac{R_{0}}{L_{0}}i_{q}-\omega_{e}i_{d }-\frac{\psi_{e}\omega_{e}}{L_{0}}+f_{q}\\ f_{d}=\left(\frac{1}{L_{s}}-\frac{1}{L_{0}}\right)u_{d}-\left(\frac{R_{0}}{L_{ s}}-\frac{R_{0}}{L_{0}}\right)i_{d}\\ f_{q}=\left(\frac{1}{L_{s}}-\frac{1}{L_{0}}\right)u_{q}-\left(\frac{R_{0}}{L_{ s}}-\frac{R_{0}}{L_{0}}\right)i_{q}-\left(\frac{\psi_{e}}{L_{s}}-\frac{\psi_{ e}}{L_{0}}\right)\omega_{e}\end{array}\right. \tag{15}\]  
where \(f_{d}\) and \(f_{q}\) denote disturbances owing to parameter mismatches.  
The main idea of this compensation control method is that the output voltage is divided into two parts, \(u=u_{0}+u_{1}\), where \(u_{0}\) denotes the predictive voltage of the delayed compensation DPCC, which is the main control quantity, and \(u_{1}\) denotes the compensation voltage, which suppresses any disturbance.  
In the case of a very short current control period, the current error is set to the following:  
\[\left[\begin{array}{c}e_{d}\\ e_{q}\end{array}\right]=\left[\begin{array}{c}i_{d}(k+1)-i_{q}^{*}(k)\\ i_{q}(k+1)-i_{q}^{*}(k)\end{array}\right] \tag{16}\]  
where \(i_{d}^{*}(k)\) and \(i_{q}^{*}(k)\) are the \(dq\)-axis reference currents at the \(k\)th moment. Therefore, the sliding mode surface is designed as follows:  
\[\left[\begin{array}{c}S_{d}\\ S_{q}\end{array}\right]=\left[\begin{array}{c}e_{d}+m\rho\\ e_{q}+m\rho\end{array}\right] \tag{17}\]  
By combining (2) and (15) to take the derivative for the sliding mode surface, we obtain the following:  
\[\overset{\bullet}{S}=\left[\begin{array}{c}\overset{\bullet}{S}_{d}\\ \overset{\bullet}{S}_{q}\end{array}\right]=\left[\begin{array}{c}\frac{u_{ d}-u_{d0}}{L_{0}}+f_{d}+m\overset{\bullet}{\rho}_{d}\\ \frac{u_{q}-u_{d0}}{L_{0}}+f_{q}+m\overset{\bullet}{\rho}_{q}\end{array}\right] =\left[\begin{array}{c}\frac{u_{d1}}{L_{0}}+f_{d}+m\overset{\bullet}{\rho}_{ d}\\ \frac{u_{d1}}{L_{0}}+f_{q}+m\overset{\bullet}{\rho}_{q}\end{array}\right] \tag{18}\]  
The reaching law adopts the following, which adds the absolute value of the sliding mode surface to the isochronous reaching term of the conventional exponential reaching law. When the motion state of a system is further away from the surface, the reaching speed is higher, and when it is closer to the surface, the speed is lower; this can enable a smoother entry into the sliding mode surface and simultaneously weaken chattering [22].  
\[\overset{\bullet}{S}=-\varepsilon|S|\text{sgn}(S)-\alpha S \tag{19}\]  
where \(\varepsilon>0\) and \(\alpha>0\).  
By combining (18) and (19), we obtain the control law, which is also the \(dq\)-axis compensation voltage:  
\[u_{1}=\left[\begin{array}{c}u_{d1}\\ u_{q1}\end{array}\right]=\left[\begin{array}{c}L_{0}(-\varepsilon|S_{d}| \text{sgn}(S_{d})-\alpha S_{d})-L_{0}m\overset{\bullet}{\rho}_{d}\\ L_{0}(-\varepsilon|S_{q}|\text{sgn}(S_{q})-\alpha S_{q})-L_{0}m\overset{ \bullet}{\rho}_{q}\end{array}\right] \tag{20}\]