## 2 Mathematical Methods and Experimental Details
### Design of the DSVC+DDC  
A GS considering disturbance and parameter perturbation is expressed as follows:  
\[X(k+1)=\Psi X(k)+\Delta\Psi X(k)+\Gamma u(k)+f(k) \tag{8}\]  
where \(X(k)=\left[ \begin{array}{c}\theta(k)\\ \omega(k)\end{array} \right]\), and \((\Psi,\ \Gamma)\) is controllable. **Assumption 1**. : _The parameter perturbation and disturbance satisfy \(\Delta\Psi=\Gamma\widetilde{\Psi}\), \(f=\Gamma\widetilde{f}\), where \(\widetilde{\Psi}\), \(\widetilde{f}\) are uncertain._  
Then, Equation (8) can be rewritten as:  
\[X(k+1)=\Psi X(k)+\Gamma(u(k)+d(k)) \tag{9}\]  
where \(d(k)\) represents the total external disturbance and \(d(k)=\widetilde{\Psi}X(k)+\widetilde{f}(k)\), \(d(k)\leq\zeta\), in which \(\zeta\) is a constant. The switching function \(s(k)\) is defined as:  
\[s(k)=Ce^{T}X_{e}(k),\ Ce=\left[c,\ 1\right]^{T},\ c>0 \tag{10}\]  
where \(X_{e}(k)=X(k)-R(k)\) and \(R(k)=\left[ \begin{array}{c}\theta_{r}(k)\\ \omega_{r}(k)\end{array} \right]\), where \(\theta_{r}(k),\omega_{r}(k)\) separately represent the expected position and expected rotational speed of the rotor at time \(k\). The DDC is used for the online estimation of and compensation for the external disturbance. For the system shown in Equation (8), the switching function in Equation (10) is selected on the condition of meeting Assumption 1. Then, according to [16; 17; 18; 19; 20; 21]:  
\[u(k)=\Big{(}Ce^{T}\Gamma\Big{)}^{-1}\Big{[}Ce^{T}R(k+1)-Ce^{T}\Psi X(k)+\alpha s (k)-\beta\text{sgn}(s(k))\Big{]}-\hat{d}(k) \tag{11}\]  
\[\hat{d}(k)=\hat{d}(k-1)+\Big{(}Ce^{T}\Gamma\Big{)}^{-1}g[s(k)-\alpha s(k-1)+ \beta\text{sgn}(s(k-1))] \tag{12}\]  
Figure 2: The linear dynamic model pertaining to the operation of the motor. where \(\alpha\), \(\beta\), and \(g\) are positive real numbers; \(\hat{d}(k)\) is the estimated external disturbance at time \(k\); and \(R(k+1)\) can be obtained by linear extrapolation. The decrease of chattering is a focus of much research on sliding-mode variable-structure control, and various methods that aim at this have been proposed [22; 23; 24; 25; 26]. [PERSON] et al. [16; 18] further weakened the chattering of sliding-mode controllers during computer numerical control (CNC) machining using a saturation function. Despite this, the method does not improve the performance of such controllers when used in GSs and even degrades the dynamic performance of step-responses in some cases. Therefore, the saturation function is not used to replace the symbolic functions in Equations (11) and (12). By combining Equations (9)-(11), it follows that:  
\[ \begin{array}{c}s(k+1)=Ce^{T}X_{e}(k+1)\\ =Ce^{T}X(k+1)-Ce^{T}R(k+1)\\ =Ce^{T}\Psi X(k)+Ce^{T}\Gamma u(k)+Ce^{T}\Gamma d(k)-Ce^{T}R(k+1)\\ =\alpha s(k)-\beta\text{sgn}(s(k))+Ce^{T}\Gamma\tilde{d}(k)\end{array}  \tag{13}\]  
where \(\tilde{d}(k)=d(k)-\hat{d}(k)\), and \(\tilde{d}(k)\) is the error of disturbance estimation. The following can be obtained by substituting Equations (12) and (13) into \(\tilde{d}(k)=d(k)-\hat{d}(k)\):  
\[ \begin{array}{c}\tilde{d}(k+1)=d(k+1)-\hat{d}(k+1)\\ =d(k+1)-\hat{d}(k)\\ -\left(Ce^{T}\Gamma\right)^{-1}g(s(k+1)-\alpha s(k)+\beta\text{sgn}(s(k)))\\ =d(k+1)-d(k)+\hat{d}(k)-\left(Ce^{T}\Gamma\right)^{-1}g(Ce^{T}\Gamma)\tilde{d}( k)\\ =d(k+1)-d(k)+(1-g)\tilde{d}(k)\end{array}  \tag{14}\]  
For the convergence of Equation (14), there is a positive constant \(m\). If \(\left|d(k+1)-d(k)\right|<m\), then there is \(k_{0}\); when \(k>k_{0}\), \(\tilde{d}(k)<\frac{m}{g}\) is established, in which \(0<g<1\). In the case where (a)-(c) are satisfied at the same time, the closed-loop system is stable and:  
When \(\left|s(k)\right|\geq Ce^{T}\Gamma\frac{m}{g}+\beta\),  
\(s(k+1)^{2}<s(k)^{2}\),  
When \(\left|s(k)\right|<Ce^{T}\Gamma\frac{m}{g}+\beta\),  
\(\left|s(k+1)\right|<Ce^{T}\Gamma\frac{m}{g}+\beta\). (a) \(0\leq\alpha\leq 1,\ 0<g<1\);  
(b) \(\left|d(k+1)-d(k)\right|<m\), in which \(m\) is a positive constant;  
(c) \(Ce^{T}\Gamma\frac{m}{g}<\beta\).