# Research on Output Voltage Stability of Non-Contact Excitation Motor
## 4 Comparison of Traditional Sliding Mode Control and Improved Sliding Mode Control
### Improved Sliding Mode Control  
As shown in Figure 6, traditional sliding mode control is often used in circuit transformation. Compared with PI adjustment, the adjustment time of sliding mode control is shorter, and there is no overshoot. However, traditional sliding mode control can no longer meet the actual conditions, increasing control accuracy and robustness requirements. Therefore, to improve the convergence speed and robustness of the current, conventional sliding mode control should be used. Through the state space method, the state equation of the buck circuit is derived. The traditional sliding mode control has many shortcomings, so the improved sliding mode control method is adopted. Two variables are designed in the system, in which the output voltage is set to \(x_{1}\), and \(x_{2}\) is the derivative of the output voltage, which is expressed with the following formula:  
\[x_{1}=v_{c} \tag{11}\]  
\[x_{2}=\dot{x}_{1}=\frac{dv_{c}}{dt}=\frac{1}{c}\dot{t}_{c}=\frac{1}{c}(\dot{t} _{L}-\frac{v_{c}}{R}) \tag{12}\]  
According to Equations (11) and (12), it can be calculated  
\[\dot{t}_{L}=\int\frac{uE-v_{c}}{L}dt \tag{13}\]  
\[x_{2}=\dot{x}_{1}=\int\frac{uE-v_{c}}{LC}-\frac{v_{c}}{RC}dt \tag{14}\]  
Further, the derivative operation is performed on Equation (14) to obtain the derivative of \(x_{2}\). \[\dot{x}_{2}=-\frac{v_{c}}{LC}-\frac{1}{RC}\frac{dv_{c}}{dt}+\frac{uE}{LC}=- \frac{1}{LC}x_{1}-\frac{1}{RC}x_{2}+\frac{E}{LC}u \tag{15}\]  
In summary, at this time, the state equation of the system can be expressed as  
\[\left\{ \begin{array}{c}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=-\frac{1}{LC}x_{1}-\frac{1}{RC}x_{2}+\frac{E}{LC}u\end{array} \right. \tag{16}\]  
where \(T\) is the sampling period, using the discrete rules, the state equation is rewritten as  
\[\left\{ \begin{array}{c}x_{1,k+1}=x_{1,k}+Tx_{2,k}\\ x_{2,k+1}=x_{2,k}-\frac{T}{LC}x_{1,k}-\frac{T}{RC}x_{2,k}+\frac{ET}{LC}u_{k} \end{array} \right. \tag{17}\]  
To facilitate the subsequent calculation, the formula is transformed  
\[M(x_{k})=x_{2,k}-\frac{T}{LC}x_{1,k}-\frac{T}{RC}x_{2,k},N(x_{k})=\frac{ET}{LC} \tag{18}\]  
After the above formulas are replaced, the state equation of the system is more concise and convenient. At this time, the new state equation is  
\[\left\{ \begin{array}{c}x_{1,k+1}=x_{1,k}+Tx_{2,k}\\ x_{2,k+1}=M(x_{k})+N(x_{k})u_{k}\end{array} \right. \tag{19}\]  
Figure 6: Power type fast discrete terminal sliding mode control. In the sliding mode control algorithm, the algorithm of the sliding mode surface is a core process, and a new sliding mode surface is designed at this time. \[s_{k}=he_{2,k}+\alpha g(e_{1,k-1})+\beta e_{1,k-1}^{\frac{q}{p}} \tag{20}\]  
\[g(e_{1,k-1})=\left\{ \begin{array}{c}g_{1}(e_{1,k-1}),if\big{|}e_{1,k-1}\big{|} \leq 1\\ g_{2}(e_{1,k-1}),if\big{|}e_{1,k-1}\big{|}>1\end{array} \right. \tag{21}\]  
\[g(e_{1,k-1})=\left\{ \begin{array}{c}g_{1}(e_{1,k-1})=bs\text{gn}e_{1,k-1}\log _{a}(2c\big{|}e_{1,k-1}\big{|}-e_{1,k-1}{}^{2}+1)\\ g_{2}(e_{1,k-1})=g_{1}(\text{sgn}e_{1,k-1})+(e_{1,k-1}-\text{sgn}e_{1,k-1})^{ \frac{1}{8}}\end{array} \right. \tag{22}\]  
According to Equations (20) and (21), we can obtain  
\[s_{k+1}=he_{2,k+1}+\alpha g(e_{1,k})+\beta e_{1,k}^{\frac{q}{p}} \tag{23}\]  
\[s_{k+1}=h(r_{2,k+1}-(M(x_{k})+N(x_{k})u_{k}))+\alpha g(e_{1,k})+\beta e_{1,k} ^{\frac{q}{p}} \tag{24}\]  
\[g(e_{1,k})=\left\{ \begin{array}{c}bs\text{gn}e_{1,k}\log_{a}(2c\big{|}e_{1, k}\big{|}-(e_{1,k}{}^{2}+1)),if\big{|}e_{1,k}\big{|}\leq 1\\ g_{1}(\text{sgn}e_{1,k})+(e_{1,k}-\text{sgn}e_{1,k})^{\frac{q}{p}},if\big{|} e_{1,k}\big{|}>1\end{array} \right. \tag{25}\]  
In the above formula, \(e_{1}\) is the output voltage tracking error, and \(e_{2}\) is the current tracking error of the capacitor. To make the system state quickly reach the PFDTSMC sliding mode surface in a limited time, the exponential reaching law was chosen.