## 4 Design of the Current Loop for the Multi-PMSM Speed Control System  
Defined the CSMC tracking error as follows:  
\[\updownarrow=\left[ \begin{array}{c}e_{d,i}\\ e_{q,i}\end{array} \right]=\left[ \begin{array}{c}i_{d,i}^{*}-i_{d,i}\\ i_{q,i}^{*}-i_{q,i}\end{array} \right]=\varsigma^{*}-\varsigma \tag{29}\]  
\[\varsigma^{*}=\left[ \begin{array}{c}i_{d,i}^{*}\\ i_{q,i}^{*}\end{array} \right];\varsigma=\left[ \begin{array}{c}i_{d,i}\\ i_{q,i}\end{array} \right].\]  
By using the mathematical model of PMSM (1), the matrix from incorporating disturbances can be obtained as follows:  
\[\dot{\varsigma}=X_{\xi}+Yu^{*}+ZF \tag{30}\]  
\[u^{*}=\left[ \begin{array}{c}u_{d,i}\\ u_{q,i}\end{array} \right];X=-R_{s,i}/L_{i};Y=1/L_{i};Z=1;F=\left[ \begin{array}[ ]{c}\omega_{r,j}i_{q,i}\\ -\omega_{r,j}i_{d,i}-\frac{\Psi_{f,j}}{L_{i}}\end{array} \right].\]  
The PMSM system can be described as follows:  
\[\dot{\varsigma}=(X_{n}+\Delta X)\varsigma+(Y_{n}+\Delta Y)u^{*}+(Z_{n}+\Delta Z )F \tag{31}\]  
\(X_{n},Y_{n},Z_{n}\) are the nominal values of \(X,Y,Z;\Delta X,\Delta Y,\Delta Z\) denote the uncertainties introduced by system parameters; \(\zeta=\Delta X\varsigma+\Delta Yu^{*}+(Z_{n}+\Delta Z)F\) is the lumped uncertainty; and \(\zeta\) is assumed to be bounded. \[|\zeta|\leq\rho \tag{32}\]  
\(\rho\) represents a control constant. The integral sliding surface \(S_{1}\) and the complementary sliding surface \(S_{2}\) are as follows:  
\[S_{1}=\updownarrow+\lambda\int_{0}^{t}\left(\updownarrow+sign^{\alpha}(S_{2} -S_{1}))d\tau \tag{33}\]  
\[S_{2}=\updownarrow-\lambda\int_{0}^{t}\left(\updownarrow+sign^{\alpha}(S_{2} -S_{1})\right)d\tau \tag{34}\]  
\(\lambda\) and \(\alpha\) represent the given positive constant; \(sign(S_{2}-S_{1})\) represent the sign function, defined as:  
\[sign(S_{2}-S_{1})=\left\{ \begin{array}{c}1,\ (S_{2}-S_{1})>0\\ 0,\ (S_{2}-S_{1})=0\\ -1,\ (S_{2}-S_{1})<0\end{array} \right.\]  
By substituting (29)-(31) into (33) and (34) and taking the derivatives, the following equations can be obtained:  
\[ \begin{array}{l}\dot{S}_{1}\\ =\updownarrow+\lambda[\updownarrow+sign^{\alpha}(S_{2}-S_{1})]\\ =\dot{\varsigma}^{*}-X_{n}\varsigma-Y_{n}u^{*}-\zeta+\lambda\updownarrow+ \lambda sign^{\alpha}(S_{2}-S_{1})\end{array}  \tag{35}\]\[ \begin{array}{ll}\dot{S}_{2}&=\dot{\updownarrow}-\lambda[\updownarrow+sign^{a}(S_{2 }-S_{1})]\\ &=\dot{\upxi}^{*}-X_{n}\varsigma-Y_{n}u^{*}-\zeta-\lambda\updownarrow-\lambda sign ^{a}(S_{2}-S_{1})\end{array}  \tag{36}\]  
The relationship between \(S_{1}\) and \(S_{2}\) is defined as follows:  
\[ \begin{array}{ll}S(\updownarrow)=S_{1}(\updownarrow)+S_{2}(\updownarrow) \end{array}  \tag{37}\]  
The following equation can be obtained by using (33)-(36)  
\[ \begin{array}{ll}\dot{S}_{2}&=\dot{S}_{1}-\lambda(S_{1}+S_{2})-2\lambda sign ^{a}(S_{2}-S_{1})\\ &=\dot{S}_{1}-\lambda S-2\lambda sign^{a}(S_{2}-S_{1})\end{array}  \tag{38}\]  
The control law \(u^{*}\) of the CSMC is defined as follows:  
\[u^{*}=u^{*}_{eq}+u^{*}_{hit} \tag{39}\]  
\[u^{*}_{eq}=\left[ \begin{array}{c}u^{*}_{deg}\\ u^{*}_{qeq}\end{array} \right]=\left[ \begin{array}{c}\frac{1}{\tau_{n}}\left( \dot{\upxi}^{*}-X_{n}\varsigma+\lambda\updownarrow+\lambda S_{1}\right)\\ \frac{1}{\tau_{n}}\left(\dot{\upxi}^{*}-X_{n}\varsigma+\lambda\updownarrow+ \lambda S_{1}\right)\end{array} \right] \tag{40}\]  
\[u^{*}_{hit}=\left[ \begin{array}{c}u^{*}_{dhit}\\ u^{*}_{dhit}\end{array} \right]=\left[ \begin{array}{c}\frac{1}{\tau_{n}} \left[\rho sst(\frac{S}{\varphi})\right]\\ \frac{1}{\tau_{n}}\left[\rho sst(\frac{S}{\varphi})\right]\end{array} \right] \tag{41}\]  
\[sat(\frac{S}{\varphi})=\left\{ \begin{array}{c}1,S\geq\varphi\\ \frac{S}{\varphi},-\varphi<S<\varphi\\ -1,S\leq-\varphi\end{array} \right. \tag{42}\]  
Define \(u^{*}_{eq}\) as the equivalent control law; \(u^{*}_{hit}\) as the switching control rate; \(sat(\cdot)\) as the saturation function; and \(\varphi\) as the boundary layer thickness.