## 3 Principles and Modeling of an Adaptive Control Method for PMSMs Based on Super-Helical SMC
### Model-Referenced Adaptive Control System Based on Super-Helical Sliding Mode Control  
This research proposes a super-helical sliding mode control to replace the PI adaptive control parameters in Model Reference Adaptive Systems (MRASs), as discussed in the previous section. This approach aims to address the issue of weak robustness associated with traditional MRASs. The basic form of this super-helical sliding mode control is presented below:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=-k_{1}|x_{1}|^{1/2}\text{sgn}(x_{1})+x_{2}\\ \dot{x}_{2}=-k_{2}\text{sgn}(x_{1})\end{array} \right. \tag{26}\]  
The sign function, sgn(), is a discontinuous function, and its presence leads to chattering issues in sliding mode control. However, as shown in Equation (26), super-twisting sliding mode control (STSMC) essentially acts as a second-order sliding mode controller. By introducing continuous terms before the sgn() function and placing sgn() in the higher-order derivative, the traditional chattering problem in sliding mode control is alleviated to a certain extent. Moreover, STSMC also exhibits robust performance under various disturbances and uncertainties. Combined with Equation (26), the formulated slip film surface is as follows:  
\[s=i_{d}\dot{i}_{q}-\dot{i}_{d}\dot{i}_{q}-\frac{\psi_{f}}{L_{d}}(i_{q}-\dot{i} _{q}) \tag{27}\]  
The derivation of Equation (27) involves the application of the basic principle of sliding mode variable structure. Specifically, when the system enters the sliding mode surface, \(s=s^{\prime}=0\), the equivalent velocity expression is as follows:  
\[\omega_{eq}=\omega_{e}+\frac{(\frac{R}{L_{d}}+\frac{R}{L_{d}})(\dot{i}_{d} \dot{i}_{q}-\dot{i}_{d}\dot{i}_{q})+\frac{u_{q}}{L_{q}}(i_{d}-\dot{i}_{d})+( \frac{R\psi_{f}}{L_{d}\dot{i}_{q}}-\frac{u_{d}}{L_{d}})(i_{q}-\dot{i}_{q})}{ \frac{L_{d}}{L_{d}}i_{d}\dot{i}_{d}+\frac{L_{d}}{L_{d}}i_{q}\dot{i}_{q}+\frac{ \psi_{f}}{L_{d}}(i_{d}+\dot{i}_{d})+\frac{\psi_{f}^{2}}{L_{d}\dot{i}_{d}}} \tag{28}\]  
Based on the equivalent velocity and sliding mode surface, the velocity observer is designed in conjunction with the super-helical sliding mode control. Equation (29) presents the velocity prediction expression and the rotor position estimate is obtained by integrating Equation (29). \[\hat{\omega}_{e}=k_{1}|s|^{1/2}\text{sgn}(s)+\int_{t}^{0}k_{2}\text{sgn}(s)dt +\hat{\omega}_{e}(0) \tag{29}\]  
System Stability Proof  
According to the Lyapunov stability theorem, the state vector is selected as follows:  
\[\mathbf{Z}= \begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix} ^{\mathsf{T}}= \begin{bmatrix}| x_{1}|^{\frac{1}{2}}\text{sgn}(x_{1})&x_{2}\end{bmatrix} ^{\mathsf{T}} \tag{30}\]  
In this case, \(Z_{1}\) and \(Z_{2}\) are selected as the state variables. From Equation (30), it is evident that if the state variables \(Z_{1}\) and \(Z_{2}\) can be shown to converge to zero in finite time, then x\({}_{1}\) and x\({}_{2}\) will also converge to zero in finite time. Combined with Equation (26), this confirms that the system states will reach the sliding mode surface and converge to zero in finite time. Taking the time derivative of the state vector Z, combined with Equation (26), yields the following:  
\[ \begin{array}{l}\dot{\mathbf{Z}}=\left[\begin{array}{c}\frac{1}{2|x_{1}|^{ 2}}\dot{x}_{1}\\ \dot{x}_{2}\end{array}\right]=\left[\begin{array}{c}\frac{1}{2|x_{1}|^{ 2}}\begin{bmatrix}-k_{1}|x_{1}|^{\frac{1}{2}}\text{sgn}(x_{1})+x_{2}\\ -k_{2}\text{sgn}(x_{1})\end{bmatrix}\\ =\frac{1}{|x_{1}|^{2}}\begin{bmatrix}-\frac{k_{1}}{2}&\frac{1}{2}\\ -k_{2}&0\end{bmatrix}\end{array}\right]\left[\begin{array}{c}|x_{1}|^{\frac{ 1}{2}}\text{sgn}(x_{1})\\ x_{2}\end{array}\right]\\ =\frac{1}{|x_{1}|^{2}}\mathbf{F}\mathbf{Z}\end{array}  \tag{31}\]  
In this context, \(\mathbf{F}= \begin{bmatrix}-\frac{k_{1}}{2}&\frac{1}{2}\\ -k_{2}&0\end{bmatrix} \). We construct the following Lyapunov function, where \(\mathbf{N}=\frac{1}{2} \begin{bmatrix}k_{1}^{2}+4k_{2}&-k_{1}\\ -k_{1}&2\end{bmatrix} \).