## 3 Principles and Modeling of an Adaptive Control Method for PMSMs Based on Super-Helical SMC  
### MRAS-Based PMSM Position Sensorless Control System  
The principle behind Model Reference Adaptive Systems (MRAS) involves treating the parameters to be identified as adjustable parameters. Here, the expression containing these parameters serves as the adjustable model, while the expression devoid of unknown parameters acts as the reference model. Ensuring the physical significance of \(x\) and \(\hat{x}\), the difference between them forms the generalized error \(e\). Designing a suitable adaptive law  
Figure 1: Equivalent circuit diagram in d-q coordinates. facilitates the adjustment of the parameters in the adjustable model, thereby minimizing \(e\) and ensuring the asymptotic stability of the system output. Ultimately, this process enables accurate identification of the system parameters. For the embedded PMSM, the voltage equation in the synchronous rotating coordinate system is represented by Equations (1) and (2). To facilitate analysis and design, it is rewritten as follows:  
\[\left\{ \begin{array}{l}\frac{d}{dt}I_{d}=-\frac{R_{e}}{L_{d}}I_{d}+\omega_{e} \frac{L_{q}}{L_{q}}I_{q}+\frac{1}{L_{d}}V_{d}\\ \frac{d}{dt}I_{q}=-\frac{R_{e}}{L_{q}}I_{q}-\omega_{e}\frac{L_{q}}{L_{q}}I_{d}- \frac{\Psi_{f}}{L_{q}}\omega_{e}+\frac{1}{L_{q}}V_{q}\end{array} \right. \tag{3}\]  
To obtain an adjustable model, Equation (3) can be changed to the following:  
\[\frac{d}{dt}\mathbf{i}^{\prime}=\mathbf{A}\mathbf{i}^{\prime}+\mathbf{B}\mathbf{u}^{\prime} \tag{4}\]  
where \(\mathbf{i}^{\prime}=\left[ \begin{array}{c}I_{d}^{\prime}\\ I_{q}^{\prime}\end{array} \right]\), \(\mathbf{u}^{{}^{\prime}}=\left[ \begin{array}{c}V_{d}^{\prime}\\ V_{q}^{\prime}\end{array} \right]\), \(\mathbf{A}=\left[ \begin{array}{cc}-\frac{R_{u}}{L_{d}}&\omega\frac{L_{q}}{L_{d} }\\ -\omega\frac{L_{d}}{L_{q}}&-\frac{R_{e}}{L_{q}}\end{array} \right]\) and \(\mathbf{B}=\left[ \begin{array}{cc}\frac{1}{L_{d}}&0\\ 0&\frac{1}{L_{q}}\end{array} \right]\). Expressing Equation (4) in terms of estimates can be abbreviated as follows:  
\[\frac{d}{dt}\mathbf{\hat{i}}^{\prime}=\mathbf{\hat{A}}\mathbf{\hat{i}}^{\prime}+\mathbf{B}\bm {u}^{{}^{\prime}} \tag{5}\]  
Included among these, \(\mathbf{\hat{i}}^{\prime}=\left[ \begin{array}{c}\hat{i}^{\prime}\\ \hat{i}_{q}^{\prime}\end{array} \right]\), \(\mathbf{\hat{A}}=\left[ \begin{array}{cc}-\frac{R_{u}}{L_{d}}&\omega_{e}\frac{L_ {q}}{L_{d}}\\ -\omega_{e}\frac{L_{q}}{L_{q}}&-\frac{R}{L_{q}}\end{array} \right]\). Since the matrices \(A\) and \(\mathbf{\hat{A}}\) both have rotor information, it is possible to use Equation (5) as the adjustable model, Equation (3) as the reference model, and \(\omega\) as the adjustable parameter. The generalized error is defined as \(\mathbf{e}=\mathbf{i}^{{}^{\prime}}-\mathbf{\hat{i}}^{{}^{\prime}}\), which is obtained by subtracting Equations (4) and (5), shown in the following:  
\[\frac{d}{dt} \begin{bmatrix}e_{d}\\ e_{q}\end{bmatrix} = \begin{bmatrix}-\frac{R_{u}}{L_{d}}&\omega\frac{L_{q}}{L_ {d}}\\ -\omega\frac{L_{d}}{L_{q}}&-\frac{R_{e}}{L_{q}}\end{bmatrix}\begin{bmatrix}e_{d} \\ e_{q}\end{bmatrix} -J(\hat{\omega}-\omega) \begin{bmatrix}\hat{i}^{\prime}_{q} \\ \hat{i}_{q}^{\prime}\end{bmatrix}  \tag{6}\]  
It can be simplified as follows:  
\[\left\{ \begin{array}{l}\dot{\mathbf{e}}=\mathbf{A}\mathbf{e}-\mathbf{W}\\ \mathbf{V}=\mathbf{C}\mathbf{e}\end{array} \right. \tag{7}\]  
Figure 2 illustrates the standard feedback system derived from Equation (7). In the upper half of the dashed box, the forward channel represents a linear time-invariant system, with C denoting a linear compensator. Meanwhile, the lower half of the dashed box represents a nonlinear time-varying feedback channel. In this research project, the Popov super stability theory is employed to demonstrate that the transfer function matrix G(s) of the linear time-invariant forward channel becomes a strictly positive real matrix when the linear compensator conforms to Equation (8).