## 3 Principles and Modeling of an Adaptive Control Method for PMSMs Based on Super-Helical SMC
### MRAS-Based PMSM Position Sensorless Control System
Additionally, it is established that the nonlinear time-invariant feedback channel satisfies the Popov integral inequality under the rotational speed estimation formula given by Equation (9). \[\mathbf{C}=\left[ \begin{array}{cc}\frac{L_{q}}{L_{q}}&0\\ 0&\frac{L_{q}}{L_{d}}\end{array} \right] \tag{8}\]  
\[\hat{\omega}=(\frac{K_{i}}{s}+K_{p})\left[ \begin{array}{cc}I_{d}\hat{I}_{q} -\hat{I}_{d}\hat{i}_{q}-\frac{\Psi_{f}}{L_{d}}(i_{q}-\hat{i}_{q})\end{array}  \right]+\hat{\omega}(0) \tag{9}\]The model-referenced adaptive rotor position and speed estimation system based on stator currents are shown in Figure 3.  
The reference model initially receives the estimated electrical angle obtained from the adaptive rate and generates the three-phase current (\(\mathrm{i_{abc}}\)) and voltage (\(\mathrm{u_{abc}}\)) parameters using reference model equations. These parameters are then transformed from the natural coordinate system to the rotating coordinate system through coordinate transformation. Concurrently, the adjustable model receives the estimated rotational speed via the adaptive law. Utilizing Equation (6), the adjustable model computes the voltage values (\(\mathrm{u_{d}}\) and \(\mathrm{u_{q}}\)) passed by the reference model, producing an estimated alternating shaft current. Subsequently, this current is fed into the adaptive law. The adaptive law, in turn, receives both the actual alternating shaft currents (\(\mathrm{i_{d}}\) and \(\mathrm{i_{q}}\)) from the reference model, as well as the estimated alternating shaft currents (\(\mathrm{i_{d}}\) and \(\mathrm{i_{q}}\)) from the adaptive law's rotational speed estimation, as defined by Equation (9). By integrating these inputs, the adaptive law computes the estimated rotational speed \(\omega\). This estimated speed is further processed by the adaptive law speed estimation Equation (9) to produce the estimated rotor position angle \(\theta\). This angle, representing the estimated rotor speed \(\omega\), is then fed back to the reference model, thereby enabling closed-loop control of the motor system without the need for a position sensor. Figure 3: Planetary gear and working modes. Figure 2: Standardized feedback systems. System Stability Proof  
As per the Popov hyperstability theory, to ensure the asymptotic stability of the system, the following two conditions must be satisfied:  
(1) As per the Positive Real Lemma, the transfer function matrix \(\mathbf{G}(s)=\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\) of the linear time-invariant forward path is a strictly positive real matrix. For the following state-space representation of the linear time-invariant system, the conditions of the lemma apply:  
\[ \begin{cases}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t)\\ \mathbf{y}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{u}(t)\end{cases}  \tag{10}\]  
The necessary and sufficient condition for its transfer function matrix \(\mathbf{G}(s)=\mathbf{D}+\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}\) to be a strictly positive real matrix is the existence of a symmetric positive definite matrix \(\mathrm{P}\) and real matrices \(\mathrm{K}\) and \(\mathrm{L}\), with positive real numbers, or a symmetric positive definite matrix \(\mathrm{Q}\) satisfying:  
\[\left\{ \begin{array}{l}\mathbf{P}\mathbf{A}+\mathbf{A}^{\mathrm{T}}\mathbf{P}=-\mathbf{L}\mathbf{L}^ {\mathrm{T}}-2\lambda\mathbf{P}=-\mathbf{Q}\\ \mathbf{B}^{\mathrm{T}}\mathbf{P}+\mathbf{K}^{\mathrm{T}}\mathbf{L}^{\mathrm{T}}=\mathbf{C}\\ \mathbf{K}^{\mathrm{T}}\mathbf{K}=\mathbf{D}+\mathbf{D}^{\mathrm{T}}\end{array} \right. \tag{11}\]  
For Equation (7), since \(\mathrm{B}=\mathrm{I}\) and \(\mathrm{D}=0\), Equation (11) can be simplified to:  
\[\left\{ \begin{array}{l}\mathbf{P}\mathbf{A}+\mathbf{A}^{\mathrm{T}}\mathbf{P}=-\mathbf{Q}\\ \mathbf{P}=\mathbf{C}\end{array} \right. \tag{12}\]  
Choose:  
\[\mathbf{C}= \begin{bmatrix}\frac{L_{d}}{L_{d}}&0\\ 0&\frac{L_{d}}{L_{d}}\end{bmatrix} \]  
Then, according to Equation (12), \(\mathrm{P}=\mathrm{C}\) is also a symmetric positive definite matrix.