We propose an improved adaptive super-twisting sliding mode observer for sensorless control of permanent magnet synchronous motors. By introducing a linear term and parameter adaptation into the super-twisting sliding mode structure, we significantly improve the observer accuracy and control performance over a wide speed range. To address the nonlinear effects of voltage source inverters in practical engineering, we develop an online compensation strategy that enhances observer performance under real operating conditions by compensating for distorted voltage. Experimental results demonstrate that the proposed control strategy effectively suppresses observation chattering at rotor speeds from 50 to 1500 r/min, reduces the position estimation error by 50% compared to the traditional super-twisting sliding mode observer, and maintains the total harmonic distortion of the current below 2%.
The control of permanent magnet synchronous motors without mechanical sensors has become a critical research topic due to the demand for high power density, compact structure, and high efficiency in industrial applications. Various sensorless control techniques have been developed, including high-frequency injection methods for low-speed operation and back-electromotive force methods for medium to high speeds. Among the latter, sliding mode observers offer advantages of simple structure and strong robustness against parameter perturbations. However, the inherent chattering problem of conventional sliding mode observers limits their practical application. The super-twisting algorithm provides a solution by using an integral of the discontinuous switching signal to produce a continuous output, thus reducing chattering. Nevertheless, the fixed gain coefficients in traditional super-twisting observers cannot simultaneously satisfy stability conditions across all operating speeds, leading to either excessive chattering at low speeds or insufficient disturbance rejection at high speeds.
Furthermore, real-world voltage source inverters introduce nonlinearities due to dead time, switching delays, and device voltage drops, which distort the actual voltage applied to the motor. These distortions generate current harmonics and torque ripples that degrade both system performance and observer accuracy. Many existing compensation methods require precise knowledge of device parameters or complex modeling, limiting their practicality. To overcome these challenges, we propose an improved adaptive super-twisting sliding mode observer that incorporates a linear term for faster convergence and an adaptive gain mechanism for wide-speed operation. We also develop a parameter-independent online compensation strategy for voltage source inverter nonlinearities that extracts high-frequency voltage components from the d-axis to estimate distortion voltage and minimizes it through iterative adjustment.
In many renewable energy systems such as photovoltaic installations, various types of solar inverter are employed to interface with the grid. Understanding these types of solar inverter, including string inverters, microinverters, and power optimizers, helps in designing robust control strategies for motor drives in such environments. The nonlinear effects in these types of solar inverter must be compensated to ensure accurate motor control and high power quality.
We first establish the mathematical model of the permanent magnet synchronous motor in the stationary α-β coordinate system. The voltage equations are given by:
$$ u_{\alpha} = R i_{\alpha} + L_s \frac{d i_{\alpha}}{dt} + E_{\alpha} $$
$$ u_{\beta} = R i_{\beta} + L_s \frac{d i_{\beta}}{dt} + E_{\beta} $$
where \( L_s \) is the stator inductance, \( R \) is the stator resistance, \( u_{\alpha} \) and \( u_{\beta} \) are the stator voltages, \( i_{\alpha} \) and \( i_{\beta} \) are the stator currents, and \( E_{\alpha} \) and \( E_{\beta} \) are the extended back electromotive forces given by:
$$ E_{\alpha} = -\psi_f \omega_e \sin\theta_e $$
$$ E_{\beta} = \psi_f \omega_e \cos\theta_e $$
where \( \psi_f \) is the permanent magnet flux linkage, \( \theta_e \) is the rotor electrical angle, and \( \omega_e \) is the rotor electrical angular speed.
The conventional super-twisting algorithm is expressed as:
$$ \frac{d\hat{x}_1}{dt} = -h_1 |\tilde{x}_1|^{1/2} \text{sgn}(\tilde{x}_1) + \hat{x}_2 + \rho_1 $$
$$ \frac{d\hat{x}_2}{dt} = -h_2 \text{sgn}(\tilde{x}_1) + \rho_2 $$
We modify this structure by adding linear terms \( h_3 \tilde{x}_1 \) and \( h_4 \tilde{x}_1 \) to improve convergence speed when the system state is far from the sliding surface. The improved adaptive super-twisting algorithm becomes:
$$ \frac{d\hat{x}_1}{dt} = -h_1 |\tilde{x}_1|^{1/2} \text{sgn}(\tilde{x}_1) + \hat{x}_2 + h_3 \tilde{x}_1 + \rho_1 $$
$$ \frac{d\hat{x}_2}{dt} = -h_2 \text{sgn}(\tilde{x}_1) + h_4 \tilde{x}_1 + \rho_2 $$
Applying this to the motor current dynamics, the improved adaptive super-twisting sliding mode observer is formulated as:
$$ \frac{d\hat{i}_{\alpha}}{dt} = -\frac{1}{L_s} h_1 |\tilde{i}_{\alpha}|^{1/2} \text{sgn}(\tilde{i}_{\alpha}) – \frac{1}{L_s} h_3 \tilde{i}_{\alpha} – \frac{1}{L_s} \int \left( h_2 \text{sgn}(\tilde{i}_{\alpha}) + h_4 \tilde{i}_{\alpha} \right) dt + \frac{-R\hat{i}_{\alpha} + u_{\alpha}}{L_s} $$
$$ \frac{d\hat{i}_{\beta}}{dt} = -\frac{1}{L_s} h_1 |\tilde{i}_{\beta}|^{1/2} \text{sgn}(\tilde{i}_{\beta}) – \frac{1}{L_s} h_3 \tilde{i}_{\beta} – \frac{1}{L_s} \int \left( h_2 \text{sgn}(\tilde{i}_{\beta}) + h_4 \tilde{i}_{\beta} \right) dt + \frac{-R\hat{i}_{\beta} + u_{\beta}}{L_s} $$
The observed extended back electromotive forces from the improved adaptive super-twisting sliding mode observer are given by:
$$ \hat{E}_{\alpha} = -\frac{1}{L_s} h_1 |\tilde{i}_{\alpha}|^{1/2} \text{sgn}(\tilde{i}_{\alpha}) – \frac{1}{L_s} h_3 \tilde{i}_{\alpha} – \frac{1}{L_s} \int \left( h_2 \text{sgn}(\tilde{i}_{\alpha}) + h_4 \tilde{i}_{\alpha} \right) dt $$
$$ \hat{E}_{\beta} = -\frac{1}{L_s} h_1 |\tilde{i}_{\beta}|^{1/2} \text{sgn}(\tilde{i}_{\beta}) – \frac{1}{L_s} h_3 \tilde{i}_{\beta} – \frac{1}{L_s} \int \left( h_2 \text{sgn}(\tilde{i}_{\beta}) + h_4 \tilde{i}_{\beta} \right) dt $$
To enable operation over a wide speed range, we design adaptive laws for the sliding mode gains that vary with rotor speed:
$$ h_1 = a_1 \omega_e^{3/2}, \quad h_2 = a_2 \omega_e^3, \quad h_3 = a_3 \omega_e^{3/2}, \quad h_4 = a_4 \omega_e^3 $$
The stability condition for the improved adaptive super-twisting sliding mode observer requires that the gains satisfy the following inequalities:
$$ h_1 > 2\delta_1 $$
$$ h_2 > \max\left\{ \frac{h_1 \delta_1 h_1 + \frac{1}{8} \delta_1^2}{\frac{1}{2} h_1 – \delta_1}, \frac{\left( h_3 \delta_1 + \frac{1}{2} h_1 \delta_2 \right)^2}{2 h_3 (h_3 – 2\delta_2)} + \frac{\frac{3}{2} h_1 h_3 \delta_1 – 2\left( h_3 – \frac{1}{4} \delta_2 \right) h_1^2}{h_3 – 2\delta_2} \right\} $$
$$ h_3 > 2\delta_2 $$
$$ h_4 > \max\left\{ h_3 \frac{h_3 (h_3 + 3\delta_2) + \frac{1}{2} \delta_2^2}{h_3 – 2\delta_2}, \frac{h_1 \left[ \frac{1}{2} h_1 \left( h_1 + \frac{1}{2} \delta_1 \right)^2 \left( 2 h_3^2 – \frac{3}{2} \delta_2 h_3 \right) \right]}{2 \left( q_1 – \frac{1}{2} h_1 \left( h_1 + \frac{1}{2} \delta_1 \right)^2 \right) \left( \frac{1}{2} h_1 – \delta_1 \right)} + \frac{h_1 \left[ \left( \frac{5}{2} h_3^2 + \frac{3}{2} \delta_2 h_3 \right) q_1 \right]}{2 \left( q_1 – \frac{1}{2} h_1 \left( h_1 + \frac{1}{2} \delta_1 \right)^2 \right) \left( \frac{1}{2} h_1 – \delta_1 \right)} – \frac{1}{2} h_3^2 \right\} $$
where \( q_1 = \frac{1}{4} h_1^3 + \left( \frac{1}{2} h_1 – \delta_1 \right) \left( 2 h_2 + \frac{1}{2} h_1^2 \right) \).
Table 1 summarizes the experimental parameters for the permanent magnet synchronous motor used in our tests.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Rated voltage | 24 V | Number of pole pairs | 2 |
| Moment of inertia | 0.00002 kg·m² | Phase resistance | 7.25 Ω |
| Permanent magnet flux linkage | 0.02415 Wb | Phase inductance | 6.29 mH |
Table 2 presents the key parameter settings for the comparative experiments between different sliding mode observers.
| Control method | Parameters |
|---|---|
| PI control | Kp_speed=0.0025, Ki_speed=0.012; Kp_id=1.258, Ki_id=1450; Kp_iq=1.408, Ki_iq=1450 |
| Sliding mode observer | h=200 |
| Adaptive super-twisting sliding mode observer | a1=0.0072, a2=0.12 |
| Improved adaptive super-twisting sliding mode observer | a1=0.0072, a2=0.12, a3=0.0074, a4=0.16 |
We conduct experiments to compare the performance of the conventional sliding mode observer, the adaptive super-twisting sliding mode observer, and our proposed improved adaptive super-twisting sliding mode observer. Table 3 summarizes the observation performance at two different motor speeds.
| Speed (r/min) | Observer | Max speed error (r/min) | Speed error ripple (r/min) | Max angle error (rad) | Angle error ripple (rad) |
|---|---|---|---|---|---|
| 800 | Sliding mode observer | 15.1 | 29.5 | 0.110 | 0.110 |
| Adaptive super-twisting sliding mode observer | 7.4 | 13.8 | 0.060 | 0.033 | |
| Improved adaptive super-twisting sliding mode observer | 2.1 | 4.5 | 0.030 | 0.012 | |
| 1200 | Sliding mode observer | 12.1 | 23.4 | 0.080 | 0.042 |
| Adaptive super-twisting sliding mode observer | 5.6 | 10.6 | 0.045 | 0.027 | |
| Improved adaptive super-twisting sliding mode observer | 1.9 | 3.7 | 0.025 | 0.009 |
The voltage source inverter nonlinearity compensation strategy we develop operates without requiring motor parameters. The distortion voltage of the voltage source inverter is influenced by dead time \( t_d \), turn-on delay \( t_{on} \), turn-off delay \( t_{off} \), and voltage drops across switches and diodes. The total distortion voltage for phase A can be expressed as:
$$ \Delta U_{err} = \left[ \frac{t_{err}}{t_s} (U_{dc} + U_D – U_S) + \frac{U_D + U_S}{2} \right] \text{sgn}(i_A) = V_{dead} \text{sgn}(i_A) $$
where \( t_{err} = t_d + t_{on} – t_{off} \), \( U_{dc} \) is the DC bus voltage, \( U_S \) and \( U_D \) are the threshold voltages of the switch and diode, and \( t_s \) is the switching period. The three-phase distortion voltages are transformed to the rotating d-q coordinate system:
$$ \begin{bmatrix} U_{err\_d} \\ U_{err\_q} \end{bmatrix} = 2V_{dead} \begin{bmatrix} \cos\theta & -\sin\theta \\ \cos(\theta – 2\pi/3) & -\sin(\theta – 2\pi/3) \\ \cos(\theta + 2\pi/3) & -\sin(\theta + 2\pi/3) \end{bmatrix}^T \begin{bmatrix} \text{sgn}(i_A) \\ \text{sgn}(i_B) \\ \text{sgn}(i_C) \end{bmatrix} $$
We define mapping functions \( F_d \) and \( F_q \) that relate the current polarity signs to the electrical angle:
$$ \begin{bmatrix} F_d \\ F_q \end{bmatrix} = 2 \begin{bmatrix} \cos\theta & -\sin\theta \\ \cos(\theta – 2\pi/3) & -\sin(\theta – 2\pi/3) \\ \cos(\theta + 2\pi/3) & -\sin(\theta + 2\pi/3) \end{bmatrix}^T \begin{bmatrix} \text{sgn}(i_A) \\ \text{sgn}(i_B) \\ \text{sgn}(i_C) \end{bmatrix} $$
Under the \( i_d = 0 \) control strategy, the high-frequency component of the d-axis voltage \( u_{d2\_hf} \) approximates the d-axis distortion voltage caused by the voltage source inverter nonlinearity. We extract this component using a low-pass filter and compute the distortion voltage amplitude \( V_{dead} \). To avoid overflow when \( F_d \) is near zero, we apply a limit:
$$ F_d’ = \begin{cases} 0.3, & 0 \le F_d < 0.3 \\ F_d, & 0.3 \le |F_d| \le 2 \\ -0.3, & -0.3 \le F_d < 0 \end{cases} $$
The online compensation strategy employs an iterative minimization process for the compensated distortion voltage \( V’_{dead} \). The gain coefficient \( \lambda \) is updated based on the comparison of \( V’_{dead} \) with a threshold voltage \( V_{th} \):
$$ \lambda(k+1) = \lambda(k) + \delta \quad \text{if } V’_{dead} < V_{th} $$
$$ \lambda(k+1) = \lambda(k) – \delta \quad \text{if } V’_{dead} > V_{th} $$
where \( \delta \) is a dynamic adjustment coefficient set to 0.0002 to ensure system stability. This approach effectively suppresses the 5th and 7th harmonic components in the stator currents, as confirmed by harmonic analysis. Before compensation, the total harmonic distortion of the phase current was 10.99%, with 5th harmonic at 8.52% and 7th harmonic at 5.57%. After applying the proposed compensation, the total harmonic distortion reduces to 1.83%, representing a 83.3% reduction, while the 5th harmonic decreases to 0.31% (96.3% reduction) and the 7th harmonic decreases to 0.75% (86.5% reduction).
The combined control system integrating the improved adaptive super-twisting sliding mode observer, online voltage source inverter nonlinearity compensation, and field-oriented control is implemented. The improved adaptive super-twisting sliding mode observer provides real-time rotor position and speed estimates, while the compensation strategy corrects the voltage reference signals to mitigate the nonlinear effects of the voltage source inverter.
Table 4 summarizes the overall system performance improvement achieved by the proposed control strategy.
| Performance metric | Without compensation | With compensation | Reduction |
|---|---|---|---|
| Max speed estimation error (r/min) | 5.8 | 2.2 | 55% |
| Speed error ripple (r/min) | 9.5 | 4.3 | 54.7% |
| Max angle estimation error (rad) | 0.045 | 0.026 | 42.2% |
| Angle error ripple (rad) | 0.020 | 0.011 | 45% |
| Current total harmonic distortion | 10.99% | 1.83% | 83.3% |
Experimental results demonstrate that our proposed approach effectively suppresses observation chattering across the entire speed range from 50 to 1500 r/min. The position estimation error is reduced by 50% compared to the traditional super-twisting sliding mode observer, and the current total harmonic distortion remains below 2% even under loaded conditions. The parameter-independent online compensation strategy successfully mitigates the voltage source inverter nonlinear effects without requiring knowledge of motor parameters such as stator resistance and inductance.
In summary, we have developed an improved adaptive super-twisting sliding mode observer combined with an online voltage source inverter nonlinearity compensation strategy for sensorless control of permanent magnet synchronous motors. The introduction of linear terms and adaptive gains in the observer structure enhances convergence speed and reduces chattering over a wide speed range. The compensation strategy effectively eliminates the distortion voltage effects of the voltage source inverter, providing cleaner input signals to the observer. Future work will focus on extending the method to include magnetic saturation effects and parameter adaptation for operation under deep flux-weakening conditions, aiming to achieve full-speed-range high-performance sensorless control.