In this paper, we present a comprehensive study on the sensorless control of permanent magnet synchronous motors (PMSM) using an improved adaptive super-twisting sliding mode observer (IAST-SMO) combined with an online compensation strategy for voltage source inverter (VSI) nonlinearity. Our work is particularly relevant to various types of solar inverters, including single-phase and three-phase string inverters, microinverters, and hybrid inverters, where high-performance motor drives are increasingly integrated for tracking, cooling, or auxiliary systems. The proposed method significantly enhances estimation accuracy and suppresses chattering over a wide speed range, making it suitable for demanding applications in renewable energy systems.
1. Introduction
PMSMs are widely used in modern servo systems due to their high power density and efficiency. Sensorless control techniques eliminate the need for mechanical position sensors, reducing cost and improving reliability. Among the various approaches, sliding mode observers (SMO) are popular for their robustness to parameter variations. However, conventional SMOs suffer from inherent chattering. The super-twisting algorithm (STA) alleviates chattering by integrating discontinuous terms, but its fixed gains limit performance across different speed ranges. In addition, the nonlinear effects of VSIs—common in all types of solar inverters—distort the voltage applied to the motor, degrading observer accuracy. To address these issues, we propose an IAST-SMO that incorporates a linear term and adaptive gains, along with a parameter-independent online VSI nonlinearity compensation method. Our experimental results demonstrate that the combined scheme reduces position estimation error by 50% and achieves current total harmonic distortion (THD) below 2%, outperforming traditional STA-based observers.
2. Problem Formulation
The PMSM model in the stationary α-β reference frame is given by:
$$
\begin{aligned}
u_\alpha &= R i_\alpha + L_s \frac{di_\alpha}{dt} + E_\alpha, \\
u_\beta &= R i_\beta + L_s \frac{di_\beta}{dt} + E_\beta,
\end{aligned}
$$
where \(u_\alpha, u_\beta\) and \(i_\alpha, i_\beta\) are stator voltages and currents, \(R\) and \(L_s\) are stator resistance and inductance, and \(E_\alpha, E_\beta\) are extended back-EMFs:
$$
\begin{aligned}
E_\alpha &= -\psi_f \omega_e \sin\theta_e, \\
E_\beta &= \psi_f \omega_e \cos\theta_e.
\end{aligned}
$$
Here, \(\psi_f\), \(\omega_e\), and \(\theta_e\) denote the permanent magnet flux linkage, electrical angular speed, and rotor position. Accurate estimation of \(\omega_e\) and \(\theta_e\) is essential for sensorless field-oriented control (FOC).
3. Improved Adaptive Super-Twisting Sliding Mode Observer
3.1 Conventional Super-Twisting SMO
The standard ST-SMO for PMSM current estimation is:
$$
\begin{aligned}
\frac{d\hat{i}_\alpha}{dt} &= \frac{1}{L_s}\left( -h_1 |\tilde{i}_\alpha|^{1/2} \text{sgn}(\tilde{i}_\alpha) – h_2 \int \text{sgn}(\tilde{i}_\alpha) dt \right) + \frac{-R\hat{i}_\alpha + u_\alpha}{L_s}, \\
\frac{d\hat{i}_\beta}{dt} &= \frac{1}{L_s}\left( -h_1 |\tilde{i}_\beta|^{1/2} \text{sgn}(\tilde{i}_\beta) – h_2 \int \text{sgn}(\tilde{i}_\beta) dt \right) + \frac{-R\hat{i}_\beta + u_\beta}{L_s}.
\end{aligned}
$$
where \(\tilde{i}_\alpha = \hat{i}_\alpha – i_\alpha\) and \(\tilde{i}_\beta = \hat{i}_\beta – i_\beta\) are current estimation errors. The estimated back-EMFs are:
$$
\begin{aligned}
\hat{E}_\alpha &= h_1 |\tilde{i}_\alpha|^{1/2} \text{sgn}(\tilde{i}_\alpha) + h_2 \int \text{sgn}(\tilde{i}_\alpha) dt, \\
\hat{E}_\beta &= h_1 |\tilde{i}_\beta|^{1/2} \text{sgn}(\tilde{i}_\beta) + h_2 \int \text{sgn}(\tilde{i}_\beta) dt.
\end{aligned}
$$
While STA suppresses chattering compared to sign-function-based SMOs, its fixed gains \(h_1, h_2\) cause sluggish convergence when the system deviates from the sliding surface, and poor robustness at low speeds.
3.2 Proposed IAST-SMO
We introduce linear terms \(h_3 \tilde{i}_{\alpha\beta}\) and \(h_4 \int \tilde{i}_{\alpha\beta} dt\) to accelerate convergence. The improved observer is:
$$
\begin{aligned}
\frac{d\hat{i}_\alpha}{dt} &= \frac{1}{L_s}\left( -h_1 |\tilde{i}_\alpha|^{1/2} \text{sgn}(\tilde{i}_\alpha) – h_3 \tilde{i}_\alpha – h_2 \int \text{sgn}(\tilde{i}_\alpha) dt – h_4 \int \tilde{i}_\alpha dt \right) + \frac{-R\hat{i}_\alpha + u_\alpha}{L_s}, \\
\frac{d\hat{i}_\beta}{dt} &= \frac{1}{L_s}\left( -h_1 |\tilde{i}_\beta|^{1/2} \text{sgn}(\tilde{i}_\beta) – h_3 \tilde{i}_\beta – h_2 \int \text{sgn}(\tilde{i}_\beta) dt – h_4 \int \tilde{i}_\beta dt \right) + \frac{-R\hat{i}_\beta + u_\beta}{L_s}.
\end{aligned}
$$
The estimated back-EMFs become:
$$
\begin{aligned}
\hat{E}_\alpha &= -\frac{1}{L_s}\left( h_1 |\tilde{i}_\alpha|^{1/2} \text{sgn}(\tilde{i}_\alpha) + h_3 \tilde{i}_\alpha + \int \left( h_2 \text{sgn}(\tilde{i}_\alpha) + h_4 \tilde{i}_\alpha \right) dt \right), \\
\hat{E}_\beta &= -\frac{1}{L_s}\left( h_1 |\tilde{i}_\beta|^{1/2} \text{sgn}(\tilde{i}_\beta) + h_3 \tilde{i}_\beta + \int \left( h_2 \text{sgn}(\tilde{i}_\beta) + h_4 \tilde{i}_\beta \right) dt \right).
\end{aligned}
$$
To ensure stability over a wide speed range, we design adaptive gains proportional to rotor speed:
$$
\begin{aligned}
h_1 &= a_1 \omega_e^{3/2}, \quad h_2 = a_2 \omega_e^3, \\
h_3 &= a_3 \omega_e^{3/2}, \quad h_4 = a_4 \omega_e^3,
\end{aligned}
$$
where the coefficients \(a_1, a_2, a_3, a_4\) satisfy the sufficient conditions derived from Lyapunov analysis:
$$
\begin{aligned}
a_1 &> 2\eta_1, \\
a_2 &> \max\left( \frac{a_1\left(\eta_1 a_1 + \frac{1}{8}\eta_1^2\right)}{a_1 – 2\eta_1}, \; \frac{(a_3\eta_1 + \frac{1}{2}a_1\eta_2)^3}{2a_3(a_3-2\eta_2)} + \frac{\frac{3}{2}a_1 a_3\eta_1 – 2(a_3-\frac{1}{4}\eta_2)a_1^2}{a_3-2\eta_2} \right), \\
a_3 &> 2\eta_2, \\
a_4 &> \max\left( a_3\frac{a_3(a_3+3\eta_2)+\frac{1}{2}\eta_2^2}{a_3-2\eta_2}, \; \text{(a lengthy expression)} \right).
\end{aligned}
$$
Here, \(\eta_1, \eta_2\) bound the perturbation term: \(|\rho_1| \le \eta_1 \omega_e^{3/2} |\tilde{i}|^{1/2} + \eta_2 \omega_e^{3/2} |\tilde{i}|\). We experimentally tuned \(\eta_1 = 0.1, \eta_2 = 0.2\) to balance robustness and convergence.
3.3 Stability Verification
Table 1 lists the key parameters used in our comparative experiments.
| Controller/Observer | Parameters |
|---|---|
| PI speed & current | \(K_{p\_speed}=0.0025,\; K_{i\_speed}=0.012;\) \(K_{p\_id}=1.258,\; K_{i\_id}=1450;\) \(K_{p\_iq}=1.408,\; K_{i\_iq}=1450\) |
| Conventional SMO | \(h=200\) |
| ST-SMO (fixed gains) | \(a_1=0.0072,\; a_2=0.12\) |
| IAST-SMO (proposed) | \(a_1=0.0072,\; a_2=0.12,\; a_3=0.0074,\; a_4=0.16\) |
Experimental results at 800 r/min and 1200 r/min (Table 2) demonstrate that the IAST-SMO reduces the peak speed estimation error \(\tilde{n}_{\max}\) and its ripple \(\tilde{n}_r\) by more than 50% compared to ST-SMO, and the position error \(\tilde{\theta}_{\max}\) is halved.
| Speed (r/min) | Observer | \(\tilde{n}_{\max}\) (r/min) | \(\tilde{n}_r\) (r/min) | \(\tilde{\theta}_{\max}\) (rad) | \(\tilde{\theta}_r\) (rad) |
|---|---|---|---|---|---|
| 800 | SMO | 15.1 | 29.5 | 0.110 | 0.110 |
| ST-SMO | 7.4 | 13.8 | 0.060 | 0.033 | |
| IAST-SMO | 2.1 | 4.5 | 0.030 | 0.012 | |
| 1200 | SMO | 12.1 | 23.4 | 0.080 | 0.042 |
| ST-SMO | 5.6 | 10.6 | 0.045 | 0.027 | |
| IAST-SMO | 1.9 | 3.7 | 0.025 | 0.009 |
4. Online Compensation of VSI Nonlinearity
4.1 Effect of VSI Nonlinearities
In all types of solar inverters, dead time, turn-on/off delays, and device voltage drops cause voltage distortion. For a three-phase VSI, the distorted voltage in the rotating dq frame is:
$$
\begin{aligned}
\begin{bmatrix} U_{\text{err},d} \\ U_{\text{err},q} \end{bmatrix} = 2 V_{\text{dead}} \begin{bmatrix} \cos\theta & -\sin\theta \\ \cos(\theta-\frac{2\pi}{3}) & -\sin(\theta-\frac{2\pi}{3}) \\ \cos(\theta+\frac{2\pi}{3}) & -\sin(\theta+\frac{2\pi}{3}) \end{bmatrix}^T \begin{bmatrix} \text{sgn}(i_A) \\ \text{sgn}(i_B) \\ \text{sgn}(i_C) \end{bmatrix},
\end{aligned}
$$
where \(V_{\text{dead}} = \frac{t_{\text{err}}}{t_s}U_{dc} + \frac{U_D+U_S}{2}\) with \(t_{\text{err}} = t_d + t_{\text{on}} – t_{\text{off}}\). The functions \(F_d\) and \(F_q\) map the current polarities:
$$
\begin{aligned}
F_d &= 2 \begin{bmatrix} \cos\theta & -\sin\theta \\ \cos(\theta-\frac{2\pi}{3}) & -\sin(\theta-\frac{2\pi}{3}) \\ \cos(\theta+\frac{2\pi}{3}) & -\sin(\theta+\frac{2\pi}{3}) \end{bmatrix}^T \begin{bmatrix} \text{sgn}(i_A) \\ \text{sgn}(i_B) \\ \text{sgn}(i_C) \end{bmatrix}_{(1)} ,\\
F_q &= \text{(similar)}.
\end{aligned}
$$
\(F_d\) exhibits a dominant 6th-order harmonic with zero average, while \(F_q\) contains a DC component. Under \(i_d=0\) control, the high-frequency component of the d-axis reference voltage is approximately equal to the d-axis distortion voltage \(V_{\text{dead}}F_d\).
4.2 Proposed Compensation Scheme
Our compensation strategy does not rely on motor parameters \(R, L_d, L_q\). We extract the high-frequency d-axis voltage component via a low-pass filter, obtaining \(u_{d2\_\text{hf}}\). Then \(V_{\text{dead}}\) is computed as:
$$
V_{\text{dead}} = \frac{|u_{d2\_\text{hf}}|}{|F_d’|},
$$
where \(F_d’\) is a clamped version of \(F_d\) to avoid division by zero:
$$
F_d’ = \begin{cases} 0.3, & 0 \le F_d < 0.3; \\ F_d, & 0.3 \le |F_d| \le 2; \\ -0.3, & -0.3 \le F_d < 0. \end{cases}
$$
The threshold of 0.3 was chosen to minimize current THD. To prevent performance degradation due to model uncertainties, we adapt the compensation gain \(\lambda\) online by minimizing the compensated d-axis high-frequency component \(\tilde{V}_{\text{dead}}’\). The update rule is:
$$
\lambda(k+1) = \begin{cases} \lambda(k) + \delta, & \text{if } |V_{\text{dead}}| > V_{\text{th}}, \\ \lambda(k) – \delta, & \text{else if } |V_{\text{dead}}| < V_{\text{th}}, \\ \lambda(k), & \text{otherwise}, \end{cases}
$$
with \(\delta = 0.0002\) for stability. The final compensation voltages applied to \(u_d^*\) and \(u_q^*\) are \(\lambda V_{\text{dead}} F_d\) and \(\lambda V_{\text{dead}} F_q\), respectively.
4.3 Current Harmonic Reduction
Figure 1 shows the experimental results at 1000 r/min. Before compensation, the A-phase current THD was 10.99% with 5th harmonic 8.52% and 7th harmonic 5.57%. After compensation, THD reduced to 1.83% (a 83.3% reduction), 5th harmonic to 0.31% (96.3% reduction), and 7th harmonic to 0.75% (86.5% reduction). The dq-axis currents became significantly cleaner. This confirms the effectiveness of our method for various types of solar inverters where low THD is critical for grid compliance.
5. Combined System and Experimental Validation
We integrated the IAST-SMO and the VSI compensation into a standard FOC framework. The block diagram is shown conceptually: the IAST-SMO estimates back-EMFs, from which a phase-locked loop extracts speed and position. The adaptive gains \(h_1\)–\(h_4\) are updated based on \(\hat{\omega}_e\). The VSI compensation block computes \(V_{\text{dead}}\) and \(\lambda\), adding corrective voltages to the dq-axis references.
Experiments were conducted on a surface-mounted PMSM with parameters: rated voltage 24 V, 2 pole pairs, \(R=7.25\,\Omega\), \(L_s=6.29\,\text{mH}\), \(\psi_f=0.02415\,\text{Wb}\), inertia \(0.00002\,\text{kg·m}^2\). A photo of our setup is omitted.
5.1 Speed and Position Estimation Improvement
Table 3 summarizes the effect of VSI compensation on the IAST-SMO performance at 1000 r/min.
| Metric | Without compensation | With compensation | Improvement |
|---|---|---|---|
| Peak speed error (r/min) | 5.8 | 2.2 | ≈62% |
| Speed ripple (r/min) | 9.5 | 4.3 | ≈55% |
| Peak position error (rad) | 0.045 | 0.026 | ≈42% |
| Position error ripple (rad) | 0.020 | 0.011 | ≈45% |
The compensation reduces the adverse effects of VSI nonlinearities, which are present in all types of solar inverters (e.g., two-level, three-level, and multilevel topologies). Our scheme is particularly beneficial for single-phase inverters where current zero-crossing distortion is severe.
5.2 Dynamic Performance
We tested the IAST-SMO under acceleration/deceleration (600–1200 r/min) and load step (0 to 0.2 N·m) conditions. The observer closely tracked the actual speed with a maximum error of 4 r/min during transients and settled within 0.2 s. The position error remained below 0.03 rad. The back-EMF waveforms were smooth and free of high-frequency noise, validating the robustness of the combined method.
6. Conclusion
We have developed an improved adaptive super-twisting sliding mode observer with online VSI nonlinearity compensation for PMSM sensorless control. The IAST-SMO uses linear terms and speed-adaptive gains to achieve fast convergence and low chattering over a wide speed range. The VSI compensation scheme, independent of motor parameters, effectively cancels distortion voltages, reducing current THD to below 2%. The combined strategy reduces position estimation error by more than 50% compared to conventional ST-SMO. The method is applicable to various types of solar inverters, including those used in photovoltaic systems, battery energy storage, and off-grid applications. Future work will extend the approach to very low speeds and incorporate magnetic saturation effects for full-speed-range operation.
Our experimental results confirm that the proposed observer and compensation technique significantly enhance the performance and reliability of sensorless PMSM drives, making them suitable for integration into modern types of solar inverters where high efficiency and low harmonic distortion are paramount.