1. Introduction
In the context of global energy transition and the strategic objective of carbon peak and carbon neutrality, enhancing the efficiency and reliability of electric car drive systems has become an academic hotspot. As the core drive component of electric cars, the permanent magnet synchronous motor (PMSM) has been widely adopted due to its advantages of high power density, high efficiency, and wide speed regulation range. However, traditional mechanical position sensors, such as resolvers or encoders, exhibit some non-negligible drawbacks in electric car applications: they increase the system cost and physical volume, reduce environmental adaptability, and even become a fragile point in the system under severe conditions of high temperature, high humidity, or strong vibration. In order to meet the strict performance and robustness requirements of electric cars, the research on sensorless control technology of PMSM has become increasingly important in the fields of motor control. This thesis focuses on the surface-mounted permanent magnet synchronous motor (SPMSM) and carries out a systematic study on the full-speed-range sensorless control technology, aiming at solving the issues of low accuracy in the low-speed domain, significant chattering in the medium-high speed domain, and poor stability during the switching process.
In the zero-to-low speed domain, the back electromotive force (EMF) of the motor is too small to be accurately used for position observation. Thus, the high-frequency signal injection (HFI) method, especially the pulsating high-frequency voltage injection method, is commonly used to extract the rotor position information by utilizing the saturation saliency effect of the motor. However, the traditional signal processing methods usually employ multiple band-pass filters (BPF) and low-pass filters (LPF), which generate phase lag and reduce the bandwidth of the control loop, consequently degrading the estimation accuracy and dynamic performance. In order to solve this problem, I propose a rotor position observer based on a second-order generalized integrator (SOGI). The improved method substitutes the traditional filters with two cascaded SOGI blocks: the first one is used to extract the high-frequency current component, and the second one is used to dynamically obtain the envelope of the position error signal. Through theoretical derivation, this method guarantees smaller position estimation error and better dynamic response than classical filter-based methods in the low-speed region.
In the medium-to-high speed range, the back EMF becomes significant enough for position observation. Among various model-based methods, the sliding mode observer (SMO) is very attractive owing to its strong robustness, structural simplicity, and fast dynamic response. Nevertheless, the conventional SMO adopts the sign function as the switching function, which causes severe high-frequency chattering and leads to large ripples in the estimated back EMF and position signals. In my research, I designed an improved exponential reaching law SMO to suppress chattering. By introducing a polynomial compensation term f(s) to adjust the reaching speed and replacing the discontinuous sign function with a continuous and smooth sigmoid-type function, the high-frequency chattering problem is effectively alleviated. Combined with a normalized phase-locked loop (PLL), the back EMF demodulation precision is further improved. Experimental tests showed that the improved algorithm significantly improves the operational efficiency, anti-interference capability, and stability of the system.
In order to realize the full-speed-range sensorless operation, a valid switching strategy must be considered in the transition region. I adopted a weighted switching control method in this thesis. In the low-speed range of 0–300 r/min, the improved pulsating high-frequency voltage injection method based on SOGI is adopted. In the transition region of 300–400 r/min, the position information is smoothly migrated by weighting functions. When the motor speed exceeds 400 r/min, the improved SMO is used. Simulation and experimental results confirm that the proposed composite control scheme is able to realize smooth switching for SPMSM with acceptable speed and position errors.
2. Mathematical Model and Vector Control Strategy of PMSM
2.1 Basic Structure and Mathematical Model
In this thesis, the research object is the three-phase SPMSM. The stator is composed of three-phase symmetrical windings, while the rotor permanent magnets are mounted on the rotor core surface. Therefore, the air gap is uniform and the direct-axis inductance \(L_d\) is equal to the quadrature-axis inductance \(L_q\). In order to simplify analysis, I make the following assumptions: ignoring the core saturation, eddy current losses, and hysteresis losses; ignoring the temperature variation effect of resistance; and assuming that the stator windings are supplied with ideal three-phase sinusoidal currents.
First, I establish the mathematical model in the three-phase stationary coordinate system. The voltage equation of PMSM in the natural coordinate frame can be expressed as:
$$
\begin{bmatrix}
u_A \\
u_B \\
u_C
\end{bmatrix}
=
R_s
\begin{bmatrix}
i_A \\
i_B \\
i_C
\end{bmatrix}
+
\frac{d}{dt}
\begin{bmatrix}
\psi_A \\
\psi_B \\
\psi_C
\end{bmatrix}
$$
where \(u_A\), \(u_B\), \(u_C\), \(i_A\), \(i_B\), \(i_C\) are the stator phase voltages and currents, \(R_s\) is the stator resistance, and \(\psi_A\), \(\psi_B\), \(\psi_C\) represent the flux linkages of each phase. Because the analytical model in the three-phase coordinate system contains nonlinear and time-varying factors, I transform the model into the stationary \(\alpha\beta\) coordinate frame and the synchronous rotating \(dq\) coordinate frame.
The transformations are realized by Clark and Park transformations. The Clark transformation matrix is given as follows:
$$
C_{3s/2s}
=
\frac{2}{3}
\begin{bmatrix}
1 & -\frac{1}{2} & -\frac{1}{2} \\
0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2}
\end{bmatrix}
$$
The Park transformation matrix is:
$$
C_{2s/2r}
=
\begin{bmatrix}
\cos\theta_e & \sin\theta_e \\
-\sin\theta_e & \cos\theta_e
\end{bmatrix}
$$
Considering the SPMSM with \(L_d = L_q = L_s\), the voltage equations in the synchronous rotating frame are described as follows:
$$
\begin{bmatrix}
u_d \\
u_q
\end{bmatrix}
=
R_s
\begin{bmatrix}
i_d \\
i_q
\end{bmatrix}
+
\frac{d}{dt}
\begin{bmatrix}
L_s i_d + \psi_f \\
L_s i_q
\end{bmatrix}
+
\omega_e
\begin{bmatrix}
-L_s i_q \\
L_s i_d + \psi_f
\end{bmatrix}
$$
where \(u_d\), \(u_q\) and \(i_d\), \(i_q\) are the stator voltages and currents in the \(dq\) frame, \(\psi_f\) is the permanent magnet flux linkage, and \(\omega_e\) is the electrical angular speed. The electromagnetic torque can be derived as:
$$
T_e = \frac{3}{2} p_n \psi_f i_q
$$
in which \(p_n\) denotes the number of pole pairs. It can be observed that when \(i_d =0\), the electromagnetic torque has a linear relationship with \(i_q\). Therefore, the vector control strategy with \(i_d=0\) is adopted in this study. The system keeps \(i_d=0\) through closed-loop current regulators, so that all the stator current is utilized to generate torque, realizing a similar control characteristic to a separately excited DC motor. The control block diagram of this vector control strategy is constructed on the theoretical basis that the rotor position angle must be obtained in real time.
2.2 Voltage Vector and SVPWM Principle
Space-vector pulse width modulation (SVPWM) technology is widely used in the electric car drive system because of its high DC bus utilization and low harmonic distortion. The typical three-phase voltage source inverter has six power switches. Different switching states can produce eight fundamental voltage vectors. Based on the average equivalent principle, the reference voltage vector can be synthesized by two adjacent active vectors and zero vectors in one switching period.
According to the relationship between the sector number \(N\) and the components \(u_\alpha\) and \(u_\beta\) in the stationary frame, SVPWM is implemented as a cyclic procedure of sector identification, adjacent vector duration calculation, and switching time arrangement. Defining
$$
\begin{aligned}
X &= \frac{\sqrt{3}T_s u_\beta}{U_{dc}} \\
Y &= \frac{T_s}{U_{dc}}\left( \frac{3}{2}u_\alpha + \frac{\sqrt{3}}{2}u_\beta \right) \\
Z &= \frac{T_s}{U_{dc}}\left( -\frac{3}{2}u_\alpha + \frac{\sqrt{3}}{2}u_\beta \right)
\end{aligned}
$$
the acting time of active vectors for each sector can be obtained. The universal relationship of the switching time threshold points and the corresponding sector number is illustrated in Table 1.
| N | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Tcm1 | Tb | Ta | Ta | Tc | Tc | Tb |
| Tcm2 | Ta | Tc | Tb | Tb | Ta | Tc |
| Tcm3 | Tc | Tb | Tc | Ta | Tb | Ta |
Through SVPWM, the voltage linear modulation range is expanded by 15.47% compared with SPWM, which is quite valuable in the electric car application where the battery bus voltage is relatively low.
3. Improved Pulsating High-Frequency Voltage Injection Method in Low-Speed Range
3.1 Principle of Pulsating High-Frequency Voltage Injection
For the SPMSM, because the rotor has no visible saliency, and the higher frequency excitation can induce the magnetic saturation nonlinearity, the saturation saliency effect appears. When a high-frequency voltage is injected into the estimated direct axis, the direct-axis high-frequency magnetomotive force overlaps with the permanent magnet magnetic field and produces different degrees of magnetic saturation. I define the estimated rotor position angle as \(\hat{\theta}_{e}\), and the actual position angle is \(\theta_e\), so that the position error angle is \(\Delta\theta = \theta_e – \hat{\theta}_e\). The high-frequency voltage is applied to the direct axis in the estimated \(dq\) frame:
$$
\begin{bmatrix}
u_{dh} \\
u_{qh}
\end{bmatrix}
=
\begin{bmatrix}
U_h \cos(\omega_h t) \\
0
\end{bmatrix}
$$
Under the condition that the motor operates at low speed, the influence of the stator resistance is ignored, and the high-frequency current response in the quadrature axis can be derived as:
$$
i_{qh} \approx -\frac{\omega_h \Delta L}{Z_d Z_q} U_h \sin (\omega_h t) \sin(2\Delta\theta)
$$
where \(\Delta L = L_q – L_d\) represents the incremental saliency, although zero in the magnetic model, it is effectively created by saturation, and \(Z_d\), \(Z_q\) are the high-frequency impedances. Since \(i_{qh}\) contains the information of \(\sin(2\Delta\theta)\), it can be demodulated to acquire the rotor position error signal.
3.2 Traditional Demodulation Scheme and Its Deficiencies
In the traditional method, the \(q\)-axis current is first filtered by a BPF to obtain the high-frequency carrier component \(i_{qh}\). Then, the demodulation is accomplished by multiplying \(i_{qh}\) by \(\sin(\omega_h t)\) and passing the product through an LPF. The resulting position error signal is expressed as:
$$
f(\Delta\theta) = LPF\left[ i_{qh} \cdot \sin(\omega_h t) \right] \approx k \sin(2\Delta\theta)
$$
When \(\Delta\theta\) is small enough, \(k \sin(2\Delta\theta) \approx 2k\Delta\theta\). The block diagram of the traditional rotor position observer is shown in Fig. X (the observer architecture uses a BPF and an LPF before the PI regulator).
However, the traditional scheme suffers from several inherent shortcomings. On the one hand, the cut-off frequency of the BPF cannot be sufficiently high if a satisfactory dynamic response is to be maintained, which results in a quantity of phase lag and weak suppression of low-order harmonic components; on the other hand, the LPF in the demodulation loop also introduces considerable group delay, lowering the speed-loop bandwidth. Because of these limitations, the real-time accuracy of the estimation is degraded, especially during speed variations and sudden load changes in electric car operation.
3.3 Proposed SOGI-Based Observer
To solve the aforementioned issue, I propose a new rotor position observer based on the second-order generalized integrator. The SOGI has excellent characteristics in signal extraction. Its transfer function for the band-pass output is:
$$
G_{BP}(s) = \frac{v_{out}(s)}{v_{in}(s)} = \frac{k_1 \omega_1 s}{s^2 + k_1 \omega_1 s + \omega_1^2}
$$
where \(\omega_1\) is the resonant frequency, and \(k_1\) is the damping coefficient. In the proposed scheme, the traditional BPF used to extract the high-frequency current component is replaced by the first SOGI block. The Bode diagram analysis shows that the SOGI can accurately extract the high-frequency component while showing a better filtering characteristic with small parameter settings. Figure 16 compares the frequency response with different coefficient \(k_1\) values.
In the demodulation part, I adopt a second SOGI structure instead of the LPF. The transfer function of this SOGI block is:
$$
G_{NF}(s) = \frac{f(\Delta\theta)(s)}{i'(s)} = \frac{s^2 + k_2 \omega_2 s}{s^2 + k_2 \omega_2 s + \omega_2^2}
$$
The improved method has clear advantages:
(1) The structural parameters of the SOGI are simple and physically clear.
(2) The signal processing chain is much simpler, which decreases the phase lag significantly.
(3) The dynamic response of the whole position observer is accelerated, and the bandwidth of the current and speed loop can be enlarged.
Meanwhile, because the PMSM saturation effect shows that the north and south poles are not symmetrical, after the estimated angle converges, the rotor magnet polarity should be identified. The north pole and south pole can be distinguished by detecting the DC component in the direct-axis high-frequency current response. If the DC current is negative, a compensation angle \(\pi\) should be added to the estimated angle.
3.4 Simulation and Analysis of the Low-Speed Algorithm
I built a simulation platform in Matlab/Simulink environment using the same motor parameters as the experimental motor, where the relevant parameters are: DC bus voltage 24 V, rated current 4 A, rated speed 3000 r/min, stator resistance 1.02 \(\Omega\), and stator inductance 0.59 mH. The injected high-frequency voltage has an amplitude of 2 V and frequency of 1 kHz.
To evaluate the observation performance under different operating conditions, I examined the starting process, speed variation process, and load torque step process. The comparison of the traditional method and improved method are summarized in Table 2 and Table 3.
| Operating conditions | Method | Maximum speed error (r/min) | Settling time |
|---|---|---|---|
| Uniform speed, no load start at 100 r/min | Traditional | 12.3 | 0.3 s |
| Improved | 7.7 | 0.2 s | |
| Speed step from 100 to 200 r/min | Traditional | 20.0 | — |
| Improved | 17.8 | — | |
| Load torque step at 100 r/min | Traditional | 12.3 | 0.2 s |
| Improved | 4.2 | 0.1 s |
| Operating conditions | Method | Maximum position error (rad) |
|---|---|---|
| Uniform speed, no load start at 100 r/min | Traditional | 0.202 |
| Improved | 0.150 | |
| Speed step from 100 to 200 r/min | Traditional | 0.257 |
| Improved | 0.201 | |
| Load torque step at 100 r/min | Traditional | 0.033 |
| Improved | 0.025 |
It is evident from the simulation results that the proposed SOGI-based method can significantly improve the position observation precision and dynamic performance of the sensorless system in the low-speed region. When the motor is subjected to speed changes or load disturbances, the improved method shows smaller position and speed estimation errors, which is essential for the driving comfort and stability of electric cars.
4. Improved Sliding Mode Observer in Medium-High Speed Range
4.1 Mathematical Model and Traditional SMO
In the medium-high speed domain, I use the back EMF-based SMO to estimate the rotor position and speed. The voltage equation of SPMSM in the stationary \(\alpha\beta\)-axis frame is:
$$
\begin{bmatrix}
u_\alpha \\
u_\beta
\end{bmatrix}
=
R_s
\begin{bmatrix}
i_\alpha \\
i_\beta
\end{bmatrix}
+
L_s
\frac{d}{dt}
\begin{bmatrix}
i_\alpha \\
i_\beta
\end{bmatrix}
+
\begin{bmatrix}
e_\alpha \\
e_\beta
\end{bmatrix}
$$
where \(e_\alpha\) and \(e_\beta\) denote the extended back EMF in the stationary frame:
$$
\begin{bmatrix}
e_\alpha \\
e_\beta
\end{bmatrix}
=
\omega_e \psi_f
\begin{bmatrix}
-\sin\theta_e \\
\cos\theta_e
\end{bmatrix}
$$
According to the measurable voltages and currents, I design the SMO as follows:
$$
\frac{d}{dt}
\begin{bmatrix}
\hat{i}_\alpha \\
\hat{i}_\beta
\end{bmatrix}
=
\frac{1}{L_s}
\begin{bmatrix}
u_\alpha \\
u_\beta
\end{bmatrix}
–
\frac{R_s}{L_s}
\begin{bmatrix}
\hat{i}_\alpha \\
\hat{i}_\beta
\end{bmatrix}
–
\frac{1}{L_s}
\begin{bmatrix}
v_\alpha \\
v_\beta
\end{bmatrix}
$$
in which \(v_\alpha\), \(v_\beta\) are the outputs of the switching control law, classically expressed as:
$$
\begin{aligned}
v_\alpha &= k \cdot \operatorname{sgn}(i_\alpha – \hat{i}_\alpha) \\
v_\beta &= k \cdot \operatorname{sgn}(i_\beta – \hat{i}_\beta)
\end{aligned}
$$
After the controller reaches the sliding surface, the equivalent values of the switching functions contain continuous back EMF information, so that the rotor position can be calculated by an inverse tangent function. However, the discontinuity of the sign function is the main reason that the chattering appears.
4.2 Design of the Novel Sliding Mode Observer
To suppress chattering while maintaining strong robustness, I improve the exponential reaching law from two aspects. First, a polynomial term \(f(s)\) is introduced into the constant-speed term, which makes the system speed self-adaptive. The reaching process can be accelerated when the state is far away from the sliding surface, and the inertia when crossing the sliding surface is decreased when the state is near the sliding surface. Second, I substitute the continuous and smooth sigmoid-type function for the conventional sign function. The designed reaching law is:
$$
\dot{s} = – \varepsilon f(s) \cdot \operatorname{sigmoid}(s) – q s
$$
where
$$
f(s) = \frac{s^2 + 1 + \ln \left( a s + b + \sqrt{s^2 + 1} \right)}{s^2 + 1}, \quad \varepsilon>0, \ q>0, \ a>0, \ b>0
$$
such that when \(|s|\) is very large, \(f(s)\) is large enough to accelerate the reaching speed, while when \(|s|\) approaches zero, \(f(s)\) is less than one, which reduces the chattering amplitude.
The sigmoid function is:
$$
\operatorname{sigmoid}(x)=\frac{2}{1+e^{-a x}}-1
$$
where \(a>0\) adjusts the steepness of the curve near zero. The Lyapunov stability condition is used to prove the convergence. I define the Lyapunov function as \(V = \frac{1}{2} \left( \tilde{i}_\alpha^2 + \tilde{i}_\beta^2 \right)\), and after calculation the following condition must be satisfied:
$$
\varepsilon > \max\{ |E_\alpha|, |E_\beta| \}
$$
The estimated back EMF after the smoothing treatment can be expressed as:
$$
\begin{aligned}
\hat{e}_\alpha &= \varepsilon f(\tilde{i}_\alpha) \cdot \operatorname{sigmoid}(\tilde{i}_\alpha) + q \tilde{i}_\alpha \\
\hat{e}_\beta &= \varepsilon f(\tilde{i}_\beta) \cdot \operatorname{sigmoid}(\tilde{i}_\beta) + q \tilde{i}_\beta
\end{aligned}
$$
4.3 Normalized PLL for Rotor Position Extraction
To further avoid the phase-lag problem caused by the traditional low-pass filter and the arctan function, I use a normalized PLL to extract the rotor position and speed from the estimated back EMF signals. The structure of the normalized PLL is shown in Fig. X. Defining the error signal as:
$$
\varepsilon = – \hat{e}_\alpha \cos\hat{\theta}_e – \hat{e}_\beta \sin\hat{\theta}_e
$$
Then, by normalizing this error using the magnitude of the estimated back EMF \(\sqrt{\hat{e}_\alpha^2 + \hat{e}_\beta^2}\), the speed-adaptive effect can be eliminated. The normalized error becomes:
$$
\varepsilon_n \approx \sin(\theta_e – \hat{\theta}_e) \approx \theta_e – \hat{\theta}_e
$$
The transfer function from \(\theta_e\) to \(\hat{\theta}_e\) is derived as:
$$
G(s) = \frac{K_p s + K_i}{s^2 + K_p s + K_i}
$$
4.4 Simulation Results and Analysis
Simulation studies are carried out on the improved SMO. Key parameters in the reaching law are selected as \(\varepsilon = 150\), \(q = 300\), \(a = 50\), \(b = 10\). The comparison results are presented in Table 4 and Table 5.
| Operating conditions | Method | Maximum speed error (r/min) |
|---|---|---|
| No load start at 1000 r/min (startup transient) | Traditional SMO | 90.6 |
| Improved SMO | 38.2 | |
| No load start at 1000 r/min (steady state) | Traditional SMO | 8.0 |
| Improved SMO | 2.8 | |
| Load torque step at 800 r/min | Traditional SMO | 8.5 |
| Improved SMO | 4.0 |
| Operating conditions | Method | Position error range or maximum value (rad) |
|---|---|---|
| No load start at 1000 r/min | Traditional SMO | 0.483 |
| Improved SMO | 0.165 | |
| Speed step from 800 to 1000 r/min | Traditional SMO | 0.198 |
| Improved SMO | 0.148 | |
| Load torque step at 800 r/min | Traditional SMO | 0.122 |
| Improved SMO | 0.115 |
From the simulation results, I find that the improved SMO reduces the speed and position error obviously in the medium-high speed range. Compared with the traditional SMO, the improved one exhibits smaller chattering, which means less acoustic noise and mechanical vibration in practical electric car driving conditions.
5. Full-Speed-Range Composite Control Strategy
5.1 Comparison of Switching Strategies
The full-speed-range sensorless control strategy should integrate the advantages of different algorithms. There are several commonly-used switching schemes: direct switching method, hysteresis switching method, and weighted switching method.
(a) Direct switching method: The algorithm is switched at a certain threshold speed. It is simple to implement but the sudden change of the observed position and speed may lead to current spikes and torque disturbance.
(b) Hysteresis switching method: Two thresholds are used to avoid the chattering between different control modes. However, the choice of the hysteresis band is empirical, and it cannot ensure smooth changes of the observed variables because the algorithm is still switched abruptly.
(c) Weighted switching method: A weighting coefficient \(\lambda\) is adjusted dynamically according to the motor speed. The observed variables are synthesized by both algorithms in the transition speed region. This method can guarantee the continuity of the position and speed outputs.
5.2 Selection of the Weighted Switching Method
In my research, I choose the weighted switching method because of its excellent stable behavior during the transition process. The weighting coefficient \(\lambda\) is defined as follows:
$$
\lambda =
\begin{cases}
0, & \text{if } \hat{\omega}_m < \omega_1 \\
\dfrac{\hat{\omega}_m – \omega_1}{\omega_2 – \omega_1}, & \text{if } \omega_1 \le \hat{\omega}_m \le \omega_2 \\
1, & \text{if } \hat{\omega}_m > \omega_2
\end{cases}
$$
where \(\omega_1 = 300\) r/min and \(\omega_2 = 400\) r/min. The estimated rotor speed and position are then expressed by:
$$
\begin{aligned}
\hat{\omega}_m &= \lambda \hat{\omega}_{SMO} + (1-\lambda) \hat{\omega}_{HFI} \\
\hat{\theta}_e &= \lambda \hat{\theta}_{SMO} + (1-\lambda) \hat{\theta}_{HFI}
\end{aligned}
$$
In the low-speed region below 300 r/min, the HFI-based observer is used as a standalone observer. In the speed region above 400 r/min, the SMO method is used. Within the transition region, the outputs of both observers are linearly weighted, which makes the dynamic switching process smoother. From the perspective of electric car application, this transition behavior is important because the wheel drive motor frequently accelerates and decelerates in the city driving condition, and a bump-free transition can protect the drive train components.
5.3 Simulation Results of the Full-speed-range Strategy
A complete simulation model based on the weighted switching algorithm is built. I apply a gradually increasing speed reference from zero up to 1000 r/min. The simulation results show that the composite speed estimate tracks the actual speed with minor deviation. During the mode transition, the transient speed error is within 12.9 r/min, and the steady-state error at high speed is about 6.5 r/min. The position estimation error is controlled within a certain range, which renders the whole sensorless system stable.
| Indicator | Value |
|---|---|
| Low-speed region | 0–300 r/min (HFI) |
| Transition region | 300–400 r/min (HFI+SMO) |
| Medium-high speed region | above 400 r/min (SMO) |
| Maximum speed error in switching process | 12.9 r/min |
| Steady-state speed error | 6.5 r/min |
6. Software and Hardware Design
6.1 Overall Hardware Platform
In order to verify the effectiveness of the theoretic analysis, I design a drive and control platform based on the STM32F446RET6 microcontroller. The hardware system contains the main control section, power supply circuit, analog amplification circuit, gate drive circuit, three-phase inverter bridge, magnetic isolation circuit, key-scanning circuit, and rotor position sensor circuit. The experimental motor is a small SPMSM whose parameters are identical to the simulation motor model. The power stage is supplied by a 24 V DC power source, and the logic circuits are powered by 5 V and 3.3 V generated by the onboard voltage regulators. The PWM signals from the microcontroller are isolated and amplified to drive the N-channel MOSFETs.
6.2 Main Software Flow
The software is compiled in the Keil uVision5 environment. In the main interrupt service routine, the analog-to-digital converter obtains the phase currents. The FOC algorithm calculates the direct and quadrature axis voltages and produces the SVPWM signals. The software flowchart of the full-speed-domain control is designed as: (1) start and initialize the system; (2) acquire three-phase currents and perform Clarke and Park transformations; (3) judge the operating speed region; (4) execute the corresponding speed and position estimation algorithm; (5) close the current PI loops and the speed PI loop; (6) output the SVPWM signal; (7) wait for the next interrupt cycle.
For the low-speed region, the improved pulsating HFI algorithm structure is executed in the software. For the medium-high speed region, the improved SMO algorithm is employed. In the software framework, the weighting module computes the coefficient based on the estimated mechanical speed, and the position and speed outputs are synthesized to drive the FOC algorithm.
6.3 Experimental Tests and Results
I performed several experiments to evaluate the performance of the proposed sensorless algorithms on the practical platform. The experimental platform is composed of the self-designed motor control board, a loading device, a DC power supply, an oscilloscope, and the SPMSM under test. The J-Scope tool is used to record the internal variables of the algorithms.
First, the low-speed test is carried out at 100 r/min. The machine is started without load. The results show that the motor can accelerate to the target speed in about 0.7 s. The estimated angle can track the actual angle well after the startup transient. When a sudden 2 N·m load is applied, the maximum dynamic speed error is around 23.2 r/min, and the system recovers to the target speed after about 0.6 s. The position waveform also verifies that the improved HFI scheme maintains a high tracking accuracy under load disturbance.
Next, I test the full-speed-range switching behavior. The motor is accelerated from standstill to 700 r/min. During acceleration, the control algorithms are changed according to the weighted switching strategy. I observe a minor speed fluctuation at the switching moment, and the estimated position tracks the actual position with an acceptable error. The weighted switching method successfully ensures a smooth transition of the sensorless scheme in the whole speed range.
Then, the load torque disturbance is tested in the full-speed range. At the reference speed of 700 r/min, a sudden 2 N·m load is applied. The traditional method causes a maximum dynamic speed error of about 210 r/min and takes about 0.7 s to recover, whereas the improved method yields less disturbance (maximum error around 110 r/min) and shorter recovery time (about 0.5 s). The corresponding rotor position tracking is more stable as well.
Finally, speed reversal tests are carried out. The motor runs at 600 r/min and then is commanded to -600 r/min. The experimental results prove that the motor can realize smooth reversal operation via the full-speed-range strategy. The estimated position angle keeps tracking the actual angle during the reversing process, thereby verifying the feasibility of the complete sensorless scheme in practical electric car applications.
7. Conclusion and Future Work
Focusing on the sensorless control of SPMSM drives for electric cars, this thesis investigates the key techniques over the entire speed range and makes several improvements:
(1) In the low-speed region, I propose a position observer based on cascaded second-order generalized integrators for the pulsating high-frequency injection method. Compared with the traditional multiple-filter scheme, the improved method reduces the signal processing complexity, suppresses the phase lag, and improves the estimation accuracy. Simulation results under no-load startup, speed variation, and load step conditions confirm its better dynamic tracking performance and disturbance rejection ability for electric-car applications.
(2) In the medium-high speed domain, I design an SMO with an improved exponential reaching law in which a polynomial compensation term is introduced into the constant-speed term, and the discontinuous sign function is replaced by a smooth sigmoid-type function. The improved SMO significantly reduces the chattering phenomenon. Furthermore, a normalized PLL is used for rotor position demodulation to avoid the error brought by the low-pass filter. The simulation and experimental results demonstrate that the proposed scheme is more robust and accurate.
(3) For the full-speed-range composite control, I apply the weighted switching method to smoothly overlap the outputs of the improved high-frequency injection observer and the improved SMO. The switching regions are set between 300 and 400 r/min. By dynamically adjusting the weighting coefficient, speed and position jumps are effectively suppressed.
(4) Finally, I construct a hardware test platform based on STM32F446RET6 and verify the proposed algorithms through experiments in the low-speed region, full-speed-range switching, load disturbance, speed step and reversal process. All of the experimental results corroborate the validity of the theoretical analysis. The sensorless control system can meet the fundamental requirements of stable and accurate operation in many practical driving scenarios of electric cars.
There are still some issues to be addressed in future work. First, the current PI parameters of the speed and current loops are tuned by a trial-and-error method. The parameters can be improved by using some self-tuning, intelligent or model-based tuning strategies to obtain better dynamic behavior and robustness across the whole operating envelope. Second, the overcurrent protection on my current platform is purely carried out by software. In order to guarantee the safety of the power electronics, a hardware-based overcurrent protection circuit should be added in future designs. Third, the integration level of the hardware prototype is insufficient, and the optimization design for mass production in electric cars needs further exploration. For instance, one could migrate the algorithm into dedicated motor-control devices or field programmable gate arrays to increase the computational efficiency and reduce the cost, so as to promote the practical application in electric car drive systems.