In the rapidly evolving field of electric vehicle technology, the permanent magnet synchronous motor (PMSM) stands as a core component of the powertrain, directly influencing the performance, efficiency, and reliability of the entire vehicle. As an essential part of the electric vehicle car system, the PMSM must operate under complex and dynamic conditions, including frequent start-stop cycles, sudden load changes, and varying environmental factors. These conditions often introduce nonlinearities and uncertainties into the motor’s behavior, making precise control a significant challenge. The electric vehicle car industry demands control strategies that ensure accurate speed regulation, high efficiency, and robust performance across diverse operating scenarios. In this context, we explore an advanced control methodology designed to address these challenges, focusing on the integration of sliding mode variable structure control with adaptive theory to enhance the control precision of PMSMs in electric vehicle car applications.
Traditional control approaches for PMSMs, such as proportional-integral (PI) regulators, often rely heavily on sensor data for three-phase currents and speeds. However, these methods frequently lack deep utilization and dynamic optimization of current vector coordinate transformations, leading to suboptimal control accuracy, especially under transient or uncertain conditions. Several studies have attempted to improve PMSM control through various techniques. For instance, some researchers have incorporated event-triggered mechanisms with neural networks to reduce computational load, but the discontinuous nature of event triggering can result in non-real-time weight updates, compromising control precision. Others have employed radial basis function neural networks (RBFNNs) to estimate and compensate for system uncertainties; however, the approximation accuracy of RBFNNs depends heavily on the sufficiency and representativeness of training samples, which may not adapt well to the complex and varying states of an electric vehicle car motor. Parameter estimation methods, such as those using least squares algorithms, dynamically adjust controller parameters based on estimated motor parameters, but inaccuracies in estimation can lead to poor current tracking and reduced control performance. Additionally, data-driven approaches combined with sliding mode compensation have been proposed, but measurement errors and data acquisition delays can affect state estimation, thereby limiting control accuracy. These limitations highlight the need for a more robust and adaptive control strategy tailored for the electric vehicle car environment.
In this work, we propose an adaptive control method for PMSMs based on sliding mode variable structure theory. Our approach combines adaptive control principles with sliding mode control to estimate uncertain parameters, such as load torque and moment of inertia, and design a control law that ensures precise speed and torque regulation. The method is specifically developed for electric vehicle car applications, where motors operate under frequent disturbances and nonlinearities. By leveraging current vector coordinate transformations and integrating them with adaptive sliding mode techniques, we aim to achieve superior control performance, including minimal speed fluctuations and enhanced robustness. The electric vehicle car’s drive cycle often involves rapid accelerations, decelerations, and load variations, making our method particularly relevant for real-world implementation. Throughout this article, we emphasize the importance of control strategies in electric vehicle car systems, as they directly impact energy consumption, driving range, and overall vehicle dynamics.
Current Vector Coordinate Transformation for PMSM in Electric Vehicle Car Systems
To achieve precise current control and rapid response to load changes in an electric vehicle car, it is essential to decouple the complex interrelationships among the motor’s physical quantities. The PMSM in an electric vehicle car typically has three-phase stator windings carrying currents \(i_A\), \(i_B\), and \(i_C\). These currents are coupled in the three-phase stationary coordinate system, making direct control challenging. Therefore, we first apply the Clarke transformation to convert the three-phase currents into a two-phase stationary coordinate system, achieving preliminary decoupling. The transformation is given by:
$$
\begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} = \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} i_A \\ i_B \\ i_C \end{bmatrix}
$$
where \(i_{\alpha}\) and \(i_{\beta}\) represent the current components in the two-phase stationary coordinate system. This step simplifies the control structure by reducing the number of variables and mitigating coupling effects inherent in the electric vehicle car motor’s operation.
However, the decoupled currents in the stationary frame still cannot directly control the motor’s torque and flux. To further decouple the current components and align them with the rotor’s magnetic field, we employ the Park transformation to convert \(i_{\alpha}\) and \(i_{\beta}\) into the \(dq\)-rotating coordinate system. This transformation is crucial for electric vehicle car applications, as it enables independent control of torque (via the \(q\)-axis current) and flux (via the \(d\)-axis current). The Park transformation is expressed as:
$$
\begin{bmatrix} i_d \\ i_q \end{bmatrix} = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix}
$$
where \(\theta\) is the electrical angle of the rotor flux, and \(i_d\) and \(i_q\) are the direct-axis and quadrature-axis currents, respectively. By operating in the \(dq\)-frame, our control strategy can more effectively manage the motor’s dynamics, which is vital for the responsive performance required in an electric vehicle car. The coordinate transformations form the foundation for subsequent control law design, allowing us to address nonlinearities and uncertainties in the electric vehicle car motor system.
Estimation of Load Torque and Moment of Inertia Using Sliding Mode Variable Structure
In an electric vehicle car, the PMSM often faces varying load conditions due to changes in road gradient, vehicle weight, or driving maneuvers. Accurate estimation of load torque and moment of inertia is essential for adaptive control. We design a sliding mode surface based on the speed error in the two-phase rotating coordinate system, combining the error and its integral term. This approach enhances robustness against disturbances common in electric vehicle car operations.
First, we define the speed error \(e\) as the difference between the desired speed \(\omega_d\) and the actual speed \(\omega\) of the motor:
$$
e = \omega_d – \omega
$$
The actual speed \(\omega\) is measured in real-time using a speed sensor mounted on the motor shaft, a standard practice in electric vehicle car systems. Based on this error, we design the sliding mode surface \(s\) as:
$$
s = e + \lambda \int_0^t e \, d\tau
$$
where \(\lambda\) is a design parameter that adjusts the dynamic characteristics of the sliding surface. The integral term \(\int_0^t e \, d\tau\) accumulates persistent speed errors over time, which is particularly useful for electric vehicle car scenarios where steady-state accuracy is critical. If the electric vehicle car experiences prolonged speed deviations, this term increases, prompting the control system to adjust its strategy and eliminate the error.
Next, we estimate the load torque \(\hat{T}_L\) and moment of inertia \(\hat{J}_L\) using sliding mode theory and integral operations. The estimation errors are defined as \(\tilde{T}_L = T_L – \hat{T}_L\) and \(\tilde{J}_L = J – \hat{J}_L\), where \(T_L\) and \(J\) are the true values. The adaptive estimation laws are formulated as:
$$
\hat{T}_L = \hat{T}_L(0) – \alpha_1 \int_0^t s \cdot \text{sgn}(s) \, d\tau
$$
$$
\hat{J}_L = \hat{J}_L(0) – \alpha_2 \int_0^t s \cdot \text{sgn}(s) \cdot \omega \, d\tau
$$
where \(\hat{T}_L(0)\) and \(\hat{J}_L(0)\) are initial estimates set at the start of the control system, \(\alpha_1\) and \(\alpha_2\) are adaptive gains, and \(\text{sgn}(\cdot)\) is the sign function. These estimation laws leverage the sliding surface information to continuously update the parameters, ensuring adaptability to changing conditions in the electric vehicle car. The use of integral operations helps smooth out noise and improve estimation accuracy, which is beneficial for the electric vehicle car’s motor control under real-world driving cycles.
Design of Adaptive Sliding Mode Control Law for Electric Vehicle Car PMSM
With the estimated parameters and sliding mode surface, we design an adaptive control law to regulate the \(q\)-axis current, which directly controls the electromagnetic torque and, consequently, the motor speed. This design is pivotal for maintaining precise control in an electric vehicle car, where torque demand can vary rapidly.
The voltage equations of the PMSM in the \(dq\)-coordinate system are derived from the motor’s dynamic model. Let \(R\) be the stator resistance, \(\psi_f\) the permanent magnet flux linkage, and \(L_d\) and \(L_q\) the \(d\)-axis and \(q\)-axis inductances, respectively. The voltage equations are:
$$
\begin{aligned}
u_d &= R i_d + L_d \frac{di_d}{dt} – \omega L_q i_q \\
u_q &= R i_q + L_q \frac{di_q}{dt} + \omega L_d i_d + \omega \psi_f
\end{aligned}
$$
where \(u_d\) and \(u_q\) are the \(d\)-axis and \(q\)-axis voltage components. These equations describe the electrical dynamics of the motor, which are influenced by the operating conditions of the electric vehicle car.
Based on the sliding surface \(s\) and the estimated parameters \(\hat{T}_L\) and \(\hat{J}_L\), we design the control input for the \(q\)-axis current \(i_q\) as:
$$
i_q = \frac{\hat{J}_L}{p \psi_f} \left( \frac{d\omega_d}{dt} – \lambda e – \alpha_3 \text{sgn}(s) \right) + \frac{\hat{T}_L}{p \psi_f}
$$
where \(p\) is the number of pole pairs, \(\alpha_3\) is a control gain that adjusts the convergence speed of the sliding mode control, and \(\frac{d\omega_d}{dt}\) is the derivative of the desired speed. This control law dynamically adjusts the electromagnetic torque by regulating \(i_q\), enabling precise speed tracking even under uncertainties. The term \(\alpha_3 \text{sgn}(s)\) introduces a discontinuous control action that ensures robustness against disturbances, a key requirement for electric vehicle car motors facing unpredictable road conditions. By integrating adaptive estimation with sliding mode control, our method achieves high control precision and adaptability, essential for the efficient operation of an electric vehicle car.
The overall control structure is summarized in the following table, highlighting the key components and their roles in the electric vehicle car PMSM control system:
| Component | Description | Role in Electric Vehicle Car PMSM Control |
|---|---|---|
| Clarke Transformation | Converts three-phase currents to two-phase stationary frame | Decouples currents for simplified control in electric vehicle car motor |
| Park Transformation | Converts stationary frame currents to \(dq\)-rotating frame | Enables independent torque and flux control for electric vehicle car dynamics |
| Sliding Mode Surface | Defined as \(s = e + \lambda \int e \, d\tau\) | Provides robust error tracking for electric vehicle car speed regulation |
| Parameter Estimation | Estimates load torque and moment of inertia via adaptive laws | Adapts to varying loads in electric vehicle car driving conditions |
| Control Law for \(i_q\) | \(i_q = \frac{\hat{J}_L}{p \psi_f} \left( \frac{d\omega_d}{dt} – \lambda e – \alpha_3 \text{sgn}(s) \right) + \frac{\hat{T}_L}{p \psi_f}\) | Regulates electromagnetic torque for precise speed control in electric vehicle car |
Experimental Validation for Electric Vehicle Car PMSM Control
To evaluate the effectiveness of our proposed method, we conducted experiments using a TYD series single-phase permanent magnet low-speed synchronous motor, specifically the TYD150-1 model. This motor features a three-phase stator winding and a rotor with embedded permanent magnets, making it representative of motors used in electric vehicle car applications. The rotor structure is designed to provide high torque density and efficiency, which are critical for electric vehicle car performance. Below is an image illustrating a typical electric vehicle car motor system, highlighting the integration of such motors in the vehicle’s powertrain:
The experimental setup involved two distinct operating conditions to simulate real-world electric vehicle car scenarios: constant load condition and load mutation condition. These conditions test the control method’s ability to handle steady-state and transient behaviors, which are common in electric vehicle car driving cycles.
The key parameters of the PMSM used in the experiments are listed in the following table, which are essential for understanding the motor’s characteristics in the context of an electric vehicle car:
| Parameter Name | Value | Relevance to Electric Vehicle Car |
|---|---|---|
| Rated Power (kW) | 30 | Determines the power output for electric vehicle car propulsion |
| Rated Voltage (V) | 320 | Typical voltage level in electric vehicle car battery systems |
| Rated Speed (r/min) | 3000 | Influences the electric vehicle car’s top speed and acceleration |
| Maximum Speed (r/min) | 8000 | Reflects high-speed capability for electric vehicle car performance |
| Number of Pole Pairs | 4 | Affects torque production and control dynamics in electric vehicle car |
| Stator Resistance (Ω) | 0.15 | Impacts efficiency and thermal management in electric vehicle car |
| d-axis Inductance (mH) | 1.2 | Influences flux weakening and high-speed operation for electric vehicle car |
| q-axis Inductance (mH) | 2.5 | Affects torque control and dynamic response in electric vehicle car |
| Permanent Magnet Flux Linkage (Wb) | 0.2 | Key parameter for torque generation in electric vehicle car motor |
| Initial Moment of Inertia (kg·m²) | 0.01 | Determines acceleration characteristics of the electric vehicle car |
In the constant load condition, the desired speed was set to 200 r/min, and a constant load torque of 10 N·m was applied at startup. This simulates an electric vehicle car accelerating from rest under a steady load, such as on a flat road. We recorded the speed response, current response, and load torque estimation to assess control performance. In the load mutation condition, the motor initially operated at 2000 r/min with a 10 N·m load torque. After 5 seconds, the load torque suddenly increased to 20 N·m for 5 seconds, then decreased to 5 N·m. This mimics an electric vehicle car encountering sudden changes, like climbing a hill or braking, testing the control method’s adaptability.
We compared our method with two conventional approaches: an event-triggered adaptive neural network control method and an RBFNN-based adaptive speed control method. These are commonly used in electric vehicle car motor control research. The comparison focused on speed fluctuation ranges, which directly indicate control precision. The results from both conditions are summarized below, emphasizing the performance in electric vehicle car contexts:
| Operating Condition | Control Method | Average Speed Fluctuation Range (r/min) | Implication for Electric Vehicle Car |
|---|---|---|---|
| Constant Load | Proposed Method | ±1 | Ensures smooth acceleration and stable cruising in electric vehicle car |
| Event-Triggered Neural Network | ±3 | May cause jerky movements in electric vehicle car, reducing comfort | |
| RBFNN-Based Control | ±4 | Could lead to inefficient energy use in electric vehicle car | |
| Load Mutation | Proposed Method | ±1 | Maintains speed stability during sudden load changes in electric vehicle car |
| Event-Triggered Neural Network | ±5 | Risk of speed drops or surges in electric vehicle car, affecting safety | |
| RBFNN-Based Control | ±6 | Poor adaptability to transient conditions in electric vehicle car |
The experimental results demonstrate that our proposed method achieves significantly lower speed fluctuations, with an average range within ±1 r/min in both conditions. This high precision is crucial for electric vehicle car applications, as it ensures smooth operation, enhances passenger comfort, and improves overall vehicle efficiency. In contrast, the conventional methods exhibited larger fluctuations, which could lead to suboptimal performance in an electric vehicle car, such as increased energy consumption or reduced driving range. The robustness of our method under load mutations highlights its suitability for real-world electric vehicle car driving, where unpredictable disturbances are common.
Discussion and Conclusion
Our research delves into the theoretical and practical integration of sliding mode variable structure control with adaptive control for PMSMs in electric vehicle car systems. By designing an appropriate sliding mode surface and adaptive laws, the control system can automatically adjust parameters under varying operating conditions, achieving precise tracking of motor speed and torque. The algorithm explicitly considers the nonlinearities and time-varying parameters of the PMSM, which are inherent in electric vehicle car operations, thereby enhancing adaptability to complex driving scenarios. This adaptability ensures that the electric vehicle car motor maintains efficient and stable performance across diverse environments, from urban stop-and-go traffic to highway cruising.
The key contributions of this work include the development of a coordinate transformation framework that facilitates decoupled control, an adaptive estimation mechanism for load torque and moment of inertia, and a robust sliding mode control law that minimizes speed errors. These elements collectively address the limitations of existing methods, offering a comprehensive solution for electric vehicle car motor control. The electric vehicle car industry stands to benefit from such advancements, as improved motor control can lead to longer battery life, reduced maintenance costs, and enhanced driving dynamics.
Future work could explore the integration of this control method with higher-level vehicle management systems in electric vehicle car platforms, such as energy management or regenerative braking control. Additionally, real-time implementation on embedded hardware and testing in actual electric vehicle car prototypes would validate the method’s practicality. As electric vehicle car technology continues to evolve, adaptive and robust control strategies like the one proposed here will play a pivotal role in achieving sustainable and high-performance transportation.
In summary, the proposed adaptive sliding mode control method offers a significant step forward in PMSM control for electric vehicle car applications. It combines theoretical rigor with practical applicability, ensuring that electric vehicle car motors can operate with high precision and robustness under the demanding conditions of modern driving. We believe that this approach contributes to the ongoing advancement of electric vehicle car technology, supporting the global transition toward cleaner and more efficient mobility solutions.