The rapid evolution of electric car technologies has intensified the demand for efficient and intelligent braking systems, particularly in the context of China EV market growth. Electronic braking systems, such as EBooster, are pivotal for enhancing vehicle safety and energy recovery. However, the inherent nonlinearities in hydraulic systems pose significant challenges for precise pressure control. This article explores a novel approach integrating Radial Basis Function (RBF) neural networks with sliding mode variable structure control to address these issues. By leveraging adaptive learning capabilities, the proposed method aims to improve the robustness and accuracy of hydraulic pressure regulation in electric car braking systems, ensuring compatibility with advanced driving scenarios like adaptive cruise control (ACC) and autonomous emergency braking (AEB).
In the realm of electric car development, China EV manufacturers are increasingly focusing on intelligent braking solutions to meet stringent safety and efficiency standards. Traditional braking systems rely on vacuum boosters, which are less adaptable to the dynamic requirements of modern electric vehicles. The EBooster system, an electro-hydraulic brake unit, offers a promising alternative by enabling both manual and automated braking functions. This system utilizes an electric motor to generate hydraulic pressure, allowing for precise control without direct human input. However, the nonlinear and time-varying nature of hydraulic dynamics complicates pressure regulation, leading to issues such as overshoot, jitter, and steady-state errors. To overcome these limitations, this study proposes an RBF network-based sliding mode control strategy, which autonomously adjusts controller parameters in response to hydraulic load variations. The integration of this advanced control method not only enhances the performance of electric car braking systems but also supports the broader adoption of China EV technologies in global markets.
The hydraulic system in an electric car braking setup typically comprises components like the master cylinder, wheel cylinders, brake lines, and electronic stability control (ESC) units. Modeling this system requires simplifying its complex structure into an equivalent circuit that accounts for pressure dynamics and leakage. The fundamental equation governing hydraulic flow can be expressed as:
$$ A\dot{x} = \frac{V(p,t)}{K_e(p,t)} \dot{p} + K_l(p,t) p $$
where \( A \) represents the piston area, \( \dot{x} \) is the velocity of the piston, \( V(p,t) \) denotes the volume as a function of pressure and time, \( K_e(p,t) \) is the effective bulk modulus, and \( K_l(p,t) \) signifies the leakage coefficient. This model highlights the nonlinear dependencies on pressure and time, which necessitate adaptive control strategies. For instance, at low pressures, leakage coefficients are high, but as pressure increases, they vary proportionally. In electric car applications, especially in China EV models, accurate pressure control is crucial for maintaining braking efficiency and energy recovery during regenerative braking cycles.
To address the nonlinearities, an RBF neural network is employed due to its universal approximation capabilities and efficient learning structure. The RBF network consists of an input layer, a hidden layer with radial basis functions, and an output layer. For hydraulic pressure control, the input is the system pressure \( p \), and the hidden layer uses Gaussian functions defined as:
$$ h_j(p) = \exp\left(-\frac{\|p – c_{Nj}\|^2}{2b_{Nj}^2}\right), \quad j = 1, 2, \ldots, 5 $$
where \( c_{Nj} \) is the center of the j-th neuron, and \( b_{Nj} \) is its width. The outputs approximate the nonlinear functions \( f(p) \) and \( g(p) \) related to system dynamics:
$$ \hat{f}(p) = W^T h(p), \quad \hat{g}(p) = V^T h(p) $$
Here, \( W \) and \( V \) are the weight vectors optimized through adaptive laws. This network enables real-time adjustment of sliding mode controller parameters, reducing reliance on extensive calibration. The sliding mode control law is designed to ensure system stability and robustness against disturbances. For example, the sliding surface \( s \) is defined as:
$$ s = c_1 e + c_2 \dot{e} $$
where \( e \) is the pressure error, and \( c_1 \), \( c_2 \) are positive constants. The control input \( u \) is derived using a reaching law that incorporates the RBF approximations:
$$ u = \frac{1}{\hat{g}(p)} \left( -\hat{f}(p) – k s – q \cdot \text{sat}(s) \right) $$
with \( k \) and \( q \) as tuning parameters, and \( \text{sat}(s) \) as a saturation function to mitigate chattering. This combination allows the electric car braking system to achieve precise pressure tracking even under varying operational conditions, which is essential for China EV applications where reliability and safety are paramount.
Simulation studies were conducted to validate the proposed control strategy. The parameters for the传动机构 and hydraulic system are summarized in Table 1. These values are representative of typical electric car braking systems, aligning with China EV design standards.
| Parameter | Value |
|---|---|
| Piston Area \( A \) (m²) | 4.2 × 10⁻⁴ |
| Transmission Ratio \( r \) | 3.2 |
| Lead \( h \) (m) | 87.6 × 10⁻³ |
| Moment of Inertia \( J_M \) (kg·m²) | 8.8 × 10⁻⁵ |
| Efficiency \( \eta \) | 0.9 |
| Spring Stiffness \( K \) (kN/m) | 4.5 |
The hydraulic load characteristics were modeled using polynomial fits based on experimental data, as shown in the P-V relationship. Under sinusoidal excitation with a target pressure of 3 MPa and amplitude of 2 MPa, the RBF sliding mode controller demonstrated superior performance compared to traditional PID control. The pressure tracking error for the proposed method was within ±0.1 MPa, whereas PID control resulted in errors up to ±0.3 MPa, representing a reduction of over 66%. This improvement is critical for electric car systems, where precise pressure control enhances braking responsiveness and energy efficiency. The RBF network’s ability to adapt to hydraulic load variations ensured minimal steady-state error and eliminated overshoot, making it highly suitable for China EV implementations.
Further analysis involved evaluating the control system under different operational scenarios. The RBF network parameters, such as the centers \( c_{Nj} \) and widths \( b_{Nj} \), were initialized based on typical hydraulic system behaviors. The adaptive laws for updating the weights \( W \) and \( V \) are given by:
$$ \dot{W} = -\gamma_1 s h(p), \quad \dot{V} = -\gamma_2 s h(p) u $$
where \( \gamma_1 \) and \( \gamma_2 \) are learning rates. These laws ensure that the controller continuously learns from system dynamics, improving its performance over time. In simulations, the RBF sliding mode controller achieved a convergence time of less than 0.5 seconds for step pressure changes, which is essential for electric car safety systems like AEB. Additionally, the control strategy maintained stability under parametric uncertainties, such as variations in brake fluid properties or temperature changes, which are common in China EV operating environments.
Experimental validation was carried out using a rapid prototyping test bench designed for electric car braking systems. The setup included real-time control hardware, sensors for pressure and motor angle measurement, and a PC interface for monitoring. The test conditions were selected to mimic real-world driving scenarios, such as ACC and AEB, with pressure gradients up to 3 MPa/s. Table 2 outlines the controller parameters used in the experiments, which were tuned to optimize performance for China EV applications.
| Parameter | Value |
|---|---|
| Sliding Surface Constants \( c_1 \), \( c_2 \) | 7.2, 3.6 |
| Reaching Law Parameters \( k \), \( q \) | 15.2, 0.09 |
| Boundary Layer \( \delta \) | 1 |
| RBF Center \( c_{Nj} \) | 0 |
| RBF Width \( b_{Nj} \) | 0.16 |
| Adaptive Law Gains \( \gamma_1 \), \( \gamma_2 \) | 6.8, 7.6 |
In ramp tests with a gradient of 3 MPa/s, the actual pressure closely followed the target without overshoot or jitter, as illustrated in Figure 6. The steady-state error remained below 0.1 MPa, with a relative error of less than 1%. This performance is vital for electric car systems, where smooth pressure transitions are necessary to avoid passenger discomfort and ensure safety. Similarly, under sinusoidal conditions with a bias of 1.5 MPa and amplitude of 3 MPa, the maximum pressure error did not exceed 0.1 MPa. Although slight delays occurred at peak and trough points due to system inertia, the overall tracking was accurate, demonstrating the controller’s effectiveness in handling dynamic braking demands in China EV models.
The motor angle tracking further confirmed the control strategy’s reliability. In both ramp and sinusoidal tests, the motor angle adhered to the reference signals with minimal deviation. The adaptive nature of the RBF network compensated for delays caused by hydraulic hysteresis, ensuring consistent performance. For instance, the motor angle error during sinusoidal tests was within ±0.5 degrees, which is acceptable for electric car braking actuators. These results underscore the potential of RBF-based sliding mode control to enhance the operational efficiency of electronic braking systems in the rapidly growing China EV sector.
Comparative studies with conventional methods, such as PID and fuzzy control, highlighted the advantages of the proposed approach. While PID controllers require extensive calibration and exhibit larger errors under nonlinear conditions, the RBF sliding mode controller autonomously adapts to changes, reducing the need for manual tuning. Fuzzy controllers, though robust, often depend on expert knowledge for rule definition, which can limit their accuracy. In contrast, the RBF network’s data-driven learning mechanism ensures precise control across diverse operating points, making it ideal for electric car applications where conditions vary widely.
Future work will focus on extending this control strategy to integrated vehicle dynamics systems, such as combined braking and steering control for autonomous electric cars. Additionally, the integration of machine learning techniques could further enhance the RBF network’s adaptability, particularly for predictive maintenance in China EV fleets. The scalability of this approach also supports its application in commercial vehicles, where braking performance directly impacts operational costs and safety.
In conclusion, the RBF network-based sliding mode variable structure control offers a robust solution for hydraulic pressure regulation in electric car electronic braking systems. Through simulations and experiments, this method demonstrated significant improvements in tracking accuracy, stability, and adaptability compared to traditional approaches. As the China EV market continues to expand, such advanced control strategies will play a crucial role in ensuring vehicle safety and efficiency, paving the way for wider adoption of intelligent braking technologies globally.