In recent years, the rapid advancement of electric vehicle (EV) technologies has driven significant innovations in braking systems, particularly with the transition from traditional vacuum-assisted brakes to electro-hydraulic systems like the Electronic Braking System (EBooster). As a key component in modern electric vehicle designs, EBooster enhances braking efficiency and supports intelligent driving functions such as adaptive cruise control (ACC) and autonomous emergency braking (AEB). The integration of these systems is crucial for improving the safety and performance of electric vehicles, especially in the context of China’s growing EV market. However, the inherent nonlinearities and time-varying characteristics of hydraulic systems pose challenges for precise pressure control. This study focuses on developing an advanced control strategy using a Radial Basis Function (RBF) neural network combined with sliding mode variable structure control to address these issues. By leveraging the adaptive capabilities of RBF networks, the proposed approach autonomously adjusts controller parameters in response to hydraulic load variations, ensuring robust and accurate pressure regulation. The following sections detail the system modeling, control design, simulation, and experimental validation, highlighting the effectiveness of this method in enhancing the operational performance of electric vehicle electronic braking systems.
The hydraulic system in an electric vehicle’s EBooster comprises components like the master cylinder, wheel cylinders, brake lines, and storage tank, which collectively generate braking force. To simplify the modeling process, the system is treated as a series connection of these elements, with an equivalent pressure leakage model capturing the dynamic behavior. The flow equation governing the system is expressed as:
$$ A \dot{x} = \frac{V(p, t)}{K_e(p, t)} \dot{p} + K_l(p, t) \cdot p $$
where \( A \) represents the piston area, \( x \) is the displacement, \( p \) denotes the pressure, \( V(p, t) \) is the volume, \( K_e(p, t) \) is the equivalent bulk modulus, and \( K_l(p, t) \) is the leakage coefficient. These parameters are pressure- and time-dependent, introducing nonlinearities that complicate control. For instance, at low pressures, leakage coefficients are small, but they increase with pressure, leading to complex dynamics. To achieve rapid response and minimize overshoot, a combined feedforward and feedback control strategy is employed. This approach compensates for nonlinearities and ensures stable pressure tracking, which is critical for applications in electric vehicles like China EV models, where braking performance directly impacts energy recovery and safety.
The RBF neural network is chosen for its ability to approximate nonlinear functions with high accuracy and minimal computational overhead. Unlike traditional backpropagation networks, RBF networks employ localized basis functions, enabling efficient learning and adaptation. The network structure used here consists of an input layer (system pressure), a hidden layer with Gaussian activation functions, and an output layer that estimates the system dynamics. The hidden layer output for the \( j \)-th neuron is given by:
$$ h_j(x_N) = \exp\left( -\frac{\| x_N – c_{Nj} \|^2}{2b_{Nj}^2} \right) $$
where \( x_N \) is the input vector, \( c_{Nj} \) is the center of the \( j \)-th neuron, and \( b_{Nj} \) is its width. The overall output approximates the nonlinear functions \( f(\cdot) \) and \( g(\cdot) \) as:
$$ \hat{f}(x_N) = W^T h(x_N), \quad \hat{g}(x_N) = V^T h(x_N) $$
Here, \( W \) and \( V \) are the optimal weight vectors, and \( h(x_N) \) is the hidden layer output vector. This configuration allows the RBF network to adaptively estimate system parameters, reducing the impact of uncertainties in the electric vehicle braking environment. For the sliding mode controller, a sliding surface 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 law incorporates a reaching law to ensure system stability:
$$ u = \frac{1}{\hat{g}(p)} \left( -\hat{f}(p) – k s – q \cdot \text{sat}(s) \right) $$
with \( k \) and \( q \) as tuning parameters, and \( \text{sat}(s) \) as a saturation function to mitigate chattering. The adaptive laws for updating the RBF network weights are derived using Lyapunov stability theory, guaranteeing convergence and robustness in the face of hydraulic disturbances common in electric vehicle systems.
| Parameter | Value |
|---|---|
| Sliding surface constants \( c_1 \), \( c_2 \) | 7.2, 3.6 |
| Reaching law parameters \( k \), \( q \) | 15.2, 0.09 |
| Boundary layer thickness \( \delta \) | 1 |
| RBF center coordinates \( c_{Nj} \) | 0 |
| RBF width \( b_{Nj} \) | 0.16 |
| Network approximation error bound \( \varepsilon \) | 0.01 |
| Adaptive law gains \( \gamma_1 \), \( \gamma_2 \) | 6.8, 7.6 |
Simulation studies were conducted to evaluate the performance of the RBF-based sliding mode controller under sinusoidal pressure tracking scenarios. The target pressure was set to a 3 MPa bias with a 2 MPa amplitude, and comparisons were made with a conventional PID controller. The results demonstrated that the RBF sliding mode controller achieved a pressure tracking error within ±0.1 MPa, significantly lower than the ±0.3 MPa error observed with PID control. This represents a reduction of over 66% in error magnitude, underscoring the superiority of the proposed method for electric vehicle applications. The RBF network’s ability to autonomously adjust to hydraulic load variations contributed to this enhanced performance, ensuring precise control even under dynamic conditions. Key parameters for the simulation, such as the piston area and transmission ratio, are summarized in the table below, which are typical for electric vehicle braking systems like those in China EV models.
| 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 |
| Small gear inertia (kg·m²) | 2.2 × 10⁻⁶ |
| Large gear inertia (kg·m²) | 2.0 × 10⁻⁴ |
| Ball screw inertia (kg·m²) | 3.9 × 10⁻⁴ |
| Translational mass \( m \) (kg) | 0.5 |
| Transmission efficiency \( \eta \) | 0.9 |
| Spring stiffness \( K \) (kN/m) | 4.5 |
Experimental validation was performed using a rapid prototyping test platform designed for electric vehicle electronic braking systems. The platform enabled real-time control and monitoring, with pressure sensors and motor encoders providing feedback for the RBF sliding mode algorithm. Tests included ramp and sinusoidal pressure profiles to simulate realistic driving conditions, such as ACC and AEB scenarios in electric vehicles. For a ramp input with a gradient of 3 MPa/s and amplitude of 5 MPa, the results showed no overshoot or chattering, with steady-state errors maintained within 0.1 MPa (relative error below 1%). Similarly, under sinusoidal conditions with a 1.5 MPa bias and 3 MPa amplitude, the maximum steady-state error did not exceed 0.1 MPa, and the pressure tracking exhibited minimal delay at peak and trough points. These findings confirm that the control strategy meets the stringent requirements of electric vehicle braking systems, particularly in the context of China EV advancements, where reliability and precision are paramount.
The integration of RBF neural networks with sliding mode control offers a robust solution for managing the nonlinearities in electric vehicle hydraulic braking systems. By continuously adapting to changes in hydraulic load, the controller maintains stability and accuracy across diverse operating conditions. Future work could explore the application of this approach to other EV subsystems, such as regenerative braking, to further enhance energy efficiency. In conclusion, this study establishes a theoretical foundation for advanced pressure control in EBooster systems, contributing to the ongoing evolution of electric vehicle technologies worldwide, with specific implications for the growing China EV market.
In summary, the proposed RBF-based sliding mode control strategy significantly improves the performance of electric vehicle electronic braking systems by reducing pressure tracking errors and enhancing adaptability. Through rigorous simulation and experimental testing, this approach has proven effective for critical functions like ACC and AEB, ensuring that electric vehicles, including those in China’s expanding EV sector, achieve higher levels of safety and efficiency. The use of adaptive neural networks represents a promising direction for future research in electric vehicle dynamics and control.