The pursuit of energy efficiency and extended range is a paramount challenge in modern electric car development. Regenerative braking, which recovers kinetic energy during deceleration and converts it back into electrical energy, represents a critical technology in this endeavor. However, a significant conflict arises when the Anti-lock Braking System (ABS) is activated. Coordinating the conventional friction brake (often hydraulic or pneumatic) with the regenerative brake from the electric motor to ensure both vehicle stability and maximum energy recovery is a complex control problem. This paper addresses this challenge by proposing a novel hierarchical cooperative control strategy integrated with an electromechanical brake-by-wire (E-BBW) system specifically designed for an electric car.
The core innovation lies in a three-layer control architecture that seamlessly blends the rapid, precise response of the regenerative brake with the high-capacity, stable force of the pneumatic brake. This strategy aims to maintain wheel slip near its optimal value for maximum road adhesion (ensuring ABS functionality and vehicle stability) while simultaneously maximizing the portion of braking torque handled by the regenerative system to improve energy recuperation. The proposed system and its control logic were validated using a sophisticated hardware-in-the-loop (HIL) simulation platform, demonstrating superior performance in both straight-line and cornering braking scenarios on surfaces with varying friction coefficients.
System Modeling for the Electric Car Braking System
To design an effective controller, a comprehensive model of the electric car’s braking dynamics and the proposed electromechanical brake-by-wire system is essential.
Electromechanical Brake-by-Wire (E-BBW) System Layout
The proposed E-BBW system for the electric car features a redundant architecture prioritizing safety. The brake pedal unit incorporates two proportional valves and a displacement sensor. The valves generate a control air pressure proportional to the pedal stroke, while the sensor measures the driver’s deceleration demand. A key component is the dual-check valve, which selects the higher control pressure from either the pedal or an active brake valve (connected to the air reservoir) and directs it to a relay valve. The relay valve acts as a safety amplifier and fail-safe; it magnifies the control signal into a high-flow supply pressure for the wheel brakes. Under normal electronic control, the active brake valve opens based on controller commands, and the ABS modulators at each wheel precisely regulate the chamber pressure. In case of an electronic failure, the controller is deactivated, and the driver’s mechanical pedal input directly controls the relay valve, ensuring basic braking capability.
For the electric car’s drivetrain, the rear wheels are driven. The braking torque on a driven wheel ($i = RL, RR$) is the sum of the regenerative torque from the electric motor (EM), transmitted via the gearbox and differential, and the pneumatic torque: $T_{bi} = T_{ei} + T_{pi}$. For non-driven front wheels ($i = FL, FR$), only pneumatic torque is present: $T_{bi} = T_{pi}$. The pneumatic torque is modeled as linearly proportional to the brake chamber pressure: $T_{pi} = K_i P_i$, where $K_i$ is the brake gain coefficient.
Vehicle and Wheel Dynamics Model
The longitudinal dynamics of the electric car during braking are described by:
$$ m \dot{u}_x = \sum_{i} F_{xi}(\lambda_i, F_{zi}) $$
where $m$ is the vehicle mass, $u_x$ is the longitudinal velocity, $F_{xi}$ is the longitudinal tire force for wheel $i$, and $F_{zi}$ is the vertical load. The wheel slip ratio $\lambda_i$, a critical state for ABS control, is defined for braking as:
$$ \lambda_i = \frac{u_x – R_i \omega_i}{u_x} $$
where $R_i$ is the wheel radius and $\omega_i$ is the wheel angular velocity. The rotational dynamics of each wheel are given by:
$$ J_i \dot{\omega}_i = F_{xi}(\lambda_i, F_{zi}) R_i – T_{bi} $$
where $J_i$ is the wheel’s moment of inertia.
The electric motor and driveline dynamics are simplified to a first-order response for control design. The torque transmitted to the driven axle, $T_d$, is related to the motor torque $T_e$ and the gear ratios. The differential distributes this torque equally to the two driven wheels under steady-state assumptions, a key point for the control allocation strategy: $T_{eRL} = T_{eRR} = T_d / 2$.
Simplified Pneumatic ABS Actuator Model
The pneumatic ABS modulator is a critical actuator. Its detailed internal dynamics involving solenoid valves and diaphragms can be significantly simplified for controller design by focusing on its input-output behavior. Experimental characterization showed that the pressure change $\Delta P$ in one control cycle $\tau_c$ is a function of the current chamber pressure $P_i$, the commanded state $u_i$ (increase, hold, decrease), and the duration $t_a$ of that command within the cycle:
$$ \Delta P = f(P_i, u_i, t_a) $$
This relationship is piecewise-linear and can be represented by a set of discrete possible pressure changes $\Omega_a$ corresponding to specific command durations $\Gamma_a$. This simplification captures the essential hysteresis and quantized nature of the pneumatic actuator’s response, which is much slower and coarser than the near-instantaneous torque control of the electric car’s motor.
Design of the Hierarchical Cooperative Controller
The proposed hierarchical controller has three layers: a top-level Slip Rate Controller (SMC) that determines the total braking torque needed per wheel/axle, a mid-level Braking Torque Distribution Controller that optimally splits this demand between the pneumatic and regenerative systems, and a low-level Actuator Controller that executes the commands.
| Parameter | Symbol | Typical Value / Range |
|---|---|---|
| Vehicle Mass | $m$ | 4850 kg |
| Wheel Radius | $R$ | 0.369 m |
| Max Regenerative Torque (per motor) | $T_{e}^{max}$ | 100 Nm |
| Pneumatic Control Cycle | $\tau_1$ | 30 ms |
| Regen Control Cycle | $\tau_2$ | 1 ms |
| Front Brake Gain | $K_f$ | 8500 Nm/MPa |
| Rear Brake Gain | $K_r$ | 9400 Nm/MPa |
Top Layer: Sliding Mode Slip Ratio Controller
This controller’s objective is to make the actual wheel slip $\lambda_i$ track a desired optimal slip ratio $\lambda_{di}$. The optimal slip $\lambda_{di}$ is determined based on estimated road adhesion. For the low-$\mu$ side during split-$\mu$ braking, it is set for maximum force. For the high-$\mu$ side, it is adjusted to limit the yaw moment for stability: $\lambda_{di} = f^{-1}(\mu_H)$, where $\mu_H = \mu_L + \frac{4M_{thre}}{N L}$ and $M_{thre}$ is the allowable yaw moment threshold.
A sliding surface $s_i$ with proportional-integral (PI) form is defined to eliminate steady-state error:
$$ s_i = e_i + \alpha \int e_i dt, \quad e_i = \lambda_i – \lambda_{di} $$
Using the vehicle dynamics, the derivative of the sliding surface is derived. A control law with a saturation function to reduce chattering is chosen:
$$ \dot{s}_i = -\epsilon_1 s_i – \epsilon_0 sat(s_i / \phi) $$
Solving for the total desired braking torque $T_{bi}^d$ yields the SMC law:
$$ T_{bi}^d = \frac{J_i}{R_i} \left[ \frac{R_i}{J_i} F_{xi} – \frac{1}{m} \sum F_{xi} + \dot{u}_x (\alpha \lambda_i – \alpha \lambda_{di} + \dot{\lambda}_{di}) – u_x (\epsilon_1 s_i + \epsilon_0 sat(s_i / \phi)) \right] $$
To ensure vehicle stability during braking, especially in a turn, a direct constraint is applied to the torque commands for left and right wheels: $|T_l^d – T_r^d| \leq T_{diff}$, where $T_{diff}$ is the maximum allowable torque difference to prevent excessive yaw moment.
Middle Layer: Optimal Braking Torque Distribution
This layer distributes the total desired torque $T_{bi}^d$ for the driven rear wheels between the regenerative ($T_{ei}$) and pneumatic ($T_{pi}$) systems. It exploits the different response times: the pneumatic system is slow ($\tau_1=30$ms) with coarse control, while the regenerative system is fast ($\tau_2=1$ms) and precise. The fast regenerative torque is used to compensate for the errors and lag in the pneumatic system’s response, improving overall tracking performance and energy recovery.
For the driven axle, the optimization cost function $J$ is defined to minimize tracking error for both wheels while maximizing regenerative braking toward an optimal value $T_d^{opt}$:
$$ J = (T_{bRL}^d – (T_{pRL}+T_d/2))^2 + (T_{bRR}^d – (T_{pRR}+T_d/2))^2 + \beta (T_d – T_d^{opt})^2 $$
Here, $T_d$ is the total regenerative torque requested from the motor (before the differential splits it). The optimal regenerative torque $T_d^{opt}$ is set close to the motor’s maximum capability but is bounded by the total driver demand to prevent over-braaking: $T_d^{opt} = min(T_d^{max}, 2T_{bRL}^d, 2T_{bRR}^d) – T_{res}$, where $T_{res}$ is a small reserve.
Minimizing $J$ with respect to $T_d$ yields an analytical solution for the desired regenerative torque. This desired $T_d$ is then used, along with the predicted pneumatic torque based on the possible pressure changes $\Omega_a$, to calculate the optimal pressure adjustment commands $\{P_{aRL}, P_{aRR}\}$ for the upcoming control cycle that will minimize the future cost $J(k+1)$.
Low Layer: Actuator Controllers
Regenerative Torque Controller: The desired regenerative torque $T_d^d$ from the middle layer is sent directly to the electric car’s motor controller. Given the fast response, it is typically achieved within one control cycle ($\tau_2$).
Pneumatic Pressure Controller: This logic block translates the desired pressure change $\Delta P_i^d$ into specific commands for the ABS modulator valve. Based on the simplified model and the set of possible changes $\Omega_a$, it selects the command state $u_i$ (increase/hold/decrease) and its duration $t_a$ within the fixed $\tau_1$ cycle to achieve the closest possible pressure adjustment. If the desired change is smaller than the minimum achievable step, a “hold” command is issued. This predictive model-based control accounts for the actuator’s hysteresis and quantization.
Experimental Validation and Results
The proposed hierarchical control strategy for the electric car was tested on a comprehensive Hardware-in-the-Loop (HIL) simulation platform replicating the full E-BBW system and vehicle dynamics.
Pressure Regulation Performance
First, the low-level pneumatic actuator’s ability to track pressure commands was validated. The controller successfully tracked both step and sinusoidal desired pressure profiles. The main error source was the quantization of the pressure change per cycle; the steady-state error for a 0.7 MPa step was about 0.03 MPa, and the maximum tracking error for the sinusoid was 0.06 MPa. This performance was deemed sufficient for the higher-level braking control, as the regenerative system can compensate for these small, low-frequency errors.
Straight-Line ABS Braking on Different Surfaces
The integrated system was tested for emergency braking from 70 km/h on two road surfaces: low adhesion ($\mu \approx 0.2$) and high adhesion ($\mu \approx 0.8$).
On the low-$\mu$ surface, the controller maintained wheel slip near the optimal range, preventing lock-up. The wheel speed oscillation for the rear driven wheels was slightly larger (7.2 km/h) than for the front wheels (0.6 km/h), but remained stable. Crucially, the actual braking torque on the rear wheels (which had regenerative compensation) tracked the desired torque more accurately than on the front wheels (pneumatic only). The regenerative torque was maximized in the initial phase and then maintained at a high level, with fine adjustments compensating for pneumatic errors, demonstrating effective cooperative control in the electric car.
On the high-$\mu$ surface, the electric car decelerated more rapidly and stably. The torque tracking performance showed the same trend, with rear wheel torque tracking being superior due to the combined action of the regenerative and pneumatic systems.
| Test Scenario | Max Wheel Speed Oscillation | Slip Ratio Stability | Key Observation |
|---|---|---|---|
| Low-$\mu$ Straight Line | ~7.2 km/h (Rear) | Stable, near optimal | Regen torque compensates pneumatic lag, improving rear wheel torque tracking. |
| High-$\mu$ Straight Line | Reduced | Excellent | Rapid deceleration, stable slip control maintained. |
| High-$\mu$ Cornering (220m radius) | N/A | N/A | Vehicle closely followed desired path with control; diverged without ABS. |
| Low-$\mu$ Cornering (90m radius) | N/A | N/A | Proposed control improved vehicle steerability compared to uncontrolled braking. |
Cornering Braking for Stability Validation
To test robustness and stability enhancement, braking was applied during cornering on split-$\mu$ surfaces. The desired slip ratio was fixed at 0.15 for simplicity. On a high-$\mu$ road with a 220m radius, the electric car with the proposed control strategy accurately followed the desired circular path. In contrast, braking without ABS control caused the vehicle to diverge from the path. On a more demanding low-$\mu$ road with a 90m radius, the proposed controller still allowed the vehicle to follow the path, whereas uncontrolled braking led to a loss of steerability. This demonstrates that the hierarchical strategy not only manages energy recovery but also actively enhances the turning stability of the electric car during aggressive braking.
Conclusion
This research successfully developed and validated a hierarchical cooperative control strategy for an electromechanical brake-by-wire system in an electric car. The strategy elegantly resolves the conflict between ABS intervention and regenerative braking by employing a three-layer architecture: a Sliding Mode Controller for robust slip regulation, an Optimal Distribution Controller that leverages the fast response of the regenerative system to compensate for the slower pneumatic system and maximize energy recovery, and efficient low-level actuator controllers.
Experimental results from a hardware-in-the-loop platform confirm the strategy’s effectiveness. It provides precise wheel slip control near the stability optimum during ABS activation, ensures vehicle stability even in challenging split-$\mu$ cornering maneuvers, and significantly enhances energy regeneration by prioritizing the use of the electric motor for braking torque. The proposed system offers a practical and cost-effective solution for integrating advanced stability control with high-efficiency regenerative braking in modern electric cars, contributing directly to improved safety and extended driving range.