The dynamic stability and energy recovery efficiency are two pivotal performance metrics for a pure electric vehicle car. Dynamic stability, often reflected in ride comfort, and energy recovery efficiency, which directly influences driving range, are critical for widespread adoption. Active suspension control can significantly enhance vehicle comfort; however, it is inherently a vertical control problem that is coupled with the vehicle’s longitudinal motion during maneuvers such as braking. This work addresses the integrated challenge of improving both the dynamic ride quality and the energy recuperation capability of an electric vehicle car during braking events.
This paper presents a comprehensive control methodology (CCM) for an electric vehicle car. The core of the approach involves establishing a coupled longitudinal-vertical interaction model that captures the mutual influences during braking, such as load transfer affecting tire forces and vice-versa. To manage this interaction, the CCM combines two advanced control strategies: Model Predictive Control (MPC) for the vertical dynamics to ensure stability and comfort, and a Neuro-Fuzzy Control (NFC) strategy for the longitudinal dynamics to maximize regenerative braking energy. The NFC is trained using a proposed optimization framework that explicitly considers the constraints imposed by vehicle vertical motion on the attainable braking torque. The performance of the proposed CCM is evaluated through simulation and compared against a baseline Dual-Loop Multi-Stage Control (DL-MSC) strategy, demonstrating significant improvements in both energy recovery and dynamic stability for the electric vehicle car.
Longitudinal-Vertical Interaction Model for Electric Vehicle Car
System Interaction and Vertical Dynamics
The integrated behavior of an electric vehicle car under braking is governed by the interaction between its longitudinal and vertical motions. The vertical model describes the dynamics of the sprung and unsprung masses, while the longitudinal model governs the vehicle’s deceleration. These are coupled because braking intensity causes dynamic load transfer, altering the vertical tire forces, which in turn limits the maximum longitudinal braking force that can be applied by each tire.
A five-degree-of-freedom (5-DOF) half-car model is adopted for the electric vehicle car, considering symmetry. This model includes heave and pitch motions of the sprung mass and vertical motions of the front and rear unsprung masses. The distinctive feature is the inclusion of equivalent forces generated due to braking intensity, which act on the sprung mass. The dynamics are derived as follows:
The forces at the front and rear suspensions, including active control forces, are:
$$F_f = K_{sf}(z_1 – z_2) + C_{sf}(\dot{z}_1 – \dot{z}_2) + F_{fc}$$
$$F_r = K_{sr}(z_3 – z_4) + C_{sr}(\dot{z}_3 – \dot{z}_4) + F_{rc}$$
where \(K_{sf}, C_{sf}, K_{sr}, C_{sr}\) are suspension stiffness and damping coefficients, \(z_1, z_3\) are unsprung mass displacements, \(z_2, z_4\) are sprung mass displacements at front and rear, and \(F_{fc}, F_{rc}\) are active control forces.
The equations of motion for the sprung mass (heave and pitch) and unsprung masses are:
$$m_b \ddot{z}_b = F_f + F_r$$
$$I_b \ddot{\theta}_b = -aF_f + bF_r – aF_{mf} – bF_{mr}$$
$$m_{wf} \ddot{z}_1 = -K_{tf} z_1 – F_f$$
$$m_{wr} \ddot{z}_3 = -K_{tr} z_3 – F_r$$
with the geometric relations:
$$z_2 = z_b – a\theta_b, \quad z_4 = z_b + b\theta_b$$
The equivalent braking forces \(F_{mf}\) and \(F_{mr}\) acting on the sprung mass due to braking intensity \(z\) are:
$$F_{mf} = zm_b g h_g / L, \quad F_{mr} = -zm_b g h_g / L$$
where \(h_g\) is the sprung mass center of gravity height and \(L = a+b\) is the wheelbase.
The actual vertical tire loads, crucial for longitudinal force generation, are then:
$$F_{z1} = G b / L + K_{tf} z_1, \quad F_{z2} = G a / L + K_{tr} z_3$$
where \(G = (m_b + m_{wf} + m_{wr})g\). The active suspension control forces \(F_{fc}, F_{rc}\) are generated by rotary actuators. The actuator dynamics, considering a permanent magnet synchronous motor (PMSM) model, and its associated power consumption \(P_{ele\_sus}\) are modeled to account for the energy used by the active system.
Longitudinal Dynamics and Powertrain of Electric Vehicle Car
The electric vehicle car under consideration is a four-wheel-drive (4WD) configuration with independent front and rear motors. The longitudinal motion during braking on a straight path is given by Newton’s second law:
$$m \delta \dot{v}_x = F_b + F_f + F_w + F_i$$
where \(F_b\) is the total braking force, \(F_f\) is rolling resistance, \(F_w\) is aerodynamic drag, and \(F_i\) is grade resistance. The braking intensity \(z\) is defined as:
$$z = -\dot{v}_x / g$$
The braking torque at each wheel is a combination of regenerative torque from the electric motors and hydraulic friction brake torque:
$$T_{d1} = T_{hf} + T_{md1}, \quad T_{d2} = T_{hr} + T_{md2}$$
$$T_{hf} = \frac{\pi}{4} p_{hf} D_f^2 R_f K_f, \quad T_{hr} = \frac{\pi}{4} p_{hr} D_r^2 R_r K_r$$
The motor torques at the wheels are related to the motor shaft torques through the final drive ratios and efficiencies:
$$T_{m1} = T_{md1} i_1 / \eta_1, \quad T_{m2} = T_{md2} i_2 / \eta_2$$
The wheel rotational dynamics are:
$$J_{w1} \dot{\omega}_1 = -T_{d1} – F_{b1} R_t, \quad J_{w2} \dot{\omega}_2 = -T_{d2} – F_{b2} R_t$$
The longitudinal tire force \(F_{bi}\) for each wheel is modeled using a simplified Pacejka magic formula:
$$F_{bi} = F_{zi} D \sin(C \arctan(B s_i – E(B s_i – \arctan(B s_i))))$$
The battery is modeled using an internal resistance model. The total recovered electrical power from the powertrain during braking, considering both drivetrain and suspension actuators, is a key metric:
$$P_{b\_total} = P_b + \sum_{j=1}^{2} P_{ele\_sus\_j} = (E – (I_p+I_{sus})r)(I_p+I_{sus})$$
where \(P_b\) is the regenerative power from the traction motors, and \(P_{ele\_sus\_j}\) is the power from the suspension motors (positive for consumption, negative for regeneration). The energy recovered over a period \(t\) is \(Q_b = \int_0^t E I_p \, dt\).
Comprehensive Control Methodology (CCM)
The proposed CCM for the electric vehicle car decomposes the control problem into two coordinated layers: vertical control using MPC and longitudinal control using NFC.
Model Predictive Control for Vertical Dynamics
The MPC controller is designed to improve the dynamic stability and ride comfort of the electric vehicle car by regulating the active suspension forces. It treats the braking intensity \(z\) as a measured disturbance. The state-space model derived from the 5-DOF half-car model is discretized with sampling time \(T_s\). The control variable vector is \(\mathbf{u} = [z, F_{fc}, F_{rc}]^T\), but the braking intensity \(z\) is treated as an external input determined by the driver or upper-level controller, leaving \(F_{fc}\) and \(F_{rc}\) as the manipulable variables for suspension control. The state vector is \(\mathbf{x} = [z_1, z_2, z_3, z_4, \dot{z}_1, \dot{z}_2, \dot{z}_3, \dot{z}_4]^T\).
The discrete-time prediction model is:
$$\boldsymbol{\xi}(t+1|t) = \mathbf{A}_a \boldsymbol{\xi}(t|t) + \mathbf{B}_a \Delta \mathbf{u}(t|t)$$
$$\boldsymbol{\eta}(t|t) = \mathbf{C}_a \boldsymbol{\xi}(t|t)$$
where \(\boldsymbol{\xi}\) is the augmented state vector. The control objective is to minimize a cost function \(J(t)\) over the prediction horizon \(N_p\) and control horizon \(N_c\):
$$J(t) = \sum_{i=1}^{N_p} \| \boldsymbol{\eta}(t+i|t) – \boldsymbol{\eta}_{ref}(t+i|t) \|^2_{\mathbf{Q}} + \sum_{i=0}^{N_c-1} \| \Delta \mathbf{u}(t+i|t) \|^2_{\mathbf{R}} + \rho \epsilon^2$$
where \(\mathbf{Q}\) and \(\mathbf{R}\) are weighting matrices, \(\rho\) is a weight on the slack variable \(\epsilon\), and the reference \(\boldsymbol{\eta}_{ref}\) is typically zero, aiming to minimize body accelerations and suspension travel. The optimization is subject to constraints on the control increments and the suspension force magnitudes, ensuring actuator limits are respected.
Neuro-Fuzzy Control for Longitudinal Energy Recovery
The longitudinal controller for the electric vehicle car aims to distribute the total required braking torque between the front and rear regenerative motors and the hydraulic brakes to maximize energy recovery while adhering to safety constraints (e.g., wheel lock-up prevention, adherence to ECE regulations). A neuro-fuzzy system is employed due to its ability to model complex, nonlinear relationships and adapt based on data.
Constraints: The brake force distribution must satisfy several conditions related to vehicle stability and actuator limits:
1. Adherence to ECE regulation curve (minimum rear axle braking):
$$\beta \ge \frac{0.07 + 0.85 z}{z L / (g h_g) + 0.07}$$
2. Prevention of premature rear wheel lock-up:
$$\beta \le 1 – \frac{0.05 + 0.45 z}{z L / (g h_g) – 0.05}$$
3. Ensuring sufficient total braking force relative to adhesion limit:
$$\beta \ge \frac{g h_g}{z L} (b/L – z)$$
4. Motor torque saturation limits:
$$T_{m1} \le \min(\eta_1 \gamma_1 \frac{R_t}{i_1} \mu_0 F_{z1}, \quad T_{m1max})$$
$$T_{m2} \le \min(\eta_2 \gamma_2 \frac{R_t}{i_2} \mu_0 F_{z2}, \quad T_{m2max})$$
where \(\beta\) is the brake force distribution coefficient defined as the ratio of front axle braking force to total braking force.
Optimization Framework and NFC Training: The core objective for the longitudinal controller of the electric vehicle car is to maximize the recovered battery energy \(Q_{bt}\) over a sampling period, which is derived from the motor regenerative power:
$$Q_{bt} = \int_0^{T_s} \left( \eta_{m1} \frac{T_{m1} n_{m1}}{9550} + \eta_{m2} \frac{T_{m2} n_{m2}}{9550} – I_p^2 r \right) dt$$
An optimization framework is designed to generate training data for the NFC. For a given set of inputs—braking intensity \(z\), front wheel speed \(\omega_1\), and rear wheel speed \(\omega_2\)—the algorithm searches over a discretized range of the distribution coefficient \(\beta\). For each candidate \(\beta_i\), it calculates the feasible motor torques \(T_{m1i}, T_{m2i}\) and hydraulic pressures \(p_{hfi}, p_{hri}\) using the constraints and vehicle models, and then computes the corresponding recovered energy \(Q_{bi}\). The optimal \(\beta_{opt}\) that yields the maximum \(Q_{bt}\) is selected. This process generates a dataset mapping \((z, \omega_1, \omega_2)\) to \(\beta_{opt}\).
This dataset is used to train a neuro-fuzzy inference system (ANFIS). The inputs are fuzzified using membership functions (e.g., Small, Medium, Large for \(z\), \(\omega_1\), \(\omega_2\)), and a set of fuzzy rules is learned. The trained NFC then serves as the real-time controller, taking current vehicle states and outputting the optimal brake force distribution coefficient \(\beta\) for the electric vehicle car. The final motor and hydraulic torque commands are derived from \(\beta\), \(z\), and the actuator models.
Simulation Verification and Discussion
The performance of the proposed CCM for the electric vehicle car is validated through simulation under a combined braking profile, which includes progressive and constant braking phases, with an initial speed of 80 km/h. The proposed CCM is compared against a baseline Dual-Loop Multi-Stage Control (DL-MSC) strategy. The DL-MSC uses a classical double-loop PID controller for active suspension and a rule-based, multi-stage brake force distribution map for longitudinal control. Key vehicle parameters for the electric vehicle car are listed in the table below.
| Parameter | Value |
|---|---|
| Wheelbase, \(L\) | 2.90 m |
| Sprung mass CoG height, \(h_g\) | 0.8 m |
| Front Motor: Rated Power / Max Torque | 45 kW / 242 Nm |
| Rear Motor: Rated Power / Max Torque | 50 kW / 283 Nm |
| Battery Capacity / Voltage | 234 Ah / 320 V |
| Final Drive Ratio (\(i_1, i_2\)) | 9.01 |
| MPC: Prediction Horizon (\(N_p\)) / Control Horizon (\(N_c\)) | 20 / 10 |
| Sampling Time (\(T_s\)) | 0.02 s |
The simulation results demonstrate the advantages of the CCM for the electric vehicle car. The motor torques under both control strategies remain within their saturation limits, confirming the effectiveness of the constraint handling. More importantly, the motor efficiency, particularly for the rear motor, is higher under CCM than under DL-MSC, indicating better utilization of the regenerative braking potential.
The most significant result is the total recovered energy. The final state of charge (SOC) gain from regenerative braking is approximately 8.33% higher with the CCM strategy compared to the DL-MSC strategy for the electric vehicle car. This substantial improvement stems from the NFC’s ability to optimally distribute braking torque in real-time, considering the dynamic tire load variations predicted by the MPC-controlled suspension.
Regarding vertical dynamics, the CCM shows superior performance in stabilizing the electric vehicle car’s body. The maximum vertical velocities of the front and rear parts of the sprung mass are reduced by 69.51% and 72.38%, respectively, compared to DL-MSC. The MPC controller quickly drives the vertical motion towards the target (zero acceleration) during the constant braking phase, significantly improving ride comfort. This superior performance requires higher active suspension control forces from the CCM, as expected. The energy consumed by the suspension actuators under CCM is detailed below.
| Control Method | Front Suspension (J) | Rear Suspension (J) |
|---|---|---|
| CCM | 280.9 | 278.9 |
| DL-MSC | 77.4 | 62.9 |
Although the CCM consumes more energy for active suspension control, this additional consumption is minor compared to the substantial gain in battery energy recovered from the powertrain. Therefore, the net effect of the CCM is a significant overall improvement in the energy efficiency of the electric vehicle car, achieved concurrently with a major enhancement in dynamic stability and ride comfort.
Conclusion
This research has addressed the integrated control challenge for an electric vehicle car by developing a Comprehensive Control Methodology (CCM). A coupled longitudinal-vertical interaction model was established, accurately capturing the mutual influences such as load transfer and its effect on tire forces during braking. The CCM synergistically combines Model Predictive Control (MPC) for vertical dynamics and Neuro-Fuzzy Control (NFC) for longitudinal dynamics. The MPC effectively manages the active suspension to suppress body oscillations induced by braking, treating braking intensity as a disturbance. The NFC, trained via a proposed optimization framework that accounts for vertical motion constraints, optimally allocates braking torque between the front and rear regenerative motors and hydraulic brakes to maximize energy recovery. Comparative simulations against a baseline Dual-Loop Multi-Stage Control (DL-MSC) strategy demonstrate that the proposed CCM simultaneously and significantly enhances both the dynamic stability (reducing body vertical velocity by over 69%) and the energy recovery efficiency (increasing recovered energy by 8.33%) of the electric vehicle car. This work provides a viable and effective co-design framework for the braking and suspension systems of advanced electric vehicle cars.