In recent years, the advancement of battery EV car technology has revolutionized the automotive industry, particularly with the emergence of distributed drive electric vehicles (DDEVs). These battery EV car platforms utilize in-wheel motors to independently control each wheel, offering enhanced maneuverability and stability. However, challenges such as excessive tire wear and slip loss due to aggressive direct yaw-moment control (DYC) interventions persist, compromising energy efficiency and driving comfort in battery EV car systems. To address these issues, I propose a novel chassis collaborative control strategy that integrates torque coordination and active rear steering (ARS) for a distributed drive battery EV car. This strategy employs a hierarchical architecture with an upper-level controller using an adaptive discrete reference sliding-mode trajectory with higher-order disturbance compensation, based on discrete-time sliding mode predictive control (DSMPC), and a lower-level controller that allocates control signals between ARS and DYC systems. Through comprehensive simulations, this approach demonstrates significant improvements in handling stability and reduction in tire longitudinal slip losses, showcasing its potential for enhancing the performance of battery EV car applications.
The importance of battery EV car development lies in its ability to reduce carbon emissions and improve energy sustainability. In a distributed drive battery EV car, the independent control of each wheel enables precise torque vectoring, but it also introduces complexities in maintaining yaw stability. Traditional control methods, such as DYC, often lead to frequent torque interventions, causing tire slip and energy loss in battery EV car operations. This not only affects the longevity of drive motors but also reduces overall efficiency in battery EV car systems. Therefore, there is a pressing need for advanced control strategies that balance stability and efficiency in battery EV car designs. My work focuses on integrating ARS and DYC through a DSMPC-based framework to mitigate these drawbacks, ensuring smoother and more sustainable battery EV car performance.
To establish a foundation for control design, I first develop vehicle models that capture the essential dynamics of a battery EV car. The 2-DOF reference model is used to generate desired responses under ideal linear conditions, which serves as a benchmark for the controller. For a battery EV car, this model simplifies lateral and yaw motions, ignoring longitudinal dynamics and nonlinearities. The equations are given by:
$$ m(\dot{V}_y + V_x \omega_r) = (k_f + k_r)\frac{V_y}{V_x} + (l_f k_f – l_r k_r)\frac{\omega_r}{V_x} – k_f \delta_f $$
$$ I_z \dot{\omega}_r = (l_f k_f – l_r k_r)\frac{V_y}{V_x} + (l_f^2 k_f – l_r^2 k_r)\frac{\omega_r}{V_x} – l_f k_f \delta_f $$
Here, \( m \) is the mass of the battery EV car, \( I_z \) is the yaw moment of inertia, \( k_f \) and \( k_r \) are the cornering stiffnesses of the front and rear tires, \( \omega_r \) is the yaw rate, \( V_x \) and \( V_y \) are longitudinal and lateral velocities, \( l_f \) and \( l_r \) are distances from the center of gravity to the front and rear axles, and \( \delta_f \) is the front steering angle. This model provides the reference yaw rate and sideslip angle for the battery EV car controller.
For the predictive model in DSMPC, I use a 3-DOF vehicle dynamics model that includes longitudinal, lateral, and yaw motions, which is more suitable for real-time control in a battery EV car. The state equations are:
$$ \dot{V}_x = \omega_r V_y + \frac{1}{m} \sum F_x $$
$$ \dot{V}_y = -\omega_r V_x + \frac{1}{m} \sum F_y $$
$$ \dot{\omega}_r = \frac{1}{I_z} \sum M_z $$
Where \( \sum F_x \), \( \sum F_y \), and \( \sum M_z \) are the total longitudinal force, lateral force, and yaw moment, respectively. These forces and moments are derived from tire interactions in the battery EV car. To linearize and discretize this model for DSMPC, I apply Taylor expansion and sampling, resulting in:
$$ x(k+1) = A x(k) + B u(k) + B_d d(k) $$
$$ y(k) = C x(k) $$
With state vector \( x(k) = [V_x(k), V_y(k), \omega_r(k)]^T \), control input \( u(k) = [\sum F_x(k), \sum F_y(k), \sum M_z(k)]^T \), and disturbance \( d(k) \) accounting for unmodeled dynamics in the battery EV car. The matrices \( A \), \( B \), \( B_d \), and \( C \) are defined based on the sampling period \( T \). This discrete model forms the basis for the predictive control in the battery EV car system.
The control system is designed with a hierarchical architecture to manage the high degrees of freedom in a distributed drive battery EV car. The upper-level controller utilizes DSMPC to compute generalized forces and moments, while the lower-level controller coordinates ARS and DYC for optimal allocation. This structure ensures robustness and efficiency in battery EV car operations. Key components include the reference state design, disturbance observer, and DSMPC algorithm.
For reference states, the desired longitudinal velocity \( v_{xd} \) is derived from the driver’s acceleration command, and the desired yaw rate \( \omega_d \) is calculated using the 2-DOF model with saturation limits based on road friction. For a battery EV car, this ensures stability under varying conditions. The equations are:
$$ \omega_d = \min \left( \frac{v_x / L}{1 + K v_x^2} \delta_f, \left| \frac{\mu g}{v_x} \right| \text{sgn}(\delta_f) \right) $$
$$ K = \frac{m (l_f / k_r – l_r / k_f)}{L^2} $$
Where \( L \) is the wheelbase, \( \mu \) is the road friction coefficient, and \( g \) is gravity. The desired sideslip angle \( \beta_d \) is set to zero to minimize lateral deviation in the battery EV car.
To handle uncertainties in the battery EV car model, such as parameter variations and external disturbances, I design a nonlinear extended state observer (NESO). This observer estimates the total disturbance \( d \) by treating it as an extended state. For the lateral dynamics:
$$ \dot{V}_y = -\omega_r V_x + \frac{1}{m} \sum F_y + d $$
$$ \dot{d} = \kappa(t) $$
The observer equations are:
$$ e_1 = z_1 – V_y $$
$$ \dot{z}_1 = z_2 – \beta_1 e_1 – \omega_r V_x + \frac{1}{m} \sum F_y $$
$$ \dot{z}_2 = -\beta_2 \text{fal}(e_1, a_1, \varepsilon) $$
Where \( z_1 \) and \( z_2 \) are estimates of \( V_y \) and \( d \), \( \beta_1 \) and \( \beta_2 \) are correction coefficients, and \( \text{fal} \) is a nonlinear function. This enhances the robustness of the battery EV car control system.
The core of the upper-level controller is the DSMPC algorithm, which integrates sliding mode control with model predictive control to reduce chattering and improve tracking in the battery EV car. I propose an adaptive discrete reference sliding-mode trajectory with higher-order disturbance compensation. The sliding surface is defined as:
$$ s(k) = c_e e(k) $$
$$ c_e = \text{diag}(c_{e1}, c_{e2}, c_{e3}), \quad c_{ei} > 0 $$
Where \( e(k) \) is the tracking error between the predicted output and reference. The reference trajectory \( s_r(k) \) is designed using a novel reaching law:
$$ s_r(k+1) = (1 – \eta(s(k)) T) s(k) + \delta_k $$
$$ \eta(s(k)) = \frac{\beta_1}{\varepsilon + (1 + 1/|s(k)| – \varepsilon) e^{-\beta |s(k)|}} + \beta_2 \frac{\ln(|s(k)|^{1-\alpha} + 1)}{|s(k)|^{1-\alpha}} $$
With \( \delta_k = D_k – 2D_{k-1} + D_{k-2} \) as the disturbance compensation from NESO, and parameters \( \beta_1, \beta_2, \beta, \alpha, \varepsilon \) tuned for the battery EV car. This law ensures smooth convergence and minimizes chattering.
The predictive model for the sliding surface over a horizon \( N_p \) is:
$$ S_m(k) = \Gamma \xi(k) + \Psi \Delta U(k) + \Phi d(k) – C_e R_{des}(k) $$
Where \( \Delta U(k) \) is the control increment vector, and \( \Gamma, \Psi, \Phi, C_e \) are matrices derived from the system model. The objective function for DSMPC in the battery EV car is formulated as a quadratic programming problem:
$$ J(k) = (S_m(k) – S_r(k))^T Q (S_m(k) – S_r(k)) + \Delta U(k)^T W \Delta U(k) + \rho \psi^2(k) $$
Subject to constraints on control increments and outputs. Here, \( Q \) and \( W \) are weighting matrices, and \( \psi(k) \) is a slack variable. Solving this yields the optimal control increments, which are applied to the battery EV car system.
The lower-level controller coordinates ARS and DYC to allocate the generalized forces and moments computed by DSMPC. This coordination is crucial for reducing tire slip loss in the battery EV car. The total yaw moment \( M_{zd} \) is distributed as:
$$ M_{ARS} = (1 – \rho) (M_{zd} – M_{AFS}) $$
$$ M_{DYC} = \rho (M_{zd} – M_{AFS}) $$
Where \( \rho \) is a weighting coefficient that dynamically adjusts based on vehicle stability. To design \( \rho \), I analyze the sideslip angle-phase plane stability region for the battery EV car. An instability factor \( I_{unified} \) is defined:
$$ I_{unified} = \max(I_\beta, I_\omega) $$
$$ I_\beta = 1 – \text{sgn}((\beta_{max} – \beta)(\beta – \beta_{min})) \times \frac{\min(|\beta_{max} – \beta|, |\beta – \beta_{min}|)}{(\beta_{max} – \beta_{min})/2} $$
$$ I_\omega = 1 – \text{sgn}((\omega_{max} – \omega)(\omega – \omega_{min})) \times \frac{\min(|\omega_{max} – \omega|, |\omega – \omega_{min}|)}{(\omega_{max} – \omega_{min})/2} $$
This factor partitions the phase plane into stable, critical, and unstable regions for the battery EV car. The weighting function \( \tau(I_{unified}) \) is:
$$ \tau(I_{unified}) =
\begin{cases}
0, & 0 \leq I_{unified} < 0.3 \\
\frac{1}{2} \left(1 – \cos\left(\pi \frac{I_{unified} – 0.3}{1 – 0.3}\right)\right), & 0.3 \leq I_{unified} < 1 \\
1, & I_{unified} \geq 1
\end{cases} $$
Additionally, a yaw rate tracking error index \( I_e \) is considered:
$$ I_e =
\begin{cases}
\frac{|\omega – \omega_{ref}|}{\sigma}, & \frac{|\omega – \omega_{ref}|}{\sigma} < 1 \\
1, & \frac{|\omega – \omega_{ref}|}{\sigma} \geq 1
\end{cases} $$
The final weighting coefficient \( \rho \) is \( \max(\tau(I_{unified}), I_e) \), ensuring that DYC intervenes more during instability in the battery EV car.
For DYC control, the tire force distribution minimizes tire load ratio to reduce slip in the battery EV car. The optimization problem is:
$$ \min_{F_{xi}} f(x) = \frac{1}{2} u_a^T W_a u_a $$
$$ \text{s.t.} \quad \sum F_x = F_{x1} + F_{x2} + F_{x3} + F_{x4} $$
$$ \sum F_y = F_{yd} – \sum_{i=1}^4 F_{yi} \cos \delta_i = \sum_{i=1}^4 F_{xi} \sin \delta_i $$
$$ M_{DYC} = (F_{x2} + F_{x4})b – (F_{x1} + F_{x3})b $$
$$ u_{a,min} \leq u_a \leq u_{a,max}, \quad u_a = [F_{x1}, F_{x2}, F_{x3}, F_{x4}]^T $$
Where \( W_a = \text{diag}(1/(\mu F_{zi})^2) \) and constraints consider motor torque and friction limits for the battery EV car. For ARS control, a similar quadratic programming approach is used to compute rear tire lateral forces \( F_{y3} \) and \( F_{y4} \), which are then converted to rear steering angles \( \delta_3 \) and \( \delta_4 \) using the Dugoff tire inverse model:
$$ \alpha_i =
\begin{cases}
\arctan\left( \frac{F_{yi} C_{si}}{C_{\alpha i} (C_{si} – F_{xi})} \right), & H \geq 1 \\
\arctan\left( \frac{C_{si} \mu^2 F_{yi} F_{zi}^2}{4 C_{si} C_{\alpha i} Q (\mu F_{zi} – Q) – C_{\alpha i} \mu^2 F_{xi} F_{zi}^2} \right), & H < 1
\end{cases} $$
$$ H = \frac{\mu F_{zi}}{2 \sqrt{F_{xi}^2 + F_{yi}^2}}, \quad Q = \sqrt{F_{xi}^2 + F_{yi}^2} $$
And the steering angles are:
$$ \delta_3 = \arctan\left( \frac{V_y – \omega_r l_r}{V_x – \omega_r b} \right) – \alpha_3 $$
$$ \delta_4 = \arctan\left( \frac{V_y – \omega_r l_r}{V_x + \omega_r b} \right) – \alpha_4 $$
This ensures precise lateral control for the battery EV car.
To validate the proposed strategy, I conduct co-simulations using CarSim/Simulink for a battery EV car under double-lane change maneuvers on both dry and wet roads. The battery EV car parameters are listed in Table 1, which are essential for modeling the distributed drive system.
| Parameter | Value |
|---|---|
| Vehicle mass (kg) | 1412 |
| Wheelbase (m) | 2.910 |
| Distance from CG to front axle (m) | 1.015 |
| Distance from CG to rear axle (m) | 1.895 |
| Wheel radius (m) | 0.354 |
| Yaw moment of inertia (kg·m²) | 1536.7 |
| Front tire cornering stiffness (N/rad) | – |
| Rear tire cornering stiffness (N/rad) | – |
The control parameters for DSMPC include a prediction horizon \( N_p = 10 \), control horizon \( N_u = 5 \), sampling time \( T = 0.01 \, \text{s} \), and reaching law parameters \( \beta_1 = \beta_2 = 1 \), \( \beta = 0.5 \), \( \alpha = 2/3 \), \( \varepsilon = 0.1 \). I compare the proposed DSMPC-based coordinated control (DSMPC-ARS+DYC) with other strategies: DSMPC-DYC (only DYC), DSMPC-ARS (only ARS), and ISMC-DYC (integral sliding mode control with DYC). This comparison highlights the benefits for the battery EV car.
On a wet road with low friction coefficient (\( \mu = 0.3 \)) at 80 km/h, the battery EV car trajectories show that all strategies except DSMPC-ARS closely follow the reference path, but DSMPC-ARS+DYC offers the best stability. The tracking errors for yaw rate and sideslip angle are minimized, as summarized in Table 2. The battery EV car with DSMPC-ARS+DYC reduces longitudinal slip energy loss by 15.68% compared to DSMPC-DYC, demonstrating efficiency gains. Moreover, the DSMPC algorithm suppresses chattering in motor torques, enhancing comfort in the battery EV car.
| Control Strategy | Max Sideslip Angle Error \( e_\beta \) (rad) | Max Yaw Rate Error \( e_\omega \) (rad/s) |
|---|---|---|
| DSMPC-ARS+DYC | 0.0113 | 0.0018 |
| DSMPC-DYC | 0.0155 | 0.0022 |
| DSMPC-ARS | 0.0120 | 0.0051 |
| ISMC-DYC | 0.0158 | 0.0028 |
On a dry road with high friction coefficient (\( \mu = 0.8 \)) at 108 km/h, the battery EV car performance further improves. The DSMPC-ARS+DYC strategy achieves the smallest path deviation and phase plane area, indicating superior stability. Table 3 shows that the maximum errors are significantly lower, and the longitudinal slip energy loss is reduced by 41.65% compared to DSMPC-DYC. This underscores the effectiveness of the coordinated approach in optimizing battery EV car dynamics.
| Control Strategy | Max Sideslip Angle Error \( e_\beta \) (rad) | Max Yaw Rate Error \( e_\omega \) (rad/s) |
|---|---|---|
| DSMPC-ARS+DYC | 0.0041 | 0.0012 |
| DSMPC-DYC | 0.0136 | 0.0017 |
| DSMPC-ARS | 0.0053 | 0.0071 |
| ISMC-DYC | 0.0142 | 0.0039 |
The coordination weight \( \rho \) varies dynamically based on the instability factor, as shown in simulation plots. When the battery EV car is in the stable region (\( I_{unified} < 0.3 \)), \( \rho \) is near zero, favoring ARS for smooth control. In critical or unstable regions, \( \rho \) increases to engage DYC for corrective yaw moments. This adaptive allocation ensures that the battery EV car maintains stability without excessive tire slip.
In terms of computational efficiency, the DSMPC algorithm involves solving a QP problem online, which is feasible for real-time implementation in modern battery EV car systems. The use of a hierarchical structure reduces complexity, and the disturbance observer enhances robustness against uncertainties common in battery EV car operations.
In conclusion, the proposed DSMPC-based coordinated control of ARS and DYC offers a significant advancement for distributed drive battery EV car applications. By integrating an adaptive sliding mode reference trajectory with predictive control, it achieves precise tracking of yaw rate and sideslip angle while minimizing tire longitudinal slip loss and suppressing controller chattering. The coordination strategy dynamically balances ARS and DYC based on stability criteria, ensuring optimal performance under diverse road conditions. Future work will focus on optimizing the algorithm for reduced computational load and extending it to heterogeneous battery EV car fleets. This research contributes to the development of smarter, more efficient battery EV car technologies that prioritize both stability and sustainability.