A Dual-Layer Control System for Steering Stability of Distributed-Drive Electric Vehicle

Abstract

In addressing the limitations of traditional steering stability control strategies applied to distributed-drive electric vehicles (DDEVs)—which primarily focus on measuring yaw rate and sideslip angle and may result in loss of control during steering maneuvers—this study conducts a more comprehensive analysis of DDEVs’ steering control stability. It specifically investigates the relationships among the lateral positions of both the front and rear wheels, the slip ratios, and the angular orientation of the vehicle’s body during steering processes. Furthermore, a dual-layer steering stability control system aimed at enhancing the steering stability performance of DDEVs is introduced. This control system consists of two components: a lateral controller and a longitudinal controller. The lateral controller aims to establish clear linkages among four key variables, the front and rear wheel sideslip angles, yaw rate, and sideslip angle, and then to compute the necessary active front wheel steering angle and corresponding yaw moment based on the current vehicle body attitude. The findings indicate that, in comparison to the conventional DDEV controller, the proposed two-layer controller achieves substantially closer alignment to the reference curve during steering, with the accuracy increased by a factor of approximately 5 to 20. These results unequivocally affirm the efficacy and viability of the proposed approach.

1. Introduction

The DDEV is the focus of current research due to the fact that its motor is directly installed in the hub or in close proximity to the wheel edge. This configuration offers a number of advantages, including high transmission efficiency, rapid response, and low power requirements [1,2]. In recent years, the primary concerns in balancing the stability of DDEV steering have been the yaw rate and the centroid bias angle. While these two parameters directly impact DDEV stability and handling and are critical to vehicle stability studies, factors such as slip rate and front and rear wheel slip angles also affect steering stability [3,4,5]. Consequently, this paper considers all these aspects comprehensively. In this study, a dual-controller system was designed to track the requisite lateral and longitudinal motions. The system aims to accurately track the required yaw torque, front and rear wheel angles, and angular speed, thereby enhancing steering stability and control and preventing vehicle imbalance in DDEVs, which could have serious consequences [6].

The distributed model predictive control (DMPC) method has been employed extensively in the field of vehicle steering control [7,8]. Recent research has combined Pareto optimal game theory with the DMPC algorithm to propose an integrated control framework for torque vectoring (TV) and active front steering (AFS) systems. This framework enables distributed parallel control, thereby facilitating coordinated system management and improving vehicle stability during steering operations [9]. A substantial proportion of the recent literature has focused on the pivotal factors influencing DDEV stability. As elucidated in the work reported in [10], a direct yaw moment controller based on a linear quadratic regulator (LQR) is used to calculate the desired longitudinal traction force and yaw moment as virtual inputs. Subsequently, the authors put forth a model predictive control (MPCA) methodology for the reasonable distribution of the aforementioned virtual inputs among the four hub motors, with the objective of optimizing vehicle stability performance. In the work reported in [11], a novel Fuzzy Proportional Integral Derivative Sliding Mode Control–Genetic Algorithm (FPIDSMC-GA) for controlling the EPS system is introduced. The efficacy of the control system is evaluated through numerical simulations, which demonstrate improvements in vehicle stability and driving comfort under complex conditions. In the work reported in [12], a dual-loop algorithm is introduced which integrates proportional–integral–derivative (PID) and sliding mode control (SMC) in order to optimize the controller’s parameters. The objective of this approach is to improve vehicle smoothness during dynamic motion. In the work reported in [13], a risk assessment algorithm and a hierarchical model predictive control approach were employed to identify the optimal steering angle. This method facilitated dynamic trajectory replanning while enhancing the vehicle’s motion stability. In the work reported in [14], a control strategy for a four-wheel steering system based on a disturbance observer was proposed, taking into account the influence of the vehicle’s lateral slip angle and yaw rate on steering stability. Meanwhile, the work reported in [15] addresses the issue of handling stability in electric vehicles with distributed drive systems. This paper proposes a model reference adaptive yaw torque controller based on Lyapunov stability theory. The controller provides additional yaw torque in a dynamic manner, contingent on the vehicle’s state, thereby ensuring enhanced handling stability. The work reported in [16,17] employed a hierarchical control approach, considering factors such as vehicle speed and tire slip ratio in order to determine the required yaw rate and sideslip angle for maintaining stable vehicle operation. Additionally, they calculated the optimal torque distribution strategy. In the work reported in [18], the authors employ the classical two-degrees-of-freedom (2DOF) vehicle model to establish two initial yaw rate references, with the objective of analyzing the vehicle’s handling characteristics in terms of both maneuverability and stability. A direct yaw moment control (DYC) system was employed with the objective of achieving lateral dynamic control, thereby improving the vehicle’s stability performance. The results of these studies indicate that the proposed methods have the potential to enhance vehicle stability in certain scenarios. It is important to note, however, that an assessment of DDEV steering stability based solely on yaw rate and sideslip angle does not provide a comprehensive overview. The crux of the matter lies in the complex challenge of adjusting the vehicle’s posture during steering, which has a significant impact on the overall quality of steering.

This paper introduces a dual-layer steering stability control system, with the objective of enhancing the steering stability of DDEVs. The lateral controller explicitly establishes intricate relationships involving four key variables: the slip angles of the front and rear wheels, yaw rate, and sideslip angle, while considering the current attitude of the vehicle body. This analytical process enables the precise computation of the necessary active front wheel steering angle and corresponding yaw moment, which are essential for achieving optimal steering stability. Concurrently, the longitudinal controller, which is based on the yaw moment information derived from the lateral controller and enhanced by longitudinal slip ratio control, transforms this yaw moment into drivetrain and braking moments that the vehicle can regulate in a systematic manner. This strategic transformation results in a discernible improvement in the steering stability performance of the DDEV.

The dual-layer steering stability control system proposed in this paper emphasizes the relationship of the front and rear wheel slip angles and slip ratios with the vehicle’s posture during the steering process. This approach results in a more sophisticated dynamic model, thereby ensuring a comprehensive evaluation of steering stability and effectively addressing potential shortcomings in vehicle posture adjustment. Through the coordinated manipulation of the active front wheel steering angle and yaw moment, this system significantly enhances the steering stability of DDEVs. To demonstrate its efficacy, the performance of this dual-layer steering stability control system is rigorously validated through extensive simulations conducted within the Simulink environment.

The following is a description of the structure of this paper: Section 2 presents a nonlinear model for the DDEV system and elucidates the control issues pertaining to DDEV steering stability. Section 3 presents the model predictive control method for the distribution of DDEV steering stability control. Section 4 presents a comparative analysis of simulation results with conventional model controllers, thereby validating the effectiveness of the proposed controller. The final section, Section 5, presents a summary of the preceding content. In addition, Appendix A provides detailed information on certain specific formulas.

2. DDEV Steering Stability Dynamic Model

Traditional models for the stability of DDEV steering tend to disregard the considerable influence exerted by the angles of tire slip and the ratios of longitudinal slip. However, when a DDEV is subjected to steering maneuvers, the tires display additional lateral and longitudinal slip movements. It is therefore essential to examine the complex relationships between the slip angles and slip ratios of the front and rear tires, as well as the dynamic position of the vehicle, within the context of a comprehensive tire model.

2.1. Vehicle Model

An eight-degree-of-freedom vehicle dynamics model will be constructed to characterize the yaw and lateral motion [19], as depicted in Figure 1.

A simplified vehicle dynamics model is constructed based on the fundamental principles of vehicle dynamics theory [20], as follows:

$$\dot{\beta }={\displaystyle \frac{F_{yf}+F_{yr}}{m\cdot v}}-\gamma$$

(1)

$$\dot{\gamma }={\displaystyle \frac{L_f{F}_{yf}-L_r{F}_{yr}+M_z}{I_z}}$$

(2)

where “β” represents the sideslip angle, and “γ” represents the vehicle’s yaw rate. The symbols “F_yf ” and “F_yr” represent the lateral forces exerted on the front and rear wheels, respectively. In this context, “m” represents the mass of the vehicle, “v” represents the speed of the vehicle, and “L_f” and “L_r” represent the distances from the center of mass to the front and rear axles, respectively. Lastly, the term “I_z” indicates the vehicle’s yaw moment of inertia.

2.2. Tire Model

The longitudinal force exerted by a tire “F_x” responds dynamically to the rate of wheel slip “ω”. The dynamic equation for “F_x” is provided in Equation (3), and the additional yaw moment “M_Z” is detailed in Equation (4). The wheel longitudinal slip rate is a critical variable that characterizes the wheel’s longitudinal performance. Therefore, in light of the preceding analyses of wheel dynamics, the slip rate “ω” is modeled as a nonlinear response to torque. The dynamic equation for the slip rate “ω” is detailed in Equation (5) [21].

$$F_x=C_k\cdot \omega$$

(3)

$$M_z={\displaystyle \frac{d}{2}}\cdot \left(-F_{xfl}+F_{xfr}-F_{xrl}+F_{xrr}\right)$$

(4)

$${\stackrel{\cdot }{\omega }}_i={\displaystyle \frac{R_e\cdot T_i}{J\cdot v}}-\left({\displaystyle \frac{4\cdot \left(1+{\omega }_i\right)}{m\cdot v}}+{\displaystyle \frac{R_e^2}{J\cdot v}}\right)\cdot C_i\cdot \omega$$

(5)

where “C_k” represents the longitudinal stiffness of the tire, “J” denotes the moment of inertia, “T_i” (i = f_l, f_r, r_l, r_r) represents the torque of the motor, “R_e” is the tire radius, and the symbol “v” represents the velocity of the vehicle.

In general, the lateral force generated by a tire, denoted as “F_y”, is a reaction to the tire’s slip angle “α”. When the tire load remains constant, an increase in the slip angle leads to an increase in the lateral force. In the case of small lateral slip angles (≤5°), “F_y” demonstrates a linear response to the tire slip angle “α”, as delineated by the kinematic relationship in Equation (6). The equations governing the tire dynamics for the front and rear tire slip angles, “α_f” and “α_r”, are Equations (7) and (8), respectively.

$$\left\{\begin{array}{l}{F}_{yf}=2C_f\cdot {\alpha }_f\\ F_{yr}=2C_r\cdot {\alpha }_r\end{array}\right.$$

(6)

$${\dot{\alpha }}_f=\dot{\beta }+\gamma +{\displaystyle \frac{L_f}{v}}\cdot \gamma -\dot{\delta }$$

(7)

$${\dot{\alpha }}_r=\dot{\beta }+\gamma +{\displaystyle \frac{L_r}{v}}\cdot \gamma$$

(8)

In the provided equations, “C_f” and “C_r” are employed to denote the lateral stiffness of the front and rear tires, respectively. The term “$\dot{\delta }$” is utilized to represent the angular velocity of the tire during steering, with “α_f” designating the slip angle of the front tire and “α_r” signifying the slip angle of the rear tire.

2.3. Stability Reference Model

In the previous steering stability studies, the yaw rate and the sideslip angle have been considered key variables [22]. This paper presents a comprehensive analysis of the relationship between the front and rear wheel slip angles, yaw rate, and sideslip angle, and the active front wheel steering angle and yaw moment, of DDEVs. To achieve stable steering performance, this analysis, from the perspective of DDEV stability controller design, is based on the previously discussed vehicle and wheel dynamics. The stability design reference model of a DDEV is described in terms of three aspects: the vehicle’s longitudinal characteristics, lateral characteristics, and yaw characteristics. A two-degree-of-freedom vehicle model, known as the “bicycle model”, serves as the reference model, as illustrated in Figure 2.

From the perspective of frequency response analysis, by analyzing the vehicle model and combining Equations (1), (2), (6) and (7), the second-order system response of the sideslip angle “β”, the yaw rate “γ”, the front wheel slip angle “α_f”, and the rear wheel slip angle “α_r” to the driver’s steering input “δ” can be obtained. Four reference models are designed, including the desired sideslip angle “β_d”, the desired yaw rate “γ_d”, the desired front wheel slip angle “α_fd”, and the desired rear wheel slip angle “α_rd”, as described by Equations (9)–(12).

$${\displaystyle \frac{\beta (s)}{\gamma (s)}}={\displaystyle \frac{{\omega }_1^2{G}_{r1}(s)}{s^2+2{\omega }_1{\epsilon }_1\cdot s+{\omega }_1^2}}$$

(9)

$${\displaystyle \frac{\gamma (s)}{\delta (s)}}={\displaystyle \frac{{\omega }_2^2{G}_{r2}(s)}{s^2+2{\omega }_2{\epsilon }_2\cdot s+{\omega }_2^2}}$$

(10)

$${\displaystyle \frac{{\alpha }_f(s)}{\delta (s)}}={\displaystyle \frac{{\omega }_3^2{G}_{r3}(s)}{s^2+2{\omega }_3{\epsilon }_3\cdot s+{\omega }_3^2}}$$

(11)

$${\displaystyle \frac{{\alpha }_r(s)}{\delta (s)}}={\displaystyle \frac{{\omega }_4^2{G}_{r4}(s)}{s^2+2{\omega }_4{\epsilon }_4\cdot s+{\omega }_4^2}}$$

(12)

For a comprehensive account of the parameters incorporated into the formulas, refer to Appendix A.

3. Steering Stability Dual-Layer Controller Based on Dynamic Model Predictive Control

This paper conducts an exhaustive examination of the intricate relationships among several key parameters in DDEVs, including “α_f” and “α_r”, “ω”, “γ”, “β”, “δ_f”, and “M_z”. Subsequently, “M_z” undergoes a transformation process to convert it into drivable or braking torque, thereby facilitating the control of steering stability in DDEVs [23]. Expanding on this conceptual framework, this paper introduces a pioneering dual-layer control system aimed at enhancing the steering stability of DDEVs, grounded in a dynamic model predictive controller. A schematic representation of this system is illustrated in Figure 3.

The dual-layer steering stability control system consists of a lateral controller and a longitudinal controller. The lateral controller utilizes the desired “α_fd” and “α_rd”, desired “γ_d”, and desired “β_d” to compute the requisite “δ_f” and “M_z” needed to ensure the steering stability of DDEVs. This computation is facilitated by a dynamic model predictive controller, which effectively manages optimization constraints. Concurrently, the longitudinal controller modulates “M_z” based on the “ω” output and further transforms “M_z” into driving or braking torque “F_d” for the execution of appropriate driving or braking actions [24].

Accordingly, the dual-layer control system considers all the variables that affect the posture of the DDEV. The adjustment of the torque and steering angle through the use of two actuators significantly enhances the stability of the steering mechanism of the DDEV.

3.1. Lateral Stability Controller Based on Dynamic Model Predictive Control

In order to address the constrained optimization problem inherent to the DDEV steering stability control system, a state-feedback dynamic model predictive controller is meticulously developed. This is designed to handle the optimization of variables such as β, γ, α_f, α_r, and ω. This presents a multifaceted optimization challenge. The primary objectives of the proposed controller are to determine the necessary δ_f, the $\stackrel{\cdot }{{\delta }_f}$, and the Mz in order to ensure optimal vehicle steering stability. In designing the controller, the system is reformulated using Equations (1), (2) and (5)–(8) in order to derive a state-space model for stability control. The state variables x and control signals u are set as follows:

$$x=\left[\begin{array}{l}\beta \\ \gamma \\ {\alpha }_f\\ {\alpha }_r\end{array}\right],u=\left[\begin{array}{l}{\delta }_f\\ \stackrel{\cdot }{{\delta }_f}\\ M_z\end{array}\right]$$

By employing the Euler method for the discretization of the DDEV system, the state equation of the controller can be expressed in a concise manner as follows:

$$x_1(k+1)=({\displaystyle \frac{2\left(l_f\cdot C_f-l_r\cdot C_r\right)-v\cdot u_2}{m\cdot v^2}})\cdot T_s+({\displaystyle \frac{2\left(C_f+C_r\right)}{m\cdot v}}\cdot T_s+1)\cdot x_1(k)$$

(13)

$$x_2(k+1)=({\displaystyle \frac{2(l_f\cdot C_f-l_r\cdot C_r)\cdot x_1(k)-2l_f\cdot C_f\cdot u_1}{I_z}}-{\displaystyle \frac{2l_r\cdot C_r\cdot u_3}{I_z}})\cdot T_s+({\displaystyle \frac{2C_r\cdot \left(l_f^2+l_r^2\right)}{I_z\cdot v}}\cdot T_s+1)\cdot x_2(k)$$

(14)

$$x_3\left(k+1\right)=\left({\displaystyle \frac{2C_r}{m\cdot v}}-{\displaystyle \frac{V}{l_f+l_r}}{\displaystyle \frac{2l_f\cdot l_r\cdot C_r}{v\cdot I_z}}\right)T_s\cdot x_4\left(k\right)+(({\displaystyle \frac{2C_r}{m\cdot v}}-{\displaystyle \frac{v}{l_f+l_r}}+{\displaystyle \frac{2l_f^2\cdot C_f}{v\cdot I_z}})\cdot T_s+1)\cdot x_3\left(k\right)+(-{\displaystyle \frac{v}{l_f+l_r}}\cdot u_1-u_2+{\displaystyle \frac{l_f}{v\cdot I_z}}\cdot u_3)\cdot T_s$$

(15)

$$x_4\left(k+1\right)=(({\displaystyle \frac{2C_f}{m\cdot v}}\cdot {\displaystyle \frac{v}{l_f+l_r}}{\displaystyle \frac{2l_f\cdot l_r\cdot C_f}{V{I}_z}})+({\displaystyle \frac{2C_r}{m\cdot v}}+{\displaystyle \frac{v}{l_f+l_r}}+{\displaystyle \frac{2l_r^2\cdot C_r}{v\cdot I_z}})T_s+1)\cdot x_4\left(k\right)\cdot T_s\cdot x_3\left(k\right)+(-{\displaystyle \frac{v}{l_f+l_r}}\cdot u_1-{\displaystyle \frac{l_f}{v\cdot I_z}}\cdot u_3)\cdot T_s$$

(16)

A quadratic objective function grounded in performance indicators is formulated, encompassing state responses such as “β_d”, “γ_d”, “α_fd”, and “α_rd”:

$$J={‖\beta -{\beta }_d‖}_Q^2+{‖\gamma -{\gamma }_d‖}_R^2+{‖{\alpha }_f-{\alpha }_{fd}‖}_F^2+{‖{\alpha }_r-{\alpha }_{rd}‖}_S^2+‖u_M^2‖$$

(17)

In systems (13) to (16), the model predictive control law, founded on the corresponding objective Function (17), leverages a quadratic programming algorithm [25] to compute the predictive control sequence within each prediction time horizon, represented as $u={[{\delta }_f,{\dot{\delta }}_f,M_z]}^T$.

3.2. Longitudinal Controller Based on Dynamic Model Predictive Control

The longitudinal controller, based on dynamic model predictive control, primarily focuses on the allocation of torque to the four wheels, based on the interplay between slip ratio and torque. Subsequently, the allocated torque is converted into its resultant effect on the yaw moment.

$$x_n\left(k+1\right)=\left(\left({\displaystyle \frac{4\cdot \left(1-x_n\left(k\right)\right)}{m\cdot v}}+{\displaystyle \frac{r^2}{J\cdot v}}\right)\cdot C_{kn}\cdot T_s+1\right)\cdot x_n\left(k\right)-T_s\cdot {\displaystyle \frac{u_n\left(k\right)\cdot r}{J\cdot v}},n=1,2,3,4$$

(18)

By employing a state-feedback dynamic model predictive control law tailored to the system described in Equation (18) and its associated objective functions, a multi-parameter quadratic programming approach is utilized to calculate the predictive control sequence ${u=[{\omega }_{fl},{\omega }_{fr},{\omega }_{rl},{\omega }_{rr}]}^T$ at each sampling time point within this controller. The state-feedback-based dynamic model predictive controller has been comprehensively detailed in the author’s prior research. It is noteworthy that the active front wheel steering angle has already been transmitted to the DDEV by the lateral controller.

Subsequently, the optimized torque signals, as per Equation (4), undergo a transformation process to yield the “M_z”.

To facilitate more precise execution of driving and braking maneuvers, within the longitudinal controller employing dynamic model predictive control, the yaw moment “M_z” is converted into control inputs for adjusting the driving or braking pressure applied to the wheels “s_i”, where $i=f_l,f_r,r_l,r_r$, thereby implementing the requisite control actions.

Initially, the yaw moment “M_z” generated by the upper-level controller undergoes a conversion process into a longitudinal force variation “F_d” specific to an individual wheel “s_i”, where $i=f_l,f_r,r_l,r_r$. Following this step, employing the wheel motion model, “F_d” is further translated into a variation in driving or braking pressure “P_w”. Derived from Equation (3), the representation of the yaw moment “M_z” in terms of the single-side longitudinal force “F_x” can be expressed as follows:

$$M_z={\displaystyle \frac{1}{2}}{F}_{xfr}{D}_f+{\displaystyle \frac{1}{2}}{F}_{xrr}{D}_{r\hspace{0.33em}}$$

(19)

Let “F_d” represent the desired longitudinal driving or braking force for an individual wheel. Therefore, for the front right and rear right wheels, we have $F_{xfr}=F_{xrr}=F_d$. Equation (19) can be subsequently expressed as

$$M_z={\displaystyle \frac{1}{2}}{F}_d(D_{f}cos{\delta }_f+D_r)$$

(20)

Due to $cos{\delta }_f\approx 1$, we have $F_d={\displaystyle \frac{2M_z}{D_f+D_r}}$, and the individual wheel motion model is as follows:

$$J\cdot \dot{\lambda }=T-F_x\cdot R_e$$

(21)

In this equation, “λ” represents the angular velocity of the wheel, “J” denotes the moment of inertia of the wheel, “T” stands for the driving or braking torque, and “R_e” signifies the radius of the wheel. This leads to the calculation of the longitudinal force as follows:

$$F_x=-{\displaystyle \frac{1}{R_e}}(J\cdot \dot{\omega }-T)$$

(22)

Based on the drive/brake actuator model, the incremental longitudinal force “F_d” can be expressed as

$$F_d=-{\displaystyle \frac{1}{R_e}}(J\cdot \dot{\omega }+C{P}_w)$$

(23)

Within this equation, $C=A_w{u}_b{R}_b$, where “A_w” signifies the drive or brake surface area, “u_b” represents the coefficient of friction, “R_b” denotes the distance between the drive or brake pad and the wheel center, and “P_w” stands for the variation in drive or brake pressure applied to effect control actions on the DDEV.

4. Analysis of Simulation Results

This paper devised a dual-tier steering stability control system within the Simulink simulation platform. The simulation was conducted with a sampling period of $T_s=0.005 \mathrm{s}$. The DDEV parameters from reference [26] were used, with the configuration details shown in

Table 1. Parameters of electric vehicles.

Parameter Names	Parameter Values (Units)	Parameter Names	Parameter Values (Units)
Vehicle Mass ($m$)	1359.8 (kg)	Front Tire Lateral Stiffness ($C_f$)	23,540 (N/rad)
Yaw Moment of Inertia ($I_z$)	1992.54 (kgm2)	Rear Tire Lateral Stiffness ($C_r$)	23,101 (N/rad)
Distance from CG to Front Axle ($L_f$)	1.0628 (m)	Center of Gravity Height ($h$)	0.512 (m)
Distance from CG to Rear Axle ($L_r$)	1.4852 (m)	Vehicle Speed ($V$)	60 (km/h)
Front Axle Length ($D_f$)	1.0828 (m)	Wheel Moment of Inertia ($J$)	0.3534 (kgm2)
Rear Axle Length ($D_r$)	1.0828 (m)	Road Friction Coefficient ($\mu$)	0.6 (-)

[26].

Figure 4 shows the simulation comparison of the steering stability control system and conventional model predictive controller under working conditions. Figure 4 depicts the experimental outcomes at a steering angle of 0.12 rad. Figure 4a depicts the outcomes associated with the β. During the non-steering phase, both the proposed dual-layer controller and the conventional controller demonstrate a high degree of alignment with the reference value β_d. Upon initiation of steering by the DDEV, a change in the value of β is observed. However, the proposed dual-layer controller demonstrates the capacity to effectively track the β_d curve throughout the test, exhibiting no significant sudden changes, but rather, minor fluctuations and adjustments, resulting in a maximum error of only 0.0006 rad. In contrast, the conventional controller displays clear limitations in its ability to maintain tracking of the reference curve, resulting in a notable deviation from β_d, with a maximum deviation of 0.01 rad. The accuracy of the dual-layer controller increases by approximately 17 times compared to conventional controllers. This suggests that when the DDEV undergoes sudden changes in motion, its tracking control capability is insufficient. Figure 4b illustrates the results pertaining to the yaw rate. In the absence of steering, the value of γ remains constant, with all models exhibiting a high degree of alignment with the reference model. During the steering process, the vehicle’s γ begins to change; however, the proposed dual-layer controller continues to effectively track the trajectory of the reference model, resulting in a maximum error of only 0.01 rad per second. This allows the DDEV to maintain stability. Conversely, while the conventional controller displays a broadly similar overall profile, the maximum deviation from γ_d reaches 0.15 rad, which shows that the accuracy of the dual-layer controller increases by approximately 15 times compared to conventional controllers.

Figure 4c depicts the outcomes of the α_f. In the case of minimal changes in steering angle, both models demonstrate effective tracking of the reference model α_fd. However, as the angle increases, the proposed dual-layer control system continues to align closely with the trajectory of the reference model, responding rapidly with a maximum error of merely 0.006 rad. In contrast, the conventional controller maintains a consistent trend but exhibits a considerable error of 0.03 rad and demonstrates some response delay, which is deleterious to the stability of the DDEV. The accuracy of the dual-layer controller increases by approximately 5 times compared to conventional controllers. Similarly, the α_r, as illustrated in Figure 4d, demonstrates that both models effectively track the reference model αrd when the DDEV angle changes minimally. As the degree of change increases, the dual-layer controller examined in this paper continues to demonstrate a high degree of alignment with the trajectory of the reference model, exhibiting a maximum error of merely 0.006 rad. In contrast, the conventional controller maintains a consistent trend; however, the difference is significant, reaching 0.06 rad. The accuracy of the dual-layer controller increases by approximately 10 times compared to conventional controllers.

Figure 4e illustrates that the active front wheel steering angle “δ_f” exhibits a close correlation with the driver’s input “δ”. when the steering stability control system is engaged. Although the conventional model predictive controller can follow the trend to a certain extent, its fluctuations still result in instability during the steering process. Figure 4f illustrates the distribution of torque across all four wheels. It can be observed that the variation in torque across the wheels aligns with the operational characteristics of the DDEV, thereby ensuring the requisite steering stability throughout the steering process. Figure 4g depicts the four-wheel slip ratio “ω”, with its values remaining within the stability range. Furthermore, the curve aligns with the torque distribution pattern.

5. Conclusions

(1) Previous studies did not consider the steering state of DDEVs in sufficient depth. This paper presents a further analysis of the relationships among the front and rear wheel slip angles, slip ratio, and vehicle posture during steering. This analysis enhances the assurance of steering stability.

(2) Based on dynamic model predictive control, the dual-layer steering stability control system for DDEVs proposed in this paper effectively addresses the challenges associated with the four variables: front and rear wheel slip angles, longitudinal slip ratio, yaw rate, and sideslip angle, concerning the active front wheel steering angle and yaw moment.

(3) The simulation results validate the advantages of the dual-layer system for DDEV steering stability control. In typical operating conditions, the system effectively ensures the steering stability of DDEVs.

(4) The field of research pertaining to intelligent electric vehicles has attracted considerable global interest. The present study focuses on the issue of steering stability in the context of distributed-drive electric vehicles (DDEVs). Subsequent studies will place greater emphasis on coordinated control strategies, with the objective of reducing energy consumption while maintaining stability and enhancing the economic efficiency of DDEVs. The objective is to mitigate range anxiety and enable the efficient and stable operation of DDEVs.