Intelligent weighted hierarchical direct yaw moment control for distributed drive electric vehicles

Abstract

Distributed Drive Electric Vehicles (DDEVs) equipped with four-wheel independent drive (4WID) systems possess the capability to independently generate both driving and braking torques. However, in the context of Direct Yaw Moment Control (DYC), the dynamic coupling between yaw rate and sideslip angle remains a significant challenge for achieving effective coordinated control. To address this issue, a hierarchical control architecture is adopted, offering advantages such as structural flexibility and modularity. Within this framework, this paper proposes a Fast Adaptive Super-Twisting Sliding Mode Control (FASTSMC) strategy for DYC, and constructs an error model based on yaw rate and sideslip angle. To improve adaptability under varying driving conditions, a data-driven neural network with online updating based on phase plane stability parameters is integrated. This approach dynamically adjusts the weight coefficients in real time according to phase plane theory, thereby ensuring stable yaw moment control. Simulation results demonstrate that the proposed strategy reduces the sideslip angle by 54.7% under step input conditions and decreases the yaw rate by 22.4% during double-lane change conditions. These results confirm the effectiveness and superiority of the proposed control method.

Introduction

Motivation and technical challenge

With the continuous advancement of the automotive industry, substantial improvements have been achieved in vehicle speed and dynamic performance. Under high-speed driving and complex operating conditions, enhancing vehicle lateral stability and ensuring driving safety have become critical challenges that intelligent driving systems must urgently address. Meanwhile, driven by the global energy crisis and increasingly stringent environmental regulations, new energy vehicles—particularly electric vehicles—have garnered widespread attention and undergone rapid development. Among them, Distributed Drive Electric Vehicles (DDEVs) have emerged as a preferred platform for high-performance electric vehicles, owing to their inherent advantages in torque distribution and vehicle dynamic control.^1,2

Literature review

DDEVs are capable of independently controlling the driving force of each wheel, enabling vehicle motion regulation through the generation of additional yaw moments—commonly referred to as Direct Yaw Moment Control (DYC). This flexible drive architecture significantly enhances maneuverability and stability under complex operating scenarios.^3–6 To further improve vehicle controllability under extreme conditions, researchers have proposed various DYC strategies. These approaches aim to actively adjust the driving torques of individual wheels to produce the required additional yaw moment, thereby maintaining the vehicle’s stability. Compared to conventional Electronic Stability Control (ESC) systems, DYC leverages the rapid response characteristics of electric drive motors, in addition to braking systems, to achieve faster and more precise lateral control performance.^7–9

In the upper control design of DYC systems, prevailing approaches primarily target the regulation of either the yaw rate, the sideslip angle, or a coordinated combination of both.^10–12 The yaw rate characterizes the vehicle’s behavior around its center of mass, while the sideslip angle reflects the deviation between the vehicle’s heading and velocity directions. These two variables are dynamically coupled: changes in yaw rate affect vehicle orientation, thereby altering the sideslip angle; conversely, excessive sideslip angles can induce nonlinear changes in yaw rate, leading to potential instability. Therefore, incorporating both parameters simultaneously in the upper DYC controller is essential to achieve coordinated control and improve overall lateral performance.^13–16

To this end, several control strategies have been proposed. Wang et al. ¹⁷ proposed a control method for identifying vehicle lateral stability categories based on an improved brain emotional learning network (GA-BEL). This approach classifies vehicle states into three categories—stable, approaching stability, and difficult to stabilize—using a clustering algorithm. Zanten ¹⁸ proposed a Proportional-Integral-Derivative (PID)-based DYC method for regulating both yaw rate and sideslip angle. Although this approach is relatively simple and provides basic improvements in stability, its effectiveness diminishes under nonlinear and complex conditions. To overcome this limitation, Sliding Mode Control (SMC) techniques have been introduced. Subroto et al. ¹⁹ proposed a novel sliding mode controller to address stability issues caused by actuator failures and road friction uncertainties. Zhou et al. ²⁰ proposed a direct yaw moment control strategy based on the super-twisting sliding mode control theory to address the system chattering issue. However, this approach only considers the yaw rate as the control variable. Ma et al. ²¹ combined SMC with adaptive control theory, introducing a hierarchical DYC framework based on Adaptive Super-Twisting Sliding Mode Control (ASTSMC). Meanwhile, Liang et al. ²² adopted a robust H∞ control method based on Takagi-Sugeno (T-S) fuzzy modeling to address control failure issues caused by vehicle nonlinear characteristics. Jin et al. ²³ established a hierarchical control structure for multi-mode chassis coordination, utilizing Model Predictive Control (MPC) to simultaneously compute the front-wheel steering angle and additional yaw moment. In contrast, Wei et al. ²⁴ proposed a Deep Reinforcement Learning (DRL)-based DYC approach, which leverages DRL algorithms to approximate complex vehicle dynamics and bypass the intensive computation required by traditional nonlinear control. However, DRL-based methods demand extensive training, high-quality data, and substantial computational resources.

At the lower control level, effective and efficient allocation of the additional yaw moment to each independently driven wheel remains a key research focus. Hu et al. ²⁵ proposed an optimal torque vectoring control strategy based on a two-layer allocation formulation, which can resolve the conflict between stability and energy efficiency. Liu et al. ²⁶ proposed an MPC-based method to predict vehicle states in real-time and optimize wheel torques accordingly. Although this approach offers high accuracy, it suffers from high computational complexity. To improve allocation efficiency, Hu et al. ²⁷ applied a Quadratic Programming (QP)-based optimization algorithm, which reduces computation time but exhibits limited robustness under extreme conditions. Li et al. ²⁸ proposed a multi-objective torque distribution strategy based on the dung beetle optimizer (DBO) algorithm to address the energy consumption limitations of traditional average and load-based distribution methods, aiming to determine the optimal torque distribution scheme. However, the algorithm’s structure is relatively complex, and its real-time performance requires further testing.

Although the above methods have improved lateral control to a certain extent, challenges remain-particularly in terms of control accuracy and adaptability to varying conditions. To address these limitations, this paper proposes a novel DYC strategy based on a hierarchical control architecture. In the upper controller, a Fast Adaptive Super-Twisting Sliding Mode Control (FASTSMC) algorithm is developed. Separate control laws are constructed based on yaw rate and sideslip angle errors, and the additional yaw moment is computed using a vehicle dynamics model. To further enhance system adaptability, a data-driven neural network is integrated to update weighting coefficients online. This mechanism dynamically adjusts the contributions of the two control objectives based on the current vehicle stability state, thereby achieving more reasonable yaw moment allocation under diverse driving conditions. In the lower controller, a QP-based torque allocation strategy is designed to ensure both real-time responsiveness and optimal distribution of the total yaw moment among the four in-wheel motors.

Main contributions

In summary, this study proposes a hierarchical control strategy that improves the lateral stability of DDEVs by considering additional yaw moments derived from both yaw rate and sideslip angle errors. By dynamically adjusting the control weightings in response to real-time vehicle states, the proposed method ensures optimal handling performance across a range of operating scenarios. The main contributions of this paper are as follows:

• (1) A FASTSMC strategy is introduced to address the dynamic coupling between yaw rate and sideslip angle in DYC systems. Coupling mechanisms during vehicle motion are analyzed, and error models for both yaw rate and sideslip angle are established. The proposed controller computes the additional yaw moment based on these errors, enabling coordinated control of the two variables.

• (2) A data-driven online adaptation mechanism is developed using a backpropagation (BP) neural network informed by phase-plane stability indicators. This approach enables dynamic adjustment of the weighting coefficients between yaw rate and sideslip angle control, improving robustness and adaptability under varying operating conditions.

• (3) A QP-based torque allocation strategy is implemented to ensure the real-time performance. By embedding equality constraints into the objective function, the proposed method eliminates the need for complex iterations, enabling efficient and real-time torque distribution among the four in-wheel motors.

Paper organization

The remainder of this paper is organized as follows. Section “Dynamical model” introduces the vehicle dynamics modeling. Section “Direct yaw moment control design” details the proposed hierarchical control strategy. Section “Simulation results and analyses” presents validation results from CarSim/Simulink co-simulations. Finally, Section “Conclusions” summarizes the conclusions of this work.

Dynamical model

Vehicle 7-dof dynamics model

In order to comprehensively consider various factors and simplify the influence of unnecessary factors, this paper establishes a seven degree of freedom vehicle dynamics model as shown in Figure 1, which can well reflect the longitudinal, lateral, and yaw motion of the vehicle. ²⁹

$$\{\begin{array}{c}m({\stackrel{·}{v}}_x-v_y\gamma )=(F_{\mathrm{xfl}}+F_{\mathrm{xfr}})\cos {\delta }_f-(F_{\mathrm{yfl}}+F_{\mathrm{yfr}})\sin {\delta }_f+F_{\mathrm{xrl}}+F_{\mathrm{xrr}}\hfill \\ m({\stackrel{·}{v}}_y+v_x\gamma )=(F_{\mathrm{xfl}}+F_{\mathrm{xfr}})\sin {\delta }_f+(F_{\mathrm{yfl}}+F_{\mathrm{yfr}})\cos {\delta }_f+F_{\mathrm{yrl}}+F_{\mathrm{yrr}}\hfill \\ I_z\stackrel{·}{\gamma }=[(F_{\mathrm{xfl}}+F_{\mathrm{xfr}})\sin {\delta }_f+(F_{\mathrm{yfl}}+F_{\mathrm{yfr}})\cos {\delta }_f]·a-(F_{\mathrm{yrl}}+F_{\mathrm{yrr}})·b\hfill \\ +[(F_{\mathrm{yfl}}-F_{\mathrm{yfr}})\sin {\delta }_f-(F_{\mathrm{xfl}}-F_{\mathrm{xfr}})\cos {\delta }_f]·0.5B_w-(F_{\mathrm{xrl}}-F_{\mathrm{xrr}})·0.5B_w\hfill \end{array}$$

(1)

where, $m$ is the mass of the vehicle; $v_x$ is the longitudinal speed of the vehicle; $v_y$ is the lateral speed of vehicle; $\gamma$ is the yaw rate; $\stackrel{·}{\gamma }$ is the yaw rate acceleration of the vehicle; $a$ is the distance from the center to the front axle; $b$ is the distance from the center to the rear axle; $I_z$ is the moment inertia of the OZ axis; ${\delta }_f$ is the front wheel angle; $B_w$ is the vehicle track width; $F_{\mathrm{xij}}$ and $F_{\mathrm{yij}}$ represent longitudinal force and lateral force, respectively. The first lower $i=f$ or $r$ represents the front or rear axle, and the second lower $j=f$ or $r$ represents the left or right.

Figure 1.

Vehicle dynamics model.

The dynamic equation for the rotation of four wheels is expressed as:

$$J_w{\omega }_{\mathrm{ij}}=T_{\mathrm{dij}}-F_{\mathrm{xij}}·R$$

(2)

where, $J_w$ is the moment of inertia of the wheel, $R$ is the wheel radius, ${\omega }_{\mathrm{ij}}$ and $T_{\mathrm{dij}}$ represent the angular velocity of the four wheels and the four-wheel torque.

Reference model

In order to achieve the ideal control objective, as shown in Figure 2, this paper designs a two degree of freedom model that can well reflect the lateral motion and yaw motion of the vehicle. ³⁰ The relevant dynamic equations are represented as (3), where $\beta$ is the sideslip angle; $k_f$ is the cornering stiffness of the front tire and $k_r$ is the cornering stiffness of the rear tire.

$$\{\begin{array}{c}m{\nu }_x(\stackrel{·}{\beta }+\gamma )=(k_f+k_r)\beta +\frac{a{k}_f-b{k}_r}{{\nu }_x}\gamma -k_f{\delta }_f\hfill \\ \multicolumn{1}{c}{I_z\stackrel{·}{\gamma }=(a{k}_f-b{k}_r)\beta +\frac{a^2{k}_f+b^2{k}_r}{{\nu }_x}\gamma -a{k}_f{\delta }_f}\end{array}$$

(3)

Figure 2.

Vehicle 2-dof dynamics model.

By setting ${[\stackrel{·}{\beta },\stackrel{·}{\gamma }]}^T=0$ , the ideal yaw rate can be calculated as:

$${\gamma }_{\mathrm{model}}=\frac{{\nu }_x·{\delta }_f}{l·(1+K·{\nu }_x^2)}$$

(4)

The ideal sideslip angle is:

$${\beta }_{\mathrm{model}}=\frac{b+\mathrm{ma}{v_x}^2/(k_r·l)}{l(1+K·{v_x}^2)}·{\delta }_f$$

(5)

where, $K$ is the vehicle stability factor, which is an important parameter for characterizing the steady-state response of the vehicle. $K$ defined as:

$$K=\frac{m}{l^2}(\frac{a}{k_f}-\frac{b}{k_r})$$

(6)

where, when $K>0$ , it is in the deficient direction; when $K=0$ , it is in the neutral direction; and when $K<0$ , it is in the degree direction.

Tire model

During the operation of the DYC system, the mechanical response of tires exhibits significant nonlinear characteristics. Additionally, vehicle motion is highly dependent on the forces exerted on the tires. To realistically simulate the interaction between longitudinal and lateral forces, this paper places special emphasis on the effect of tire slip ratio on lateral forces and the influence of tire sideslip angle on longitudinal forces. In this work, a magic formula is employed to model the nonlinear characteristics of the tires ³¹ :

$$F_{x0}=D_x\sin \{C_x\arctan [B_x{x}_1-E_x(B_x{x}_1-\arctan \left(B_x{x}_1\right))]\}+S_{\nu 1}$$

(7)

where, $F_{x0}$ represents the tire longitudinal force; $x_1$ indicate tire longitudinal slip ratio or the tire sideslip angle under pure-slip conditions. $B_x$ is the stiffness factor, $B_x=B_x{C}_x{D}_x/(C_x·D_x)$ ; $C_x$ is the shape factor, $C_x=a_0$ ; $D_x$ is the peak factor, $D_x=a_1{F}_z^2+a_2{F}_z$ ; $E_x$ is the curvature factor. $F_z$ is the tire vertical force. $a_0,a_1,a_2$ are fitting parameters, $a_0=1.399$ , $a_1=-0.006249$ , $a_2=0.9901$ .

Driving system model

This paper adopts a permanent magnet synchronous wheel hub motor, which has a simple structure, high transmission efficiency, and good controllability. The torque of each motor can be independently controlled. The external characteristic curve of the wheel hub motor is shown in Figure 3.

Figure 3.

External characteristic curve of wheel hub motor.

Formula for motor external characteristic curve:

$$T=\{\begin{array}{c}{T}_{e}n\le n_0\hfill \\ \frac{9550P_e}{n}n>n_0\hfill \end{array}$$

(8)

where, $T$ is the motor output torque, $P_e$ is the motor rated power, $n_0$ is the motor base speed, and $T_e$ is the motor rated torque.

When simplifying the wheel hub motor, considering that the dynamic response speed of the motor is much faster than that of the wheel, the actual output torque of the motor can be obtained by transmitting the torque command to the motor controller. Meanwhile, considering the possibility of communication delay during transmission, a delay element was introduced on this basis. ³² The transfer function between the two is:

$$G\left(s\right)=\frac{T_m}{T_d}=\frac{1}{2{\xi }^2{s}^2+2\xi s+1}·e^{-\tau s}$$

(9)

where, $\tau$ is the motor response time.

Direct yaw moment control design

DYC is commonly employed for vehicle stability control, with the real-time calculation of optimal additional yaw moment under complex road conditions being crucial for controller design. To obtain the optimal additional yaw moment across various operating conditions, this paper adopts a hierarchical control architecture. In the upper controller, an error model based on yaw rate and sideslip angle is established, utilizing a FASTSMC algorithm to compute the additional yaw moments for yaw rate and sideslip angle respectively. For the lower controller, a torque distribution controller is developed using tire load rate as the optimization objective, ensuring the system maintains excellent real-time control performance. The overall structure of the DYC system is illustrated in Figure 4.

Figure 4.

Overall structure of DYC system.

In vehicle lateral dynamics, the yaw rate and sideslip angle serve as critical state variables characterizing vehicle stability, exhibiting significant dynamic coupling. In DYC, the controller directly adjusts the vehicle’s yaw acceleration by applying an additional yaw moment, thereby influencing the yaw rate. However, since changes in the yaw rate further affect the evolution of the sideslip angle—which in turn alters tire slip angles and lateral force distribution—the resulting feedback ultimately impacts yaw motion. Therefore, DYC must comprehensively account for the coupling characteristics of these two variables to achieve effective vehicle stability control.

Upper controller design

Considering the influence of additional yaw moment, further express equation (3) as:

$$\stackrel{·}{\gamma }=\frac{a{k}_f-b{k}_r}{I_z}·\beta +\frac{a^2{k}_f+b^2{k}_r}{I_z{\nu }_x}\gamma -\frac{a{k}_f}{I_z}·{\delta }_f+\frac{\mathrm{\Delta }{M}_z}{I_z}$$

(10)

$$\stackrel{·}{\beta }=\frac{k_f+k_r}{m{\nu }_x}·\beta +(\frac{a{k}_f-b{k}_r}{m{\nu }_x^2}-1)·\gamma -\frac{k_f}{m{\nu }_x}·{\delta }_f$$

(11)

where, $\mathrm{\Delta }{M}_z$ is the additional yaw moment.

Calculation of additional yaw moment based on yaw rate error

The tracking error and sliding surface of vehicle yaw rate are defined as:

$$e_{\gamma }=\phi -{\phi }_d;{\stackrel{·}{e}}_{\gamma }=\gamma -{\gamma }_d;{\stackrel{··}{e}}_{\gamma }=\stackrel{·}{\gamma }-{\stackrel{·}{\gamma }}_d$$

(12)

$$s_{\gamma }={\lambda }_1(\gamma -{\gamma }_d)$$

(13)

where, ${\lambda }_1$ is the weight coefficient; ${\stackrel{·}{\gamma }}_d$ is the ideal yaw rate acceleration of the vehicle. On this basis, the expression for the derivative of the sliding mode surface in time is given by:

$${\stackrel{·}{s}}_{\gamma }={\lambda }_1(\stackrel{·}{\gamma }-{\stackrel{·}{\gamma }}_d)$$

(14)

$${\stackrel{·}{s}}_{\gamma }={\lambda }_1(\frac{a{k}_f-b{k}_r}{I_z}·\beta +\frac{a^2{k}_f+b^2{k}_r}{I_z{\nu }_x}\gamma -\frac{a{k}_f}{I_z}·{\delta }_f+\frac{\mathrm{\Delta }{M}_z}{I_z}-{\stackrel{·}{\gamma }}_d)$$

(15)

Based on the FAST algorithm theory, the control law is designed as follows ³³ :

$$\{\begin{array}{c}{u}_0=u_{\mathrm{eq}}+u_s\hfill \\ \multicolumn{1}{c}{u_s=-k_3{\varphi }_1\left(s\right)+u_1}\\ \multicolumn{1}{c}{{\stackrel{·}{u}}_1=-k_4{\varphi }_2\left(s\right)}\end{array}$$

(16)

where, $k_3,k_4$ is the parameter to be designed; ${\varphi }_1\left(s\right),{\varphi }_2\left(s\right)$ is a sliding mode variable function. Among them.

$$\{\begin{array}{c}{\varphi }_1\left(s\right)=\mid s{\mid }^{1/2}\mathrm{sgn}\left(s\right)+s\hfill \\ {\varphi }_2\left(s\right)=\frac{1}{2}\mathrm{sgn}\left(s\right)+\frac{3}{2}\mid s{\mid }^{1/2}\mathrm{sgn}\left(s\right)+s\hfill \end{array}$$

(17)

According to (15), the equivalent sliding mode control input $u_{\mathrm{eq}}$ is:

$$u_{\mathrm{eq}}={\lambda }_1(\frac{a{k}_f-b{k}_r}{I_z}·\beta +\frac{a^2{k}_f+b^2{k}_r}{I_z{\nu }_x}\gamma -\frac{a{k}_f}{I_z}·{\delta }_f+\frac{\mathrm{\Delta }{M}_z}{I_z}-{\stackrel{·}{\gamma }}_d)$$

(18)

Due to the fact that the sliding surface fluctuates within the domain around 0, resulting in large parameter variations $k_3,k_4$ , the system becomes unstable. Therefore, rewrite $k_3,k_4$ as:

$$\{\begin{array}{c}{\stackrel{·}{k}}_3=\{\begin{array}{c}\frac{4(2m+n)}{1+n^2}{\varphi }_1^2\left(s\right){\varphi }_1^{\prime }\left(s\right)\mathrm{sgn}(\mid s\mid -{\epsilon }_2)\left|s\right|>0\hfill \\ \multicolumn{1}{c}{0\left|s\right|=0}\end{array}\hfill \\ k_4=\frac{n}{2}{k}_3+\frac{4m+n^2}{2}-\frac{n}{2c}-\frac{n(4m+n^2)}{4d}\hfill \end{array}$$

(19)

where, ${\epsilon }_2$ is a very small positive number. The formula of $k_3$ contains ${\varphi }_1^{\prime }\left(s\right)$ and ${\varphi }_1^2\left(s\right)$ , so that it can be dynamically adjusted according to the real-time changes in the slip surface $s_{\gamma }$ . This ensures that when the error is large, $k_3$ increases to improve the ability to correct the error; while when the error is small, $k_3$ automatically decreases to avoid over-control of the system.

According to (18) and (19), the external yaw moment calculated based on yaw rate error can be obtained as:

$$\begin{array}{c}\mathrm{\Delta }{M}_{\gamma }=(b{k}_r-a{k}_f)\beta -\frac{a^2{k}_f+b^2{k}_r}{{\nu }_x}\gamma +a{k}_f·{\delta }_f+{\stackrel{·}{\gamma }}_d\hfill \\ +\frac{I_z}{{\lambda }_1}[k_3{\varphi }_1\left(s\right)-\int k_4{\varphi }_2\left(s\right)]\hfill \end{array}$$

(20)

Calculation of additional yaw moment based on sideslip angle error

$$e_{\beta }=\beta -{\beta }_d;{\stackrel{·}{e}}_{\beta }=\stackrel{·}{\beta }-{\stackrel{·}{\beta }}_d;{\stackrel{··}{e}}_{\beta }=\stackrel{··}{\beta }-{\stackrel{··}{\beta }}_d$$

(21)

$$s_{\beta }={\lambda }_2(\stackrel{·}{\beta }-{\stackrel{·}{\beta }}_d)$$

(22)

where, ${\lambda }_2$ is the weight coefficient. On this basis, the derivative expression of sliding mode with respect to time is:

$${\stackrel{·}{s}}_{\beta }={\lambda }_2(\stackrel{··}{\beta }-{\stackrel{··}{\beta }}_d)$$

(23)

Based on (1), the yaw motion of the vehicle can be further expressed as:

$$\stackrel{·}{\gamma }=\frac{1}{I_z}(F+\mathrm{\Delta }{M}_z)$$

(24)

where, $F$ represents the influence of tire force and $\mathrm{\Delta }{M}_z$ represents additional yaw moment, which are respectively represented as:

$$\{\begin{array}{c}F=0.5B_w(F_{\mathrm{yfl}}-F_{\mathrm{yfr}})\sin {\delta }_f+a(F_{\mathrm{yfl}}+F_{\mathrm{yfr}})\cos {\delta }_f-b(F_{\mathrm{yrl}}+F_{\mathrm{yrr}})\hfill \\ \mathrm{\Delta }{M}_z=a(F_{\mathrm{xfl}}+F_{\mathrm{xfr}})\sin {\delta }_f-0.5B_w(F_{\mathrm{xfl}}-F_{\mathrm{xfr}})\cos {\delta }_f-0.5B_w(F_{\mathrm{xrl}}-F_{\mathrm{xrr}})\hfill \end{array}$$

(25)

The sliding surface expression can be further expressed as:

$${\stackrel{·}{s}}_{\beta }={\lambda }_2[\frac{k_f+k_r}{m{\nu }_x}·\stackrel{·}{\beta }+(\frac{a{k}_f-b{k}_r}{m{\nu }_x^2}-1)·\stackrel{·}{\gamma }-\frac{k_f}{m{\nu }_x}·{\stackrel{·}{\delta }}_f-{\stackrel{··}{\beta }}_d]$$

(26)

According to (24), (25), and (26), the external yaw moment calculated based on the sideslip angle error can be ultimately expressed as:

$$\begin{array}{c}\mathrm{\Delta }{M}_{\beta }=\frac{I_z}{{\lambda }_2(\frac{a{k}_f-b{k}_r}{m{{\nu }_x}^2}-1)}\hfill \\ [-{\lambda }_2(\frac{k_f+k_r}{m{\nu }_x}·\stackrel{·}{\beta }-\frac{k_f}{m{\nu }_x}·{\stackrel{·}{\delta }}_f-{\stackrel{··}{\beta }}_d)+k_3{\varphi }_1+\int k_4{\varphi }_2\left(s\right)]-F\hfill \end{array}$$

(27)

The stability of the FASTSMC control system will be proved next, and the following necessary theorems are first given to aid in the derivation of stability.

Theorem: If the Lyapunov function of the system satisfies $\stackrel{·}{V}\le -r{V}^{1/2}(r>0)$ , the system is finite time stable.

The Lyapunov function is defined as follows:

$$V=V_0+\frac{1}{2{\gamma }_1}{(k_1-k_1^{*})}^2+\frac{1}{2{\gamma }_2}{(k_2-k_2^{*})}^2$$

(28)

where, $k_1^{*},k_2^{*}$ is an unknown constant, assuming $k_1^{*}>k_1,k_2^{*}>k_2$ . Since Lyapunov stability only requires proving the existence of a Lyapunov function, it is always possible to find such a reasonable assumption.

$$\begin{array}{l}\stackrel{·}{V}={\stackrel{·}{V}}_0+\frac{1}{{\gamma }_1}\left(k_1-k_1^{\ast }\right){\stackrel{·}{k}}_1+\frac{1}{{\gamma }_2}\left(k_2-k_2^{\ast }\right){\stackrel{·}{k}}_2\\ \le -r{V}_0^{1/2}+\frac{1}{{\gamma }_1}\left(k_1-k_1^{\ast }\right){\stackrel{·}{k}}_1+\frac{1}{{\gamma }_2}\left(k_2-k_2^{\ast }\right){\stackrel{·}{k}}_2\\ \le -n{V}^{1/2}+\frac{1}{{\gamma }_1}\left(k_1-k_1^{\ast }\right){\stackrel{·}{k}}_1+\frac{1}{{\gamma }_2}\left(k_2-k_2^{\ast }\right){\stackrel{·}{k}}_2\\ +\frac{w_1}{\sqrt{2{\gamma }_1}}\left|k_1-k_1^{\ast }\right|+\frac{w_2}{\sqrt{2{\gamma }_2}}\left|k_2-k_2^{\ast }\right|\\ \le -n{V}^{1/2}+\left(\frac{w_2}{\sqrt{2{\gamma }_2}}-\frac{{\stackrel{·}{k}}_2}{{\gamma }_2}\right)·\left|k_2-k_2^{\ast }\right|\end{array}$$

(29)

According to the theorem, the system has stability.

$$\stackrel{·}{V}\le -n{V}^{\frac{1}{2}}+\eta \le 0$$

(30)

From the above stability theory analysis and proof process, it can be seen that the FASTSMC algorithm designed in this paper has theoretical feasibility.

Data-driven neural network online updating methodology

Phase plane-based vehicle stability analysis

The primary objective of vehicle stability control systems is to maintain the lateral dynamics within stable operational boundaries. Consequently, establishing appropriate stability criteria becomes particularly crucial for both system design and performance evaluation. Currently, there are two main phase planes used for analyzing vehicle lateral stability: $\beta -\gamma$ and $\beta -\stackrel{·}{\beta }$ . The first method is limited to vehicle stability analysis on high-adhesion road surfaces, where the vehicle is prone to side-slip or rear-end phenomena on low-adhesion surfaces. ³⁴ To overcome these limitations, this paper adopts the $\beta -\stackrel{·}{\beta }$ phase plane method for evaluating and analyzing vehicle stability.

From Figure 5 it can be seen that many phase trajectories converge to the phase point and two black lines are drawn to include the main phase trajectories returned to the phase point. Therefore, the stable region surrounded by two parallel black lines can be expressed as:

$$|\stackrel{·}{\beta }+B_1\beta |\le B_2$$

(31)

where $B_1$ and $B_2$ are stability boundary coefficients. Based on the previous conclusions and simulation data, the boundary coefficient of the stable region can be expressed as:

$$B_1=-4.28{\mu }^2+13.83\mu -0.1283$$

(32)

$$B_2=0.2197{\mu }^2+0.5814\mu +0.03167$$

(33)

The first performance criterion for determining the lateral stability of the vehicle involves the ratio of the yaw rate tracking error to the yaw rate deviation thresholds, as shown in Table 1. The performance indicator $J_1$ is defined as:

$$J_1=\frac{|\gamma -{\gamma }_d|}{\theta }$$

(34)

Figure 5.

Division of phase plane region.

Table 1.

Threshold value of yaw rate deviation the stability region.

Velocity (km/h)	The threshold value $\mathit{\theta }$ (°/s)
60	1.432
90	1.611
120	1.731

Secondly, this paper selects the $\beta -\stackrel{·}{\beta }$ phase plane to analyze the lateral stability of vehicles and provides an expression for the stable region of the phase plane. Therefore, performance indicators $J_2$ can be obtained, which are expressed as follows:

$$J_2=\frac{\stackrel{·}{\beta }+B_1\beta }{B_2}$$

(35)

Online parameter updating based on BP neural network

Since the two optimal external yaw moments based on the yaw rate and sideslip angle tracking errors have been calculated, the research work of this section will focus on how to determine the weight coefficient of the two calculated values. The final external yaw moment based on the optimal weight coefficient is expressed as:

$$\mathrm{\Delta }{M}_z=q\mathrm{\Delta }{M}_{\gamma }+(1-q)\mathrm{\Delta }{M}_{\beta }$$

(36)

where, q is the weight coefficient of two additional yaw moments, $q\in [0,1]$ . From (36), it can be seen that when $q\to 1$ , the additional yaw moment is entirely determined by $\mathrm{\Delta }{M}_{\gamma }$ ; conversely, when $q\to 0$ , it is fully governed by $\mathrm{\Delta }{M}_{\beta }$ .

Therefore, to obtain additional yaw moment, the determination of q is very important. This article adopts the weighting coefficient q of the BP neural network.

The BP neural network is a multi-layer feedforward neural network trained using the error backpropagation algorithm. Its architecture typically consists of an input layer, hidden layer, and an output layer. The learning process involves two phases: forward propagation of signals and backward propagation of errors. ³⁵ The BP neural network employed in this study underwent offline training for the weight model using MATLAB. During the training process, scripts were first developed based on MATLAB’s Neural Network Fitting Toolbox to generate a comprehensive dataset through extensive simulations under various operating conditions, capturing performance metrics $J_1$ and $J_2$ along with weight coefficient q. This approach established a solid foundation for the network’s generalization capability. As illustrated in Figure 6, the designed BP neural network features an input dimension of 2, a hidden layer with 32 neurons, and an output dimension of 1. The dataset was partitioned into training, validation, and test sets with ratios of 80%, 10%, and 10% respectively, while the sigmoid function was adopted as the activation function.

Figure 6.

BP neural network model.

According to the BP neural network model shown in Figure 6, the input $X_{\mathrm{input}}$ and the output $Y_{\mathrm{ouput}}$ are as follows:

$$\{\begin{array}{c}{X}_{\mathrm{input}}=[\begin{array}{cc}{J}_1\hfill & J_2\hfill \end{array}]\hfill \\ Y_{\mathrm{ouput}}=q\hfill \end{array}$$

(37)

The forward propagation process, in which $J_1,J_2$ serve as the input signal and are passed through the hidden layer to generate the output weight coefficient $q$ , can be described by the following equation:

$$\{\begin{array}{c}\zeta \left(x\right)=\frac{e^x-e^{-x}}{e^x+e^{-x}}\hfill \\ H_j=\zeta [(w_{J_1·k}{X}_{\mathrm{input}}+b_{J_1·k})+(w_{J_2·k}{X}_{\mathrm{input}}+b_{J_2·k})],k=1,2,\dots ,32\hfill \\ Y_{\mathrm{ouput}}=\zeta \left(\sum _{k=1}^{32}(w_{q·i}{H}_j+b_{q·i})\right),i=1\hfill \\ \mathrm{err}=\frac{1}{2}{(Y_{\mathrm{des}}-Y_{\mathrm{output}})}^2\hfill \end{array}$$

(38)

where, $H_j$ is the output moment descent of the hidden object; $w_{J_1·k},w_{J_2·k}$ is the weight input to the hidden object; $w_{q·i}$ is the weight hidden to the output house; $b_{J_1·k},b_{J_2·k}$ is the national value input to the hidden object; $b_{q·i}$ is the national value hidden to the output house; $\mathrm{err}$ is the error between output signal and expected output variable.

The Levenberg-Marquardt algorithm is selected for neural network backpropagation. ³⁶

$$\{\begin{array}{c}w(\tau +1)=w\left(\tau \right)-{[J^{T}J+\iota I]}^{-1}{J}^{T}E\hfill \\ \multicolumn{1}{c}{b(\tau +1)=b\left(\tau \right)-{[J^{T}J+\iota I]}^{-1}{J}^{T}E}\end{array}$$

(39)

Where, $I$ is the identity matrix, $J$ is the Jacobian matrix, $E$ is the error matrix, $w\left(\tau \right)$ and $b\left(\tau \right)$ are the matrices composed of the weights and thresholds of the $\tau$ iterations, respectively, and $\iota$ is the learning rate.

Neural network data training

Table 2 lists the size and weight coefficient range of the training data, with a total of 154 training samples. Figure 7 illustrates the training performance of the BP neural network model, showing that the optimal validation performance of 0.0000151 is reached at epoch 4. The mean square error (MSE) stabilizes toward the end, indicating that the model has been effectively trained and that the training data is reliable.

Table 2.

Introduction to the training date.

Characteristic	Data volume and scope
Number of samples	154
q	(0, 1)
Training sets	154 × 80%
Validation sets	154 × 10%
Test sets	154 × 10%
Number of hidden layer neurons	32

Figure 7.

Training performance of BP model MSE value.

The training data were divided into training set, validation set, and test set according to the proportion, and the data fitting effect graph of BP neural network model was obtained as shown in Figure 8, in which the correlation coefficient R can truly reflect the correlation between the output signal and the target signal.

Figure 8.

The fitting effect of the BP model.

Lower controller design

To ensure four-wheel actuators jointly produce the external yaw moment calculated by the above two FASTSMC controllers, taking the tire load rate as the optimization target, the torque distribution controller for the lower layer is designed in this paper. The controller not only ensures high-precision torque distribution but also meets real-time requirements. Since the yaw moment can be directly adjusted by controlling the motor to modify the tire’s longitudinal force, this paper does not consider the influence of lateral force. Thus, the optimal objective function expression is as follows:

$$\min J=\sum _{i=1}^4\frac{F_{\mathrm{xij}}^2}{{\mu }_{\mathrm{ij}}^2{F}_{\mathrm{zij}}^2}$$

(40)

Restrictive condition

• (1) The four-wheel torque should meet the longitudinal force requirements and additional lateral torque requirements of the driver during normal driving:

$$F_{\mathrm{xtotal}}=F_{\mathrm{xfl}}+F_{\mathrm{xfr}}+F_{\mathrm{xrl}}+F_{\mathrm{xrr}}$$

(41)

$$\mathrm{\Delta }{M}_z=(F_{\mathrm{xfl}}-F_{\mathrm{xfr}})·\frac{B_w}{2}+(F_{\mathrm{xrl}}-F_{\mathrm{xrr}})·\frac{B_w}{2}$$

(42)

• (2) The tire force must satisfy the constraint condition of the attached ellipse, and the expression after ignoring the influence of lateral force is:

$$-\mu F_{\mathrm{zij}}\le F_{\mathrm{zij}}\le \mu F_{\mathrm{zij}}$$

(43)

• (3) The longitudinal force assigned to the four wheels should also consider the limitation of the maximum output torque of the wheel hub motor, including:

$$-\frac{T_{\max }}{R}\le F_{\mathrm{xij}}\le \frac{T_{\max }}{R}$$

(44)

From the above analysis, it can be seen that the objective function is a quadratic function and the constraint function is a linear function, which is a typical quadratic programming problem. Its general form is as follows:

$$\mathrm{minJ}=\frac{1}{2}{x}^T\mathrm{Hx}+f^{T}xs.t.\{\begin{array}{c}\mathrm{Ax}\le b\hfill \\ A_{\mathrm{eq}}·x=b_{\mathrm{eq}}\hfill \\ \mathrm{LB}\le x\le \mathrm{UB}\hfill \end{array}$$

(45)

where, $H$ is a symmetric matrix, $f$ is a primary term coefficient. $A$ denotes the coefficient of the inequality constraint, $b$ denotes the upper bound of the inequality constraint. $A_{\mathrm{eq}}$ denotes the coefficient of the equation constraint and $b_{\mathrm{eq}}$ denotes the value of the equation constraint. $\mathrm{UB},\mathrm{LB}$ denote the upper and lower limits of $x$ .

Simulation results and analyses

The effectiveness of the proposed control strategy was evaluated through co-simulation analysis using the CarSim/Simulink platform. To comprehensively examine the strategy’s performance under realistic operating conditions, four distinct driving conditions are designed based on typical activation conditions of the DYC system, with comparative evaluations performed against uncoordinated FASTSMC control, fixed-weight SMC, and the No Control (NC). The parameters of the DDEVs used in the simulation experiments are listed in Table 3.

Table 3.

Vehicle simulation parameters.

Description	Parameter	Value
Vehicle mass	$m$	1412 kg
Gravitational acceleration	$g$	9.8 m/s2
Moment of inertia around OZ axis	$I_z$	1536.7 kg m2
Distance from the center of mass to front axle	$a$	1.015 m
Distance from the center of mass to the rear axle	$b$	1.895 m
Tread	$B_w$	1.675 m
Tire radius	$R$	0.325 m

Step steering angle input conditions

Considering real-world driving conditions, this section employs a step steering input to validate the effectiveness of the proposed control strategy. To comprehensively assess its performance, the step input test is conducted under three representative scenarios: (1) extreme low adhesion, (2) medium-speed low adhesion, and (3) high-speed high adhesion. Specifically, the extreme low adhesion scenario is tested at a vehicle speed of 45 km/h with a road friction coefficient of 0.1; the medium-speed low adhesion scenario is tested at 60 km/h with a coefficient of 0.25; and the high-speed high adhesion scenario is tested at 120 km/h with a coefficient of 0.85.

Extreme low adhesion condition

Figure 9 presents the simulation results under extremely low-adhesion conditions. As shown in Figure 9(a), without control, the vehicle fails to maintain stability under extremely low-adhesion conditions, exhibiting severe chattering. With control enabled, the proposed strategy demonstrates superior tracking performance for the sideslip angle reference compared to other control methods, achieving 47.2% higher accuracy than the FASTSMC strategy. Figure 9(b) compares the yaw rate responses, where the uncontrolled case still shows significant chattering. At 1 s, the proposed control strategy exhibits minimal overshoot and maintains the highest tracking accuracy after vehicle stabilization, confirming its effectiveness. Figure 9(c) displays the vehicle speed variation during the maneuver, where sudden steering causes deceleration while the speed remains near the target value of 45 km/h. The dynamic adjustment of the weighting coefficient q is illustrated in Figure 9(d). During the 0–2 s period when the vehicle undergoes step steering, the weight coefficient q changes significantly. After 2 s, as the vehicle stabilizes, the weight coefficient remains constant, demonstrating its excellent real-time performance.

Figure 9.

Comparative simulation results under step-steer condition at extreme low adhesion condition: (a) sideslip angle, (b) yaw rate, (c) vehicle speed, and (d) weight coefficient q.

Medium-speed low-adhesion condition

Figure 10 presents the simulation results under medium-speed low-adhesion conditions. As shown in Figure 10(a) and (b), compared with other control strategies, the proposed approach enables faster convergence and stabilization of both yaw rate and sideslip angle with smaller overshoot. Specifically, it reduces the peak value of the sideslip angle by 54.7% and improves yaw rate performance by 19.1% compared to the SMC algorithm. Figure 10(c) demonstrates the vehicle speed tracking performance of the proposed control strategy, showing that the speed remains consistently close to the target value, indicating that the control strategy maintains stability without significant speed loss. Figure 10(d) illustrates the real-time adjustment of parameter q by the data-driven neural network, which dynamically optimizes the weighting of additional yaw moment based on vehicle states during operation.

Figure 10.

Comparative simulation results under step-steer condition at medium-speed low-adhesion condition: (a) sideslip angle, (b) yaw rate, (c) vehicle speed, and (d) weight coefficient q.

High-speed high-adhesion condition

The simulation results of different control strategies under the high-speed, high-adhesion condition are presented in Figure 11. Figure 11(a) and (b) illustrate the tracking performance of the vehicle sideslip angle and yaw rate for the four control strategies. The results indicate that all three strategies achieve better reference tracking performance compared to the uncontrolled case. However, the proposed control strategy exhibits superior stability enhancement by accounting for the coupling between yaw rate and sideslip angle through an online-updated, data-driven neural network. Notably, around the 2-s mark, although the yaw rate successfully tracks the reference value, the sideslip angle still shows noticeable oscillations. At this moment, the weight coefficient q primarily adjusts to improve the tracking of the sideslip angle, enabling it to converge rapidly to the reference value. Figure 11(c) and (d) show the vehicle speed tracking performance and the variation of the online-updated parameters in the proposed strategy. The dynamic adjustment of the weight coefficients further demonstrates that the proposed strategy ensures vehicle stability without inducing excessive speed reduction. Moreover, the parameters adaptively modify the weight of the additional yaw moment in real time based on the current vehicle states.

Figure 11.

Comparative simulation results under step-steer condition at high-speed high-adhesion condition: (a) sideslip angle, (b) yaw rate, (c) vehicle speed, and (d) weight coefficient q.

Double lane change condition

To further analyze the performance of the control strategy under different driving conditions, this section conducts experiments under a typical DLC condition with a vehicle speed of 90 km/h and a road adhesion coefficient of 0.85. Figure 12 presents the experimental results of different control strategies in the DLC condition. As shown in Figure 12(a) and (b), the uncontrolled vehicle exhibits significant oscillations in both sideslip angle and yaw rate, particularly with the sideslip angle reaching its maximum value during 3–5 s and the yaw rate peaking around 5 s, which further demonstrates the necessity of the proposed control strategy. Compared to the fixed-weight sliding mode control strategy, the proposed control strategy achieves smoother overall performance, reducing the peak sideslip angle by 35.75% and improving the peak yaw rate by 22.4%, thereby tracking the reference values more closely.

Figure 12.

Comparison of simulation results under DLC condition: (a) sideslip angle, (b) yaw rate, (c) vehicle speed, (d) weight coefficient q, and (e) four-wheel torques.

As shown in Figure 12(c), the vehicle speed remains generally stable around the target speed. Figure 12(d) presents the variation curve of the online-updated weight coefficient q. During the DLC condition, the changes in weight coefficient q indicate that the additional yaw moment primarily tracks the sideslip angle while secondarily tracking the yaw rate, a conclusion supported by the simulation results in Figure 12(a) and (b). Therefore, the magnitude of the weight coefficient q directly determines the dominant factor in vehicle additional yaw moment, playing a leading role in the overall control strategy and fulfilling the design objectives of this study. Figure 12(e) displays the four-wheel torque distribution results, demonstrating the effectiveness of the lower-layer torque allocation method.

Conclusions

This paper proposes an intelligent weighted hierarchical direct yaw moment control strategy based on distributed drive electric vehicles. Specifically, the upper-layer controller establishes an error model for yaw rate and sideslip angle, deriving the additional yaw moments for both parameters using the FASTSMC algorithm. To address the coordination between yaw rate and sideslip angle in stability control, a data-driven neural network online update method based on phase plane stability parameters is introduced to enhance controller robustness. This data-driven approach integrates phase plane stability parameters with a BP neural network, dynamically updating the weighting coefficient q between the additional yaw moments for yaw rate and sideslip angle, thereby improving the controller’s adaptability across various operating conditions. In the lower-layer controller, a quadratic programming-based torque distribution strategy is proposed, incorporating equality constraints into the objective function to eliminate the need for cumbersome iterative computations. The feasibility and effectiveness of the proposed control strategy are validated through CarSim/Simulink simulations. Results demonstrate a 54.7% reduction in sideslip angle under step conditions and an 22.4% decrease in yaw rate during double-lane change maneuvers, significantly enhancing vehicle stability and handling. Notably, the proposed control strategy exhibits some dependency on system model errors, and the inclusion of nonlinear functions and integral terms in the algorithm may necessitate further consideration of real-time performance in practical applications. Additionally, this study does not account for the influence of vertical dynamics on the control strategy, and future work will incorporate vertical motion while conducting real-vehicle validation.