An Adaptive Compound Control Strategy of Electric Vehicles for Coordinating Lateral Stability and Energy Efficiency

Abstract

To enhance the balance between lateral stability and energy efficiency, we propose an adaptive compound controller based on phase plane analysis for four-wheel independent drive electric vehicles (4WID-EVs). The adaptive stability and energy-saving controller (SEC) is designed with a three-layer structure. The upper-layer controller employs model predictive control (MPC) to compute the external yaw moment based on the desired yaw rate and side slip angle derived from a reference model. The adaptive-layer controller utilizes a phase plane diagram to evaluate vehicle stability and reduces unnecessary external yaw moment consumption by accounting for the vehicle’s steering state and battery’s state-of-charge (SOC) level. The lower-layer controller implements an optimal torque distribution algorithm to minimize an objective function that considers tire workload, energy consumption, and smooth motor control. Numerical simulations are performed in MATLAB/Simulink using three distinct steering angles to evaluate the performance of the proposed control strategy. At each steering angle, the SEC’s stability and energy efficiency are compared to those of the energy-saving controller (EC) and stability controller (SC) under varying battery charge levels. The results indicate that, at small steering angles, the vehicle operates in a highly stable state, enabling a reduction in the external yaw moment to achieve substantial energy savings. As the steering angle increases, the vehicle approaches a critical stability state, where the external yaw moment is applied to maintain lateral stability. Furthermore, as the SOC decreases, the SEC strategy will increasingly prioritize energy savings. Simulation results verify that the SEC strategy effectively balances lateral stability and energy savings while maintaining consistent performance across a range of operating conditions.

1. Introduction

As an environment-friendly alternative to internal combustion engine vehicles, electric vehicles (EVs) have gained significant attention due to the high efficiency of electric motors and their minimal greenhouse gas emissions [1,2]. Safety and sustainability are the primary objectives in the development of electric vehicles. Stability control aims to keep the vehicle’s driving state within a controllable range, minimizing the risk of accidents due to insufficient driver control. However, accurately estimating a vehicle’s stability remains a challenging task [3,4,5], which is a main focus of the present study. Improving vehicle range primarily depends on either increasing battery capacity or reducing energy consumption. Given the limitations of battery capacity, optimizing both vehicle stability and energy efficiency through strategic motor torque distribution offers an effective approach to reducing energy consumption.

In EVs, hierarchical control is commonly employed to study lateral stability, where the upper-layer controller computes the required external yaw moment, and the lower-layer controller allocates the torque accordingly. The complex dynamic control capabilities of the four-wheel independent drive electric vehicle (4WID-EV) offer significant advantages over other powertrain layouts in EVs, particularly in terms of lateral stability control and energy efficiency [6,7,8]. Lateral stability is a crucial component of vehicle stability control. By independently distributing torque to each of the four motors, the vehicle’s stability is improved through the generation of external yaw moment [9]. Additionally, implementing appropriate torque distribution can reduce energy consumption. The 4WID-EV uses a hierarchical control structure [10] to enhance lateral stability control by simplifying the optimization process. The upper-layer controller employs direct yaw moment control (DYC) to generate external yaw moment, thereby achieving the desired state through control methods such as the sliding mode control (SMC) [11,12], linear quadratic regulator (LQR) [13,14,15], proportional-integral-derivative (PID) [16], and model predictive control (MPC) methods [17]. Zhang et al. [18] proposed a switching strategy based on a stability criterion that involved real-time observation of side slip angle. They also implemented a steering support system for the operation of the active steering system. To minimize actuator usage, the DYC method based on ${\mathrm{H}}_{\infty }$ control theory was employed for lateral stability control. Guo et al. [19] introduced an adaptive sliding mode control method for vehicle stability, where switching surfaces were designed using linear matrix inequalities, and fuzzy logic was applied to dynamically adjust the uncertain term and the switching control gain. Chen et al. [20] utilized a dynamic sliding mode control to compute the external yaw moment and proposed an adaptive energy-efficient control allocation method based on the Karush–Kuhn–Tucker conditions for optimal torque distribution. Liu et al. [21] introduced a hierarchical control concept to balance energy efficiency and vehicle motion, with a concise and modular framework that enabled superior dynamic trade-offs.

The lower-layer controller typically applies the optimal torque distribution algorithm [22] to provide precise control and achieve the desired steering performance, including dynamic load distribution [23] and average distribution [24]. According to refs. [25,26], the use of an appropriate torque control allocation method can help reduce energy consumption. In ref. [27], a torque control allocation method based on experimentally measured efficiency maps of motors was proposed to enhance motor efficiency. Additionally, Wang et al. [28] considered torque distribution under tire–road friction conditions. Zhai et al. [29] used tire workload usage as the objective function for torque distribution and compared it with dynamic load distribution and average load distribution algorithms. Zheng et al. [30] identified several optimization objectives in torque distribution, such as vehicle stability, energy savings, and tire wear energy consumption, with the weight coefficient adapted to different working conditions using fuzzy logic theory. Ma et al. [31] also developed a torque distribution strategy to reduce tire slip energy.

Minimizing energy usage during turns, in addition to lateral stability, is also a key research focus. This is primarily accomplished by operating the motors in efficient ranges or reducing motor losses. Liu et al. [32] used the power of four motors as the cost function and employed an exhaustive search method to obtain the front-to-rear torque distribution ratio. Lin et al. [33] introduced a slip energy factor into the objective function to minimize unnecessary tire slip and reduce tire slip energy. Jing et al. [34] combined DYC with active front steering and incorporated a minimum driving energy objective function in the lower-layer controller, enabling dual-layer energy-saving control. Han et al. [35] designed a yaw stability controller to minimize external yaw torque, achieving energy savings without the application of economic optimization algorithms. Zhai et al. [36] developed a continuous steering stability controller based on an energy-saving torque distribution algorithm, achieving a balance between lateral stability and energy savings.

Since the 4WID-EV is a typical over-actuated system [37], multiple possible torque distribution schemes are possible among the four wheels during steering. These systems require sufficient torque for both driving and generating external yaw moment. This paper proposes an adaptive compound controller based on a hierarchical control structure. The objective of this work is to integrate the battery’s state of charge and vehicle stability to dynamically modify the external yaw moment, thereby balancing lateral stability and energy efficiency. The proposed approach provides a novel solution for enhancing energy efficiency in 4WID-EVs.

The rest of this paper is structured as follows: Section 2 presents the seven-degree-of-freedom (7-DOF) nonlinear dynamic model and driver model of the 4WID-EV. Section 3 introduces the adaptive stability and energy-saving control method. Section 4 discusses the main simulation results. Conclusions are summarized in Section 5.

2. System Overview and Vehicle Model

2.1. System Overview

Figure 1 depicts a control system that integrates a vehicle model, a motor model, a battery model, and an adaptive compound controller. The controller improves lateral stability and energy efficiency by dynamically modifying the external yaw moment. The modified yaw moment is determined through a hierarchical structure consisting of a reference model, an upper-layer controller, an adaptive-layer controller, and a lower-layer controller. The upper-layer controller uses MPC to generate the external yaw moment based on the reference model. The adaptive-layer controller evaluates vehicle stability using phase plane diagrams and modifies the external yaw moment accordingly. Finally, the lower-layer controller optimizes tire workload and minimizes total power loss across the four motors using nonlinear torque distribution methods.

2.2. Vehicle Model

The nomenclature used in this paper is presented in

Table 1. Nomenclature.

Definition	Symbol	Definition	Symbol
vehicle mass	$m$	vertical loads at each wheel	$F_{z,i}$
steering angle	$\delta$	gravitational acceleration	$g$
front wheel steering angle	${\delta }_f$	steering ratio	$i_t$
front right wheel steering angle	${\delta }_{fr}$	rolling friction coefficient	$f$
front left wheel steering angle	${\delta }_{fl}$	wheel rolling inertia	$I_{\omega }$
front to rear axle distance	$L$	yaw moment of inertia	$I_z$
distance from CG to front axle	$l_f$	motor torque on each wheel	$T_i$
distance from CG to rear axle	$l_r$	motor rotational speed	$n_i$
front track width	$b_f$	motor efficiency	${\eta }_i$
rear track width	$b_r$	angel velocity of each wheel	${\omega }_i$
vehicle longitudinal velocity	$v_x$	wheel rolling radius	$r$
vehicle lateral velocity	$v_y$	road friction coefficient	$\mu$
yaw rate in vehicle body	$\gamma$	cornering stiffness of total tires	$K$
steady-state yaw rate	${\gamma }_{\mathrm{ss}}$	cornering stiffness of the front tires	$K_f$
side slip angle in vehicle body	$\beta$	cornering stiffness of the rear tires	$K_r$
steady-state side slip angle	${\beta }_{\mathrm{ss}}$	external yaw moment	$M_z$
tire longitudinal forces	$F_{x,i}$	modified external yaw moment	$M_z^{\prime }$
tire lateral forces	$F_{y,i}$	total longitudinal torque	$T_{\mathrm{req}}$
total lateral forces of front tires	$F_{y,f}$	maximum torque on each wheel	$T_{\max }$
total lateral forces of rear tires	$F_{y,r}$	state of charge	$\mathrm{SOC}$

. In this section, we present a 7-DOF dynamic model of an electric vehicle for the design of the adaptive-layer controller to enhance both the lateral stability and battery energy savings, as shown in Figure 2. The longitudinal, lateral, and yaw motions of the vehicle body are considered with the pitch, roll, and vertical motions being ignored. The rotational motion of the front left, front right, rear left, and rear right wheels are indexed by fl, fr, rl, and rr, respectively. Let $O$ denote the origins of the vehicle body frame. Note that the origin $O$ is tied to the center of gravity (CG) of the vehicle body. In the vehicle body frame, $l_f$ and $l_r$ represent the distances from the origin $O$ to the front and rear axles, resulting in a wheelbase $L=l_f+l_r$. Let $b_f$ and $b_r$ represent the front and rear track width, and $\delta$ and $i_t$ represent the steering angles and the steering ratio, respectively.

The Ackermann steering characteristics are employed to model the kinematics of the vehicle, which means that all wheels possess an instantaneous center of rotation. Based on geometric correlations, the following relation holds:

$${\delta }_f={\displaystyle \frac{\delta }{i_t}}$$

(1)

$${\delta }_{fl}=\arctan \left({\displaystyle \frac{2\tan {\delta }_{f}L}{2L-b_r\tan {\delta }_f}}\right)$$

(2)

$${\delta }_{fr}=\arctan \left({\displaystyle \frac{2\tan {\delta }_{f}L}{2L+b_r\tan {\delta }_f}}\right)$$

(3)

The longitudinal, lateral, and yaw motions of the vehicle can be represented by the following differential Equations (4)–(6):

$$m\left({\dot{v}}_x-\gamma v_y\right)=\left(F_{x,fl}\sin {\delta }_{fl}+F_{x,fr}\sin {\delta }_{fr}\right)-\left(F_{y,fl}\cos {\delta }_{fl}+F_{y,fr}\cos {\delta }_{fr}\right)+\left(F_{x,rl}+F_{x,rr}\right)$$

(4)

$$m\left({\dot{v}}_y+\gamma v_x\right)=\left(F_{y,fl}\cos {\delta }_{fl}+F_{y,fr}\cos {\delta }_{fr}\right)+\left(F_{x,fl}\sin {\delta }_{fl}+F_{x,fr}\sin {\delta }_{fr}\right)+\left(F_{y,rl}+F_{y,rr}\right)$$

(5)

$$\begin{array}{c}{I}_z\dot{\gamma }={\displaystyle \frac{b_f}{2}}\left(F_{x,fr}\cos {\delta }_{fr}-F_{x,fl}\cos {\delta }_{fl}+F_{y,fl}\sin {\delta }_{fl}-F_{y,fr}\sin {\delta }_{fr}\right)+{\displaystyle \frac{b_r}{2}}\left(F_{x,rr}-F_{x,rl}\right)\\ +l_f\left(F_{x,fr}\sin {\delta }_{fr}+F_{x,fl}\sin {\delta }_{fl}+F_{y,fr}\cos {\delta }_{fr}+F_{y,fl}\cos {\delta }_{fl}\right)-l_r\left(F_{y,rl}+F_{y,rr}\right)\end{array}$$

(6)

where $v_x$, $v_y$, and $\gamma$ denote the longitudinal velocity, lateral velocity, and the yaw rate, $m$ and $I_z$ denote the vehicle mass and yaw moment of inertia, and $F_{x,i}$ and $F_{y,i}$ denote the longitudinal and lateral forces exerted on tire $i$ ($i=fl,fr,rl,rr$), respectively.

The kinematic equation for the rotational motion of the wheel is as follows:

$$I_{\omega }{\dot{\omega }}_i=T_i-F_{z,i}fr-F_{x,i}r$$

(7)

where $I_{\omega }$ denotes the wheel moment of inertia, $T_i$ and $F_{z,i}$ denote motor torque and vertical force exerted on the ith wheel, and $f$ and $r$ denote the rolling friction coefficient and wheel dynamic rolling radius. The slip angle of each wheel is denoted by

$${\alpha }_{fl}=\arctan \left({\displaystyle \frac{v_y+l_f\gamma }{v_x-b_f\gamma }}\right)-{\delta }_{fl}$$

(8)

$${\alpha }_{fr}=\arctan \left({\displaystyle \frac{v_y+l_f\gamma }{v_x+b_f\gamma }}\right)-{\delta }_{fr}$$

(9)

$${\alpha }_{rl}=\arctan \left({\displaystyle \frac{v_y-l_r\gamma }{v_x-b_r\gamma }}\right)$$

(10)

$${\alpha }_{rr}=\arctan \left({\displaystyle \frac{v_y-l_r\gamma }{v_x+b_r\gamma }}\right)$$

(11)

The lateral tire force $F_{y,i}$ can be calculated using the Magic Formula tire model [38] through curve fitting:

$$F_{y,i}=e_1\sin \left[e_2\arctan \left\{e_3{\alpha }_i-e_4\left(e_3{\alpha }_i-\arctan \left(e_3{\alpha }_i\right)\right)\right\}\right]$$

(12)

where $e_1=\mu F_{z,i}$ represents the peak factor, $\mu$ is the road surface friction coefficient, and $e_2$, $e_3$, and $e_4$ represent the stiffness factor, the shape factor, and the curvature factor, respectively.

In this study, the 4WID-EV is driven by four independently operating permanent magnet synchronous motors, as shown in Figure 3. The input power of the motor is determined by the output power $P_{\mathrm{out}}$ and the power losses $P_{\mathrm{loss}}$, which are mainly caused by the copper loss, iron loss, and mechanical loss. Based on the motor’s torque, speed, and efficiency characteristics, $P_{\mathrm{in}}$ and $P_{\mathrm{out}}$ can be calculated as follows:

$$P_{\mathrm{out}}={\displaystyle \frac{T_i{n}_i}{9550}},P_{\mathrm{loss}}=P_{\mathrm{out}}\left(1-{\eta }_i\right),P_{\mathrm{in}}=P_{\mathrm{out}}+P_{\mathrm{loss}}$$

(13)

where ${\eta }_i$ is the efficiency, based on the torque $T_i$ and speed $n_i$ of the motor.

As shown in Figure 4, a speed tracking driver model is developed using a proportional-integral (PI) controller. The difference between the desired velocity $v_d$ and the current velocity $v$ serves as the input for the controller, which determines the total required torque.

In this study, we design an adaptive compound controller to balance lateral stability and energy efficiency. The controller consists of high-precision sensors, actuators, and a control unit to facilitate its implementation. A high-precision gyroscope, radar speed sensor, inertial measurement unit, and optical encoder measure the yaw rate, longitudinal velocity, lateral velocity, and steering wheel angle, respectively.

3. Design of an Adaptive Compound Controller for Reference Model

3.1. Dynamics of the Reference Vehicle Model

A 2-DOF single-track vehicle model is developed in Equations (14) and (15), which includes both the lateral and yaw motions, as shown in Figure 5. The steering angles of the front left and front right wheels are assumed identical and denoted by ${\delta }_f$. This model provides the desired yaw rate and the side slip angle for the predictive model. Under the assumption of a small side slip angle at the CG, we have $\beta =\arctan (v_y/v_x)\approx v_y/v_x$. Therefore, the equations governing the vehicle dynamics can be written as

$$m{v}_x(\dot{\beta }+\gamma )=F_{y,f}\cos ({\delta }_f)+F_{y,r}$$

(14)

$$I_z\dot{\gamma }=F_{y,f}\cos ({\delta }_f)l_f-F_{y,r}{l}_r$$

(15)

Assuming that the front wheel steering angle is small, we have $\cos ({\delta }_f)\approx 1$. The lateral forces acting on tires and tire side slip angles for the front and rear axles can be expressed as follows:

$$F_{y,f}=K_f{\alpha }_f$$

(16)

$$F_{y,r}=K_r{\alpha }_r$$

(17)

$${\alpha }_f=\beta +l_f\gamma /v_x-{\delta }_f$$

(18)

$${\alpha }_r=\beta -l_r\gamma /v_x$$

(19)

where $K_f$ and $K_r$ denote the cornering stiffness of the front and rear tires, $K=K_f+K_r$ represents the total cornering stiffness of the vehicle, and ${\alpha }_f$ and ${\alpha }_r$ represent the tire slip angles for the front and rear axles, respectively.

Given that the maximum lateral acceleration is constrained by road friction conditions, the yaw rate should satisfy the condition $\left|\gamma \right|\le \mu g/v_x$ [39]. Note that high-friction road surfaces lead to a larger effective region for maintaining vehicle lateral stability. When the vehicle reaches a steady state, both the accelerations $\dot{\beta }$ and $\dot{\gamma }$ are equal to zero. The reference yaw rate and side slip angle can be computed as follows:

$${\gamma }_{\mathrm{ss}}=\min \left(\left|{\displaystyle \frac{v_x{\delta }_f}{L\left(1+\kappa v_x^2\right)}}\right|,{\displaystyle \frac{\mu g}{v_x}}\right)\mathrm{sign}({\delta }_f)$$

(20)

$${\beta }_{\mathrm{ss}}={\gamma }_{\mathrm{ss}}{v}_x\left({\displaystyle \frac{l_r}{v_x^2}}+{\displaystyle \frac{m{l}_f}{K_{r}L}}\right)$$

(21)

where $\kappa =m(l_f{K}_f-l_r{K}_r)/(L{K}_r{K}_f)$ represents the understeer gradient.

3.2. Upper-Layer Controller

In this section, the MPC is used to determine the required yaw moment for steering. The predictive model is a critical component of MPC, as it accurately captures the dynamic characteristics of the controlled system and enables real-time problem-solving. The predictive model in this section is a nonlinear 2-DOF model that accounts for the external yaw moment. As presented in Equation (22), the external yaw moment $M_z$ is incorporated into Equation (15) as follows:

$$I_z\dot{\gamma }=F_{y,f}\cos {\delta }_f{l}_f-F_{y,r}{l}_r+M_z$$

(22)

Based on the linear relationship between the tire slip angle and the lateral force, the system’s nonlinear state-space representation can be linearized as follows:

$$\dot{\mathbf{x}}=f(\mathbf{x},u)\triangleq \mathbf{A}\mathbf{x}+\mathbf{B}u+\mathbf{C}{\delta }_f$$

(23)

where

$$\mathbf{x}=\left\{\begin{array}{c}\beta \\ \gamma \end{array}\right\},u=M_z,\mathbf{A}=\left[\begin{array}{cc}{\displaystyle \frac{K_f+K_r}{m{v}_x}}& -1+{\displaystyle \frac{l_f{K}_f-l_r{K}_r}{m{v}_x^2}}\\ {\displaystyle \frac{l_f{K}_f-l_r{K}_r}{I_z}}& {\displaystyle \frac{{l_f}^2{K}_f+{l_r}^2{K}_r}{I_z{v}_x}}\end{array}\right],\mathbf{B}=\left[\begin{array}{c}0\\ {\displaystyle \frac{1}{I_z}}\end{array}\right],\mathbf{C}=\left[\begin{array}{c}-{\displaystyle \frac{K_f}{m{v}_x}}\\ -{\displaystyle \frac{l_f{K}_f}{I_z}}\end{array}\right]$$

(24)

The predictive model is discretized with an equal step size of $\Delta \tau$, retaining only the first-order terms. At each discrete step, the steering angle ${\delta }_f$ is treated as a constant value, $d$. Taking into accounting that $\Delta u(t_k)=u(t_k)-u(t_k-\Delta \tau )$, the incremental state-space representation is thus obtained as follows:

$$\zeta (t_k+\Delta \tau )=\widehat{\mathbf{A}}\zeta (t_k)+\widehat{\mathbf{B}}\Delta u(t_k)+\widehat{\mathbf{C}}d$$

(25)

$${\mathbf{x}}^T(\left.t_k+j\Delta \tau \right|t_k)=\mathbf{\Phi }{\widehat{\mathbf{A}}}^j\zeta (t_k)+\mathbf{\Phi }{\displaystyle \sum _{i=0}^{j-1}{\widehat{\mathbf{A}}}^{j-i-1}\left[\widehat{\mathbf{B}}\Delta u(t_k+i\Delta \tau )+\widehat{\mathbf{C}}d\right]},i\in [0,c],j\in [1,p]$$

(26)

where

$$\zeta (t_k)=\left\{\begin{array}{c}\mathbf{x}(t_k)\\ u(t_k-\Delta \tau )\end{array}\right\},\widehat{\mathbf{A}}=\left[\begin{array}{cc}\mathbf{A}\Delta \tau +\mathbf{I}& \mathbf{B}\Delta \tau \\ \mathbf{0}& \mathbf{I}\end{array}\right],\widehat{\mathbf{B}}=\left[\begin{array}{c}\mathbf{B}\Delta \tau \\ \mathbf{I}\end{array}\right],\widehat{\mathbf{C}}=\left[\begin{array}{c}\mathbf{C}\Delta \tau \\ \mathbf{0}\end{array}\right]$$

(27)

The predictive equation, with a prediction horizon of $p$ steps and a control horizon of $c$ steps, is defined as follows:

$$\begin{array}{cc}\hfill {\mathbf{X}}_{k,p}& ={\left[{\mathbf{x}}^T(\left.t_k+\Delta \tau \right|t_k),{\mathbf{x}}^T(\left.t_k+2\Delta \tau \right|t_k),\dots ,{\mathbf{x}}^T(\left.t_k+p\Delta \tau \right|t_k)\right]}^T\hfill \\ & ={\mathbf{S}}_c[\zeta (t_k)]+{\mathbf{S}}_u\Delta u(t_k)+{\mathbf{S}}_{d}d\hfill \end{array}$$

(28)

where

$${\mathbf{S}}_c[\zeta (t_k)]={\left[\begin{array}{cccccc}\mathbf{\Phi }\widehat{\mathbf{A}}& \mathbf{\Phi }{\widehat{\mathbf{A}}}^2& \cdots & \mathbf{\Phi }{\widehat{\mathbf{A}}}^c& \cdots & \mathbf{\Phi }{\widehat{\mathbf{A}}}^p\end{array}\right]}^T$$

(29)

$${\mathbf{S}}_u=\left[\begin{array}{cccc}\mathbf{\Phi }\widehat{\mathbf{B}}& 0& \cdots & 0\\ \mathbf{\Phi }\widehat{\mathbf{A}}\widehat{\mathbf{B}}& \mathbf{\Phi }\widehat{\mathbf{B}}& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{\Phi }{\widehat{\mathbf{A}}}^c\widehat{\mathbf{B}}& \mathbf{\Phi }{\widehat{\mathbf{A}}}^{c-1}\widehat{\mathbf{B}}& \cdots & \mathbf{\Phi }\widehat{\mathbf{A}}\widehat{\mathbf{B}}\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{\Phi }{\widehat{\mathbf{A}}}^{p-1}\widehat{\mathbf{B}}& \mathbf{\Phi }{\widehat{\mathbf{A}}}^{p-2}\widehat{\mathbf{B}}& \cdots & \mathbf{\Phi }{\widehat{\mathbf{A}}}^{p-c}\widehat{\mathbf{B}}\end{array}\right]$$

(30)

$${\mathbf{S}}_d={\left[\begin{array}{cccccc}\mathbf{\Phi }\widehat{\mathbf{C}}& \mathbf{\Phi }\widehat{\mathbf{A}}\widehat{\mathbf{C}}+\mathbf{\Phi }\widehat{\mathbf{C}}& \cdots & {\displaystyle \sum _{i=0}^{c-1}\mathbf{\Phi }{\widehat{\mathbf{A}}}^i\widehat{\mathbf{C}}}& \cdots & {\displaystyle \sum _{i=0}^{p-1}\mathbf{\Phi }{\widehat{\mathbf{A}}}^i\widehat{\mathbf{C}}}\end{array}\right]}^T$$

(31)

$$\mathbf{\Phi }=\left[\begin{array}{cc}\mathbf{I}& \mathbf{0}\end{array}\right]$$

(32)

The desired yaw rate and side slip angle are used as reference inputs. Let $\mathbf{r}={\left[{\beta }_{\mathrm{ss}},{\gamma }_{\mathrm{ss}}\right]}^T$; then, the reference sequence can be defined as follows:

$${\mathbf{R}}_{k,p}={\left[{\mathbf{r}}^T(\left.t_k+\Delta \tau \right|t_k),{\mathbf{r}}^T(\left.t_k+2\Delta \tau \right|t_k),\dots ,{\mathbf{r}}^T(\left.t_k+p\Delta \tau \right|t_k)\right]}^T$$

(33)

To ensure that the system follows the desired yaw rate ${\gamma }_{\mathrm{ss}}$ and side slip angle ${\beta }_{\mathrm{ss}}$, the optimization objective function is formally defined as follows:

$$J\left({\mathbf{u}}_{k,p}\right)={\left({\mathbf{X}}_{k,p}-{\mathbf{R}}_{k,p}\right)}^T\mathbf{Q}\left({\mathbf{X}}_{k,p}-{\mathbf{R}}_{k,p}\right)+{\mathbf{u}}_{k,p}^T\mathbf{W}{\mathbf{u}}_{k,p}$$

(34)

where $\mathbf{Q}$ and $\mathbf{W}$ represent the weighted weight coefficients, and the external yaw moment sequence is ${\mathbf{u}}_{k,p}={\left[u(t_k+\Delta \tau ),u(t_k+2\Delta \tau ),\cdots ,u(t_k+p\Delta \tau )\right]}^T$.

If we define $\mathbf{\Psi }={\mathbf{S}}_c\left[\zeta \left(t_k\right)\right]+{\mathbf{S}}_{d}d-{\mathbf{R}}_{k,p}$, then substituting Equations (28) and (33) into Equation (34) results in the following optimization problem:

$${\mathbf{u}}_{k,p}^{\ast }=J\left({\mathbf{u}}_{k,p}\right)=\arg \underset{{\mathbf{u}}_{k,p}}{\min }{\mathbf{u}}_{k,p}^T\left({\mathbf{S}}_u^T\mathbf{Q}{\mathbf{S}}_u+\mathbf{W}\right){\mathbf{u}}_{k,p}+\left(2{\mathbf{\Psi }}^T\mathbf{Q}{\mathbf{S}}_u\right){\mathbf{u}}_{k,p}$$

(35)

which is subjected to the following constraint:

$$M_{z,\min }\le u(t_k+i\Delta \tau )\le M_{z,\max },i\in [0,(p-1)]$$

(36)

3.3. Adaptive-Layer Controller

The optimization of vehicle steering angles aims to achieve a balance between stability and energy savings. The adaptive-layer controller is employed to reduce the excessive yaw moment addition in the upper-layer controller, which works in conjunction with the lower-layer controller to achieve energy savings. The vehicle stability is evaluated through a phase plane diagram. In the present analysis, we focus on the vehicle stability evaluation index $\rho$, which ranges from 0 to 1. The external yaw moment $M_z$ obtained from the upper-layer controller is adjusted based on the evaluation index $\rho$ to produce the modified external yaw moment $M_z^{\prime }$. We use the side slip angle and its rate of change for phase plane analysis, i.e., $(\beta -\dot{\beta })$, which effectively addresses accuracy issues on low-adhesion road surfaces or with large side slip angles [40,41]. The fitted boundary of the stability region can be represented as follows:

$${\lambda }_3\le {\lambda }_1\dot{\beta }+\beta \le {\lambda }_2$$

(37)

where ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ represent the stability region boundary coefficients. Figure 6 illustrates the saddle point positions and boundary curves in the phase plane diagram. Two grey dashed lines represent right and left boundaries, and a blue solid line and a green solid line represent the minimum and maximum yaw rates. The region enclosed by these lines is the stability region. The phase trajectories within the stability region will converge to the equilibrium point. The coefficients of ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ are calculated as follows:

$${\lambda }_1={\displaystyle \frac{1}{p_1{v}_x^3+p_2{v}_x^2+p_3{v}_x+p_4}}$$

(38)

$${\lambda }_2=\left(p_5\mu +p_6\right)(p_7{v}_x^{p_8}+p_9)+{\delta }_f$$

(39)

$${\lambda }_3=-\left(p_5\mu +p_6\right)(p_7{v}_x^{p_8}+p_9)+{\delta }_f$$

(40)

where ${\beta }_{rss}$ and ${\beta }_{lss}$ are the side slip angles for the right and left saddle points. The negative values in

Table 2. Parameters of the self-stable boundary.

Parameter	Value
$p_1$	−0.0005
$p_2$	0.0372
$p_3$	−1.1254
$p_4$	19.1114
$p_5$	0.1079
$p_6$	0.0001
$p_7$	79.9997
$p_8$	−1.8113
$p_9$	0.7331

are obtained by numerically fitting the stability boundaries and saddle points through a two-lines method [35,42].

The above method defines the boundary of the stability region; however, vehicle stability differs near the node and saddle boundaries. Therefore, an evaluation index $\rho$ is introduced to quantitatively describe the vehicle’s current stability state [42]:

$$I_{\beta }=1-\mathrm{sign}\left[\left({\beta }_{\max }-\beta \right)\left(\beta -{\beta }_{\min }\right)\right]{\displaystyle \frac{\min \left(\left|{\beta }_{\max }-\beta \right|,\left|\beta -{\beta }_{\min }\right|\right)}{\left({\beta }_{\max }-{\beta }_{\min }\right)/2}}$$

(41)

$$I_{\gamma }=1-\mathrm{sign}\left[\left({\gamma }_{\max }-\gamma \right)\left(\gamma -{\gamma }_{\min }\right)\right]{\displaystyle \frac{\min \left(\left|{\gamma }_{\max }-\gamma \right|,\left|\gamma -{\gamma }_{\min }\right|\right)}{\left({\gamma }_{\max }-{\gamma }_{\min }\right)/2}}$$

(42)

$$\rho =\left\{\begin{array}{l}0,0\le I_{\mathrm{actual}}<z/2\\ {\displaystyle \frac{1}{2}}\left\{1-\cos \left[\pi \left(I_{\mathrm{actual}}-z/2\right)/\left(1-z/2\right)\right]\right\},z/2\le I_{\mathrm{actual}}<1\end{array}\right.$$

(43)

where ${\beta }_{\max }={\lambda }_2-{\lambda }_1\dot{\beta }$, ${\beta }_{\min }={\lambda }_3-{\lambda }_1\dot{\beta }$, ${\gamma }_{\max }=\mu \mathrm{g}/v_x$, and ${\gamma }_{\min }=-\mu \mathrm{g}/v_x$; $z$ is the adaptive coefficient; $I_{\beta }$ is used to describe the level of the side slip angle; $I_{\gamma }$ is used to describe the level of the yaw rate; and $I_{\beta }$ and $I_{\gamma }$ are indices ranging from 0 to 1, determined by the shortest distance from the vehicle’s current state to the upper and lower boundaries of the stability region. If we define $I_{\mathrm{actual}}=\max \left\{I_{\beta },I_{\gamma }\right\}$, then $I_{\beta }$ and $I_{\gamma }$ are unified to obtain the evaluation index $\rho$ in Equation (43).

By analyzing the phase plane diagram, the vehicle’s stability level can be determined. The evaluation index $\rho$ is closely correlated with the adaptive coefficient $z$. By properly adjusting the adaptive coefficient $z$, the external yaw moment in the upper-layer controller can be regulated. The parameter tuning method is summarized as follows.

The phase plane diagram is divided into three regions based on the value of the evaluation index $\rho$: Type I $\left(0\le \rho <0.25\right)$, representing the energy-saving region; Type II $\left(0.25\le \rho <0.75\right)$, representing the transition region; and Type III $\left(0.75\le \rho <1\right)$, representing the critical stability region. These three regions correspond to three distinct control methods, designated as Type = 0, 1, and 2, respectively. In the energy-saving region, the vehicle reaches a highly stable state and does not require an external yaw moment. In the critical stability region, the vehicle’s steering maneuverability is insufficient, so the actual external yaw moment $M_z^{\prime }$ is exactly equal to the external yaw moment $M_z$ obtained from the upper-layer controller. In the transition region, the evaluation index $\rho$ is scaled between 0.25 and 0.75, with less external yaw moment being added, i.e., $M_z^{\prime }=2M_z\left(\rho -0.25\right)$. When performing phase plane analysis, it is also necessary to determine the adaptive coefficient $z$. The adaptive coefficient $z$ influences both energy consumption and vehicle stability. For a fixed SOC, with an increase in $z$, the energy-saving region expands, and the difference between $M_z^{\prime }$ and $M_z$ increases, leading to a reduction in the external yaw moment. Conversely, as $z$ decreases, the energy-saving region narrows, and the difference between $M_z^{\prime }$ and $M_z$ decreases, resulting in an increase in the external yaw moment.

In this section, a fuzzy controller is used with $I_{\mathrm{actual}}^{\ast }$ and $\mathrm{SOC}$ as inputs and $z$ as the output. The value of $I_{\mathrm{actual}}^{\ast }$ is derived by substituting ${\beta }_{\mathrm{ss}}$ and ${\gamma }_{\mathrm{ss}}$ into Equations (41) and (42). The values of $I_{\mathrm{actual}}^{\ast }$ and $\mathrm{SOC}$ are classified into three levels: L (Low), M (Medium), and H (High), while the value of $z$ is classified into five levels: VL (Very Low), L (Low), M (Medium), H (High), and VH (Very High). The fuzzy rules are presented in

Table 3. Rule base of the fuzzy logic controller.

$\mathit{z}$		$\mathbf{SOC}$
		L	M	H
$I_{\mathrm{actual}}^{\ast }$	L	VH	H	M
	M	H	M	L
	H	M	L	VL

and Figure 7, where the adaptive coefficient $z$ is determined as a function of the stability evaluation index $I_{\mathrm{actual}}^{\ast }$ and the SOC. The surface reflects the controller’s ability to dynamically balance lateral stability and energy savings.

3.4. Lower-Layer Controller

The lower-layer controller optimally distributes torque to the four motors based on the required torque and the external yaw moment. The primary goal of the torque distribution algorithm is to reduce the tire workload, that is, the required lateral and longitudinal forces based on the normal load, thereby minimizing energy loss due to the side slip. Assuming that the road surface friction coefficients are equal for each tire, the objective function $J_1$ for minimizing the workload of all four tires is calculated as follows:

$$J_1={\displaystyle \sum _i\frac{{F_{x,i}}^2+{F_{y,i}}^2}{{\left(\mu F_{z,i}\right)}^2}}$$

(44)

Additionally, a primary objective of the lower-layer controller is to minimize the energy consumed during steering. This objective function is presented in Equation (13). Based on the relationship between $F_{x,i}$ and $T_i$, the motor energy consumption can be converted into the objective function $J_2$ as follows:

$$F_{x,i}={\displaystyle \frac{T_i}{r}}$$

(45)

$$J_2={\displaystyle \sum _i\frac{F_{x,i}r{n}_i}{9550{\eta }_i}}$$

(46)

To ensure smooth motor control and minimize control increments, the sum of the squared differences between $F_{x,i}$ and $F_{x,i}^{\prime }$ for each motor is used as the objective function. Therefore, the objective function $J_3$ is defined as follows:

$$J_3={\displaystyle \sum _i{\left(F_{x,i}-F_{x,i}^{\prime }\right)}^2}$$

(47)

where $F_{x,i}^{\prime }$ represents the force provided by each motor in the previous time step.

By combining the three objective functions with the weighting coefficients $q_1$, $q_2$, and $q_3$, the final objective function $J$ is formulated. The following constraints are considered when distributing torque to the tires:

$$J=q_1{J}_1+q_2{J}_2+q_3{J}_3$$

(48)

$$\begin{array}{l}\mathrm{s}.\mathrm{t}.-{\displaystyle \frac{T_{\min }}{r}}\le F_{x,i}\le {\displaystyle \frac{T_{\max }}{r}}\\ -\sqrt{{\left(\mu F_{z,i}\right)}^2-{F_{y,i}}^2}\le F_{x,i}\le \sqrt{{\left(\mu F_{z,i}\right)}^2-{F_{y,i}}^2}\\ {\displaystyle \sum _i{F}_{x,i}=\frac{T_{\mathrm{req}}}{r}}\\ M_z^{\prime }=-{\displaystyle \frac{b_f}{2}}{F}_{x,fl}+{\displaystyle \frac{b_f}{2}}{F}_{x,fr}-{\displaystyle \frac{b_r}{2}}{F}_{x,rl}+{\displaystyle \frac{b_r}{2}}{F}_{x,rr}\end{array}$$

(49)

4. Numerical Simulations

A simulation platform is developed based on Figure 1 to verify the efficacy of the proposed adaptive compound controller in regulating lateral stability and energy savings. The platform comprises a 7-DOF vehicle dynamics model, motor model, battery model, driver model, and adaptive compound controller, all implemented in MATLAB/Simulink 2023b. Outputs from the vehicle dynamics model, such as the vehicle speed, side slip angle, yaw rate, and $\mathrm{SOC}$ from the battery model, are provided as feedback to the adaptive compound controller. The torque distribution results from the adaptive compound controller are then used as inputs to the vehicle dynamics model, forming a closed-loop control system.

This section evaluates the performance of the adaptive stability and energy-saving controller (SEC) in comparison to two limiting cases: the energy-saving controller (EC) and the stability controller (SC). In the EC, no external yaw moment control is applied, which corresponds to $M_z^{\prime }=0$ (Type I) in the SEC, whereas in the SC, no external yaw moment modification is added, which corresponds to $M_z^{\prime }=M_z$ (Type III) in the SEC.

The vehicle performs a lane-changing maneuver on a flat road surface ($\mu =0.85$) at a constant speed of 72 km/h, with the remaining vehicle parameters shown in

Table 4. Parameters of vehicle dynamics.

Parameter	Unit	Value
$m$	$\mathrm{kg}$	$1761$
$h_g$	$\mathrm{m}$	$0.5$
$I_z$	$\mathrm{kg}\cdot {\mathrm{m}}^2$	$2700$
$r$	$\mathrm{m}$	$0.354$
$l_f$	$\mathrm{m}$	$1.497$
$l_r$	$\mathrm{m}$	$1.378$
$b_f$	$\mathrm{m}$	$1.6$
$b_r$	$\mathrm{m}$	$1.6$

. Various steering angles (0.5 rad, 1.0 rad, and 1.5 rad) and initial $\mathrm{SOC}$ levels (0.2 and 0.8) are considered, with the vehicle speed maintained at 72 km/h while following the desired path under PI control. A larger steering angle generally indicates a higher likelihood of vehicle instability. Note that $\mathrm{SOC}=0.2$ represents a typical low battery charge, while $\mathrm{SOC}=0.8$ represents a high battery charge. The steering angle varies sinusoidally with an amplitude of 0.5 rad and a frequency of $\pi /2$, ranging between 1 s and 5 s, as illustrated in Figure 8.

4.1. Case 1: $\delta =0.5\mathrm{rad}$

The simulation results for a steering angle of 0.5 rad are presented in Figure 9, Figure 10 and Figure 11. Figure 9a–c depict the variations in the coefficient $z$, $\rho$, and $\mathrm{Type}$ within the adaptive-layer controller under the SEC strategy, while Figure 9d shows the time history of the modified external yaw moment $M_z^{\prime }$.

Figure 9a,b indicate that, under the SEC strategy, the initialization of $\mathrm{SOC}$ introduces variations in the adaptive coefficient $z$, leading to differences in the evaluation index $\rho$ when $\mathrm{SOC}=0.2$ (SEC-0.2) and $\mathrm{SOC}=0.8$ (SEC-0.8), respectively. In these two cases, the evaluation index $\rho$ remains within the range $0\le \rho <0.25$, corresponding to the energy-saving region described in Section 3.3. And the value of $\mathrm{Type}$ remains constant at 0, as shown in Figure 9c. Consequently, no external yaw moment is applied in this region.

Figure 10a–d show the yaw rate $\gamma$, side slip angle $\beta$, phase plane diagram $(\beta –\dot{\beta })$, and vehicle trajectory in the X-Y plane, respectively. It is worth noting that under the scenario of $\delta =0.5\mathrm{rad}$, SEC-0.2 and SEC-0.8 yield equivalent external yaw moments, leading to nearly identical vehicle trajectories, as shown in Figure 10d. The maximum yaw rates for both the EC and SEC strategies are identical when $\gamma =0.1663\mathrm{rad}/s$, with a corresponding side slip angle of $\beta =0.009\mathrm{rad}$. In contrast, the SC strategy generates a reduced maximum yaw rate at $\gamma =0.1279\mathrm{rad}/\mathrm{s}$ and a side slip angle at $\beta =0.0042\mathrm{rad}$, which indicates enhanced stability at the cost of reduced maneuverability (lateral displacement), as demonstrated in Figure 10a–d. The phase plane diagram in Figure 10c further confirms that the phase trajectories of the EC, SEC, and SC are significantly distant from the stability boundary, indicating a highly stable state.

The overall vehicle energy consumption is summarized in Figure 11, where the EC and SEC achieve energy consumptions of 5.61 kJ and 19.14 kJ, respectively, indicating significant energy savings compared to the SC.

4.2. Case 2: $\delta =1.0\mathrm{rad}$

From the analysis of Case 1, it is clear that, when the steering angle is 0.5 rad, the vehicle exhibits high lateral stability, with the SEC strategy prioritizing energy savings. As the steering angle increases from $\delta =0.5\mathrm{rad}$ to $\delta =1.0\mathrm{rad}$, the evaluation index $\rho$ ranges from 0.25 to 0.75, and the value of $\mathrm{Type}$ increases to 1, as shown in Figure 12d, which demonstrates the system’s adaptive response to the higher steering angle. In this case, the energy-saving strategy remains the primary focus.

Figure 12a shows that the adaptive coefficient $z$ for the SEC-0.2 is larger than that of the SEC-0.8. Figure 12b reveals that the SEC-0.8 achieves a higher $\rho$ value and faster dynamic response compared to the SEC-0.2. In Figure 12d, the absolute value of $M_z^{\prime }$ for the SEC lies between those of the EC and SC, and the SEC-0.2 exhibits a lower value of $M_z^{\prime }$ than the SEC-0.8. These observations highlight that the SEC-0.8 strategy maintains better stability under these conditions.

Figure 13a,b compare the maximum yaw rates and side slip angles for different control strategies. The maximum yaw rates $\gamma$ for the EC, SEC-0.2, SEC-0.8, and SC are 0.3218 rad/s, 0.2638 rad/s, 0.2576 rad/s, and 0.2358 rad/s, respectively. Similarly, the side slip angle $\beta$ gradually decreases from 0.0236 rad (EC) to 0.0085 rad (SC). Figure 13c demonstrates that the stability of the EC, SEC-0.2, SEC-0.8, and SC improves sequentially, as shown in Figure 13d. Regarding stability, the boundary of the yaw rate is constrained by $\left|\gamma \right|\le \mu g/v_x$, which can reach a maximum value of 0.4165 rad/s, while the boundary of the side slip angle is approximately 0.1 rad. Since the yaw rate $\gamma$ becomes closer to the boundary in the control process, it is considered a more critical factor influencing stability.

In Figure 13 and Figure 14, the lateral stability and energy consumption of the EC, SEC-0.8, and SC are compared. The SEC-0.8 can increase vehicle stability by 19.95% over the EC at the cost of a 21.79% increase in energy consumption, with these values computed as $\max \left({\gamma }_{\mathrm{SEC}\text{-}0.8}-{\gamma }_{\mathrm{EC}}\right)/{\gamma }_{\mathrm{EC}}$ and $\max \left(E_{\mathrm{SEC}\text{-}0.8}-E_{\mathrm{EC}}\right)/E_{\mathrm{EC}}$, respectively. Compared to the SC, the SEC-0.8 reduces vehicle stability by 9.25% but achieves 30.51% energy savings. Under identical conditions, the SEC-0.2 reduces stability by 2.41% while improving energy savings by 4.34% compared to the SEC-0.8. Notably, the SEC with an SOC of 0.1 achieves approximately 8.44% greater energy savings than the SEC with an SOC of 1.0.

4.3. Case 3: $\delta =1.5\mathrm{rad}$

Similar trends are observed for the control parameters shown in Figure 15, with the underlying physics consistent with those described for Case 2. However, a key distinction lies in the fact that the evaluation index extends into the range $0.75\le \rho <1$, indicating the critical stability region. Additionally, this scenario introduces a value of $\mathrm{Type}=2$, as demonstrated in Figure 15c,d. For $\mathrm{Type}=2$, no modification of the external yaw moment is required, i.e., $M_z^{\prime }=M_z$, while for a value of $\mathrm{Type}=1$, the external yaw moment should be modified as $M_z^{\prime }=2M_z\left(\rho -0.25\right)$. Moreover, no external yaw moment is required when $\mathrm{Type}=0$.

Due to the modification of the external yaw moment, the yaw rate $\gamma$ and the side slip angle $\beta$ under SEC control consistently fall between those of the EC and SC, as shown in Figure 16a,b. The maximum yaw rate closely matches that of the SC, thereby ensuring stability. Figure 16c indicates that the phase trajectories of SEC-controlled vehicles are closer to those of the SC, while EC-controlled vehicles show a significant deviation from the phase trajectories of both the SC and SEC. In terms of maneuverability, an SEC-controlled vehicle yields a balanced performance between the EC and SC. However, the SEC-0.2 strategy exhibits slightly reduced lateral displacement compared to the SEC-0.8, as illustrated in Figure 16d. As shown in Figure 17, the energy consumption of the SEC ranges from 59.21 kJ to 61.24 kJ, depending on the initial $\mathrm{SOC}$ level, which represents a reduction of at least 9.11% compared to the SC. These results suggest that, in scenarios approaching the maximum cornering threshold, an SEC-controlled vehicle can maintain stability and maneuverability similar to the SC, while achieving significant energy savings.

5. Conclusions

During steering maneuvers, real-time monitoring of the vehicle’s side slip angle and yaw rate is essential. To maintain lateral stability, an external yaw moment must be applied, which increases energy consumption. Therefore, balancing lateral stability with energy consumption becomes a critical challenge.

This paper presents a novel adaptive compound controller based on phase plane analysis to effectively balance vehicle stability and energy savings during the steering process. By quantifying the overall stability of the vehicle, the control framework can continuously switch between the pure energy-saving controller, the adaptive stability and energy-saving controller, and the stability controller in real time, with the obtained results summarized as follows:

• A compound control strategy, which integrates an adaptive-layer controller, is employed to preserve the advantages of hierarchical control and simultaneously improve energy efficiency.
• The proposed SEC strategy dynamically adjusts the adaptive coefficient based on the vehicle’s battery state and divides the phase plane diagram into three regions according to the evaluation index, which effectively balances lateral stability and energy savings.
• Simulation results demonstrate that the SEC strategy prioritizes energy savings as the battery’s state of charge decreases.
• These findings verify the effectiveness of the proposed adaptive compound controller in achieving a robust balance between lateral stability and energy savings.