1. Introduction
With the deep integration of intelligent connected technologies and autonomous driving systems, the conventional mechanical steering system is increasingly unable to satisfy the stringent requirements of high-level autonomous driving for steering accuracy, functional safety, and dynamic response performance. The inherent lag of mechanical transmission chains and the rigid coupling structures create unavoidable conflicts between steering agility and stability. In this context, the steer-by-wire (SBW) system has emerged as one of the most promising solutions for next-generation electric vehicles. By completely decoupling the mechanical connection between the steering wheel and the road wheels, the SBW system transmits steering commands through electrical signals, enabling precise and flexible control of the vehicle steering angle.
Electric vehicles provide an ideal platform for the application of SBW technology. Since electric vehicles do not require complex mechanical transmission layouts for internal combustion engines, the SBW system can fully exploit its advantages in structural simplification and packaging flexibility. The adoption of dual-motor drive configuration further improves the system redundancy, since a hot-backup power path can maintain basic steering functionality when one motor fails. In the dual-motor SBW system, two permanent magnet synchronous motors (PMSMs) are coupled through a gear-and-rack mechanism to realize the steering action. However, the system inevitably suffers from motor parameter perturbations, load variations, and external road disturbances, which introduce undesired tracking errors and synchronization deviation between the two motors.
This paper focuses on the synchronization control problem of a dual-motor SBW system for electric vehicles. A composite angle tracking controller combining the super-twisting second-order sliding mode control (SOSMC) algorithm with an extended state observer (ESO) is developed to replace the conventional triple closed-loop position control structure. On this basis, a synchronization controller based on the super-twisting SOSMC and a mean deviation coupling structure is designed to suppress the torque conflicts caused by inconsistent motor parameters and unbalanced load distribution. The proposed control strategies are validated through extensive simulations under typical operating conditions such as step response, sinusoidal response, double-lane change maneuvers, and alternating road adhesion conditions. The results demonstrate that the proposed methods effectively improve the tracking accuracy, robustness, and synchronization performance of the dual-motor SBW system, thus providing theoretical and technical support for the engineering application of dual-motor SBW systems in electric vehicles.
2. Modeling of Dual-Motor Steer-by-Wire System
2.1 System Architecture
The dual-motor SBW system mainly consists of the steering wheel module, the electronic control unit (ECU), and the steering actuator module. The steering wheel module includes the steering wheel, torque sensor, road feeling motor, angle sensor, and reduction gears. The actuator module integrates two PMSMs, two reduction gears, a pinion-rack mechanism, and associated mechanical linkages. The ECU receives the driver’s steering intention or the autonomous driving command, calculates the desired front wheel angle through the variable steering ratio algorithm, and sends commands to the motor controllers. Each PMSM drives the pinion through its own reducer, and the pinion engages with the rack to convert the rotational motion into transverse displacement of the rack, thereby steering the front wheels. In the case of one motor failing, the other motor can still maintain a certain level of steering capability, which is a crucial redundancy benefit of the dual-motor architecture.
2.2 Dynamic Model of Permanent Magnet Synchronous Motor
In this study, the surface-mounted PMSM is selected as the steering actuator because of its high torque density, low cogging torque, and excellent dynamic response. To simplify the analysis while preserving the essential dynamics, several assumptions are made: the three-phase stator windings are symmetrical; the magnetic saturation, hysteresis losses and eddy current losses are neglected; the induced electromotive force is ideally sinusoidal; and the slot effect is ignored. Under these assumptions, the three-phase voltage equations in the ABC reference frame can be expressed as
$$
\begin{bmatrix} u_a \\ u_b \\ u_c \end{bmatrix}
=
R_s \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix}
+
\frac{d}{dt}\begin{bmatrix} \psi_a \\ \psi_b \\ \psi_c \end{bmatrix},
$$
where \(u_a,u_b,u_c\) are the three-phase stator voltages, \(R_s\) is the stator resistance, and \(\psi_a,\psi_b,\psi_c\) are the three-phase stator flux linkages. The stator flux linkage of phase A can be written as \(\psi_a = L_s i_a + \psi_{fa}\), where \(L_s\) is the stator inductance and \(\psi_{fa}\) is the flux linkage produced by the permanent magnet in phase A. By applying the Park transformation, the voltage equations in the synchronously rotating dq reference frame become
$$
\begin{cases}
\displaystyle u_d = R_s i_d + L_d \frac{di_d}{dt} – \omega_e L_q i_q, \\[6pt]
\displaystyle u_q = R_s i_q + L_q \frac{di_q}{dt} + \omega_e \left(L_d i_d + \psi_f\right),
\end{cases}
$$
where \(\omega_e = p_n \omega_m\) is the electrical angular velocity, \(p_n\) is the number of pole pairs, \(\omega_m\) is the mechanical angular velocity, and \(\psi_f\) denotes the rotor flux linkage. For surface-mounted PMSMs, \(L_d = L_q = L_s\). The electromagnetic torque is given by
$$
T_e = \frac{3}{2}p_n\left[\psi_f i_q + \left(L_d – L_q\right)i_d i_q\right].
$$
Adopting the \(i_d = 0\) vector control strategy, the electromagnetic torque is simplified to
$$
T_e = \frac{3}{2}p_n\psi_f i_q.
$$
The mechanical dynamics of the motor shaft are described by
$$
J\frac{d\omega_m}{dt} = T_e – T_L – B\omega_m,
$$
where \(J\) is the moment of inertia, \(T_L\) is the load torque, and \(B\) is the viscous friction coefficient. The angular position \(\theta_m\) satisfies \(\dot{\theta}_m = \omega_m\).
When two PMSMs are coupled to drive the same rack, the electromagnetic torque of motor \(k\,(k=1,2)\) can be expressed as
$$
T_{ek} = \frac{3}{2}p_n\psi_f i_{qk},\qquad k=1,2.
$$
Considering parameter perturbations and external disturbances, the dynamic equation of motor \(k\) can be written as
$$
(J+\Delta J_k)\frac{d\omega_k}{dt} + (B+\Delta B_k)\omega_k = (T_{ek}-T_{Lk}) + T_{dis,k},
$$
where \(\Delta J_k\) and \(\Delta B_k\) denote the uncertain variations of inertia and damping coefficient, \(T_{Lk}\) is the load torque transmitted from the rack, and \(T_{dis,k}\) represents the lumped disturbance including road excitation, friction variation, and unmodeled dynamics. From the above equation, the angular acceleration of motor \(k\) is
$$
\ddot{\theta}_k = -\frac{B}{J}\dot{\theta}_k – \frac{1}{J}\left(T_{Lk}+T_{dis,k}\right) + \frac{p_n\psi_f}{J}i_{qk}.
$$
2.3 Pinion-Rack Dynamics
The rotary motion of each PMSM is transmitted through a reduction gear with ratio \(G_1\) to the pinion. The pinion of radius \(r_p\) engages with the rack, converting the rotation into the rack displacement \(x_r\). The dynamic equation of the rack can be established according to Newton’s second law as
$$
m_r \ddot{x}_r + B_r \dot{x}_r + F_{fric} = \frac{G_1}{r_p}(T_{e1} + T_{e2}) – \frac{T_z}{G_2},
$$
where \(m_r\) is the rack mass, \(B_r\) is the rack damping coefficient, \(F_{fric}\) represents the Coulomb friction, \(G_2\) is the transmission ratio from the pinion rotation to the front wheel angle, and \(T_z\) is the total aligning torque acting on the front wheels. The relationship between the rack displacement and the front wheel steering angle is approximated by
$$
\delta_f \approx \frac{x_r}{G_2}.
$$
The aligning torque is mainly generated by the pneumatic trail and mechanical trail of the tires, and it varies according to the road adhesion conditions, tire slip angle, and vehicle longitudinal speed. Therefore, the load torque transmitted to each motor is time-varying and subject to sudden changes during real driving.
2.4 Vehicle Dynamics Model
To investigate the vehicle-level stability performance, the two-degree-of-freedom (2-DOF) bicycle model is adopted in this study. This model describes the lateral and yaw motions of the electric vehicle while ignoring the roll, pitch, and vertical dynamics. The governing equations are
$$
\begin{cases}
\displaystyle mu\left(\dot{\beta}+\gamma\right) = k_f \alpha_f + k_r \alpha_r, \\[8pt]
\displaystyle I_z \dot{\gamma} = a k_f \alpha_f – b k_r \alpha_r,
\end{cases}
$$
where \(m\) is the vehicle mass, \(u\) is the longitudinal velocity, \(\beta\) is the sideslip angle at the vehicle center of gravity, \(\gamma\) is the yaw rate, \(I_z\) is the yaw moment of inertia, \(a\) and \(b\) are the distances from the center of gravity to the front and rear axles, and \(k_f\), \(k_r\) are the equivalent cornering stiffness values of the front and rear tires, respectively. The tire slip angles of the front and rear axles are defined as
$$
\alpha_f = \beta + \frac{a\gamma}{u} – \delta_f,
\qquad
\alpha_r = \beta – \frac{b\gamma}{u}.
$$
Substituting the slip angle expressions into the lateral dynamics yields the state-space representation
$$
\frac{d}{dt}\begin{bmatrix}\beta\\\gamma\end{bmatrix}
=
\begin{bmatrix}
\dfrac{k_f+k_r}{mu} & \dfrac{ak_f-bk_r}{mu^2}-1 \\[8pt]
\dfrac{ak_f-bk_r}{I_z} & \dfrac{a^2k_f+b^2k_r}{I_z u}
\end{bmatrix}
\begin{bmatrix}\beta\\\gamma\end{bmatrix}
+
\begin{bmatrix}
-\dfrac{k_f}{mu} \\[6pt]
-\dfrac{ak_f}{I_z}
\end{bmatrix}\delta_f.
$$
This vehicle model is implemented in the CarSim software environment and coupled with the dual-motor SBW model established in MATLAB/Simulink to form a co-simulation platform. The main parameters of the dual-motor SBW system and the selected electric vehicle are listed in Table 1 and Table 2, respectively.
Table 1. Parameters of the dual-motor steer-by-wire actuator system
| Parameter | Value |
|---|---|
| Pinion radius \(r_p\) | 0.007 m |
| Motor reducer ratio \(G_1\) | 8 |
| Pinion-to-front-wheel ratio \(G_2\) | 20 |
| Rack mass \(m_r\) | 2 kg |
| Steering arm length \(l_{fl}=l_{fr}\) | 0.36 m |
| Torsional stiffness \(k_{md}\) | 178 N·m/rad |
| Rack damping \(B_r\) | 635 N·m/(rad/s) |
Table 2. Parameters of the permanent magnet synchronous motor
| Parameter | Value |
|---|---|
| Stator resistance \(R_s\) | 0.36 Ω |
| Stator inductance \(L_s\) | 1.7×10⁻⁴ H |
| Rotor moment of inertia \(J\) | 0.00043 kg·m² |
| Damping coefficient \(B\) | 0.00023 N·m/(rad/s) |
| Rotor flux linkage \(\psi_f\) | 0.032 Wb |
| DC bus voltage \(U_{dc}\) | 12 V |
| Number of pole pairs \(p_n\) | 4 |
3. Angle Tracking Control Based on Disturbance Observer
3.1 Problem Formulation
In conventional PMSM position-servo systems, the angle control is realized by a three-loop cascade structure, namely the current loop, the speed loop, and the position loop. Although the cascade PI structure is simple and reliable in practice, its control performance deteriorates when the system is subjected to strong external disturbances and parameter uncertainties. For the SBW system of electric vehicles, the steering load can change abruptly during emergency obstacle avoidance, and the motor parameters can drift because of temperature variation and magnetic saturation. In this context, a composite angle tracking controller that integrates the super-twisting SOSMC algorithm and an ESO is designed to achieve fast and robust angle tracking.
Taking motor 1 as an example, its second-order angle dynamics can be expressed as
$$
\ddot{\theta}_1 = -\frac{B}{J}\dot{\theta}_1 – \frac{T_{L1}+T_{dis1}}{J} + \frac{p_n\psi_f}{J}i_{q1}.
$$
Let the lumped disturbance of motor 1 be defined as
$$
D_1 = \frac{T_{L1}+T_{dis1}}{J}.
$$
Then the tracking error and its derivative are defined as
$$
e_{1,d} = \theta_d – \theta_1,
\qquad
\dot{e}_{1,d} = \dot{\theta}_d – \dot{\theta}_1,
$$
where \(\theta_d\) is the desired motor angle generated by the target front wheel angle and the steering transmission ratio.
3.2 Design of Super-Twisting Second-Order Sliding Mode Angle Controller
A sliding surface for the angle tracking controller is designed as
$$
s_1 = c_1 e_{1,d} + \dot{e}_{1,d}
= c_1\left(\theta_d-\theta_1\right) + \left(\dot{\theta}_d-\dot{\theta}_1\right),
$$
where \(c_1 > 0\) is a positive constant that determines the convergence rate. Taking the derivative of \(s_1\) with respect to time yields
$$
\dot{s}_1 = c_1\left(\dot{\theta}_d-\dot{\theta}_1\right) + \ddot{\theta}_d – \ddot{\theta}_1.
$$
Substituting the dynamic expression of \(\ddot{\theta}_1\) into the above equation gives
$$
\dot{s}_1 = c_1\left(\dot{\theta}_d-\dot{\theta}_1\right) + \ddot{\theta}_d + \frac{B}{J}\dot{\theta}_1 + D_1 – \frac{p_n\psi_f}{J}i_{q1}.
$$
The equivalent control term \(u_{1,eq}\) is obtained by setting \(\dot{s}_1 = 0\):
$$
u_{1,eq} = \frac{BJ}{p_n\psi_f}\omega_1 + \frac{J}{p_n\psi_f}\left[\ddot{\theta}_d + D_1 + c_1\left(\dot{\theta}_d-\dot{\theta}_1\right)\right].
$$
To further accelerate the convergence and to suppress the chattering phenomenon, the super-twisting control term is adopted. The complete control law of the angle tracking controller is designed as
$$
u_{1,d} = \frac{BJ}{p_n\psi_f}\omega_1 + \frac{J}{p_n\psi_f}\left[\ddot{\theta}_d + D_1 + c_1\left(\dot{\theta}_d-\dot{\theta}_1\right)\right]
– \lambda_1 |s_1|^{1/2}\mathrm{sign}(s_1) + \varphi_1 s_1 – v_1,
$$
$$
\dot{v}_1 = \gamma_1\,\mathrm{sign}(s_1),
$$
where \(\lambda_1\), \(\gamma_1\), and \(\varphi_1\) are positive control gains. The term \(\varphi_1 s_1\) introduces a linear proportional action of the sliding surface that improves the reaching speed when the system state is far away from the sliding manifold.
3.3 Stability Analysis
The Lyapunov function is chosen as
$$
V_1 = \frac{1}{2}s_1^2.
$$
Differentiating \(V_1\) along the trajectory of the closed-loop system gives
$$
\dot{V}_1 = s_1\dot{s}_1
= s_1\left\{c_1\left(\dot{\theta}_d-\dot{\theta}_1\right)+\ddot{\theta}_d+\frac{B}{J}\dot{\theta}_1+D_1-\frac{p_n\psi_f}{J}u_{1,d}\right\}.
$$
Substituting the composite control law \(u_{1,d}\) into the equation yields
$$
\dot{V}_1 = s_1\left[-\lambda_1|s_1|^{1/2}\mathrm{sign}(s_1)+\varphi_1 s_1-v_1\right]
= -\lambda_1 |s_1|^{3/2} + \varphi_1 s_1^2 – s_1 \gamma_1 \int_0^t \mathrm{sign}(s_1)\,d\tau.
$$
Since \(\mathrm{sign}(s_1)s_1 = |s_1|\), the following inequality holds when \(\lambda_1,\gamma_1,\varphi_1>0\):
$$
\dot{V}_1
= -\lambda_1|s_1|^{3/2} – \gamma_1\int_0^t |s_1|\,d\tau + \varphi_1 s_1^2
\le 0.
$$
The above inequality satisfies the Lyapunov stability condition, which proves that the sliding surface \(s_1\) converges to zero asymptotically. Therefore, the designed angle tracking controller is stable and the tracking error converges to zero in finite time.
3.4 Extended State Observer Design
In practice, the total disturbance \(D_1\) cannot be measured directly. To estimate the disturbance in real time and compensate it in the feedforward path, a third-order extended state observer is designed. The state variables are selected as \(x_1=\theta_1\), \(x_2=\dot{\theta}_1\), and \(x_3=f(t)\) where \(f(t)\) denotes the total disturbance. The extended state equation is given by
$$
\begin{cases}
\dot{x}_1 = x_2, \\
\dot{x}_2 = x_3 + b_0 u_{1,d}, \\
\dot{x}_3 = \dot{f}(t),
\end{cases}
$$
where \(b_0 = p_n\psi_f/J\). The corresponding ESO is constructed as
$$
\begin{cases}
e_0 = z_1 – \theta_1, \\
\dot{z}_1 = z_2 – \beta_1 \mathrm{fal}(e_0,\alpha_{01},\delta), \\
\dot{z}_2 = z_3 – \beta_2 \mathrm{fal}(e_0,\alpha_{02},\delta) + b_0 u_{1,d}, \\
\dot{z}_3 = -\beta_3 \mathrm{fal}(e_0,\alpha_{03},\delta),
\end{cases}
$$
where \(z_1\), \(z_2\), and \(z_3\) are the estimates of \(\theta_1\), \(\dot{\theta}_1\), and the total disturbance, respectively. The nonlinear function \(\mathrm{fal}(e,\alpha,\delta)\) has the following definition:
$$
\mathrm{fal}(e,\alpha,\delta)=
\begin{cases}
|e|^{\alpha}\,\mathrm{sign}(e), & |e|>\delta, \\[4pt]
e/\delta^{1-\alpha}, & |e|\le\delta.
\end{cases}
$$
The observer gains are selected as \(\beta_1 = 3\omega_o\), \(\beta_2 = 3\omega_o^2\), \(\beta_3 = \omega_o^3\), where \(\omega_o\) is the desired bandwidth of the ESO. The observed disturbance \(\hat{D}_1 = z_3\) is then compensated in the control law through the feedforward term. The entire composite angle tracking control scheme is illustrated in the control block diagram.
3.5 Simulation Results of Angle Tracking Control
Simulations are first carried out with a square-wave angle reference signal. The amplitude of the square wave is 1 rad with a frequency of 1/4 Hz, and a step disturbance of 0.5 N·m is applied at the load side at \(t=0.1\) s. The angle tracking response and the tracking error curves are plotted.
Table 3. Performance comparison under square-wave angle input
| Controller | Error settling time (s) | First error convergence (s) | Maximum error (rad) |
|---|---|---|---|
| Composite SOSMC-ESO | 0.02 | 0.027 | < 0.03 |
| Conventional ADRC | ~0.20 | 0.040 | > 0.12 |
From the simulation results, the composite controller maintains an accurate tracking performance even when the load changes abruptly. The conventional ADRC controller exhibits a noticeable deviation at the load mutation instant. The proposed composite controller yields a faster transient response and a substantially smaller tracking error.
In the second case, a sinusoidal angle reference with amplitude of 1 rad and frequency of 10 Hz is applied, and a load step disturbance is introduced at \(t=0.1\) s. The measured absolute peak tracking error is summarized in Table 4.
Table 4. Performance comparison under sinusoidal angle input
| Controller | Absolute peak error (rad) | Disturbance rejection |
|---|---|---|
| Composite SOSMC-ESO | 0.02 | Good |
| Conventional ADRC | 0.183 | Moderate |
The simulation results demonstrate that under both square-wave and sinusoidal angle tracking tasks, the composite SOSMC-ESO angle controller significantly outperforms the conventional ADRC in terms of transient response speed, tracking accuracy, and robustness against load disturbances. This establishes a solid foundation for the subsequent dual-motor synchronization control design.
4. Design of Dual-Motor Synchronization Controller
4.1 Comparison of Synchronization Control Structures
Four widely used synchronization architectures are considered in this study, including parallel control, master-slave control, cross-coupling control, and deviation coupling control. Their characteristics are compared qualitatively in Table 5.
Table 5. Qualitative comparison of synchronization control structures
| Structure | Signal interaction | Disturbance rejection | Control complexity | Synchronization accuracy |
|---|---|---|---|---|
| Parallel | None | Poor | Low | Low |
| Master-slave | One-way | Moderate | Low | Moderate |
| Cross-coupling | Two-way | Good | Medium | Good |
| Deviation coupling | Full coupled | Excellent | High | Excellent |
In the dual-motor SBW system, deviation coupling control has the best synchronization capability because the control input of each motor depends not only on its own tracking error but also on the synchronization error with respect to the average output of both motors. Therefore, the deviation coupling structure is selected as the basis for the synchronization controller design.
4.2 Mean Deviation Coupling Control Architecture
The block diagram of the mean deviation coupling control architecture is shown in the constructed model. The average angle of the two motors is defined as
$$
\theta_m = \frac{\theta_1+\theta_2}{2}.
$$
The synchronization error of motor \(k\) is the difference between the average angle \(\theta_m\) and the actual angle of motor \(k\):
$$
e_{k,m} = \theta_m – \theta_k = \frac{\theta_j-\theta_k}{2},
\qquad k,j \in \{1,2\},\; j\neq k.
$$
For the dual-motor system, the synchronization controller is designed separately for each motor. Taking motor 2 as an example, the synchronization error can be further expressed as
$$
e_{2,m} = \theta_m – \theta_2 = \frac{\theta_1-\theta_2}{2}.
$$
Its derivative is
$$
\dot{e}_{2,m} = \frac{\dot{\theta}_1-\dot{\theta}_2}{2}.
$$
4.3 Design of Angle Synchronization Controller Based on Super-Twisting SOSMC
The sliding surface for the synchronization controller is designed as
$$
s_2 = c_2 e_{2,m} + \dot{e}_{2,m},
$$
where \(c_2 > 0\) is a constant. The derivative of \(s_2\) can be written as
$$
\dot{s}_2 = c_2\left(\dot{\theta}_m-\dot{\theta}_2\right) + \ddot{\theta}_m – \ddot{\theta}_2.
$$
Using the dynamic equations of the two motors, \(\ddot{\theta}_m\) and \(\ddot{\theta}_2\) are expressed respectively as
$$
\ddot{\theta}_m = \frac{1}{2}\left[ -\frac{B}{J}\dot{\theta}_1 -\frac{B}{J}\dot{\theta}_2 + \frac{p_n\psi_f}{J}(i_{q1}+i_{q2}) – D_1 – D_2 \right],
$$
$$
\ddot{\theta}_2 = -\frac{B}{J}\dot{\theta}_2 – D_2 + \frac{p_n\psi_f}{J}i_{q2}.
$$
The equivalent control \(u_{2,eq}\) is derived from \(\dot{s}_2=0\) as
$$
u_{2,eq} = \frac{B}{p_n\psi_f}(\dot{\theta}_1-\dot{\theta}_2)
+ \frac{2J}{p_n\psi_f}\left[\ddot{\theta}_m + c_2(\dot{\theta}_m-\dot{\theta}_2)\right]
+ \frac{2J}{p_n\psi_f}D_2.
$$
Adding the super-twisting reaching law gives the complete synchronization control law:
$$
u_{2,m} = u_{2,eq} – \lambda_2 |s_2|^{1/2}\mathrm{sign}(s_2) + \varphi_2 s_2 – v_2,
$$
$$
\dot{v}_2 = \gamma_2\,\mathrm{sign}(s_2),
$$
where \(\lambda_2\), \(\gamma_2\), and \(\varphi_2\) are positive constants. Similarly, the synchronization controller for motor 1 can be derived in a parallel fashion. The proposed synchronization controller compensates for the angle deviation between the two motors, thereby suppressing the torque conflict caused by inconsistent parameters and unbalanced load distribution.
4.4 Stability Analysis of the Synchronization Controller
The Lyapunov candidate function for motor 2 is selected as
$$
V_2 = \frac{1}{2}s_2^2.
$$
Differentiating \(V_2\) yields
$$
\dot{V}_2 = s_2\dot{s}_2
= s_2\left[ -\lambda_2|s_2|^{1/2}\mathrm{sign}(s_2)+\varphi_2 s_2-v_2\right]
= -\lambda_2|s_2|^{3/2}+\varphi_2 s_2^2-\gamma_2 \int_0^t |s_2|\,d\tau.
$$
Given that \(\lambda_2\), \(\gamma_2\), and \(\varphi_2\) are all positive, \(\dot{V}_2\le 0\) holds. The same stability proof applies to the motor-1 synchronization controller. Consequently, the designed synchronization controller is asymptotically stable and the synchronization error converges to zero.
4.5 Simulation Results of Dual-Motor Synchronization Control
To evaluate the proposed synchronization structure, a sinusoidal sweeping-angle input is applied to the dual-motor system. Three synchronization structures are compared: master-slave control, cross-coupling control, and mean deviation coupling control. The obtained maximum synchronization errors are presented in Table 6.
Table 6. Maximum synchronization error under different structures
| Synchronization structure | Maximum synchronization error (rad) |
|---|---|
| Master-slave control | 0.150 |
| Cross-coupling control | 0.151 |
| Mean deviation coupling control | 0.078 |
The mean deviation coupling structure reduces the maximum synchronization error by approximately 48% compared with both the master-slave structure and the cross-coupling structure. This indicates that the full-coupled deviation compensation mechanism effectively improves the synchronization accuracy of the dual-motor system.
Furthermore, the performance of three different control algorithms is compared under the same synchronization structure: the super-twisting SOSMC, the ADRC, and the conventional PI control. The adjustment time and the synchronization error of each algorithm are listed in Table 7.
Table 7. Performance comparison under different control algorithms
| Algorithm | Adjustment time (s) | Overshoot | Synchronization error trend |
|---|---|---|---|
| Super-twisting SOSMC | 0.45 | Smallest | Smallest |
| ADRC | 0.57 | Medium | Medium |
| Conventional PI | 1.36 | Largest | Largest |
Compared with ADRC, the adjustment time under the proposed super-twisting SOSMC is reduced by about 21%; compared with conventional PI control, it is reduced by about 66.9%. These results prove that the proposed scheme achieves both rapid response and excellent synchronization performance, which can effectively solve the torque conflict problem in the dual-motor SBW system of electric vehicles.
5. Vehicle-Level Stability Simulation Analysis
5.1 Steering on High Adhesion Road
In this section, the vehicle-level control performance is investigated by the CarSim-Simulink co-simulation platform. A double-lane-change maneuver is selected as the typical driving scenario to evaluate the path-following ability and lateral stability of the electric vehicle. The vehicle speed is set to 70 km/h and the road adhesion coefficient is 0.85. The vehicle response under both the conventional ADRC and the proposed composite SOSMC-ESO controller is compared in terms of motor angle tracking error, dual-motor synchronization error, lateral trajectory tracking error, and yaw rate error.
The quantitative comparison for the high-adhesion road condition is summarized in Table 8.
Table 8. Vehicle response comparison under high-adhesion road condition
| Performance index | ADRC | Composite SOSMC-ESO | Improvement |
|---|---|---|---|
| Maximum motor angle tracking error (rad) | 0.32 | 0.15 | 53.1% |
| Dual-motor synchronization error | Large | Small | Significant |
| Trajectory tracking accuracy | Lower | Higher | Significant |
| Yaw rate peak deviation | Large | Small | Significant |
It can be observed that the composite angle tracking controller considerably improves the motor angle tracking accuracy and the synchronization performance. Moreover, the vehicle equipped with the proposed controller follows the desired double-lane-change trajectory with smaller lateral deviations, which indicates that the vehicle handling stability is effectively enhanced.
5.2 Steering on Low Adhesion Road
To evaluate the robustness of the proposed controller under harsh road conditions, a low-adhesion road case is simulated. The vehicle speed is reduced to 40 km/h, and the road adhesion coefficient is set to 0.3, representing a wet or icy road surface. The same double-lane-change maneuver is performed, and the simulation results are summarized in Table 9.
Table 9. Vehicle response comparison under low-adhesion road condition
| Performance index | ADRC | Composite SOSMC-ESO | Improvement |
|---|---|---|---|
| Maximum motor angle tracking error (rad) | 0.89 | 0.35 | 60.7% |
| Dual-motor synchronization error | Large | Small | Significant |
| Trajectory tracking accuracy | Lower | Higher | Significant |
| Yaw rate tracking deviation | Large | Small | Significant |
On the low-adhesion road, the maximum angle tracking error of the conventional ADRC reaches 0.89 rad, while the proposed composite controller restricts the error to only 0.35 rad. This corresponds to a reduction of 60.7%. The low-adhesion simulation confirms that the composite controller retains robust disturbance rejection capability and yields better stability performance even when the tire-road friction limit is significantly reduced.
5.3 Steering under Sudden External Disturbance
To simulate a sudden road excitation such as a pothole or a speed bump, a step steering angle command is applied while the vehicle is traveling at a constant speed of 60 km/h. At \(t=0.5\) s, an external load disturbance of 0.5 N·m is added to motor 1 and then removed at \(t=0.55\) s. This asymmetric disturbance acts as a severe challenge to the synchronization control performance. The proposed SOSMC-ESO-based deviation coupling controller is compared with a conventional sliding-mode-based ADRC scheme.
The key comparison indices are listed in Table 10.
Table 10. Synchronization performance under sudden disturbance
| Algorithm | Maximum synchronization speed oscillation (r/min) | Variance of synchronization error (r/min) |
|---|---|---|
| Conventional sliding mode ADRC | 0.21 | 0.142 |
| Proposed SOSMC-ESO | 0.05 | 0.030 |
| Improvement | 76.2% | 78.9% |
The maximum synchronization speed oscillation is reduced by 76.2%, and the variance of the synchronization error is reduced from 0.142 r/min to 0.030 r/min. In addition, the q-axis current response under the SOSMC-ESO controller exhibits a substantially smaller fluctuation amplitude compared with the conventional sliding mode ADRC. The improved synchronization performance is attributed to the fast power reaching term and the integral compensation mechanism embedded in the super-twisting second-order sliding mode structure. These mechanisms enable the dual-motor actuator system to withstand sudden torque conflicts and maintain a stable coordinated steering action.
The simulation results collectively indicate that the proposed composite control scheme is capable of achieving excellent motor angle tracking, synchronization, and vehicle-level stability under both nominal and degraded road conditions. This provides valuable references for the further engineering development of dual-motor SBW systems in electric vehicles.
6. Conclusion
This paper has presented a comprehensive study on the synchronization control method of a dual-motor steer-by-wire system for electric vehicles. The main contributions and conclusions can be summarized as follows.
First, a high-fidelity co-simulation platform was established by combining the mathematical models of PMSMs, the pinion-rack mechanism, and the 2-DOF vehicle dynamics with the CarSim vehicle dynamics software. The dual-motor SBW architecture with two independent PMSMs and a common rack was analyzed to highlight its advantages in redundancy and reliability for electric vehicles.
Second, a composite angle tracking controller combining the super-twisting SOSMC algorithm with an ESO was designed to replace the conventional three-loop position controller. The controller achieves fast and robust angle tracking while the ESO estimates and compensates the lumped disturbance caused by parameter perturbations and road excitation. Simulation results under square-wave and sinusoidal angle references demonstrate that the composite controller offers a faster dynamic response, smaller overshoot, and significantly better disturbance rejection compared with the conventional ADRC.
Third, to solve the torque conflict problem of the dual-motor system, an angle synchronization controller based on the super-twisting SOSMC and the mean deviation coupling structure was proposed. The synchronization error between the two motors is used as an additional feedback signal to dynamically compensate the output of each motor. The simulation results show that the proposed synchronization structure reduces the maximum synchronization error by about 48% compared with master-slave and cross-coupling structures. The super-twisting SOSMC achieves a 21% shorter adjustment time than ADRC and a 66.9% shorter adjustment time than conventional PI control.
Finally, vehicle-level simulations under high-adhesion, low-adhesion, and sudden-disturbance conditions were carried out to investigate the integrated performance of the dual-motor SBW system. The composite controller improves the motor angle tracking error by 53.1% on a high-adhesion road and by 60.7% on a low-adhesion road. In the case of sudden asymmetric load disturbance, the maximum synchronization speed oscillation is reduced by 76.2%. The trajectory tracking and yaw rate response are also significantly improved, indicating that the proposed scheme effectively maintains the handling stability of electric vehicles under complex driving conditions.
The limitations of this study include the relatively simplified vehicle model and the lack of experimental validation. Future work will focus on establishing a more detailed vehicle model with nonlinear tire characteristics, implementing hardware-in-the-loop tests, and investigating fault-tolerant control strategies under sensor failures and communication delays. Furthermore, the integration of the dual-motor SBW system with the electronic stability control and autonomous driving modules will be investigated to promote the practical application of SBW technology in next-generation electric vehicles.