1. Introduction
In response to the dual challenges of energy scarcity and environmental degradation, the global automotive industry has been undergoing a profound transformation toward electrification. Among the emerging technological paradigms, the distributed drive electric vehicle (DDEV) stands out as a particularly promising architecture that fundamentally redefines the relationship between the driver, the vehicle, and the road surface. Unlike conventional centralized powertrains that rely on mechanical differentials, drive shafts, and complex transmission systems, the distributed drive electric vehicle integrates propulsion motors directly into each wheel hub or wheel side, thereby eliminating most mechanical transmission components. This architectural innovation creates an over-actuated system in which each wheel can be individually and independently controlled, enabling unprecedented agility in torque generation and force allocation.
The quadruple independent actuation of a distributed drive electric vehicle provides a natural platform for advanced vehicle dynamics control. Through precise modulation of wheel torques, one can actively generate direct yaw moments to correct understeer or oversteer tendencies, enhance lateral stability under critical maneuvers, and extend the vehicle’s operational envelope on low-friction surfaces. Furthermore, the high bandwidth and fast torque response characteristic of electric motors allow for more sophisticated control strategies compared to conventional vehicles. However, this same attribute of actuator redundancy introduces a nontrivial vulnerability: the probability of component failure increases with the number of actuators. In a distributed drive electric vehicle, any fault in a drive motor, power electronics, or associated sensor can disrupt the carefully balanced torque distribution, potentially leading to degraded handling or even catastrophic loss of stability.
Given this context, the present research focuses on two intertwined aspects of distributed drive electric vehicle dynamics: (1) the design of a robust yaw stability control framework capable of exploiting the system’s actuation redundancy under normal operating conditions, and (2) the development of a drive-system fault-tolerant control methodology that ensures the distributed drive electric vehicle retains acceptable stability and safety when one or more drive motors lose functionality. The interplay between these two objectives constitutes the central theme of this thesis.
2. Vehicle Dynamics Modeling
2.1 Seven-Degree-of-Freedom Vehicle Model
To support the development of both stability controllers and fault diagnosis observers, a mathematical representation of the vehicle’s dynamic behavior is required. I constructed a seven-degree-of-freedom (7-DOF) vehicle model, encompassing the longitudinal, lateral, and yaw motions of the sprung mass, together with the rotational dynamics of the four wheels. A key simplifying assumption adopted throughout the modeling process is the neglect of suspension-induced roll and pitch motions, which is acceptable for the predominant focus on lateral dynamics and stability control.
The equations of motion in the vehicle body-fixed coordinate system are expressed as follows:
Longitudinal dynamics:
$$m(\dot{v}_x – v_y \gamma) = (F_{xfl} + F_{xfr})\cos\delta – (F_{yfl} + F_{yfr})\sin\delta + F_{xrl} + F_{xrr} \tag{1}$$
Lateral dynamics:
$$m(\dot{v}_y + v_x \gamma) = (F_{xfl} + F_{xfr})\sin\delta + (F_{yfl} + F_{yfr})\cos\delta + F_{yrl} + F_{yrr} \tag{2}$$
Yaw dynamics:
$$I_z \dot{\gamma} = l_f[(F_{xfl} + F_{xfr})\sin\delta + (F_{yfl} + F_{yfr})\cos\delta] – l_r(F_{yrl} + F_{yrr}) + \frac{d}{2}[(F_{xfr} – F_{xfl})\cos\delta + (F_{yfl} – F_{yfr})\sin\delta + F_{xrr} – F_{xrl}] \tag{3}$$
where \( m \) is the total vehicle mass; \( v_x \) and \( v_y \) represent the longitudinal and lateral velocities at the center of gravity; \( \gamma \) denotes the yaw rate; \( \delta \) is the front-wheel steering angle; \( l_f \) and \( l_r \) are the distances from the center of gravity to the front and rear axles, respectively; \( d \) denotes the track width; and \( I_z \) is the yaw moment of inertia. The indices \( fl \), \( fr \), \( rl \), and \( rr \) refer to front-left, front-right, rear-left, and rear-right wheels, respectively. The forces \( F_{xi} \) and \( F_{yi} \) are the longitudinal and lateral tire forces at wheel \( i \).
2.2 Wheel Rotational Dynamics and Vertical Load Distribution
The rotational behavior of each wheel is governed by the equilibrium between the driving torque, the longitudinal tire force, and the rolling resistance torque:
$$I_w \dot{\omega}_i = T_i – RF_{xi} – RF_{zi} f_r, \quad i \in \{fl, fr, rl, rr\} \tag{4}$$
where \( I_w \) is the wheel moment of inertia, \( \omega_i \) is the angular velocity, \( T_i \) is the actual motor torque, \( R \) is the effective tire radius, and \( f_r \) is the rolling resistance coefficient. The vertical loads on each tire, which are affected by longitudinal and lateral load transfer, are given by:
$$F_{zfl} = mg\frac{l_r}{2L} – ma_x\frac{h_g}{2L} – ma_y\frac{l_r h_g}{L d} \tag{5}$$
$$F_{zfr} = mg\frac{l_r}{2L} – ma_x\frac{h_g}{2L} + ma_y\frac{l_r h_g}{L d} \tag{6}$$
$$F_{zrl} = mg\frac{l_f}{2L} + ma_x\frac{h_g}{2L} – ma_y\frac{l_f h_g}{L d} \tag{7}$$
$$F_{zrr} = mg\frac{l_f}{2L} + ma_x\frac{h_g}{2L} + ma_y\frac{l_f h_g}{L d} \tag{8}$$
with \( a_x \) and \( a_y \) being the measured longitudinal and lateral accelerations, \( h_g \) the center-of-gravity height, and \( L \) the wheelbase.
2.3 Nonlinear Tire Model
Owing to the central role that tire forces play in determining vehicle handling limits, I adopted the semi-empirical Magic Formula tire model to represent the nonlinear coupling between longitudinal slip, side-slip angle, and the resulting tire forces. The fundamental equation of the Magic Formula takes the form:
$$Y(x) = D \sin\{C \arctan[Bx – E(Bx – \arctan(Bx))]\} + S_v \tag{9}$$
where \( x \) represents either the tire slip ratio or side-slip angle, \( D \) is the peak factor, \( C \) is the shape factor, \( B \) is the stiffness factor, \( E \) is the curvature factor, and \( S_v \) is the vertical offset. The longitudinal force \( F_x \) is computed with extended parameter sets that depend on the vertical load, while the lateral force \( F_y \) is similarly derived from the slip angle and vertical load separately. Tables 1 and 2 list the fitting parameters used in this study.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| a0 | 1.65 | a6 | -0.021 |
| a1 | -34 | a7 | 0.7739 |
| a2 | 1250 | a8 | 0.0023 |
| a3 | 3036 | a9 | 0.013 |
| a4 | 12.8 | a10 | 0.0037 |
| a5 | 0.005 |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| b0 | 2.373 | b6 | 0.00402 |
| b1 | -9.46 | b7 | -0.0615 |
| b2 | 1490 | b8 | 1.2 |
| b3 | 130 | b9 | 0.03 |
| b4 | 276 | b10 | -0.176 |
| b5 | 0.089 |
The slip ratio and side-slip angle for each wheel are obtained from the kinematic relationships between the wheel hub velocities, steering angle, and vehicle motion states. The wheel hub velocity is first calculated based on the vehicle longitudinal/lateral velocity at the center of gravity and the yaw rate contribution: \( v_{wi} \), with contributions from the steering angle for front wheels. The slip angle for front and rear wheels is expressed as:
$$\alpha_{fl} = \delta – \arctan\left( \frac{v_y + l_f \gamma}{v_x – \frac{d}{2}\gamma} \right), \quad \alpha_{fr} = \delta – \arctan\left( \frac{v_y + l_f \gamma}{v_x + \frac{d}{2}\gamma} \right) \tag{10}$$
$$\alpha_{rl} = \arctan\left( \frac{l_r \gamma – v_y}{v_x – \frac{d}{2}\gamma} \right), \quad \alpha_{rr} = \arctan\left( \frac{l_r \gamma – v_y}{v_x + \frac{d}{2}\gamma} \right) \tag{11}$$
For the longitudinal slip ratio, the definition is:
$$s_i = \frac{\omega_i R – v_{wi}}{v_{wi}} \tag{12}$$
2.4 Drive Motor Model
Since the thesis focuses on vehicle-level control rather than internal motor design, the wheel hub motor is modeled by its steady-state torque output relationship and a second-order lag characteristic representing the motor’s finite bandwidth. The transfer function adopted in the simulation model is:
$$T_i = \frac{1}{\tau^2 s^2 + 2\tau s + 1} T_{d,i} \tag{13}$$
where \( T_{d,i} \) is the demanded torque, and \( \tau \) is the time constant.
2.5 Model Validation
To assess the fidelity of the constructed model, I conducted a comparison between the 7-DOF model and a high-fidelity CarSim vehicle model. A step-steering input of 0.5 rad was applied at the front wheels while the vehicle traveled at a constant speed of 80 km/h. The comparative responses of yaw rate and side-slip angle are summarized in Table 3.
| Response variable | 7-DOF model | CarSim | Relative error (%) |
|---|---|---|---|
| Yaw rate (rad/s) | 0.242 | 0.251 | 3.6 |
| Side-slip angle (rad) | -0.025 | -0.028 | 10.7 |
The small discrepancies are primarily attributable to the neglect of suspension kinematics and body roll in the simplified model. Nevertheless, the overall agreement substantiates the practical utility of the 7-DOF model for the subsequent design of stability control and fault diagnosis algorithms.
3. Yaw Stability Control Strategy for Distributed Drive Electric Vehicles
3.1 Stability Analysis and Control Parameter Selection
For a distributed drive electric vehicle, the evaluation of yaw stability involves simultaneous monitoring of two critical vehicle states: the yaw rate \( \gamma \) and the side-slip angle \( \beta \). These quantities uniquely characterize the vehicle’s rotational and translational lateral behavior. I made use of the \( \beta – \dot{\beta} \) phase-plane method to identify the boundary between stable and unstable vehicle motion. The stable region can be enclosed by two lines expressed as:
$$B_1 \beta + \dot{\beta} \leq B_2 \tag{14}$$
where the boundary coefficients \( B_1 \) and \( B_2 \) are functions of the road friction coefficient. Table 4 summarizes the values of these coefficients for different friction levels.
| Friction coefficient μ | B1 | B2 |
|---|---|---|
| 0.1 | 1.48 | 0.12 |
| 0.2 | 2.62 | 0.19 |
| 0.3 | 3.28 | 0.34 |
| 0.4 | 4.22 | 0.44 |
| 0.5 | 4.68 | 0.52 |
| 0.6 | 5.42 | 0.60 |
| 0.7 | 6.24 | 0.66 |
| 0.8 | 7.28 | 0.70 |
| 0.9 | 7.98 | 0.76 |
| 1.0 | 8.46 | 0.81 |
When the vehicle state lies within the stable region, the controller tracks the reference yaw rate primarily; when the state violates the stability boundary, both \( \gamma \) and \( \beta \) are jointly regulated.
3.2 Reference State Generation
The desired yaw rate and side-slip angle are derived from a linear two-degree-of-freedom bicycle model under steady-state conditions. The dynamic equations of this model are:
$$m v_x (\dot{\beta} + \gamma) = F_{yf} + F_{yr} \tag{15}$$
$$I_z \dot{\gamma} = l_f F_{yf} – l_r F_{yr} \tag{16}$$
Under linear tire assumptions, with front and rear axle cornering stiffnesses \( k_f \) and \( k_r \), the steady-state reference yaw rate and side-slip angle are:
$$\gamma_d = \frac{v_x / L}{1 + K v_x^2} \delta \tag{17}$$
$$\beta_d = \frac{l_r – \dfrac{m l_f}{2 L k_r} v_x^2}{L(1 + K v_x^2)} \delta \tag{18}$$
where \( K \) is the stability factor:
$$K = \frac{m}{L^2}\left( \frac{l_f}{k_r} – \frac{l_r}{k_f} \right) \tag{19}$$
To respect the physical limit imposed by the tire–road friction coefficient \( \mu \), the reference yaw rate is saturated as follows:
$$|\gamma_d| \leq \left| \frac{0.85 \mu g}{v_x} \right| \tag{20}$$
3.3 Upper-Level Controller: Fuzzy Sliding Mode Control
The purpose of the upper-level controller is to determine the corrective yaw moment \( \Delta M_z \) required to track the desired vehicle states. I formulated a combined sliding surface joining the yaw rate error and the side-slip angle error:
$$s = (\gamma – \gamma_d) + w(\beta – \beta_d) \tag{21}$$
with \( w \) being a weighting factor. Differentiating the sliding surface and setting \( \dot{s}=0 \) yields the equivalent control law. Considering the vehicle yaw dynamics described in Equation (3), the resulting equivalent yaw moment command is:
$$u_{eq} = (F_{yrl} + F_{yrr})l_r – (F_{yfl} + F_{yfr})l_f + I_z(\dot{\gamma}_d – w(\dot{\beta} – \dot{\beta}_d)) \tag{22}$$
To ensure reachability of the sliding surface in the presence of parameter uncertainties and external disturbances, a discontinuous switching term is added:
$$u_s = \xi \cdot \text{sat}(s) \tag{23}$$
where \( \xi \) is the switching gain. The total control output is thus:
$$M_z = u_{eq} – I_z \xi \,\text{sat}(s) \tag{24}$$
To alleviate the inherent chattering phenomenon associated with conventional first-order sliding mode control, I applied a fuzzy logic module whose inputs are the sliding variable \( s \) and its derivative \( \dot{s} \). The output of the fuzzy inference system is the switching gain \( \xi_b \). A set of seven triangular membership functions, labeled from NB to PB, is defined for the input and output domains. The fuzzy rules are designed following the logic that larger deviations from the sliding surface require stronger corrective action, while near the surface the control effort is deliberately reduced to prevent oscillation. The firing strength of the rules is aggregated using the centroid defuzzification method.
The parameters of the membership functions used in the fuzzy inference system are summarized in Table 5.
| s \ ṡ | NB | NM | NS | ZO | PS | PM | PB |
|---|---|---|---|---|---|---|---|
| NB | NB | NB | NM | NM | NM | NB | NB |
| NM | NM | NM | NS | NS | NS | NM | NM |
| NS | NS | NS | ZO | ZO | NS | NS | NS |
| ZO | ZO | ZO | ZO | ZO | ZO | NS | NS |
| PS | PS | ZO | ZO | ZO | PS | ZO | ZO |
| PM | PM | PS | PS | PS | PS | PS | PS |
| PB | PB | PM | PM | PM | PM | PM | PM |
The final sliding-mode yaw moment control law, with the fuzzy-tuned gain, is expressed as:
$$M_z = (F_{yrl}+F_{yrr})l_r – (F_{yfl}+F_{yfr})l_f + I_z\left[ \dot{\gamma}_d – w(\dot{\beta} – \dot{\beta}_d) – \xi_b\,\text{sat}(s) \right] \tag{25}$$
3.4 Longitudinal Force Control Based on PID
To maintain the driver’s demanded longitudinal motion while performing lateral stabilization, a PID controller is employed to compute the total longitudinal force \( F_x \). Given the error \( e_v = v_{des} – v_{x} \), the control output is:
$$F_x = K_p e_v + K_i \int_0^t e_v(\tau) d\tau + K_d \frac{d e_v}{dt} \tag{26}$$
The gains \( K_p \), \( K_i \), and \( K_d \) were tuned to achieve rapid convergence to the requested vehicle speed without excessive overshoot.
3.5 Lower-Level Torque Distribution
The lower-level controller receives the demanded corrective yaw moment \( \Delta M_z \) and the total longitudinal force \( F_x \), and must decide how these demands are to be distributed among the four wheel torques. The optimization formulated in this chapter aims at minimizing the maximum tire load ratio, conceptually equivalent to minimizing the sum of squared tire workload ratios. Since the lateral tire forces are difficult to on-line quantify with high accuracy, the problem is approximated by considering only the longitudinal contribution to the tire load:
$$J = \min \sum_{i=1}^{4} \frac{F_{xi}^2}{(\mu F_{zi})^2} = \min \sum_{i=1}^{4} \frac{T_i^2}{(\mu R F_{zi})^2} \tag{27}$$
The distribution problem has two equality constraints arising from the driver demand and the stability moment:
$$F_x = \frac{\cos\delta}{R}(T_{fl} + T_{fr}) + \frac{1}{R}(T_{rl} + T_{rr}) \tag{28}$$
$$\Delta M_z = \frac{d}{2R}[(T_{fr} – T_{fl})\cos\delta + (T_{rr} – T_{rl})] \tag{29}$$
In addition, the torque vector must lie within the permissible region formed by the motor capacity bound and the road-friction bound. The torque capacity constraint is:
$$-T_{\max,i} \leq T_i \leq T_{\max,i} \tag{30}$$
while the friction constraint is expressed as:
$$- \mu F_{zi} R \leq T_i \leq \mu F_{zi} R \tag{31}$$
By writing the optimization problem in standard quadratic programming (QP) form:
$$\min_x \frac{1}{2} x^T H x + f^T x, \quad \text{s.t.} \; A_{eq}x = B_{eq},\; LB \leq x \leq UB \tag{32}$$
where the Hessian matrix reads
$$H = \begin{bmatrix} \frac{2}{(\mu R F_{zfl})^2} & 0 & 0 & 0 \\ 0 & \frac{2}{(\mu R F_{zfr})^2} & 0 & 0 \\ 0 & 0 & \frac{2}{(\mu R F_{zrl})^2} & 0 \\ 0 & 0 & 0 & \frac{2}{(\mu R F_{zrr})^2} \end{bmatrix} \tag{33}$$
and the equality constraints are given by the matrix \( A_{eq} = \begin{bmatrix} \cos\delta / R & \cos\delta / R & 1/R & 1/R \\ d\cos\delta / 2R & -d\cos\delta / 2R & d/2R & -d/2R \end{bmatrix} \), with \( B_{eq} = [F_x, \Delta M_z]^T \). The \( LB \) and \( UB \) vectors are constructed from the element-wise minima/maxima of the torque saturation and the friction bound. The QP problem is solved at each time step using the built-in active-set solver in MATLAB.
3.6 Simulation Validation of Yaw Stability Control
To assess the effectiveness of the proposed yaw stability controller, I conducted simulation experiments under two representative maneuvers: a slalom maneuver and a double lane-change maneuver. Both were executed by comparing three configurations: the uncontrolled vehicle, the vehicle with conventional sliding mode control (SMC), and the vehicle with the proposed fuzzy sliding mode control (FSMC). The key numerical results for the slalom maneuver are tabulated in Table 6.
| Performance index | Uncontrolled | SMC | FSMC |
|---|---|---|---|
| Max yaw-rate error (rad/s) | 0.043 | 0.015 | 0.008 |
| Yaw-rate error reduction (%) | — | 65 | 81 |
| Max side-slip-angle error (rad) | 0.015 | 0.010 | 0.009 |
| Side-slip-angle error reduction (%) | — | 33 | 40 |
| Max lateral acceleration (m/s²) | 4.378 | 3.960 | 3.781 |
| Lateral acceleration reduction (%) | — | 10 | 14 |
A second set of experiments was performed under a double lane-change maneuver with a lower road friction coefficient, the outcomes of which are summarized in Table 7.
| Performance index | Uncontrolled | SMC | FSMC |
|---|---|---|---|
| Max yaw-rate error (rad/s) | 0.145 | 0.066 | 0.027 |
| Yaw-rate error reduction (%) | — | 54 | 81 |
| Max side-slip-angle error (rad) | 0.055 | 0.035 | 0.028 |
| Side-slip-angle error reduction (%) | — | 36 | 49 |
| Max lateral acceleration (m/s²) | 3.411 | 2.982 | 2.901 |
| Lateral acceleration reduction (%) | — | 13 | 15 |
The simulation results demonstrate that both SMC and FSMC effectively track the reference yaw rate and side-slip angle, significantly improving the handling stability of the distributed drive electric vehicle. The FSMC consistently outperforms the conventional SMC in terms of peak error suppression and lateral-acceleration reduction, confirming the benefit of fuzzy gain adaptation in mitigating chattering and improving transient accuracy.
4. Drive-System Fault Diagnosis and Failure Control
4.1 Failure Definition and Mode Analysis
In a distributed drive electric vehicle, a drive motor can fail either partially or completely. For a rigorous treatment of actuator faults in the torque-allocation framework, I introduced a motor failure factor \( \eta_i \) such that the actual torque generated by motor \( i \) is expressed as:
$$T_i = \eta_i T_{d,i} \tag{34}$$
where \( T_{d,i} \) is the torque commanded by the vehicle controller. The value of \( \eta_i \) lies in the interval \([0, 1]\), with \( \eta_i = 1 \) denoting a fully healthy motor and \( \eta_i = 0 \) a completely failed one. Partial degradation corresponds to intermediate values.
The plausible failure modes of a four-wheel distributed drive system are: single-motor failure, same-side dual-motor failure, different-side same-axle dual-motor failure, different-side different-axle dual-motor failure, three-motor failure, and four-motor failure. Among these, three-motor and four-motor failures are catastrophic events that completely eliminate the actuator redundancy needed to maintain stable driving; those scenarios demand a safety stop rather than a torque-reallocation solution. Accordingly, this thesis restricts the fault-tolerant control investigation to single-motor and dual-motor failures.
4.2 Design of Failure-Factor Observer Based on Cubature Kalman Filter
The exact value of the failure factor is not directly measurable. To estimate it in real time, I leveraged the interconnected dynamical relationships among wheel rotational acceleration, driving torque, and vehicle states. For each wheel, the rotational dynamics in Equation (4) are augmented with the failure-factor model to give the following discrete-time state-space formulation:
$$x(k+1) = f(x(k)) + T_s B u(k) + v(k) \tag{35}$$
$$y(k) = g(x(k)) + w(k) \tag{36}$$
Following the random-walk approach, the failure factors are extended as augmented states appended to the original vehicle state vector:
$$x_{\text{aug}} = [v_x, v_y, \gamma, \omega_{fl}, \omega_{fr}, \omega_{rl}, \omega_{rr}, \eta_{fl}, \eta_{fr}, \eta_{rl}, \eta_{rr}]^T \tag{37}$$
The measurement vector comprises the longitudinal acceleration \( a_x \), lateral acceleration \( a_y \), yaw rate \( \gamma \), and the four wheel rotational speeds. The nonlinear observation map \( g(\cdot) \) combines the kinematic relationships from the vehicle model with the wheel dynamics, where the motor torque commands serve as known inputs.
For state estimation of this highly nonlinear system, I adopted the cubature Kalman filter (CKF), whose key advantage lies in its numerically robust sampling rule. The CKF algorithm performs time update and measurement update steps as follows. First, the prior covariance is factorized via Cholesky decomposition \( P_{k-1|k-1} = S_{k-1|k-1} S_{k-1|k-1}^T \). Then, a set of \( 2n \) cubature points are generated from the state estimate:
$$\chi_{i,k-1|k-1} = S_{k-1|k-1}\xi_i + \hat{x}_{k-1|k-1} \tag{38}$$
where the cubature points and weights are given by \( \xi_i = \sqrt{n}[1]_i \) and \( w_i = 1/(2n) \) respectively. After propagating the cubature points through the state transition function, the predicted state and covariance are:
$$\hat{x}_{k|k-1} = \frac{1}{2n}\sum_{i=1}^{2n} f(\chi_{i,k-1|k-1}) \tag{39}$$
$$P_{k|k-1} = \frac{1}{2n}\sum_{i=1}^{2n} (f(\chi_{i,k-1|k-1}) – \hat{x}_{k|k-1})(f(\chi_{i,k-1|k-1}) – \hat{x}_{k|k-1})^T + Q \tag{40}$$
Similarly, the measurement prediction and Kalman gain are computed by propagating the cubature points through the observation function. The detailed steps follow the standard CKF recursion based on the spherical-radial cubature rule. The algorithm produces online estimates of the failure factors \( \hat{\eta}_i \), serving as inputs to the fault-tolerant control layer.
4.3 Fault Reconstruction Torque Allocation
When a motor fails, the original fixed equality constraints of the lower-level torque distribution problem may become infeasible due to the loss of one or more actuators. Therefore, I implemented a reconstruction-based fault-tolerant allocation strategy. The estimated failure factors \( \hat{\eta}_i \) are integrated into the equality constraint by modifying the effectiveness matrix. The reconstructed equality constraint is:
$$\begin{bmatrix} \frac{\eta_{fl}\cos\delta}{R} & \frac{\eta_{fr}\cos\delta}{R} & \frac{\eta_{rl}}{R} & \frac{\eta_{rr}}{R} \\[6pt] \frac{d\eta_{fl}\cos\delta}{2R} & -\frac{d\eta_{fr}\cos\delta}{2R} & \frac{d\eta_{rl}}{2R} & -\frac{d\eta_{rr}}{2R} \end{bmatrix} x = \begin{bmatrix} F_x \\ \Delta M_z \end{bmatrix} \tag{41}$$
Then, to prevent the optimizer from failing when exact satisfaction of the constraints is impossible, the equality constraints are incorporated as a weighted penalty term into the objective function. The reconstructed QP formulation is:
$$J_r = \min_x \left[ \sum_{i=1}^{4} \frac{\eta_i T_i^2}{(\mu R F_{zi})^2} + c \| A’_{eq}x – B_{eq} \|^2 \right] \tag{42}$$
where the entrywise motor torque bounds and road-friction limits become:
$$-\eta_i T_{\max,i} \leq T_i \leq \eta_i T_{\max,i} \tag{43}$$
$$-\mu F_{zi} R \leq T_i \leq \mu F_{zi} R \tag{44}$$
The weighting coefficient \( c \) is chosen high enough to prioritize demand fulfillment over tire-load optimization. Since both the quadratic cost and the penalty term are convex, the reconstructed problem remains solvable by standard QP solvers.
4.4 Verification of Failure-Diagnosis Observer
Before validating the complete fault-tolerant control system, I assessed the accuracy and responsiveness of the CKF-based fault diagnosis observer in isolation. Two simulation scenarios were conducted at a constant vehicle speed of 80 km/h on a straight road with \(\mu=0.8\). In the single-failure case, the left-front motor was forced to a complete failure at \( t=3 \) s. In the dual-failure case, the left-front motor failed at \( t=4 \) s and the right-rear motor failed at \( t=6 \) s. The observation results are summarized in Table 8.
| Scenario | Fault location | Injection time (s) | Detection delay (s) | Estimated failure factor |
|---|---|---|---|---|
| Single motor fault | FL | 3.0 | 0.02 | ≈ 0 |
| Dual motor fault | FL, RR | 4.0 / 6.0 | 0.02 / 0.02 | ≈ 0 |
In both scenarios, the remaining healthy motors maintained failure-factor estimates near unity, with only minor transient deviations produced by the mechanical coupling at the instant of fault injection.
4.5 Simulation Validation of Fault-Tolerant Control
Having verified the fault detection capability, I proceeded to evaluate the combined system in both a straight-line driving condition and a sinusoidal steering maneuver. In each scenario, I compared the uncontrolled, the optimized-allocation (without fault reconstruction), and the fault-tolerant (with reconstruction) configurations.
4.5.1 Single-Motor Failure
For the straight-line case, the distributed drive electric vehicle was accelerated to 80 km/h with a constant zero steering angle and \(\mu=0.8\); the left-front motor was commanded to fail at \( t=3 \) s. The uncontrolled vehicle promptly deviated from its intended straight-line path. The optimized-allocation scheme applying the nominal torque distributor without reconfiguration provided only partial compensation. In contrast, the proposed fault-tolerant control strategy immediately redistributed torque among the three healthy motors, compensating for the lost longitudinal driving force and the induced disturbing yaw moment. The lateral deviation of the vehicle trajectory was therefore effectively suppressed, with the most significant quantitative results shown in Table 9.
| Configuration | Max yaw-rate error (rad/s) | Max side-slip angle error (rad) | Lateral deviation (m) |
|---|---|---|---|
| Uncontrolled | 0.0037 | 0.0022 | 2.48 |
| Optimized allocation | 0.0021 | 0.0011 | 1.42 |
| Fault-tolerant control | 0.0006 | 0.0004 | 0.38 |
A more demanding sinusoidal steering maneuver with a steering amplitude of 0.5 rad, a period of 4 s, and a vehicle speed of 60 km/h was also examined. The left-front motor was failed at \( t=2 \) s. The concluding indices at the moment when the longitudinal displacement reached 160 m are summarized in Table 10.
| Configuration | Max yaw-rate error (rad/s) | Max side-slip angle error (rad) | Lateral deviation (m) |
|---|---|---|---|
| Uncontrolled | 0.085 | 0.021 | 2.36 |
| Optimized allocation | 0.048 | 0.016 | 1.61 |
| Fault-tolerant control | 0.025 | 0.011 | 0.42 |
The fault-tolerant torque re-allocation significantly reduced yaw-rate error, side-slip-angle error, and trajectory deviation. Specifically, with the fault-tolerant control, the lateral deviation was limited to 0.42 m at 160 m of longitudinal travel, whereas the uncontrolled vehicle had already drifted 2.36 m at the same point.
4.5.2 Dual-Motor Failure
The dual-motor-failure conditions were studied under both operating scenarios. In the straight-line driving test, the simulation time was 12 s, the vehicle speed was fixed at 80 km/h, and \(\mu=0.8\). The left-front motor failed at \( t=4 \) s, followed by the right-front motor at \( t=6 \) s. This particular failure combination produced a certain cancellation of the disturbance yaw moment, but the overall driving force was still significantly reduced. The tolerance results are reported in Table 11.
| Configuration | Max yaw-rate error (rad/s) | Lateral deviation at 250 m (m) |
|---|---|---|
| Uncontrolled | 0.0046 | 1.51 |
| Optimized allocation | 0.0032 | 0.98 |
| Fault-tolerant control | 0.0008 | 0.30 |
In the sinusoidal steering scenario, the distributed drive electric vehicle traveled at 60 km/h with the same steering input as before. At \( t=2 \) s, both the left-front and the right-rear motors were set to a complete failure. The worst-case deviations are summarized in Table 12.
| Configuration | Max yaw-rate error (rad/s) | Max side-slip angle error (rad) | Lateral deviation at 160 m (m) |
|---|---|---|---|
| Uncontrolled | 0.076 | 0.019 | 3.65 |
| Optimized allocation | 0.045 | 0.014 | 2.44 |
| Fault-tolerant control | 0.020 | 0.009 | 0.45 |
The fault-tolerant strategy succeeded in maintaining the distributed drive electric vehicle close to the intended path in all dual-motor-failure scenarios. The observer rapidly identified the two faulty motors and the torque-allocation optimiser promptly redistributed the remaining capacity to compensate for the lost driving forces. Across all the evaluated cases, the reconstructed allocation achieved substantial improvements in yaw-rate tracking, side-slip regulation, and path-following accuracy.
5. Conclusion
In this thesis, I systematically examined both the yaw stability control problem and the drive-system fault-tolerant control problem for distributed drive electric vehicles. The principal contributions and findings can be stated as follows.
First, a comprehensive seven-degree-of-freedom vehicle model was constructed and experimentally validated against a high-fidelity CarSim model under step-steering input. The close agreement in yaw rate and side-slip angle confirmed the validity of the vehicle model for controller design and observer development. The model was subsequently used as the theoretical foundation for all control and diagnosis strategies.
Second, a hierarchical yaw stability control architecture was formulated. The upper layer, based on a fuzzy sliding mode controller, generates a corrective yaw moment from a combined sliding surface that fuses the yaw-rate error and side-slip-angle error. An adaptive fuzzy inference strategy dynamically adjusts the switching gain, which both preserves the robustness of the sliding-mode approach and suppresses the high-frequency chattering phenomenon. The lower layer solves a constrained convex quadratic programming problem to distribute the total longitudinal force and corrective yaw moment among the four wheels while minimizing the total tire workload. Dual-maneuver simulations confirmed that the proposed fuzzy sliding-mode controller achieves significant reductions in yaw-rate and side-slip-angle errors when compared with the uncontrolled case and conventional sliding-mode control.
Third, a fault diagnosis module and a fault-tolerant torque reconstruction strategy were developed. The diagnosis module is based on a cubature Kalman filter that incorporates motor failure factors as augmented states through a random-walk model. The observer reliably and quickly estimates the failure factor of each drive motor, thereby giving precise information about the fault location and severity. The fault-tolerant control strategy redesigns both the constraint matrix and the objective function of the torque allocation problem, making the optimization feasible even after the loss of one or two motors. Extensive simulations for single-motor and dual-motor failure cases, under both straight-line driving and sinusoidal maneuvers, proved that the resulting control architecture maintains the distributed drive electric vehicle stable and close to its intended trajectory even under adverse faulty conditions.
Future research could extend the current framework by incorporating suspension dynamics and load variation due to road roughness, by integrating the steering system and the braking system as additional fault-tolerant actuators, and by verifying the control algorithms experimentally on a prototype distributed drive electric vehicle.