Static Output-Feedback Path-Tracking Controller Tolerant to Steering Actuator Faults for Distributed Driven Electric Vehicles

Abstract

The steering system plays a critical role in the vehicle’s handling and directly influences its lateral dynamics. Faults or abnormal behavior in this system can affect performance, cause vehicle instability, and even lead to accidents. Therefore, considering these potential events is essential for designing robust controllers for autonomous vehicles. For this reason, in this work, a fault-tolerant path-tracking Static Output-Feedback controller is designed to handle steering actuator faults in autonomous vehicle steering systems. The controller adopts a Linear Parameter Varying approach to effectively handle nonlinearities associated with varying vehicle speeds and tire behavior. Furthermore, it only uses information from sensors, avoiding estimation stages. This controller can operate in two modes: a no-fault mode where only the steering is controlled to follow the reference path and a fault mode where the controller manages both the steering and torque vectoring. In fault mode, torque vectoring compensates for faults in the steering actuator. The design of the controller is completed considering gain faults in the steering system. The simulation results show that the proposed controller successfully maintains vehicle stability and significantly reduces tracking errors during high-risk maneuvers, achieving reductions of up to 50.65% in lateral error and 47.26% in heading error under worst-case fault scenarios.

1. Introduction

Automated vehicles (AVs) have emerged as a transformative technology with the potential to revolutionize transport systems worldwide. These vehicles promise numerous benefits, including improved safety, increased mobility, and greater efficiency in transport networks. The convergence of AV technology with electric vehicles (EVs) and the rise of distributed energy systems have further accelerated this transition. Electric powertrains offer benefits such as lower emissions, reduced noise, and simpler mechanical designs that support more sustainable transport solutions [1,2]. EVs include several types, such as Battery Electric Vehicles (BEVs), Plug-in Hybrid Electric Vehicles (PHEVs), and Fuel Cell Electric Vehicles (FCEVs), each with unique characteristics and control requirements [3].

One of the critical tasks in automated driving is path-tracking, which involves guiding the vehicle along a predetermined path while maintaining stability and passenger comfort. Some of the most widely used methods include simple geometry-based techniques such as the pure pursuit controller and the Stanley controller [4], as well as classical control methods such as Proportional–Integral–Derivative (PID) control [4,5]. These controllers are popular for their ease of implementation. Although effective at lower speeds, these methods are unsuitable for high-speed applications where the nonlinear dynamic behavior of the vehicle can lead to tracking errors or instability. Consequently, model-based approaches that better capture vehicle dynamics are preferred in these cases [6]. More modern approaches such as Linear Quadratic Regulator (LQR) control have been introduced to improve path-following control and stability by minimizing a cost function that balances tracking accuracy and control efforts [7,8]. More recently, Model Predictive Control (MPC) has emerged as a powerful control technique for path-tracking. MPC uses predictive models to optimize future control actions over a receding horizon, providing enhanced path-tracking performance, particularly in challenging driving scenarios [9,10,11]. However, this method is computationally demanding and requires knowledge of all vehicle states, which are often difficult to measure directly or require additional state observers [12]. Ensuring the stability of the system is crucial, and Lyapunov-based stability methods are commonly used to ensure the stability of the system [13,14].

To further increase robustness to disturbances and model uncertainties, robust control techniques have been applied regarding the path-tracking problem. In [15,16,17], Sliding Mode Control (SMC) is used to ensure stability in the context of parameter variations and external disturbances in the design of path-tracking controllers. Another commonly used approach, ${\mathcal{H}}_{\infty }$ control, provides robustness by minimizing the worst-case gain between input disturbances and system outputs, effectively enhancing the system’s ability to handle uncertainties [18,19,20]. Robust Model Predictive Control (RMPC) combines the predictive capabilities of MPC with a robust design, ensuring that the controller remains effective even in the presence of significant model mismatches or disturbances [21,22,23].

Vehicle components are susceptible to faults that degrade performance and compromise stability, potentially leading to unsafe driving situations. Such effects can be mitigated by the early detection of these anomalies using model-based [24] and data-driven techniques [25,26]. Beyond detection, fault-tolerant control (FTC) incorporates fault handling into the controller design, enabling the system to adjust inputs and provide a robust dynamic response even when faults occur [27,28].

In the literature, path-tracking fault-tolerant control has been extensively studied to improve the robustness of automated vehicles against actuator faults. FTC control has been used to deal with faults in both steering actuators and wheel motors [29,30]. In [31], a second-order Kalman filter is used to estimate the steering actuator fault so that this estimation can then be used in a fault-tolerant path-tracking Sliding Mode Controller (SMC). In [32], an LQR controller is designed to deal with faults in the steering system, working in conjunction with an SMC to achieve the desired yaw rate, while [33] uses a fault-tolerant controller based on SMC, incorporating fault diagnosis based on the use of neural networks. The work presented in [34] compensates for stuck steering faults by using a differential steering approach for compensation. In [35], an interval observer is presented to recalculate the control input and mitigate the effects of steering actuator faults, similar to the work in [36], where a robust SMC is used in combination with an MPC.

Many of the aforementioned approaches rely on information that is typically not available in commercial vehicles or require additional stages to estimate faults or vehicle states, making real-world implementation difficult. The purpose of this work is to overcome these challenges and enhance the practicality of controller designs for commercial vehicles by introducing a flexible Static Output-Feedback (SOF) controller that adapts to both fault-free and fault conditions using only information available from on-board sensors. The proposed controller operates in two modes: a fault-free mode that controls the vehicle using only the front steering angle and a fault-tolerant mode that uses torque vectoring (TV) to compensate for actuator faults. This dual-mode operation ensures that vehicle stability and path-tracking performance are maintained even under fault conditions, thereby minimizing tracking degradation, along with steering control. The main contributions of this work are as follows:

• A Multi-Input–Multi-Output (MIMO) SOF controller is developed for a path-tracking system, enabling operation in both fault-tolerant and fault-free modes. This novel approach enables the system to operate in a dual mode: under normal conditions, it relies only on the steering input actuator, and, in the event of a steering system fault, torque vectoring is activated to assist the vehicle and effectively compensate for the steering actuator faults.
• Vehicle uncertainties and nonlinearities caused by varying vehicle speeds are considered in the controller design using an LPV approach. This ensures that the system remains robust and stable under a wide range of operating conditions.
• The driving automation system presented in this work only requires on-board sensors that can be found in series-production vehicles to compute the control inputs. By eliminating the need for expensive sensors for the estimation of vehicle states such as sideslip angle, the practical implementation of the method is guaranteed.

The rest of the article is organized as follows: in Section 2, the problem formulation is introduced, along with the formulation of the vehicle model considered for the controller design. Section 3 presents the controller design methodology. Section 4 discusses the results obtained via simulation. Section 5 concludes the paper.

2. Problem Formulation

2.1. Vehicle Model

The lateral dynamic behavior of the vehicle can be expressed as [37]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill m({\dot{v}}_y+r{v}_x)& =(F_{x,fl}+F_{x,fr})sin\left(\delta \right)+(F_{y,fl}+F_{y,fr})cos\left(\delta \right)+F_{y,r}-F_{y,rl}+F_{y,rr}\hfill \\ \hfill I_z\dot{r}& =l_f(F_{y,fl}+F_{y,fr})cos\left(\delta \right)-l_r(F_{y,rl}-F_{y,rr})+{\displaystyle \frac{t_w}{2}}(F_{y,fl}-F_{y,fr})sin\left(\delta \right)+\mathrm{\Delta }{M}_z\hfill \end{array}\end{array}$$

(1)

where $v_x$, $v_y$, and r are the vehicle’s longitudinal velocity, lateral velocity, and yaw rate. The vehicle parameters are described in

Table 1. Vehicle model parameters.

Symbol	Description	Value	Units
$C_{f,min}/C_{f,max}$	Cornering stiffness of the front tire	80,750/109,250	N/rad
$C_{r,min}/C_{r,max}$	Cornering stiffness of the rear tire	72,250/97,750	N/rad
$l_f$	Distance from CoG to front axis	1.49	m
$l_r$	Distance from CoG to rear axis	1.81	m
R	Wheel radius	0.465	m
$t_w$	Track width	1.9	m
m	Vehicle total mass	1700	kg
$I_z$	Moment of inertia about the yaw axis	3246.6	kgm2
a	Look-ahead bias parameter	7	m
b	Look-ahead gain parameter	0.5	s

. $\delta$ stands for the front steering angle and $\mathrm{\Delta }{M}_z$ for the yaw moment generated by the action of braking/driving on the individual controller wheels:

$$\mathrm{\Delta }{M}_z=(F_{x,rr}-F_{x,rl}){\displaystyle \frac{t_w}{2}}+F_{x,fl}(l_{f}sin\left(\delta \right)-{\displaystyle \frac{t_w}{2}}cos\left(\delta \right))+F_{x,fr}(l_{f}sin\left(\delta \right)+{\displaystyle \frac{t_w}{2}}cos\left(\delta \right))$$

(2)

where $F_{x,ij}$ denotes the longitudinal driving/braking forces and $F_{y,ij}$ the lateral tire forces, for $i\in f,r$, front and rear axes and for $j\in l,r$, left and right sides of the vehicle. The values of these forces depend on several factors, including vertical load, tire pressure, and road conditions. Despite nonlinear tire–road models existing that represent tire behavior, such as the Pacejka magic formula, the Burckhardt model, or the Dugoff model [38,39,40], in this work, a simplified linear model [41,42] is considered to simplify the tire behavior as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{y,f}& =2C_f{\alpha }_f\hfill \\ \hfill F_{y,r}& =2C_r{\alpha }_r\hfill \end{array}\end{array}$$

(3)

where ${\alpha }_f=\delta -{\displaystyle \frac{v_y-l_{f}vr}{v_x}}$ and ${\alpha }_r={\displaystyle \frac{r{l}_r-v_y}{v_x}}$ are the front and rear tire slip angles, respectively. $C_f$ and $C_r$ are the front and rear tire cornering stiffness, respectively. These parameters are functions of the vertical load and tire–road interaction. As direct measurement in real-time is not possible, they can be assumed to be bounded as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_f& \in [C_{f,min},C_{f,max}]\hfill \\ \hfill C_r& \in [C_{r,min},C_{r,max}]\hfill \end{array}\end{array}$$

(4)

This enables the nonlinearities of the tire model to be taken into account, especially at high slip angles (Figure 1).

By combining Equations (1) and (3), the lateral vehicle dynamics can be described as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{v}}_y& =-{\displaystyle \frac{2C_f+2C_r}{m{v}_x}}{v}_y+\left({\displaystyle \frac{l_r{C}_r-l_f{C}_f}{m{v}_x}}-v_x\right)r+{\displaystyle \frac{C_f}{m}}\delta \hfill \\ \hfill \dot{r}& ={\displaystyle \frac{l_r{C}_r-l_f{C}_f}{I_z{v}_x}}{v}_y-{\displaystyle \frac{l_r^2{C}_r+l_f^2{C}_f}{I_z{v}_x}}r+{\displaystyle \frac{l_f{C}_f}{I_z}}\delta +{\displaystyle \frac{1}{I_z}}\mathrm{\Delta }{M}_z\hfill \end{array}\end{array}$$

(5)

2.2. Path-Tracking Model

Consider the path defined by its reference positions and reference heading angle $\left[\begin{array}{ccc}{x}_r& y_r& {\psi }_r\end{array}\right]$, as represented in Figure 2. To track the reference path, the lateral error $e_y$ and the heading error $e_{\psi }$ must be minimized:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e_y& =(y_v-y_r)cos\left({\psi }_r\right)-(x_v-x_r)sin\left({\psi }_r\right)\hfill \\ \hfill e_{\psi }& =\psi -{\psi }_r\hfill \end{array}\end{array}$$

(6)

$\psi$ being the heading angle of the vehicle. The path-tracking error dynamics are [43]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{e}}_y& =v_y+l_a\left(v_x\right)r+v_x{e}_{\varphi }\hfill \\ \hfill {\dot{e}}_{\psi }& =r-v_x{C}_R\hfill \end{array}\end{array}$$

(7)

where $C_R$ denotes the path curvature and is assumed as unknown in this work; $l_a\left(v_x\right)$ is the look-ahead distance, which has been defined as a function of the vehicle’s speed as in [44]

$$l_a\left(v_x\right)=a+b{v}_x$$

(8)

2.3. Combined Lateral Dynamics and Path-Tracking Model

A model that captures both the vehicle lateral dynamics and the path-tracking behavior is obtained by combining (5) and (7) such that

$$\dot{x}=A_c\left(t\right)x\left(t\right)+B_{c}u\left(t\right)+E_{c}d\left(t\right)$$

(9)

where $x\left(t\right)={\left[\begin{array}{cccc}{v}_y& r& e_y& e_{\psi }\end{array}\right]}^{\top }$ is the vehicle state vector, $u\left(t\right)={\left[\begin{array}{cc}\delta & \mathrm{\Delta }{M}_z\end{array}\right]}^{\top }$ is the control input, and $d\left(t\right)=-v_x{C}_R$ is the system disturbance. The state-space matrices are

$$A_c\left(t\right)=\left[\begin{array}{cccc}{a}_{11}& a_{12}& 0& 0\\ a_{21}& a_{22}& 0& 0\\ 1& l_a\left(v_x\right)& 0& v_x\\ 0& 1& 0& 0\end{array}\right],B_c\left(t\right)=\left[\begin{array}{cc}{b}_1& 0\\ b_2& b_3\\ 0& 0\\ 0& 0\end{array}\right],E_c\left(t\right)=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]$$

(10)

with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a_{11}& =-{\displaystyle \frac{2C_f+2C_r}{m{v}_x}},a_{12}=-v_x+{\displaystyle \frac{l_r{C}_r-l_f{C}_f}{m{v}_x}}\hfill \\ \hfill a_{21}& ={\displaystyle \frac{l_r{C}_r-l_f{C}_f}{I_z{v}_x}},a_{22}=-{\displaystyle \frac{l_r^2{C}_r+l_f^2{C}_f}{I_z{v}_x}}\hfill \\ \hfill b_1& ={\displaystyle \frac{C_f}{m}},b_2={\displaystyle \frac{l_f{C}_f}{I_z}}\hfill \\ \hfill b_3& ={\displaystyle \frac{1}{I_z}}\hfill \end{array}\end{array}$$

(11)

Matrices $A_c\left(t\right)$ and $B_c\left(t\right)$ from (9) depend on the stiffness of the front and rear tires and the longitudinal velocity of the vehicle through the terms $C_f$, $C_r$, $v_x$, and $1/v_x$, with ${\underline{v}}_x\le v_x\le {\overline{v}}_x$. To overcome this problem, a polytope model method is used to handle the range of varying parameters. As a result, the time-varying parameters can be expressed as a linear combination of $2^4=16$ vertices. The terms $v_x$ and $1/v_x$ are not completely independent; a change in the variable can occur to reduce the conservatism of the problem:

$${\displaystyle \frac{1}{v_x}}={\displaystyle \frac{1}{v_1}}+{\displaystyle \frac{1}{v_2}}\mu$$

(12)

where $\mu$ is a bounded parameter that satisfies $-1\le \mu \le 1$. If we use Taylor’s approximation [43] and $v_1={\displaystyle \frac{2{\overline{v}}_x{\underline{v}}_x}{{\underline{v}}_x+{\overline{v}}_x}}$ and $v_2={\displaystyle \frac{2{\overline{v}}_x{\underline{v}}_x}{{\underline{v}}_x-{\overline{v}}_x}}$, we can obtain

$$v_x\simeq v_1\left(1-{\displaystyle \frac{v_1}{v_2}}\mu \right)$$

(13)

In this way, the number of polytope vertices is reduced to $2^3=8$ vertices:

$$\mu =\sum _{i=1}^2{\alpha }_{1i}{\rho }_{1i},C_f=\sum _{i=1}^2{\alpha }_{2i}{\rho }_{2i},C_r=\sum _{i=1}^2{\alpha }_{3i}{\rho }_{3i}$$

(14)

with

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\alpha }_{11}={\displaystyle \frac{\overline{\mu }-\mu }{\overline{\mu }-\underline{\mu }}},{\alpha }_{21}={\displaystyle \frac{\overline{C_f}-C_f}{\overline{C_f}-\underline{C_f}}},{\alpha }_{31}={\displaystyle \frac{\overline{C_r}-C_r}{\overline{C_r}-\underline{C_r}}},{\alpha }_{i2}=1-{\alpha }_{i1}\end{array}\end{array}$$

(15)

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\rho }_{11}=\underline{\mu },{\rho }_{12}=\overline{\mu },{\rho }_{21}=\underline{C_f},{\rho }_{22}=\overline{C_f},{\rho }_{31}=\underline{C_r},{\rho }_{32}=\overline{C_r}\end{array}\end{array}$$

(16)

Then, the nonlinear system in (9) can be rewritten as

$$\dot{x}\left(t\right)=\sum _{i=1}^8{\zeta }_i[A_{c,i}x\left(t\right)+B_{1c,i}u\left(t\right)+E_{c}d\left(t\right)]$$

(17)

where ${\zeta }_i>0$ represents the weights of the polytope vertices and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \sum _{i=1}^2{\zeta }_i=1,{\zeta }_{ijk}^{\prime }={\alpha }_i{\alpha }_j{\alpha }_k,\hfill \\ \hfill & {\zeta }_1={\zeta }_{11}^{\prime },{\zeta }_2={\zeta }_{112}^{\prime },{\zeta }_8={\zeta }_{222}^{\prime }\hfill \end{array}\end{array}$$

(18)

To facilitate design and implementation, the system in (17) is converted into its discrete analogue via Euler’s discretization:

$$\dot{x}\left(t\right)\simeq {\displaystyle \frac{x(k+1)-x\left(k\right)}{T_s}}$$

(19)

$T_s$ being the sampling time. The discrete-time state-space model of the system in question is

$$x(k+1)=\sum _{i=1}^8{\zeta }_i[A_i\left(k\right)x\left(t\right)+B_{i}u\left(k\right)+Ed\left(k\right)]$$

(20)

with

$$A=I+T_s{A}_c,B=T_s{B}_c,E=T_s{E}_c$$

(21)

Using on-board vehicle sensors, the vehicle’s yaw rate can be measured using the vehicle’s Inertial Measurement Unit (IMU), while lateral and heading errors are acquired using a LiDAR sensor or a video camera. Therefore, the vector of measured variables $y\left(k\right)={\left[\begin{array}{ccc}r& e_y& e_{\psi }\end{array}\right]}^{\top }$ is obtained as follows:

$$y\left(k\right)=C_{y}x\left(k\right),C_y=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(22)

In order to achieve good tracking performance, the lateral tracking error and heading tracking error must be minimized. Additionally, lateral velocity should remain as low as possible in order not to compromise driving safety. These key performance indicators are contained in the output vector $z\left(k\right)={\left[\begin{array}{ccc}{v}_y& e_y& e_{\psi }\end{array}\right]}^{\top }$, which is expressed as

$$z\left(k\right)=C_{z}x\left(k\right),C_z=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(23)

2.4. Active Steering System Actuator Fault

In this work, the fault in the ASS is modeled as a loss of effectiveness fault, which is very common in the literature [45,46]:

$${\delta }_f=\epsilon \delta$$

(24)

where $\epsilon$ stands for the loss of effectiveness factor $0\le \epsilon \le 1$ and ${\delta }_f$ is the final steering angle in the front wheels. To compensate for this fault, a yaw torque is generated when a steering actuator fault is detected. Under this consideration, system (20) is rewritten as

$$x(k+1)=\sum _{i=1}^8{\zeta }_i[A_i\left(k\right)x\left(k\right)+\epsilon B_{1,i}{u}_1\left(k\right)+B_{2,i}{u}_2\left(k\right)\sigma \left(k\right)+Ed\left(k\right)]$$

(25)

where $u_1=\delta$, $u_2=\delta M_z$, $B_1=B(:,1)$, $B_2=B(:,2)$, and $\sigma \in \{0,1\}$ is a switching variable that indicates whether the steering actuator is faulty or not and is used to activate the torque-vectoring control:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \sigma \left(k\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{fault}\text{-}\mathrm{free}\mathrm{steering}\mathrm{actuator},\epsilon =1\hfill \\ 1\hfill & \mathrm{fault}\mathrm{in}\mathrm{the}\mathrm{steering}\mathrm{actuator},\epsilon \ne 1\hfill \end{array}\right.\end{array}\end{array}$$

(26)

2.5. Torque Allocation

In order to distribute the torque to the four wheels of the vehicle, the yaw moment $\mathrm{\Delta }{M}_z$ provided by the upper-level controller is distributed as [47]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{\Delta }{T}_{fl}& =-{\displaystyle \frac{R}{4L_{f}l}}\mathrm{\Delta }{M}_z{\lambda }_l\hfill \\ \hfill \mathrm{\Delta }{T}_{fr}& ={\displaystyle \frac{R}{4L_{f}r}}\mathrm{\Delta }{M}_z(1-{\lambda }_l)\hfill \\ \hfill \mathrm{\Delta }{T}_{rl}& =-{\displaystyle \frac{R}{4L_{rl}}}\mathrm{\Delta }{M}_z{\lambda }_r\hfill \\ \hfill \mathrm{\Delta }{T}_{rr}& ={\displaystyle \frac{R}{4L_{rr}}}\mathrm{\Delta }{M}_z(1-{\lambda }_r)\hfill \end{array}\end{array}$$

(27)

with

$$\begin{array}{cc}\hfill L_{fl}& ={\displaystyle \frac{t_w}{2}}cos\left(\delta \right)-l_{f}sin\left(\delta \right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill L_{fr}& ={\displaystyle \frac{t_w}{2}}cos\left(\delta \right)+l_{f}sin\left(\delta \right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill L_{rl}& =L_{rr}={\displaystyle \frac{t_w}{2}}\hfill \end{array}$$

(29)

where $\mathrm{\Delta }{T}_{ij}$ represents the torque increment added to the current torque of the wheel motor $ij$ to avoid compromising the longitudinal dynamics, and ${\lambda }_l$ and ${\lambda }_r$ represent the trade-off between the torque in the left and right sides of the vehicle, respectively. In this work, these parameters are set to 0.5 to provide an even distribution between both sides of the vehicle.

3. Fault-Tolerant Controller Design

This section derives the fault-tolerant path-tracking controller that compensates for the steering actuator faults using TV. The control architecture is shown in Figure 3.

The controller has to maintain the stability of the vehicle and track the reference path in both fault-free and faulty conditions of the steering actuator. Furthermore, the controller may only use the information provided by commercial vehicles [48]. For this purpose, the following control laws are proposed:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u_1\left(k\right)=K_{1}y\left(k\right)\\ \hfill u_2\left(k\right)=K_{2}y\left(k\right)\end{array}\end{array}$$

(30)

where $K_1={\sum }_{i=1}^2{\theta }_i{K}_{1,i}$ and $K_2={\sum }_{i=1}^2{\theta }_i{K}_{2,i}$ are the gain control matrices to be found. Matrices $K_1$ and $K_2$ are computed online using $\mu$, as is the only polytopic parameter available, which is considered to be known as it depends on the longitudinal velocity of the vehicle (12):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\theta }_1={\displaystyle \frac{\overline{\mu }-\mu }{\overline{\mu }-\underline{\mu }}}\hfill \\ \hfill & {\theta }_2=1-{\theta }_1\hfill \end{array}\end{array}$$

(31)

Combining the control laws from (30) and state-space equation from (25), we can write the closed-loop dynamics as

$$x(k+1)=\sum _{i=1}^8{\zeta }_i\left[A_{i}x\left(k\right)+\epsilon B_{1,i}\left(\sum _{j=1}^2{\theta }_j{K}_{1,j}{C}_{y}y\left(k\right)\right)+\sigma \left(k\right)B_2\left(\sum _{j=1}^2{\theta }_j{K}_{2,j}{C}_{y}y\left(k\right)\right)+Ed\left(k\right)\right]$$

(32)

Theorem 1.

Given a definite positive symmetric matrix $Q_j=Q_j^{\top }\succ 0$, the system of (32) is asymptotically stable with an ${\mathcal{H}}_{\infty }$ performance index less than γ and for an actuator fault gain $\epsilon \in [0,1]$ if the following LMIs hold:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left[\begin{array}{cccc}-Q_j& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★\\ A_i{Q}_j+B_{1,i}{K}_{1,j}{C}_y{Q}_j& E& -Q_j& ★\\ C_z{Q}_j& 0& 0& -I\end{array}\right]\prec 0,\\ \hfill \left[\begin{array}{cccc}-Q_j& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★\\ A_i{Q}_j+\epsilon B_{1,i}{K}_{1,j}{C}_y{Q}_j+B_{2,i}{K}_{2,j}{C}_y{Q}_j& E& -Q_j& ★\\ C_z{Q}_j& 0& 0& -I\end{array}\right]\prec 0\end{array}\end{array}$$

(33)

Proof. 

Let us define a Lyapunov function

$$\mathcal{V}\left(k\right)=\sum _{j=1}^2{\varrho }_j{x}^{\top }\left(k\right)P_{j}x\left(k\right)$$

(34)

with $P_j=P_j^{\top }\succ 0$. It is assumed that the parameter $\varrho$ does not change significantly between successive instants since it depends on the longitudinal velocity, which is assumed to vary slowly. For the system in (32) to be stable, the following condition must be satisfied:

$$\mathcal{V}(k+1)-\mathcal{V}\left(k\right)<0$$

(35)

In order to minimize the influence of the external disturbance over the chosen controlled outputs, $H_{\infty }$ approach is used:

$$z{\left(k\right)}^{T}z\left(k\right)<{\gamma }^{2}d{\left(k\right)}^{T}d\left(k\right)$$

(36)

$\gamma$ being the upper bound of the worst-case gain from disturbance d to controlled outputs z. For control design, combining ${\mathcal{H}}_{\infty }$ performance in (36) and Lyapunov stability criterion in (35) yields

$$\mathcal{V}(k+1)-\mathcal{V}\left(k\right)-{\gamma }^2{d}^{\top }\left(k\right)d\left(k\right)+x^{\top }\left(k\right)C_z^{\top }{C}_{z}x\left(k\right)<0$$

(37)

According to the steering, the actuator fault indicator $\sigma \left(k\right)$ (37) can be expressed in LMI form for both non-faulty $\sigma \left(k\right)=0$ and faulty $\sigma \left(k\right)=1$ cases as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left[\begin{array}{cc}-P_j+C_z^{\top }{C}_z+{\mathrm{\Theta }}_{ij}^{\top }{P}_j{\mathrm{\Theta }}_{ij}& ★\\ E^{\top }{P}_j& -{\gamma }^{2}I+E^{\top }{P}_{j}E\end{array}\right]\prec 0\hfill \\ \hfill & \left[\begin{array}{cc}-P_j+C_z^{\top }{C}_z+{\Gamma }_{ij}^{\top }{P}_j{\Gamma }_{ij}& ★\\ E^{\top }{P}_j& -{\gamma }^{2}I+E^{\top }{P}_{j}E\end{array}\right]\prec 0\hfill \end{array}\end{array}$$

(38)

with ${\mathrm{\Theta }}_{ij}=A_i+B_{1,i}{K}_{1,j}{C}_y$ and ${\Gamma }_{ij}=A_i+\epsilon B_{1,i}{K}_{1,j}{C}_y+B_{2,i}{K}_{2,j}{C}_y$. Using Schur complement, LMIs in (38) are rewritten as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left[\begin{array}{cccc}-P_j& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★\\ P_j(A_i+B_{1,i}{K}_{1,j}{C}_y)& P_{j}E& -P_j& ★\\ C_z& 0& 0& -I\end{array}\right]\prec 0\hfill \\ \hfill & \left[\begin{array}{cccc}-P_j& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★\\ P_j(A_i+\epsilon B_{1,i}{K}_{1,j}{C}_y+B_{2,i}{K}_{2,j}{C}_y)& P_{j}E& -P_j& ★\\ C_z& 0& 0& -I\end{array}\right]\prec 0\hfill \end{array}\end{array}$$

(39)

Now, if we apply a congruent transformation by multiplying by $diag[P_j^{-1},I,P_j^{-1},I]$ on the left and by its transpose on the right, the LMIs in (39) become

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left[\begin{array}{cccc}-P_j^{-1}& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★\\ (A_i{P}_j^{-1}+B_{1,i}{K}_{1,j}{C}_y{P}_j^{-1})& E& -P_j^{-1}& ★\\ C_z{P}^{-1}& 0& 0& -I\end{array}\right]\prec 0\hfill \\ \hfill & \left[\begin{array}{cccc}-P_j^{-1}& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★\\ (A_i{P}_j^{-1}+\epsilon B_{1,i}{K}_{1,j}{C}_y{P}_j^{-1}+B_{2,i}{K}_{2,j}{C}_y{P}_j^{-1})& E& -P_j^{-1}& ★\\ C_z{P}_j^{-1}& 0& 0& -I\end{array}\right]\prec 0\hfill \end{array}\end{array}$$

(40)

By defining a matrix $Q_j=P_j^{-1}$, the LMIs (40) are equivalent to the LMIs in (33). □

The controller gains are determined by solving the following optimization problem:

$$\begin{array}{ll}min& {\gamma }^2\hfill \\ \mathrm{s}.\mathrm{t}.& Q_j=Q_j^{\top }\succ 0,\left(33\right)\hfill \end{array}$$

(41)

The LMI in (33) to be solved is a bi-linear problem as variable $Q_j$ is coupled to $K_{1j}$ and $K_{2j}$; an iterative procedure is used to find a feasible solution [44]. A full description of this method is presented in Algorithm 1.

Algorithm 1 Iterative procedure for solving bi-linear LMI problem.

Input:

Vehicle parameters from Table 1.

Output:

Controller gain matrices from (30)

1: Set $C_y=I$ and ${\tilde{K}}_{1,j}^{sf}=K_{1,j}{Q}_j$, ${\tilde{K}}_{2,j}^{sf}=K_{2,j}{Q}_j$.   2: Solve (41).   3: Obtain $K_{1,j}^{sf}$ and $K_{2,j}^{sf}$, $j\in 1,2$, with the optimal value ${\gamma }_{min}={\gamma }^{\left(0\right)}$.   4: Set $C_y$ as in (22) and set $K_{1,j}^{\left(0\right)}=K_{1,j}^{sf}(:,2:4)$ and $K_{2,j}^{\left(0\right)}=K_{2,j}^{sf}(2:4)$.   5: for $k=1:k^{max}$ do   6:   Given $K_{1,j}^{\left(k\right)}$ and $K_{2,j}^{\left(k\right)}$, solve LMIs from (41) to obtain $Q_j^{\left(k\right)}$   7:   Given $Q_j^{\left(k\right)}$, solve LMIs from (41) to obtain $K_{1,j}^{(k+1)}$, $K_{2,j}^{(k+1)}$ and ${\gamma }^{\left(k\right)}$   8:   if ${\displaystyle \frac{|{\gamma }^{\left(k\right)}-{\gamma }^{(k-1)}|}{{\gamma }^{\left(k\right)}}}<ϵ$ then   9:     stop 10:   end if 11: end for

4. Results and Discussion

In this section, the proposed fault-tolerant SOF path-tracking controller is tested using the vehicle dynamics simulation software CarSim ^® 2022.1 version. The vehicle model is configured with the parameters described in Table 1. The vehicle speed is assumed to be in the range $v_x\in [2,25]$ m/s. The path-tracking controller is designed by solving the optimization problem (41), considering a maximum gain fault of $\epsilon =0.1$, using the Robust Control Toolbox included in Matlab 2022b. By computing Algorithm 1, a feasible control solution has been found with an ${\mathcal{H}}_{\infty }$ robustness index of $\gamma =9.89$. Only two iterations were required.

Two different fault scenarios are simulated to test the proposed controller:

(A)

DLC maneuver under steering actuator degradation with $\epsilon =0.3$ and $\epsilon =0.1$.

(B)

DLC maneuver under steering actuator saturation at $|{\delta }_{max}=5^{\circ }|$ and $|{\delta }_{max}=3^{\circ }|$.

The test maneuver is a Double Lane Change (DLC), designed to simulate an obstacle avoidance situation and challenge the controller under a high-risk scenario.

4.1. Case A: Loss of Effectiveness in the Steering Actuator

To evaluate the effectiveness of the proposed methodology, three cases are compared: (1) a baseline case with no steering faults, (2) a fault case without fault-tolerant control, and (3) the proposed fault-tolerant controller. All the cases are evaluated at a constant vehicle speed of $v_x=25$ m/s under identical fault conditions.

As shown in Figure 4, the fault-tolerant controller reduces the Root Mean Square (RMS) of the lateral error from $41.13$ cm to $30.30$ cm with the proposed controller for $\epsilon =0.3$ and from $80.71$ cm to $39.83$ cm for $\epsilon =0.1$. This demonstrates that the proposed controller improves the tracking accuracy, particularly during the second turn when the vehicle returns to its original lane at 4.5s of the simulation. Evidently, the controller can reduce the tracking error of the vehicle even for the most extreme case, $\epsilon =0.1$. The proposed controller also reduces the heading error, with the maximum error dropping from $10.{08}^{\circ }$ to $8.{20}^{\circ }$ and RMS heading error from $4.{29}^{\circ }$ to $1.{86}^{\circ }$ for the most severe scenario (Figure 5).

The control inputs are shown in Figure 6, which displays the steering angle input in Figure 6a and the input torque of the wheel for the fault-tolerant case in Figure 6b. Evidently, the controller demands a higher steering angle due to the actuator’s reduced effectiveness, but this demand is limited by physical saturation, as shown in Figure 6a, and the motor torque compensates for this effect by adding extra torque in the wheels as in [36]. Finally, the path followed by the vehicle in each scenario is illustrated in Figure 7, highlighting the differences in tracking performance between the cases with and without fault-tolerant control.

4.2. Case B: Steering Actuator Saturation

In this scenario, the fault limits the steering angle to a range of $-5^{\circ }<\delta <5^{\circ }$ and $-3^{\circ }<\delta <3^{\circ }$. As shown in Figure 8, the non-fault-tolerant controller results in a maximum lateral error of $169.09$ cm for $|{\delta }_{max}|=3^{\circ }$ in the case of the most severe fault. However, the proposed controller reduces this maximum error to $107.51$, a more acceptable level. The RMS lateral error is also decreased from $46.81$ cm to $30.13$ cm and $66.43$ cm to $35.19$ cm for $|{\delta }_{max}|=5^{\circ }$ and $|{\delta }_{max}|=3^{\circ }$, respectively. Similarly, as in the previous case, the heading error is minimized, as shown in Figure 9, dropping from $11.{86}^{\circ }$ to $7.{38}^{\circ }$ for the most extreme case. The RMS of the heading error is further reduced from $2.{98}^{\circ }$ to $2.{33}^{\circ }$ and from to $4.{02}^{\circ }$ to $2.{12}^{\circ }$ for $|{\delta }_{max}|=5^{\circ }$ and $|{\delta }_{max}|=3^{\circ }$, respectively (Figure 9).

The control inputs for this case are shown in Figure 10, where the steering angle limitation caused by the fault is visible as in [30], resulting in behavior similar to the previous scenario. The path tracked by the vehicle is illustrated in Figure 11.

Table 2. Tracking errors.

Case		${\mathit{e}}_{\mathit{y}}$ (cm)		${\mathit{e}}_{\mathit{\varphi }}{(}^{\circ })$
		MAX	RMS	MAX	RMS
No fault	Reference	78.00	24.97	4.81	1.43
$\epsilon =0.3$	Non-fault-tolerant	89.60	41.13	10.96	2.86
	Fault-tolerant	93.21	30.30	6.24	1.85
$\epsilon =0.1$	Non-fault-tolerant	223.89	80.71	10.08	4.29
	Fault-tolerant	120.64	39.83	8.20	2.33
$|{\delta }_{max}=5^{\circ }|$	Non-fault-tolerant	104.55	46.81	11.32	2.98
	Fault-tolerant	92.12	30.13	6.52	1.86
$|{\delta }_{max}=3^{\circ }|$	Non-fault-tolerant	169.09	66.43	11.86	4.02
	Fault-tolerant	107.51	35.19	7.38	2.12

summarizes the results obtained.

5. Conclusions

In this work, a novel fault-tolerant path-tracking SOF controller is developed for electric DDV vehicles, specifically designed to handle actuator faults with a flexible approach. Using an LPV approach, the controller design accounts for nonlinearities associated with varying vehicle speeds and tire behavior, using ${\mathcal{H}}_{\infty }$ theory to minimize its effects on tracking performance. A key innovation of this work is the dual-mode operation: a fault-free mode and an actuator fault mode in which the TV compensates for faults. The simulation results demonstrate the effectiveness of the controller in improving vehicle stability and reducing tracking error under fault conditions in a high-risk maneuver. The controller achieves up to a 50.65% reduction in RMS lateral error and 45.69% reduction in RMS heading error in the worst-case scenario for gain steering fault and a reduction of 36.42% for RMS lateral error and 47.26% reduction for RMS heading error for the saturation fault type.

This controller provides practical benefits for vehicle designers and engineers, offering a robust solution for path-tracking in fault conditions with ease of implementation. Future work may explore extending this approach to address more complex fault types and to enhance the controller for trajectory-tracking applications.