Synergistic Guaranteed Cost and Integral Sliding Mode Fault-Tolerant Control for Steer-by-Wire Systems Subject to Multiple Uncertainties

Abstract

The actuator reliability of Steer-by-Wire (SBW) systems is critical to the functional safety of autonomous vehicles. However, existing control methods struggle to simultaneously enhance both response speed and fault-tolerant performance when facing multiple uncertainties such as parameter perturbations, external disturbances, and actuator faults. To address these issues, this paper proposes a synergistic fault-tolerant control (FTC) strategy combining guaranteed cost control (GCC) and integral sliding mode control (ISMC). First, a dynamic model of the SBW system incorporating the multiple uncertainties is established. Second, a GCC law is derived based on linear matrix inequalities (LMIs) to impose strict constraints on the system’s tracking accuracy and robustness. Building upon this, an ISMC is integrated to significantly accelerate the system’s dynamic response without sacrificing steady-state accuracy, thereby forming a synergistic fault-tolerant architecture characterized by both high precision and rapid response. The results indicate that, under typical fault modes and steering conditions, the response speed of GCC+ISMC is significantly improved compared with GCC alone, and the GCC+ISMC reduces tracking errors by approximately 35% compared to adaptive integral sliding mode control (AISMC). These findings demonstrate that the proposed approach effectively mitigates multiple system uncertainties, offering comprehensive advantages in tracking accuracy, response speed, and robustness.

1. Introduction

With the rapid advancement of automotive intelligence, traditional steering mechanisms are accelerating their evolution towards electrification and integration. As a core node of future chassis domain control, the SBW system not only breaks the physical constraints of mechanical linkages—thereby significantly enhancing vehicle layout freedom and handling response—but also serves as a critical enabler for the implementation of high-level autonomous driving. However, constrained by the intrinsic physical attributes of electronic components, the sensors, controllers, and actuators in SBW systems are highly susceptible to electromagnetic interference, thermal aging, and sudden load transients. Consequently, it is difficult to completely avoid the risks of performance drift or even system failure. In complex and dynamic driving environments, any minor anomaly in electronic components can be amplified, directly degrading the transient response and steady-state accuracy of the steering system, and in severe cases, jeopardizing vehicle safety. Therefore, researching and developing effective FTC strategies is of paramount importance for improving the functional safety and operational reliability of SBW systems.

Currently, mainstream FTC technologies can be broadly categorized into two types: Fault Detection and Isolation (FDI)-based strategies [1] and Fault Compensation (FC)-based strategies [2]. FDI-based strategies focus on accurately identifying and isolating the fault source to prevent fault propagation [3]. However, this technique relies heavily on the accuracy and real-time performance of fault diagnosis, and any delay or error in the diagnostic process will compromise fault-tolerance precision and control efficacy. In contrast, FC-based strategies maintain system functionality without interrupting operation by dynamically reallocating resources to compensate for faults in real time. Common compensation control methods, such as sliding mode control (SMC) [4], Model Predictive Control (MPC) [5], and Reinforcement Learning (RL) [6], have found certain applications across various industrial scenarios. Nevertheless, considering the highly complex operating conditions of vehicles, the strong uncertainty of external environments, and the inevitable parameter perturbations and partial failures after long-term service, the design of SBW controllers must rigorously account for both system robustness and stability.

To achieve robust and stable control under uncertainties and partial failures, various control strategies have emerged in recent years. For example, advanced sliding mode control techniques such as delay-resilient super-twisting strategies [7] and event-triggered integral sliding mode approaches [8] have been developed for uncertain dynamic systems, while fault-tolerant h-infinity stabilization methods with adaptive event-triggered mechanisms [9] have been proposed for networked control systems subject to actuator faults. Among the various fault-tolerant and robust control strategies, Prescribed Performance Control (PPC) and GCC are particularly prominent. PPC strategies constrain tracking errors within predefined boundaries by constructing time-varying performance bounds, thereby ensuring the system meets specific transient and steady-state requirements [10]. For instance, Zhou S. [11] and Zhao X. [12,13] applied PPC to address actuator faults, sensor faults, and external disturbances in steering systems, Lu L. [14] utilized PPC for actuator faults in multi-agent systems, and Wang H. [15] and Wang Z. [16] designed PPC for actuator faults in strict-feedback systems. While PPC can improve control accuracy and stability under fault conditions, the parameter selection for its performance functions often relies heavily on empirical tuning. Furthermore, PPC can be overly conservative or insufficient when handling the impact of severe uncertainties on performance boundaries, making it difficult to achieve desired outcomes in systems subject to multiple uncertainties.

In comparison, the GCC strategy exhibits distinct advantages when dealing with systems subject to multiple uncertainties. Its fundamental principle relies on constructing a control law via Lyapunov and LMIs within the allowable range of uncertain disturbances, ensuring that the closed-loop system is not only stable but also provides a provable upper bound for a specified performance index [17]. Since its inception, this method has garnered continuous attention for its ability to balance stability and measurable performance in uncertain systems [18,19,20,21], with successful extensions into applications such as spacecraft attitude control [22,23,24], multi-agent systems [25], and networked control systems [26]. In the field of vehicle control, GCC has been applied to path tracking [27] and handling stability control [28]. For example, Zhang Z. [29] proposed an intelligent vehicle trajectory tracking control strategy based on robust GCC for unreliable vehicular transmission environments, successfully attenuating the adverse effects of external disturbances and unmodeled errors. Liang Y. [30] addressed the path tracking problem of autonomous vehicles by performing multi-objective optimization on performance indices using robust GCC and Particle Swarm Optimization (PSO), yielding superior controller performance under different priority objectives. Regarding SBW fault-tolerant control, Zhou J. [31] designed a hierarchical GCC for motor open-circuit faults, enhancing torque output capability while reducing copper losses; Liu W. [32] proposed a GCC and adaptive FTC for actuator faults, maintaining certain tracking performance even when severe faults exceed the GCC limits. However, the latter study neglected the effects of parameter perturbations and external disturbances, and suffered from a relatively slow response speed.

In summary, research on GCC in the SBW domain remains relatively limited. On one hand, some existing works insufficiently consider multi-source uncertainties such as parameter perturbations and external disturbances. On the other hand, to obtain high steady-state accuracy, the controller design often introduces considerable conservatism, which can lead to sluggish steering responses. It must be emphasized that in critical scenarios such as emergency obstacle avoidance and high-speed lane changing, the dynamic response speed of the steering system is equally as crucial as steady-state accuracy, as it directly impacts the rapid execution of driving intentions and the vehicle’s maneuverability in unexpected situations. Therefore, enhancing the response speed while ensuring ultra-high steady-state accuracy of the SBW system, along with fully accounting for multiple uncertainties in actual operations, remains an urgent and challenging issue in current research.

Motivated by the above discussions, targeting the multiple uncertainties in SBW systems including parameter perturbations, external disturbances, and actuator faults, this paper establishes a comprehensive system dynamic model. Building upon this, a synergistic fault-tolerant strategy integrating GCC and ISMC is proposed. By defining a quadratic performance index and solving LMIs, the performance upper bound and the GCC law are obtained, thereby achieving strict constraints on tracking accuracy and robustness. To further enhance the system’s dynamic response capability to command signals, an ISMC mechanism is introduced, significantly reducing the time to reach the desired state while maintaining excellent steady-state performance. Simulation validations under typical vehicle steering conditions, including step, sinusoidal, and double lane change maneuvers, show that the proposed method outperforms existing schemes across various fault scenarios. The main contributions of this paper are summarized as follows:

(1)

An application-oriented GCC–ISMC fault-tolerant control framework is developed for SBW systems subject to parameter perturbations, external disturbances, and actuator faults, where the contribution lies in the integrated design for this practical engineering scenario.

(2)

A coordinated mechanism between GCC and ISMC is established, in which GCC provides performance guarantees and robustness constraints, while ISMC enhances transient response and accelerates convergence to the desired state.

The remainder of this paper is organized as follows: Section 2 presents the dynamic modeling of the SBW system. Section 3 details the design of the controller. Section 4 provides the example illustration. Section 5 concludes the paper.

2. Dynamic Modeling of the SBW System

The structure of the SBW system is shown in Figure 1.

A three-phase surface-mounted permanent magnet synchronous motor (SPMSM) serves as the steering actuator. Its dynamic equation is expressed as

$$J_{\mathrm{m}}{\ddot{\theta }}_{\mathrm{m}}+B_{\mathrm{m}}{\dot{\theta }}_{\mathrm{m}}+{\tau }_{\mathrm{mf}}={\tau }_{\mathrm{m}}$$

(1)

where θ_m is the steering motor angle, J_m is the steering motor moment of inertia, B_m is the steering motor viscous damping, τ_mf is the torque applied to the motor steering shaft from the front wheels via the rack-and-pinion steering gear, and τ_m is the steering motor electromagnetic torque. Considering the torque ripple disturbance during actual motor operation, we have

$${\tau }_{\mathrm{m}}={\tau }_{\mathrm{mr}}+{\tau }_{\mathrm{md}}$$

(2)

where τ_mr is the ideal electromagnetic torque of the steering motor, and τ_md is the torque ripple disturbance of the steering motor.

The dynamic equation of the front steering wheel is expressed as

$$J_{\mathrm{f}}{\ddot{\delta }}_{\mathrm{f}}+B_{\mathrm{f}}{\dot{\delta }}_{\mathrm{f}}+{\tau }_{\mathrm{f}}+{\tau }_{\mathrm{e}}={\tau }_{\mathrm{fm}}$$

(3)

where δ_f is the front wheel steering angle, J_f is the moment of inertia of the front wheel, B_f is the viscous damping of the front wheel, τ_f is the front wheel rotational resistance torque, τ_e is the self-aligning torque, and τ_fm is the torque applied to the front wheel by the steering motor through the rack-and-pinion steering gear.

Assuming no transmission backlash between the rack and pinion, the relationship between the front wheel angle and the steering motor angle is as follows:

$${\displaystyle \frac{{\delta }_{\mathrm{f}}}{{\theta }_{\mathrm{m}}}}={\displaystyle \frac{1}{r}}={\displaystyle \frac{{\tau }_{\mathrm{mf}}}{{\tau }_{\mathrm{fm}}}}$$

(4)

where r is the transmission ratio between the steering motor and the front steering wheel. Differentiating Equation (4) yields the following relationship:

$${\displaystyle \frac{{\delta }_{\mathrm{f}}}{{\theta }_{\mathrm{m}}}}={\displaystyle \frac{{\dot{\delta }}_{\mathrm{f}}}{{\dot{\theta }}_{\mathrm{m}}}}={\displaystyle \frac{{\ddot{\delta }}_{\mathrm{f}}}{{\ddot{\theta }}_{\mathrm{m}}}}={\displaystyle \frac{1}{r}}={\displaystyle \frac{{\tau }_{\mathrm{mf}}}{{\tau }_{\mathrm{fm}}}}$$

(5)

Based on Equations (1)–(5), the equivalent model of the steering actuator assembly of the SBW system is obtained as

$$\left\{\begin{array}{l}{J}_{\mathrm{eq}}{\ddot{\delta }}_{\mathrm{f}}+B_{\mathrm{eq}}{\dot{\delta }}_{\mathrm{f}}-r{\tau }_{\mathrm{md}}+{\tau }_{\mathrm{f}}+{\tau }_{\mathrm{e}}={\tau }_{\mathrm{eq}}\\ J_{\mathrm{eq}}=J_{\mathrm{f}}+r^2{J}_{\mathrm{m}}\\ B_{\mathrm{eq}}=B_{\mathrm{f}}+r^2{B}_{\mathrm{m}}\\ {\tau }_{\mathrm{eq}}=r{\tau }_{\mathrm{mr}}\end{array}\right.$$

(6)

where J_eq is the equivalent moment of inertia, B_eq is the equivalent viscous damping, and τ_eq is the equivalent torque.

Considering the sixth and twelfth harmonic components in the magnetic flux and the current bias at the electronic terminals, the torque disturbance of the motor can be expressed as [33]

$${\tau }_{\mathrm{md}}={\tau }_{\mathrm{md}6}\cos (6{\theta }_{\mathrm{e}})+{\tau }_{\mathrm{md}12}\cos (12{\theta }_{\mathrm{e}})+{\displaystyle \frac{3}{2}}{p}_{\mathrm{n}}{\psi }_{\mathrm{f}0}{i}_{\mathrm{dis}}$$

(7)

where τ_md6 and τ_md12 are the sixth and twelfth harmonic torques, respectively; θ_e is the motor electrical angle; P_n is the number of pole pairs; ψ_f0 is the DC average amplitude of the d-axis flux linkage; and i_dis is the current bias, which can be expressed as

$$i_{\mathrm{dis}}={\displaystyle \frac{2}{\sqrt{3}}}\sin ({\theta }_{\mathrm{e}}+{\displaystyle \frac{2}{3}}\pi )\sqrt{(\Delta i_{\mathrm{a}}^2)+\Delta i_{\mathrm{a}}\Delta i_{\mathrm{b}}+(\Delta i_{\mathrm{b}}^2)}$$

(8)

where Δi_a and Δi_b are the current biases of phase a and phase b, respectively.

The vehicle is simplified into a two-degree-of-freedom (2-DOF) model, as shown in Figure 2. The vehicle’s center of mass (CG) is taken as the origin of the vehicle coordinate system, where O_x and O_y are the longitudinal and lateral axes, respectively, and the components of the CG velocity V₁ along the O_x and O_y axes are V_x and V_y, respectively [34].

Assuming the vehicle’s longitudinal speed remains constant; neglecting its vertical, pitch, and roll motions; and assuming identical slip angles for the left and right wheels, when the side-slip angle is small, the lateral tire forces F_Y1 and F_Y2 acting on the front and rear wheels from the ground can be expressed as

$$\left\{\begin{array}{l}{F}_{\mathrm{Y}1}=-k_1{\alpha }_1\\ F_{\mathrm{Y}2}=-k_2{\alpha }_2\end{array}\right.$$

(9)

where k₁ and k₂ are the cornering stiffness of the front and rear wheels, respectively, and α₁ and α₂ are the side-slip angles of the front and rear wheels, respectively, which are expressed as follows:

$$\left\{\begin{array}{l}{\alpha }_1=\beta +{\displaystyle \frac{a{\omega }_r}{V_x}}-{\delta }_f\\ {\alpha }_2=\beta -{\displaystyle \frac{b{\omega }_r}{V_x}}\end{array}\right.$$

(10)

where β is the sideslip angle of the center of mass, ω_r is the yaw rate, and a and b are the distances from the center of mass to the front and rear axles, respectively.

Based on Equations (9) and (10), combined with the equilibrium equations and kinematic relationships, the following vehicle dynamic equations are obtained:

$$\left\{\begin{array}{l}-k_1(\beta +{\displaystyle \frac{a{\omega }_{\mathrm{r}}}{V_{\mathrm{x}}}}-{\delta }_{\mathrm{f}})-k_2(\beta -{\displaystyle \frac{b{\omega }_{\mathrm{r}}}{V_{\mathrm{x}}}})=m({\dot{V}}_{\mathrm{y}}+V_{\mathrm{x}}{\omega }_{\mathrm{r}})\\ -a{k}_1(\beta +{\displaystyle \frac{a{\omega }_{\mathrm{r}}}{V_{\mathrm{x}}}}-{\delta }_{\mathrm{f}})+b{k}_2(\beta -{\displaystyle \frac{b{\omega }_{\mathrm{r}}}{V_{\mathrm{x}}}})=I_{\mathrm{Z}}{\dot{\omega }}_{\mathrm{r}}\end{array}\right.$$

(11)

where m is the vehicle mass and I_Z is the moment of inertia of the vehicle about the z-axis.

When the tire experiences cornering, it generates a self-aligning torque about the z-axis:

$${\tau }_e=l{F}_{Y1}$$

(12)

where l is the tire pneumatic trail.

Substituting Equations (9) and (10) into (12) yields

$${\tau }_{\mathrm{e}}=-l{k}_1(\beta +{\displaystyle \frac{{\omega }_{\mathrm{r}}a}{V_{\mathrm{x}}}}-{\delta }_{\mathrm{f}})$$

(13)

The front wheel rotational resistance torque can be expressed as

$${\tau }_{\mathrm{f}}={\tau }_{\mathrm{fs}}\mathrm{sign}({\dot{\delta }}_{\mathrm{f}})$$

(14)

where τ_fs is the resistance torque coefficient and sign() is the switching function.

Substituting Equations (7), (13) and (14) into Equation (6), and combining with Equation (11), the system dynamic equation is arranged as follows:

$$\left\{\begin{array}{l}{\ddot{\delta }}_{\mathrm{f}}={\displaystyle \frac{r}{J_{\mathrm{eq}}}}u-{\displaystyle \frac{B_{\mathrm{eq}}}{J_{\mathrm{eq}}}}{\dot{\delta }}_{\mathrm{f}}+{\displaystyle \frac{l{k}_1}{J_{\mathrm{eq}}}}\beta +{\displaystyle \frac{l{k}_{1}a}{J_{\mathrm{eq}}{V}_{\mathrm{x}}}}{\omega }_{\mathrm{r}}-{\displaystyle \frac{l{k}_1}{J_{\mathrm{eq}}}}{\delta }_{\mathrm{f}}+{\displaystyle \frac{1}{J_{\mathrm{eq}}}}\left(r{\tau }_{\mathrm{md}}-{\tau }_{\mathrm{f}}\right)\\ \dot{\beta }={\displaystyle \frac{-\left(k_1+k_2\right)}{m{V}_{\mathrm{x}}}}\beta +\left({\displaystyle \frac{b{k}_2-a{k}_1}{m{V}_{\mathrm{x}}^2}}-1\right){\omega }_{\mathrm{r}}+{\displaystyle \frac{k_1}{m{V}_{\mathrm{x}}}}{\delta }_{\mathrm{f}}\\ {\dot{\omega }}_{\mathrm{r}}={\displaystyle \frac{b{k}_2-a{k}_1}{I_{\mathrm{z}}}}\beta +{\displaystyle \frac{-(a^2{k}_1+b^2{k}_2)}{I_{\mathrm{z}}{V}_{\mathrm{x}}}}{\omega }_{\mathrm{r}}+{\displaystyle \frac{a{k}_1}{I_{\mathrm{z}}}}{\delta }_{\mathrm{f}}\end{array}\right.$$

(15)

where u = τ_mr is the control input.

The tracking error of the front wheel steering angle is defined as

$$e={\delta }_{\mathrm{fr}}-{\delta }_{\mathrm{f}}$$

(16)

The state vector is constructed as

$$x(t)={\left[\begin{array}{ccccc}{\int }_0^{t}e(t) \mathrm{d}t\hfill & {\delta }_{\mathrm{f}}(t)& {\dot{\delta }}_{\mathrm{f}}(t)& \beta & \hfill {\omega }_{\mathrm{r}}\end{array}\right]}^{\mathrm{T}}$$

(17)

Combined with Equation (15), the state-space equation of the system is obtained as

$$\dot{x}(t)=A_{0}x(t)+B_{0}u(t)+C_0$$

(18)

where $A_0=\left[\begin{array}{ccccc}0\hfill & -1& 0& 0& 0\\ 0\hfill & 0& 1& 0& 0\\ 0\hfill & {\displaystyle \frac{-k_{1}l}{J_{\mathrm{eq}}}}& {\displaystyle \frac{-B_{\mathrm{eq}}}{J_{\mathrm{eq}}}}& {\displaystyle \frac{k_{1}l}{J_{\mathrm{eq}}}}& {\displaystyle \frac{a{k}_{1}l}{J_{\mathrm{eq}}{V}_{\mathrm{x}}}}\\ 0\hfill & {\displaystyle \frac{k_1}{m{V}_{\mathrm{x}}}}& 0& -{\displaystyle \frac{k_1+k_2}{m{V}_{\mathrm{x}}}}& {\displaystyle \frac{b{k}_2-a{k}_1}{m{V}_{\mathrm{x}}^2}}-1\\ 0\hfill & {\displaystyle \frac{a{k}_1}{I}}& 0& {\displaystyle \frac{b{k}_2-a{k}_1}{I}}& -{\displaystyle \frac{a^2{k}_1+b^2{k}_2}{I{V}_{\mathrm{x}}}}\end{array}\right]$, $B_0={\left[\begin{array}{ccccc}0\hfill & 0& {\displaystyle \frac{r}{J_{\mathrm{eq}}}}& 0& \hfill 0\end{array}\right]}^{\mathrm{T}}$, $C_0={\left[\begin{array}{ccccc}{\delta }_{\mathrm{fr}}\hfill & 0& {\displaystyle \frac{1}{J_{\mathrm{eq}}}}(r{\tau }_{\mathrm{md}}-{\tau }_{\mathrm{f}})& 0& \hfill 0\end{array}\right]}^{\mathrm{T}}$.

The main parameters of the above SBW system model are shown in

Table 1. Model parameters of SBW system.

Parameters	Values	Parameters	Values
Jf (kg·m2)	2.6	Pn	4
Bf (Nms/rad)	12	Ψf0 (Wb)	0.0275
Jm (kg·m2)	0.02129	Δia (A)	−0.08
Bm (Nms/rad)	0.038	Δib (A)	0.06
τfs (Nm)	2.68	Iz (kg·m2)	1300
r	200	m (kg)	2000
N	12	Vx (m/s)	35
l (m)	0.039	k1, k2 (N/rad)	45,000
a (m)	1.2	τmd6	0.022
b (m)	1.05	τmd12	0.005

.

3. Design of the Controller

3.1. GCC

The state equation of the system with linear uncertainties can be expressed in the following form [35]:

$$\dot{x}(t)=(A+\Delta A)x(t)+(B+\Delta B)u_{\mathrm{g}}(t)+\omega ,x(0)=x_0$$

(19)

where ΔA and ΔB represent the uncertainties of the system state matrix and the control input matrix, respectively; u_g is the guaranteed performance control input; and ω denotes the system disturbance, including the nonlinear torque ripple disturbance of the steering motor and the front wheel rotational resistance torque.

ΔA, ΔB and ω are norm-bounded and have the following form:

$$\left[\begin{array}{ll}\Delta A\hfill & \Delta B\hfill \end{array}\right]=DF(t)\left[\begin{array}{ll}{E}_1\hfill & E_2\hfill \end{array}\right]$$

(20)

where D, E₁ and E₂ are known constant matrices of appropriate dimensions, which reflect the structural information of the uncertainties. F(t) ∈ R^i×j is an unknown matrix, which can be time-varying and satisfies:

$$F^{\mathrm{T}}(t)F(t)\le I$$

(21)

In an actual SBW system, the nominal model parameters are subject to time-varying perturbations due to various physical factors. Since a linear model cannot fully capture the nonlinear friction components within the reducer, and the viscosity of the lubricating oil is significantly affected by temperature variations, the equivalent viscous damping exhibits considerable uncertainty. Additionally, the tire cornering stiffness is highly sensitive to environmental conditions, including tire inflation pressure fluctuations, progressive tread wear, and temperature, all leading to significant variations. These parameter uncertainties have a substantial impact on system performance. Considering the uncertainty perturbations of the equivalent viscous damping and tire cornering stiffness, as well as the torque fault of the actuator motor, assume that the uncertainty component of the equivalent viscous damping is ΔB_eq, the uncertainty component of the front wheel cornering stiffness is Δk₁, the uncertainty component of the rear wheel cornering stiffness is Δk₂, and the torque fault factor is ρ defined in the range ρ ∈ (0, 1). Then, J_eq0, B_eq0 and u_g can be expressed as

$$\left\{\begin{array}{l}{B}_{\mathrm{eq}}=B_{\mathrm{eq}0}+\Delta B_{\mathrm{eq}}\\ k_1=k_{10}+\Delta k_1\\ k_2=k_{20}+\Delta k_2\\ u_{\mathrm{g}}=(1-\rho )u_{\mathrm{c}}\end{array}\right.$$

(22)

where B_eq0 is the ideal viscous damping, k₁₀ is the ideal front wheel cornering stiffness, k₂₀ is the ideal rear wheel cornering stiffness, and u_c is the ideal input torque.

Substituting Equation (22) into (18), the matrices in Equations (19) and (20) can be obtained as follows:

$$A=\left[\begin{array}{ccccc}0\hfill & -1& 0& 0& 0\\ 0\hfill & 0& 1& 0& 0\\ 0\hfill & {\displaystyle \frac{-k_{10}l}{J_{\mathrm{eq}}}}& {\displaystyle \frac{-B_{\mathrm{eq}0}}{J_{\mathrm{eq}}}}& {\displaystyle \frac{k_{10}l}{J_{\mathrm{eq}}}}& {\displaystyle \frac{a{k}_{10}l}{J_{\mathrm{eq}}{V}_{\mathrm{x}}}}\\ 0\hfill & {\displaystyle \frac{k_{10}}{m{V}_{\mathrm{x}}}}& 0& -{\displaystyle \frac{k_{10}+k_{20}}{m{V}_{\mathrm{x}}}}& {\displaystyle \frac{b{k}_{20}-a{k}_{10}}{m{V}_{\mathrm{x}}^2}}-1\\ 0\hfill & {\displaystyle \frac{a{k}_{10}}{I}}& 0& {\displaystyle \frac{b{k}_{20}-a{k}_{10}}{I}}& -{\displaystyle \frac{a^2{k}_{10}+b^2{k}_{20}}{I{V}_{\mathrm{x}}}}\end{array}\right], B={\left[\begin{array}{ccccc}0\hfill & 0& {\displaystyle \frac{r}{J_{\mathrm{eq}}}}& 0& \hfill 0\end{array}\right]}^{\mathrm{T}},$$

$$E_1=\left[\begin{array}{ccccc}0\hfill & {\displaystyle \frac{-\Delta k_{1}l}{J_{\mathrm{eq}}}}& {\displaystyle \frac{-\Delta B_{eq}}{J_{\mathrm{eq}}}}& {\displaystyle \frac{\Delta k_{1}l}{J_{\mathrm{eq}}}}& {\displaystyle \frac{a\Delta k_{1}l}{J_{\mathrm{eq}}{V}_{\mathrm{x}}}}\\ 0\hfill & {\displaystyle \frac{\Delta k_1}{m{V}_{\mathrm{x}}}}& 0& -{\displaystyle \frac{\Delta k_1+\Delta k_2}{m{V}_{\mathrm{x}}}}& {\displaystyle \frac{b\Delta k_2-a\Delta k_1}{m{V}_{\mathrm{x}}^2}}\\ 0\hfill & {\displaystyle \frac{a\Delta k_1}{I}}& 0& {\displaystyle \frac{b\Delta k_2-a\Delta k_1}{I}}& -{\displaystyle \frac{a^2\Delta k_1+b^2\Delta k_2}{I{V}_{\mathrm{x}}}}\end{array}\right], E_2={\left[\begin{array}{ccc}{\displaystyle \frac{-r\rho }{J_{\mathrm{eq}}}}\hfill & 0& \hfill 0\end{array}\right]}^{\mathrm{T}},$$

$$\omega ={\left[\begin{array}{ccccc}{\delta }_{\mathrm{fr}}\hfill & 0& {\displaystyle \frac{1}{J_{\mathrm{eq}}}}(r{\tau }_{\mathrm{md}}-{\tau }_{\mathrm{f}})& 0& \hfill 0\end{array}\right]}^{\mathrm{T}}.$$

For system (19), the quadratic performance index is defined as

$$J={\displaystyle {\int }_0^{\infty }\left[x^{\mathrm{T}}(t)Qx(t)+u_{\mathrm{g}}^{\mathrm{T}}(t)R{u}_{\mathrm{g}}(t)\right]} \mathrm{d}t$$

(23)

where Q and R are given symmetric positive definite weighting matrices.

Lemma 1 [35].

For any matrices Z and Y with appropriate dimensions, and for any positive scalar τ, the following inequality holds:

$$Z^{\mathrm{T}}Y+Y^{\mathrm{T}}Z\le \tau Z^{\mathrm{T}}Z+{\displaystyle \frac{1}{\tau }}{Y}^{\mathrm{T}}Y$$

(24)

Theorem 1.

For the uncertain system (19) and the performance index (23), if there exists a matrix K and symmetric positive definite matrices P, along with positive scalars ε_g, ϑ_g, τ_g and γ_g, such that for all admissible uncertainties, the following holds:

$$Q+K^{\mathrm{T}}RK+P\left(A+BK\right)+{\left(A+BK\right)}^{\mathrm{T}}P+{\tau }_{\mathrm{g}}{P}^2+{\displaystyle \frac{{\epsilon }_{\mathrm{g}}}{{\tau }_{\mathrm{g}}}}I+{\vartheta }_{\mathrm{g}}PD{D}^{\mathrm{T}}P+{\displaystyle \frac{1}{{\vartheta }_{\mathrm{g}}}}{\left(E_1+E_{2}K\right)}^{\mathrm{T}}\left(E_1+E_{2}K\right)<0$$

(25)

$${\gamma }_{\mathrm{g}}-{\gamma }_{\mathrm{g}}{x}_0^{\mathrm{T}}P\gamma x_0>0$$

(26)

then u_g(t) = Kx(t) is an optimal GCC law for system (19), and the corresponding upper bound of the system performance index is J ≤ 1/γ_g.

Stability Proof: Based on the optimal guaranteed performance control law and combining Equations (19) and (20), the corresponding closed-loop system is obtained as

$$\dot{x}(t)=[A+BK+DF(E_1+E_{2}K)]x(t)+\omega$$

(27)

The Lyapunov function is chosen as V(t) = x^TPx. The positive definiteness of the matrix P implies that the Lyapunov function V(t) > 0 is positive definite. Differentiating V(t) yields

$$\dot{V}(t)=x^{\mathrm{T}}\left\{P[A+BK+DF(E_1+E_{2}K)]+{[A+BK+DF(E_1+E_{2}K)]}^{\mathrm{T}}P\right\}x+x^{\mathrm{T}}P\omega +{\omega }^{\mathrm{T}}Px$$

(28)

According to Lemma 1, it follows that

$$\dot{V}(t)\le x^{\mathrm{T}}\left\{P(A+BK)+{(A+BK)}^{\mathrm{T}}P+{\tau }_{\mathrm{g}}{P}^2+{\vartheta }_{\mathrm{g}}PD{D}^{\mathrm{T}}P+{\displaystyle \frac{1}{{\vartheta }_{\mathrm{g}}}}{(E_1+E_{2}K)}^{\mathrm{T}}(E_1+E_{2}K)\right\}x+{\displaystyle \frac{1}{{\tau }_{\mathrm{g}}}}{\omega }^{\mathrm{T}}\omega$$

(29)

Assuming that the system disturbance ω satisfies ω^Tω < ε_gx^Tx and ε > 0, it can be obtained that

$$\dot{V}(t)<x^{\mathrm{T}}\left\{P(A+BK)+{(A+BK)}^{\mathrm{T}}P+{\tau }_{\mathrm{g}}{P}^2+{\displaystyle \frac{{\epsilon }_{\mathrm{g}}}{{\tau }_{\mathrm{g}}}}I+{\vartheta }_{\mathrm{g}}PD{D}^{\mathrm{T}}P+{\displaystyle \frac{1}{{\vartheta }_{\mathrm{g}}}}{(E_1+E_{2}K)}^{\mathrm{T}}(E_1+E_{2}K)\right\}x$$

(30)

From Equations (25) and (30), it can be obtained that for all admissible uncertainties,

$$\dot{V}(t)<-x^{\mathrm{T}}(Q+K^{\mathrm{T}}RK)x<0$$

(31)

According to the Lyapunov stability theory, the closed-loop system shown in Equation (27) is robustly asymptotically stable.

Integrating Equation (31) and utilizing the asymptotic stability of the system, it can be obtained that

$$J={\displaystyle {\int }_0^{\infty }\left[x^{\mathrm{T}}(t)Qx(t)+u_{\mathrm{g}}^{\mathrm{T}}(t)R{u}_{\mathrm{g}}(t)\right]} \mathrm{d}t\le V(0)=x_0^{\mathrm{T}}P{x}_0$$

(32)

Letting V(0) < 1/γ_g yields

$$V(0)=x_0^{\mathrm{T}}P{x}_0<{\displaystyle \frac{1}{{\gamma }_{\mathrm{g}}}}$$

(33)

From Equations (30), (31) and (33), we can obtain Equations (25) and (26), thus proving Theorem 1.

Lemma 2 [35].

Schur Complement Property:

For a given symmetric matrix $\left.S=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right.\right]$, where S₁₁ is a square matrix, the following three conditions are equivalent:

$$\left\{\begin{array}{c}S<0\\ S_{11}<0,S_{22}-S_{12}^{\mathrm{T}}{S}_{11}^{-1}{S}_{12}<0\\ S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{12}^{\mathrm{T}}<0\end{array}\right.$$

(34)

According to the Lemma 2, Equations (25) and (26) transform into

$$\left[\begin{array}{ccccc}P\left(A+BK\right)+{\left(A+BK\right)}^{\mathrm{T}}P+{\tau }_{\mathrm{g}}{P}^2+{\vartheta }_{\mathrm{g}}PD{D}^{\mathrm{T}}P& {(E_1+E_{2}K)}^{\mathrm{T}}& I& K^{\mathrm{T}}& \hfill I\\ E_1+E_{2}K& -{\vartheta }_{\mathrm{g}}I& 0& 0& \hfill 0\\ I& 0& -Q^{-1}& 0& \hfill 0\\ K& 0& 0& -R^{-1}& \hfill 0\\ I& 0& 0& 0& \hfill -{\displaystyle \frac{{\tau }_{\mathrm{g}}}{{\epsilon }_{\mathrm{g}}}}I\end{array}\right]<0$$

(35)

$$\left[\begin{array}{cc}{\gamma }_{\mathrm{g}}& {\gamma }_{\mathrm{g}}{x}_0^{\mathrm{T}}\\ {\gamma }_{\mathrm{g}}{x}_0& P^{-1}\end{array}\right]>0$$

(36)

Pre-multiplying and post-multiplying inequality (35) by the matrix diag{P⁻¹, I, I, I}, and letting X = P⁻¹ and W = KP⁻¹, we obtain

$$\left[\begin{array}{ccccc}AX+BW+{(AX+BW)}^{\mathrm{T}}+{\tau }_{\mathrm{g}}I+{\vartheta }_{\mathrm{g}}D{D}^{\mathrm{T}}& {(E_{1}X+E_{2}W)}^{\mathrm{T}}& X& W^{\mathrm{T}}& \hfill X\\ E_{1}X+E_{2}W& -{\vartheta }_{\mathrm{g}}I& 0& 0& \hfill 0\\ X& 0& -Q^{-1}& 0& \hfill 0\\ W& 0& 0& -R^{-1}& \hfill 0\\ X& 0& 0& 0& \hfill -{\displaystyle \frac{{\tau }_{\mathrm{g}}}{{\epsilon }_{\mathrm{g}}}}I\end{array}\right]<0$$

(37)

$$\left[\begin{array}{cc}{\gamma }_{\mathrm{g}}& {\gamma }_{\mathrm{g}}{x}_0^{\mathrm{T}}\\ {\gamma }_{\mathrm{g}}{x}_0& X\end{array}\right]>0$$

(38)

Given the system matrices A, B, D, F, E₁ and E₂, the control matrices Q and R, and the torque peak value u_max, with ε_g, ϑ_g, τ_g and γ_g set to be greater than zero, the matrices X and K are solved such that the index 1/γ_g is minimized while satisfying the constraints of inequalities (37) and (38). Finally, the optimal state-feedback GCC law is obtained.

3.2. ISMC

Within the GCC framework, the system achieves high control accuracy. To further improve the response speed, an ISMC is incorporated on the basis of guaranteed performance control to regulate the response speed.

To improve the convergence speed, an integral sliding mode surface is introduced, which eliminates the reaching phase from the initial moment, and the integral sliding surface is designed as

$$s=\dot{e}\begin{array}{cc}+ce+\mu {\displaystyle \int e \mathrm{d}t}& \end{array}c>0,\mu >0$$

(39)

According to Equations (15) and (16), let d₁ = τ_f − rτ_md. Assuming that the upper bound of d₁ is D₁, it can be obtained that

$$\dot{s}=\ddot{e}+c\dot{e}+\mu e=({\ddot{\delta }}_{\mathrm{fr}}-{\ddot{\delta }}_{\mathrm{f}})+c\dot{e}+\mu e={\ddot{\delta }}_{\mathrm{fr}}-{\displaystyle \frac{1}{J_{\mathrm{eq}}}}\left[ru-B_{\mathrm{eq}}{\dot{\delta }}_{\mathrm{f}}-{\tau }_{\mathrm{e}}+-d_1\right]+c\dot{e}+\mu e$$

(40)

where c and μ are sliding surface parameters, both greater than zero.

The following reaching law is adopted:

$$\dot{s}=-\mathrm{ksat}(s)$$

(41)

where k is the control parameter, which is greater than zero, and sat() is the saturation function.

To verify the system stability, the Lyapunov function is designed as follows:

$$V_1={\displaystyle \frac{1}{2}}{s}^2$$

(42)

Differentiating Equation (42) yields

$${\dot{V}}_1=s\dot{s}=s\left\{{\ddot{\delta }}_{\mathrm{fr}}-{\displaystyle \frac{1}{J_{\mathrm{eq}}}}\left[r{u}_{\mathrm{s}}-B_{\mathrm{eq}}{\dot{\delta }}_{\mathrm{f}}-{\tau }_{\mathrm{e}}-d_1\right]+c\dot{e}+\mu e\right\}$$

(43)

The control law is designed as follows:

$$u_s={\displaystyle \frac{J_{\mathrm{eq}}}{r}}[{\displaystyle \frac{B_{\mathrm{eq}}}{J_{\mathrm{eq}}}}{\widehat{\dot{\delta }}}_{\mathrm{f}}+{\displaystyle \frac{1}{J_{\mathrm{eq}}}}{\tau }_{\mathrm{e}}+{\ddot{\delta }}_{\mathrm{fr}}+c\dot{e}+\mu e]+ksat(s)$$

(44)

Substituting Equation (44) into (43) yields

$${\dot{V}}_1={\displaystyle \frac{1}{J_{\mathrm{eq}}}}s(d_1-ksat(s))\le {\displaystyle \frac{1}{J_{\mathrm{eq}}}}\left|s\right|D_1-k\left|s\right| \le {\displaystyle \frac{1}{J_{\mathrm{eq}}}}\left|s\right|(D_1-k)$$

(45)

Taking D₁ < k, it can be obtained that when $\dot{V}<0$, the system satisfies the Lyapunov stability condition.

Therefore, the cooperative fault-tolerant control law of GCC+ISMC is given by

$$u=u_{\mathrm{g}}+u_{\mathrm{s}}$$

(46)

Under the cooperative framework of GCC and ISMC, the system can fully leverage the fast response capability of the ISMC during the initial response phase to rapidly track the desired angle, and subsequently rely on the inherent characteristics of GCC to achieve stable fault tolerance against controller failures. This cooperative mechanism enables the system to prioritize the fast response of ISMC during the transient phase while exploiting the robust guaranteed cost advantage of GCC during the steady-state phase, thereby achieving a unification of fast response and stable fault tolerance throughout the entire process.

4. Example Illustration

The designed control strategy was simulated and verified in MATLAB/SIMULINK R2022a, with the controller parameters shown in

Table 2. Control parameters.

Parameters	Values
D	[0, 0, 0; 0, 0, 0; 1, 0, 0; 0, 1, 0; 0, 0, 1]
F	[1, 0, 0; 0, 1, 0; 0, 0, 1]
Q	[106, 0, 0, 0, 0; 0, 10−2, 0, 0, 0; 0, 0, 10, 0, 0; 0, 0, 0, 0, 10]
R	10−6
ρ	0.5
c	100
μ	100
k	10,000
ca	100
μa	100
α	10,000
β	5

.

The simulation setup is as follows:

(1)

The simulation time interval is set to 0.001 s.

(2)

The uncertainty perturbation of the viscous damping B_eq is set to ±5%, reflecting the equivalent modeling error introduced by unmodeled nonlinear friction components under normal operating temperatures and lubrication conditions.

(3)

The uncertainty perturbations of the front and rear wheel cornering stiffness k₁ and k₂ are set to ±10%, reflecting moderate variations arising from normal tread wear, mild tire inflation pressure fluctuations, and minor changes in road surface friction coefficient during regular highway driving.

(4)

A 50% torque failure fault of the steering motor is considered, representing a severe 50% loss of torque, which is injected from the eighth second of the simulation; that is, ρ = 0 (0 < t < 8) and ρ = 0.5 (t ≥ 8), and the corresponding effective torque of the faulty motor is τ_mr (1 − ρ). This value is deliberately chosen to evaluate the fault tolerance capability of the proposed controller under demanding conditions.

It is worth noting that the parameter perturbation bounds are intentionally kept at moderate, physically realistic levels. This ensures that the control performance improvements can be unambiguously attributed to the fault-tolerant mechanism of the proposed GCC+ISMC scheme, rather than being confounded by the effects of simultaneous large-scale parameter variations.

By employing the Mincx solver in the LMI toolbox, the guaranteed cost feedback gain matrix is obtained as K= [1.0685 −0.5024 −0.0743 0.0014 −0.0080] × 10³. Based on this offline fixed gain K, the real-time GCC law u_g is obtained by multiplying K with the system state vector x(t). Simultaneously, the ISMC law u_s is computed online, and the cooperative fault-tolerant control law u is finally derived.

To demonstrate the superiority of the proposed GCC+ISMC, comparative analyses are conducted against a standalone GCC and an AISMC. The control law of AISMC is as follows:

$$\left\{\begin{array}{l}{u}_{\mathrm{s}}={\displaystyle \frac{J_{\mathrm{eq}}}{r}}[{\displaystyle \frac{B_{\mathrm{eq}}}{J_{\mathrm{eq}}}}{\widehat{\dot{\delta }}}_f+{\displaystyle \frac{1}{J_{\mathrm{eq}}}}{\tau }_{\mathrm{e}}+{\ddot{\delta }}_{\mathrm{fr}}+c_{\mathrm{a}}\dot{e}+{\mu }_{\mathrm{a}}e]+k_{\mathrm{a}}sat(s)\\ k_{\mathrm{a}}=\alpha \left|s\right|+\beta \end{array}\right.$$

(47)

where c_a and μ_a are sliding surface parameters, both greater than zero, k_a is the adaptive gain, and α and β are the gain control parameters, both of which are greater than zero.

The control performance of the SBW system is evaluated under three typical steering maneuvers, step steering, sinusoidal steering, and double lane change steering, and illustrated in Figure 3, Figure 4 and Figure 5, respectively.

As depicted in Figure 3a,b, all three control strategies can effectively track the step steering reference signal. The response speed of the proposed synergistic GCC+ISMC is comparable to that of the AISMC, both outperforming the standalone GCC. At the instant of fault occurrence, the tracking accuracy of the synergistic GCC+ISMC remains virtually unaffected, whereas the tracking errors for both standalone GCC and AISMC increase significantly. However, while the GCC can rapidly recover to its nominal tracking level, the error convergence process of the AISMC is relatively slow. Furthermore, Figure 3c illustrates that to cope with the torque failure fault, all three strategies actively regulate the control torque at the moment of the fault. Notably, the synergistic GCC+ISMC strategy achieves a significantly shorter adjustment time and a smoother transition compared to the other two strategies.

For the sinusoidal steering maneuver, Figure 4a demonstrates that all three strategies successfully track the reference signal. Here, the synergistic GCC+ISMC exhibits superior dynamic response characteristics compared to the standalone GCC. As shown in Figure 4b, the tracking errors of both GCC+ISMC and AISMC are substantially smaller than that of the GCC. Upon fault injection, the tracking precision of GCC+ISMC and GCC sustains minimal impact, whereas the AISMC presents more pronounced error fluctuations, underscoring the stronger robustness of the former two under fault conditions. It can be observed from Figure 4c that the GCC+ISMC achieves the fastest torque adjustment speed in response to faults, followed by the GCC, with the AISM being the slowest. This further corroborates the superiority of the proposed strategy in fault response.

Under the double lane change maneuver, the performance of the three control strategies are consistent with those observed in the sinusoidal maneuver. In terms of response speed, Figure 5a indicates a marked improvement for the synergistic GCC+ISMC over the standalone GCC. Regarding tracking accuracy, Figure 5b reveals that the errors associated with GCC+ISMC and AISMC are smaller than those of the GCC. Crucially, at the moment of fault injection, the error of the GCC+ISMC remains remarkably stable, whereas the error of the AISM notably increases. Moreover, regarding torque allocation, Figure 5c demonstrates that the GCC+ISMC executes the fastest adjustment speed when responding to faults.

To further quantify the control performance, the Maximum Absolute Error (MAE) and the Root-Mean-Square Error (RMSE) of the tracking errors under the step, sinusoidal, and double lane change maneuvers are calculated as quantitative evaluation metrics, with the results presented in Figure 6.

As clearly evidenced by Figure 6, the proposed synergistic GCC+ISMC strategy yields the smallest values for both error metrics. Specifically, the RMSE is reduced by an average of approximately 35% compared to the AISMC. This solidly verifies that the synergistic GCC-ISM control possesses the highest tracking accuracy.

In summary, the proposed synergistic GCC+ISMC strategy seamlessly combines rapid dynamic response, high-precision tracking, and robust fault-tolerance, effectively satisfying the stringent control demands across varied steering conditions.

5. Conclusions

Aiming at the challenge of synergistically enhancing the control performance of SBW systems subject to multiple uncertainties, this paper proposes a synergistic fault-tolerant control strategy integrating GCC+ISMC. The main contributions and results are summarized as follows:

(1)

A dynamic model of the SBW system incorporating multiple uncertain factors was formulated, providing an accurate mathematical description of the controlled plant for subsequent controller design;

(2)

By deriving the GCC law based on LMI solutions, strict constraints were imposed on the system’s tracking accuracy and robustness, ensuring that the theoretical upper bound of the steady-state performance remains controllable under the influence of uncertainties.

(3)

By integrating an ISMC mechanism, the system’s dynamic response speed was significantly accelerated without sacrificing steady-state accuracy. This resolves the intrinsic trade-off between precision and rapidity common in existing methods, formulating a synergistic fault-tolerant architecture characterized by both high accuracy and fast response.

(4)

Simulations conducted under typical steering maneuvers demonstrated that the proposed strategy effectively suppresses the adverse effects of multiple uncertainties across various fault scenarios. Quantitative analysis reveals that the tracking RMSE of GCC+ISMC is reduced by approximately 35% compared to AISMC. This thoroughly corroborates the comprehensive advantages of the proposed method in tracking accuracy, response speed, and robustness.

In conclusion, the synergistic GCC+ISMC proposed in this paper effectively addresses the challenges posed by multiple uncertainties in SBW systems. It successfully achieves a synergistic enhancement of control performance while guaranteeing functional safety, thereby providing robust theoretical support and technical reference for the reliable operation of intelligent vehicles under complex environments. Future work will focus on Hardware-in-the-Loop experimental validation and the coordinated control of multiple chassis actuators.