Design of a fractional-order sliding mode controller for lane- keeping in autonomous driving

Abstract

An improved fractional-order integral sliding mode control (FOISMC) method is proposed in this study to address the performance degradation of lane-keeping control for intelligent vehicles under complex conditions. In this approach, a fractional-order integral sliding surface is constructed by introducing a composite tracking error e, and is combined with an improved fractional-order variable power reaching law to design a lane-keeping controller based on FOISMC. Subsequently, the Lyapunov stability theory is employed to analyze the system’s stability, and a proof is provided to demonstrate that the system states can reach the fractional-order integral sliding surface in finite time. Finally, a co-simulation model was established using MATLAB/Simulink and CarSim to evaluate the proposed control strategy. The results demonstrate that, even under high-speed conditions with crosswind disturbances, the FOISMC outperforms both TISMC and FOPID. Specifically, the maximum lateral position error is reduced by 10.1% and 68.7%, respectively, while the root mean square error is decreased by 8% and 60.9%. These results confirm that the proposed approach not only achieves a significant improvement in control accuracy but also exhibits strong robustness.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-18883-1.

Introduction

The development of autonomous driving technology has not only reshaped our understanding of mobility but also demonstrated unprecedented influence in enhancing logistics efficiency, reducing transportation costs, and optimizing urban traffic structures¹. As a critical component of autonomous vehicles, lane-keeping assistance control dynamically tracks lane markings using onboard cameras. When a deviation from the intended trajectory is detected, the system immediately applies steering corrections to guide the vehicle back to the desired path, effectively preventing unintended lane departure. To improve lane-keeping performance, various control strategies have been proposed, such as LQR^2–4 and Model Predictive Control^5,6. However, due to fluctuations in longitudinal velocity, variations in road curvature, and changes in measurement conditions, the vehicle system often exhibits strong nonlinearity and model uncertainty⁷which may adversely affect the effectiveness of optimal controllers and MPC-based approaches. In contrast, Sliding Mode Control offers strong adaptability and robustness to modeling inaccuracies and external disturbances, owing to its lower dependency on accurate system models and relatively simple design procedure^8–12.

The application of Sliding Mode Control in lane-keeping systems has proven effective in mitigating the effects of system parameter variations and external disturbances^13–15. Reference¹⁶ addressed the problem of ship course-keeping and proposes as well as compares various nonlinear control strategies, including integral sliding mode (ISMC) control and integral super-twisting sliding mode control. The study demonstrates that integral super-twisting sliding mode control (IST-SMC) offers distinct advantages in terms of convergence speed, response characteristics, and energy consumption reduction. But The sliding surface in ISMC is typically designed using integer-order derivatives, which restricts the flexibility of dynamic response tuning. Compared with integer-order operators, fractional-order calculus operators enable a smoother temporal release and adjustment of control energy, thereby reducing chattering and overshoot in control responses^17,18. Fractional calculus has been widely applied in various fields, such as control systems^19–22 and epidemic spreading²³. Podlubny, in Reference²⁴ pioneered the fractional-order controller, which generalizes the integral and derivative orders to arbitrary real values. This extension introduces additional degrees of freedom, thereby enhancing the controller’s flexibility and enabling more precise shaping of system dynamics. Nevertheless, owing to its inherently model-free nature and the absence of consideration for system-specific characteristics, its robustness to external disturbances remains limited. To address these shortcomings, fractional-order sliding mode control (FOSMC) has been proposed, which integrates the flexibility of fractional calculus with the inherent robustness of sliding mode control. This combination provides improved disturbance rejection and more reliable dynamic performance^25,26.

In²⁷ this method was extended to the trajectory tracking of unmanned aerial vehicles (UAVs). The study demonstrated that the approach enabled high-precision trajectory tracking even under complex environmental conditions. To further enhance the convergence speed and dynamic performance of fractional-order sliding mode control, Reference²⁸ proposed an Improved Adaptive Fractional Order Fast Integral Terminal Sliding Mode Control (IAFOTSMC) method applied to quadrotor unmanned aerial vehicles. This approach employs a fractional-order fast integral sliding surface, which ensures finite-time stability of the tracking error while significantly improving the system’s robustness against time-varying disturbances. Reference²⁹ proposed an Embedded Adaptive Fractional-Order Sliding Mode Control (AFOSMC) to address the challenges in controlling nonlinear fractional-order systems. Reference³⁰ introduces an event-triggered fractional-order sliding mode control strategy, which utilizes event-triggered mechanisms to avoid frequent periodic executions, thus reducing computational costs while maintaining satisfactory control performance. Although fractional-order sliding mode control has achieved promising results in the aforementioned fields, chattering remains a major drawback of sliding mode control and its fractional variants. Reference³¹ proposed a “chattering-free” adaptive fuzzy sliding mode control approach for synchronizing incommensurate fractional-order chaotic systems. The core of this method is leveraging the universal approximation capability of fuzzy logic systems to estimate unknown dynamics, which allows for the construction of an inherently smooth and continuous control law. This approach eliminates the chattering mechanism by design while still achieving practical finite-time synchronization. These studies demonstrate that fractional-order sliding mode strategies, due to their unique fractional differential properties, not only effectively alleviate high-frequency chattering but also improve the dynamic response characteristics of the system. However, to date, no research has been conducted on the application of fractional-order sliding mode control in lane-keeping systems.

To achieve finite-time convergence of tracking errors and enhance the control performance of the system, this paper proposes a lane-keeping controller based on fractional-order integral sliding mode control. The main contributions of this article are as follows:

• (i)An improved fractional-order integral sliding mode control (FOISMC) method is proposed to address the lane-keeping control problem of intelligent vehicles under complex driving conditions. By introducing a composite error e, a fractional-order integral sliding surface function is constructed. And a novel fractional-order variable power reaching law is proposed to improve the dynamic performance of system state convergence to the sliding surface and to effectively suppress chattering, thereby enhancing the smoothness and stability of the system.
• (ii)Based on the Lyapunov stability theory, this paper provides a rigorous mathematical proof that the designed FOISMC system is finite-time convergent, ensuring the stability of the closed-loop system. The contribution of this analysis is not merely to verify the method’s theoretical feasibility, but more importantly, to establish that the control strategy is mathematically robust and reliable, laying a necessary theoretical foundation for its application in safety-critical domains such as autonomous driving.
• (iii)Through rigorous co-simulation under harsh conditions including crosswind disturbances, this work verifies the proposed FOISMC’s outstanding robustness and practical superiority. The results confirm that even against external disturbances, the strategy translates its theoretical advantages into quantifiable performance gains, demonstrating its strong potential for engineering applications.

Vehicle dynamics model

Two degrees of freedom vehicle dynamics model

A two degree of freedom (2-DOF) bicycle model is employed to describe the lateral dynamics of the vehicle. This model is derived by combining the two wheels on each axle into a single equivalent wheel along the vehicle’s centerline, allowing for the capture of fundamental lateral vehicle dynamics while reducing model complexity³². As shown in Fig. 1, a three-dimensional vehicle body coordinate system (x, y,z) is established with the vehicle’s center of mass as the origin, the longitudinal forward direction as the x-axis, the lateral direction as the y-axis, and the z-axis determined according to the right-hand rule. In Fig. 1, F_f and F_r are the lateral tire forces at the front and rear wheels, ψ is the yaw angle, δ is the front wheel steering angle, β is the sideslip angle at the center of mass, v is the velocity at the vehicle’s center of mass, v_x is the longitudinal velocity, l_f and l_r are the distances from the vehicle’s center of gravity to the front and rear axles, respectively, θ_vf and θ_vr are the velocity angles of the front and rear wheels, respectively. The lateral and yaw motions of road vehicles are described as follows³³:

1

2

Fig. 1.

Two degree of freedom vehicle dynamics model.

where m denotes the vehicle mass, represents the lateral acceleration, and is the yaw rate. is the moment of inertia about the z-axis, and represents the lateral slip acceleration.

When the tire slip angles are relatively small, the lateral tire forces can be expressed as³⁰:

3

4

where C_f and C_r are the total cornering stiffness stiffnesses of the front and rear wheels, respectively.

The velocity angles of the front and rear wheels can be expressed as follows³³:

5

6

where denotes the lateral velocity.

Substituting (3), (4), (5), and (6) into Eqs. (1) and (2), the resulting dynamic equations can be obtained as follows:

7

Tracking error model

For lane-keeping steering control, adopting a road-aligned error dynamic model is more effective. In³³the following definition is provided:

8

9

10

11

Where, denotes the desired yaw rate, R is the road curvature radius, represents the lateral position error, and denotes the yaw angle error.

By substituting (8) and (9) into Eqs. (1) and (2), the state equations of the tracking error variables can be derived as follows³³:

12

where

 

where

 

Fractional-order integral sliding mode control strategy

Fractional calculus

The study of computing non-integer order derivatives and integrals of functions is referred to as fractional calculus³⁴. Fractional calculus can be defined through various approaches, including the Riemann-Liouville method.

Definition 1³⁵

The fundamental definition of the fractional-order calculus operator s given as follows:

13

Where α denotes the order of the fractional derivative, . t denotes the independent variable, and represents the lower limit of this variable.

Definition 2³⁵

The Riemann–Liouville fractional-order derivative is defined as follows:

14

where. denotes the Gamma function, .

Property 1³⁶

Under the Riemann-Liouville definition of the fractional derivative, the fractional-order sign function satisfies the following relation:

15

where , .

Property 2³⁷

For fractional-order derivatives, the following equality holds:

16

17

Note

This article will be abbreviated as .

Design of a lane-keeping controller based on fractional-order integral sliding mode control

Figure 2 illustrates the structure of the intelligent vehicle lane-keeping control system. Based on the reference path and the current vehicle state information, the error calculation module calculates the lateral position error and yaw angle error , and other relevant variables, which are then transmitted to the FOISMC lane-keeping controller. The FOISMC controller calculates the front wheel steering angle δ and sends it to the Carsim vehicle model, thereby enabling real-time control of the vehicle.

Fig. 2.

The structure diagram of lane-keeping control.

To achieve better control of the lateral position error and the yaw angle error , a composite error e is introduced as follows:

18

Where are the weighting parameters for the and the , respectively, . This design couples the two most relevant tracking errors into a single control variable, enabling the sliding mode controller to simultaneously:

1. Enhance transient response – By incorporating yaw angle error, the controller can initiate corrective steering earlier based on the predicted future lateral deviation, leading to faster path convergence and reduced overshoot in lane-change or curve-entry scenarios.

2. Improve steady-state accuracy – The yaw angle term ensures that the vehicle’s orientation is aligned with the road even when lateral error is small, effectively eliminating residual offsets and maintaining the lane centerline over extended periods.

3. Increase robustness under disturbances – The integrated formulation allows the control law to react to both instantaneous displacement errors and orientation trends, which is especially beneficial when facing disturbances such as crosswinds.

A fractional-order integral sliding surface function is constructed by introducing a fractional-order calculus operator as follows:

19

where C₁ and C₂ are the gains of the fractional derivative and integral terms, respectively. α and λ the orders of the fractional derivatives. ,; is the fractional-order integral term, which serves to eliminate the steady-state error.

Taking the derivative of (19) yields:

20

Substituting (12) and (18) into (20) gives:

21

where:

To enhance the dynamic performance of system state convergence toward the sliding surface and to better suppress chattering, thereby improving the smoothness and stability of the system, an improved fractional-order variable power reaching law is proposed:

22

where k₁, ε and k₂ are the gains of the reaching law, l is the variable exponent term, β is the order of the fractional derivative, f and h are the coefficients of the variable exponent term l. k₁ > 0,. By introducing a variable exponent term, the system can adaptively adjust the exponent parameter in the reaching law according to different values of l.

By combining (21) and (22), the lane-keeping control law can be derived as:

23

Stability analysis of control systems

Theorem 1

Under the action of the lane-keeping control law (23), for the system described by (12), and using the fractional-order integral sliding surface defined by (19), the controlled system can achieve finite-time stability.

Proof Take the Lyapunov function as:

24

Differentiating the above equation and combining it with (20) yields:

25

Substituting (23) into (25) yields:

26

Since , therefore

27

Furthermore, given that ³⁸ it follows that:

28

When s > 0,

29

When s < 0,

30

The function if and only if s = 0. Consequently, satisfying the Lyapunov stability condition, and thus the controlled system can achieve finite-time stability. □.

Theorem 2

For the controlled system described by (12), the adoption of the fractional-order variable power reaching law (22) ensures that the system state can reach the fractional-order integral sliding surface within a finite time , where.

Proof From (28), it follows that:

31

32

Therefore,

33

By integrating (33) from 0 to t_r, can obtain:

34

Therefore, the time at which the system states reach the fractional-order integral sliding mode surface, namely the convergence time of the system state variables, is given by:

□

To further suppress chattering, the following continuous function is employed to replace the discontinuous sign function sgn(s) in the fractional-order variable power reaching law.

35

Where µ is the parameter of the continuous function , µ > 0.

The fractional-order variable power reaching law, with the sign function sgn(s) substituted by the continuous function , can be expressed as:

36

By combining (21) and (36), the updated lane-keeping control law is obtained as:

The continuous function provides a smooth transition region in the vicinity of the sliding surface, rather than the instantaneous switching characteristic of the sign function. This smooth transition effectively reduces high-frequency switching, thereby mitigating the chattering phenomenon.

By replacing the sign function sgn(s) with a continuous function , the proof of system stability is provided in the Appendix.

Simulation analysis

This paper establishes a co-simulation platform using Matlab/Simulink and Carsim, selecting a double-lane-change road with rapidly varying continuous curvature as the reference path. Simulations are conducted for vehicle driving scenarios on both low-speed slippery road and high-speed dry asphalt surfaces to verify the effectiveness of the lane-keeping controller based on FOISMC. Comparative studies are performed against controllers designed with traditional integral sliding mode control (TISMC) and fractional-order PID (FOPID) to demonstrate the superiority of the proposed FOISMC-based lane-keeping controller. In order to further evaluate the robustness of the FOISMC, TISMC, and FOPID controllers, this study, in addition to the representative conditions of low-speed slippery road and high-speed dry asphalt, introduces continuous crosswind disturbances to construct a more realistic and rigorous testing environment.

To facilitate a more comprehensive comparison of trajectory tracking performance, this paper adopts the maximum lateral position error (m), root mean square (RMS) of lateral position error (m), maximum yaw angle error (rad) and RMS of yaw angle error (rad) as performance evaluation metrics. The expressions for the RMS and the maximum value are given as follows:

where n denotes the number of samples, and can represent either the lateral position error or the yaw angle error .

The vehicle parameters and the relevant parameters of the proposed controller are listed in Tables 1 and 2, respectively.

Table 1.

Vehicle-related parameters.

Parameter	Value	Unit
m	1412	kg
I z	1536.7	kg•m2
l f	1.015	m
l r	1.895	m
C f	−148,970	N/rad
C r	−82,204	N/rad

Table 2.

Parameters of proposed controller.

Parameter	Value
	0.15, 0.4, 0.1
C1,C2, f	25, 0.1, 0.8
h, k1, k2	1.5, 2, 12
	0.001, 0.1, 1
	0.1

Condition 1: Low-speed and slippery road

Under slippery road condition with a surface adhesion coefficient of 0.55, a lane-keeping simulation was performed During a double lane change maneuver at a vehicle speed of 54 km/h. The corresponding simulation results are illustrated in Fig. 3.

Fig. 3.

The simulation results for Condition 1.

Figure 3(a) illustrates that, under road conditions, all three lane-keeping controllers maintained satisfactory trajectory tracking accuracy and stability. Among them, the FOISMC outperformed both TISMC and FOPID in terms of overall control performance, exhibiting higher tracking precision.

Figure 3(b) shows that the FOISMC controller exhibits a smaller overshoot in lateral position error compared to TISMC. This improvement is primarily attributed to the tunability of the fractional-order calculus operators in FOISMC, which enables more precise control of transient response characteristics. As shown in Fig. 3(c), the yaw angle errors associated with the FOISMC and TISMC controllers are nearly the same, both exhibiting substantially lower errors compared to the FOPID controller. Figure 3(d) illustrates the front wheel steering angles required by each controller for lane-keeping control. The steering angles generated by FOISMC and TISMC are nearly identical, and both effectively suppress oscillations. However, the FOISMC achieves a smaller maximum lateral position error than TISMC, indicating superior control accuracy. The FOPID controller exhibits noticeably larger steering angles during cornering. This may be due to a sudden increase in path curvature at the turning section, resulting in a larger desired steering angle. The relatively slow dynamic response of the FOPID controller could lead to cumulative tracking errors, ultimately causing higher peak steering angles.

Figure 3(e) and (f) present the dynamic response curves of the sideslip angle at the vehicle’s center of gravity and the yaw rate, respectively, for the three lane-keeping controllers. The two sliding mode control strategies demonstrate superior capability in accomplishing the lane-keeping task, effectively constraining critical stability parameters—especially the sideslip angle—within tighter bounds than those achieved by the FOPID controller. These results underscore the inherent robustness of SMC in handling the nonlinearities and uncertainties associated with vehicle dynamics.

The error metrics of each controller under Condition 1 are summarized in Table 3.

Table 3.

The error metrics of each controller under condition 1.

Controller
FOISMC	0.0274	0.0106	0.0182	0.0066
TISMC	0.0280	0.0121	0.0183	0.0067
FOPID	0.05	0.0125	0.0332	0.0104

As shown in Table 3, under low-speed and slippery road conditions, FOISMC reduces the by 2.5% and 45.4% compared to TISMC and FOPID, respectively, and reduces the by 12.3% and 17.9%, respectively. The and values of FOISMC and TISMC are nearly identical; however, compared with FOPID, FOISMC achieves a 45.2% reduction in and a 36.5% reduction in . The root mean square (RMS) value reflects both the overall average level and the dispersion of the error, indicating that, compared with TISMC and FOPID, FOISMC exhibits smaller data deviations and higher stability during the lane-keeping process.

Condition2: High-speed and dry asphalt

Under dry asphalt road conditions with a surface adhesion coefficient of 0.85, a lane-keeping simulation was performed at a vehicle speed of 90 km/h during a double lane change maneuver. The corresponding simulation results are presented in Fig. 4.

Fig. 4.

The simulation results for Condition 2.

Figure 4(a) demonstrates that while all three controllers maintain effective trajectory tracking, an overall decline in control performance is observed as the vehicle speed increases, in comparison to the low-speed and slippery road condition. This degradation is particularly evident for the FOPID controller, whose limited robustness leads to a more significant loss in control accuracy.

As shown in Fig. 4(b) and (c), the tracking errors predominantly occur during sharp turning maneuvers. The FOISMC controller exhibits smaller lateral position and yaw angle errors compared to the other two controllers. Figure 4(d) indicates that, relative to the FOPID controller, both FOISMC and TISMC demonstrate clear advantages, requiring smaller front wheel steering angles and delivering smoother control responses without evident oscillations or chattering. From Fig. 4(e) and (f), it is evident that as the vehicle speed increases, the FOPID controller leads to larger sideslip angles at the vehicle’s center of gravity and higher yaw rates, both accompanied by oscillations, indicating reduced vehicle stability. In contrast, the FOISMC and TISMC sliding mode controllers significantly reduce the yaw rate and sideslip angle while effectively suppressing oscillations, thereby exhibiting superior vehicle stability and path-tracking performance.

The error metrics of each controller under Condition 2 are summarized in Table 4.

Table 4.

The error metrics of each controller under condition 2.

controller
FOISMC	0.0614	0.0211	0.0726	0.0228
TISMC	0.0693	0.0229	0.0750	0.0249
FOPID	0.1429	0.0371	0.1687	0.0459

As shown in Table 4, under high-speed and dry-asphalt condition, FOISMC reduces by 11.4% and 57.0% compared to TISMC and FOPID, respectively, while reducing by 7.9% and 43.1%, by 3.2% and 57.0%, and by 8.4% and 50.3%, respectively. In both maximum error and root mean square (RMS) metrics, FOISMC outperforms TISMC and FOPID, underscoring the superiority of the proposed method. The reduction in simulation errors reflects not only numerical optimization but also clear physical significance. A smaller lateral position error indicates that the vehicle can maintain a more accurate alignment with the lane center, which is directly related to driving safety. Likewise, a smaller yaw angle error signifies improved vehicle attitude stability, thereby enhancing ride comfort. Furthermore, the suppression of oscillations leading to smoother front-wheel steering angle outputs not only improves driving experience but also effectively reduces mechanical wear of the steering actuators, thus extending their service life. These improvements hold considerable value for practical engineering applications.

Condition 3: Low-speed and slippery road with crosswind disturbance

A lane-keeping simulation was performed under a slippery road condition with a friction coefficient of 0.55, considering crosswind disturbance and a vehicle speed of 54 km/h during a double lane-change maneuver. Figure 5 presents the curves of crosswind heading and speed as functions of the vehicle position during motion.

Fig. 5.

Crosswind disturbance.

As shown in Fig. 5(a) and (b), the crosswind speed gradually increases from zero, reaching 55 km/h at X = 60 m. It then remains constant until X = 150 m, after which it gradually decreases and eventually returns to zero. The crosswind heading is 0° within the ranges of X = 0 ~ 50 m, X = 90 ~ 120 m, and X = 160 ~ 200 m. It changes to 90° in the range of X = 50 ~ 90 m, and to − 90° in the range of X = 120 ~ 160 m.

Figure 6 presents the simulation results for Condition 3. As shown in Fig. 6(a)-(c), although all three controllers are able to effectively track the reference path under crosswind disturbances, the FOPID exhibits noticeably larger lateral position error and yaw angle error compared with the case without crosswind, particularly around X = 150 m. In contrast, the lateral position error and yaw angle error of FOISMC and TISMC remain relatively stable. Among them, FOISMC achieves the smallest lateral position error, while its yaw angle error is nearly identical to that of TISMC, which sufficiently demonstrates the strong robustness of FOISMC. Figure 6(d) reveal that the front wheel steering angle of FOPID increase significantly near X = 150 m, whereas the steering angles of FOISMC and TISMC show minimal variation, with their curves almost overlapping. This suggests that under crosswind disturbances,

Fig. 6.

The simulation results for Condition 3.

FOPID requires a larger steering input to return the vehicle to the lane centerline, while FOISMC and TISMC achieve this with smaller steering angles, resulting in smoother control performance. From Fig. 6(e) and (f), it can be observed that, due to the presence of crosswind disturbances, the FOPID controller exhibits a relatively large vehicle sideslip angle at the center of mass, indicating poorer vehicle stability. In contrast, both FOISMC and TISMC significantly reduce the yaw rate and center-of-mass sideslip angle, demonstrating superior vehicle stability and path-tracking performance.

The error metrics of each controller under Condition 3 are summarized in Table 5.

Table 5.

The error metrics of each controller under condition 3.

controller
FOISMC	0.0275	0.0103	0.0192	0.0067
TISMC	0.0283	0.0120	0.0193	0.0067
FOPID	0.0613	0.0142	0.0434	0.0112

From Table 5, it can be seen that, compared with TISMC and FOPID, the FOISMC-based controller reduces the by 2.8% and 55.1%, and by 14.2% and 27.5%, respectively, indicating smaller deviations in lateral position and higher stability during the lane-keeping process. Regarding yaw angle errors, the FOISMC and TISMC controllers have almost identical and . Compared with FOPID, the FOISMC achieves reductions of 55.8% in and 40.2% in , demonstrating significantly improved yaw control performance.

Condition 4: High-speed and dry asphalt with crosswind disturbance

A lane-keeping simulation was performed on a dry asphalt with a friction coefficient of 0.85, incorporating the crosswind disturbances from Fig. 5 and a vehicle speed of 90 km/h during a double lane-change maneuver.

Figure 7(a)–(c) indicate that all three controllers effectively track the reference path under crosswind disturbances. However, compared with the case without crosswind, the FOPID exhibits a noticeable increase in lateral position error, particularly around X = 80 m and X = 150 m. In contrast, the lateral position and yaw angle errors of FOISMC and TISMC remain relatively stable. Among them, FOISMC achieves the smallest lateral position error, while its yaw angle error is nearly identical to that of TISMC, demonstrating the strong robustness of FOISMC under crosswind disturbances. In Fig. 7(d), it can be observed that under crosswind disturbances, the FOPID requires a noticeably larger front wheel steering angle around X = 75 m and around X = 150 m, while the steering angles of FOISMC and TISMC remain nearly unchanged, with their curves almost coinciding. Figure 7(e) and (f) reveal that, even under the influence of crosswind disturbances, both sliding mode control methods can effectively handle the lane-keeping task while maintaining stability parameters—particularly the center-of-mass sideslip angle—within a tighter range than FOPID, highlighting the inherent robustness of sliding mode control in vehicle dynamic systems.

Fig. 7.

The simulation results for Condition 4.

The error metrics of each controller under Condition 4 are summarized in Table 6.

Table 6.

The error metrics of each controller under condition 4.

controller
FOISMC	0.0595	0.0207	0.0821	0.0249
TISMC	0.0662	0.0225	0.0847	0.0272
FOPID	0.1903	0.0529	0.1515	0.0419

From Table 6, it can be concluded that, compared with TISMC and FOPID, the FOISMC-based controller reduces the by 10.1% and 68.7%, and by 8% and 60.9%, respectively. Regarding yaw angle errors, FOISMC decreases by 3% and 45.8%, and by 8.5% and 40.6%, compared with TISMC and FOPID, respectively. These results indicate that even under high-speed conditions with crosswind disturbances, FOISMC not only effectively reduces peak errors but also achieves smaller mean errors and error dispersion. This sufficiently demonstrates the superiority and strong robustness of the proposed method.

To provide a more direct statistical evaluation, the key performance indicators from Tables 3, 4, 5 and 6 are visualized in Fig. 8. Figure 8 clearly shows that, across all four conditions, the bar heights representing FOISMC are consistently the lowest. This indicates that the proposed controller significantly outperforms TISMC and FOPID in terms of both maximum error and root mean square error. These results provide strong evidence of the superiority and robustness of FOISMC.

Fig. 8.

Performance evaluation of FOISMC, TISMC, and FOPID based on tracking errors under different conditions.

Conclusion

In this study, a lane-keeping controller based on fractional-order integral sliding mode control is proposed to address the lane-keeping control problem. First, a fractional-order integral sliding surface is constructed and combined with an improved fractional-order variable power reaching law to ensure that the system states reach the predefined sliding surface within finite time. Subsequently, the Lyapunov stability theory is employed to analyze the system’s stability, and a theoretical proof is provided to demonstrate that the system states can converge to the fractional-order integral sliding surface in finite time. Finally, co-simulations conducted with MATLAB/Simulink and CarSim demonstrate that the FOISMC-based controller is able to reliably track the reference trajectory, even under high-speed conditions with crosswind disturbances. In comparison with TISMC and FOPID, the FOISMC achieves reductions in the maximum lateral position error by 10.1% and 68.7%, respectively, while the corresponding root mean square errors are reduced by 8% and 60.9%. These findings highlight the effectiveness and robustness of the proposed control strategy.

Limitations and future work

Although the proposed FOISMC demonstrates significant performance improvements in the simulated deterministic environments, this study has several limitations that provide clear directions for future research.

1. Stochastic Robustness Analysis: The current study is based on a deterministic simulation model. It does not account for stochastic uncertainties that are prevalent in the real world, such as sensor noise (from GPS, IMU, etc.), sudden environmental disturbances (e.g., road irregularities), and time-varying vehicle parameters (e.g., changes in mass or tire-road friction). Therefore, future work will focus on evaluating the controller’s stochastic robustness.

2. Vehicle Model Fidelity: The lateral control strategy was developed based on a 2-DOF vehicle dynamics model. While effective for many driving scenarios, this model neglects the influence of roll dynamics, suspension characteristics, and highly non-linear tire behavior under extreme conditions. Future research could integrate a more comprehensive, high-fidelity vehicle model to validate and refine the controller’s performance, especially during highly dynamic or emergency maneuvers.

3. Experimental Validation: The validation of the proposed controller was conducted entirely in a co-simulation environment (CarSim and Simulink). The real-time computational performance and practical effectiveness have not yet been verified on physical hardware. Consequently, the next steps will involve Hardware-in-the-Loop (HIL) testing to assess the controller’s real-time feasibility, followed by implementation and validation on a real-world autonomous vehicle platform.

4. Parameter Optimization: The tuning of certain controller parameters currently relies on engineering experience and a trial-and-error process, which can be time-consuming and may not guarantee optimality. Future work could explore the application of intelligent optimization algorithms (e.g., genetic algorithms, particle swarm optimization) to systematically tune the controller parameters, potentially further enhancing its performance and adaptability.

5. Real-time computation of fractional operators: Fractional-order operators have memory characteristics, which require high computational resources and may challenge real-time embedded systems. Future research will employ efficient numerical approximations (e.g., Oustaloup recursive filters) to achieve a balance between real-time feasibility and control accuracy.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

Weilin Wu: Conceptualization, Methodology, Investigation, Writing-Original draft preparation, Writing – Review & Editing, Funding Acquisition. Shenghua Huang: Methodology, Writing-Original draft preparation, Writing – Review & Editing, Supervision. Jincheng Qin: Writing – Review & Editing, Funding Acquisition, Validation. Haoran Yang: Validation, Software. Caixu Xu: Validation. All authors reviewed the manuscript.

Funding

This research was funded by the Guangxi Science and Technology Base and Talent Project under Grant AD23026199, by the Guangxi Zhuang Autonomous Region Youth Talent Project under Grant 301780227, by the Guangxi Science and Technology Major Project under Grant AD23023001, and by the Guangxi Minzu University Xiangsi Lake Youth Scholar Innovation Team under Grant 2023GXUNXSHQN06.

Data availability

Data are contained within the article.

Declarations

Competing interests

The authors declare no competing interests.