Enhanced performance and robustness in anti-lock brake systems using barrier function-based integral sliding mode control

Mohsin N. Hamzah; Mujtaba A. Flayyih; Taha A. Al-Gadery; Yasir K. Al-Nadawi; Shibly A. Al-Samarraie

doi:10.1515/eng-2024-0090

Artikel Open Access

Enhanced performance and robustness in anti-lock brake systems using barrier function-based integral sliding mode control

Mohsin N. Hamzah , Mujtaba A. Flayyih , Taha A. Al-Gadery , Yasir K. Al-Nadawi und Shibly A. Al-Samarraie

Veröffentlicht/Copyright: 15. November 2024

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Abstract

In anti-lock brake systems (ABS), the primary goal of the controller is to maximize vehicle deceleration by maintaining the slip ratio at an optimal level. This work presents a fresh approach that enhances ABS performance by integrating a sliding mode controller with a barrier function. This method combines integral sliding mode control with adaptive laws informed by barrier functions, effectively managing external disturbances and uncertainties in inertia. A significant benefit of this approach is that it does not require prior knowledge of the upper limits of these uncertainties and disturbances, thanks to the barrier function-based sliding mode control. The system state is initially aligned with the switching manifold, ensuring robust compensation for any uncertainties and disturbances right from the start of braking. During the sliding mode phase, dynamic properties are finely tuned to ensure that the system’s performance remains consistent. The effectiveness and reliability of the proposed controller have been demonstrated through numerical simulations conducted in MATLAB/Simulink, proving its capability across a range of road conditions.

Keywords: sliding mode control; ABS; adaptive systems; simulation; half-vehicle model

1 Introduction

The anti-lock braking system (ABS) optimizes tire–road friction to enhance driving safety by balancing both longitudinal and lateral forces [1]. As a result, it improves vehicle control, particularly in adverse conditions, significantly enhancing overall safety. ABS aims to maximize deceleration while preserving vehicle stability and steering control. The effectiveness of the braking system’s parameters is crucial in designing active braking control systems. Conventional ABS systems, which rely on hydraulic actuators, typically use rule-based control methods. These systems are designed to rapidly prevent wheel lock-up, as a result reducing stopping distances during braking, preventing skidding, and improving vehicle ride control.

The research in this field has looked at different vehicle dynamics models that can simulate braking scenarios, which are important for designing and testing ABS systems. Baslamisli et al. [2] concentrated on the quarter-car model and Harifi et al. [3] investigated the half-car model. These studies immensely helped us to understand how vehicles behave when they are breaking thus more advanced control algorithms have been built on them. There are many complications that make them difficult to handle. The presence of obscure, noisy sensor signals also makes things worse. Besides, the nature of vehicle dynamics is not linear and hence not easy to control without any problems let alone near an unstable equilibrium point for optimum controller performance. This is further aggravated by the fact that model parameters like road conditions, vehicle mass, and center of gravity vary. Nonetheless, this can be viewed as an indication that strong and adaptive control strategies are needed to ensure dependable functioning of the system [4,5]. Researchers have thoroughly explored different control strategies, especially the sliding mode control (SMC), to manage such challenges effectively. SMC is distinguished for its robustness in dealing with system uncertainties and external disturbances. Research conducted by Emanuel et al. [6], Moussa and Bakhti [7], Flayyih et al. [8,9], and Wang et al. [10] has effectively validated SMC’s capacity to regulate ABS systems in various settings and hence enhance the stability of the vehicle.

Combining the SMC with other control systems is becoming more popular as a way to boost its effectiveness and reduce its need on exact vehicle models. Lin and Hsu [11] developed a fuzzy controller in collaboration with SMC to reduce the dependency on accurate vehicle modeling. This solution takes use of SMC’s resilience to control interruptions and fuzzy logic’s flexibility to cope with ambiguity. Wang et al. [12] proposed a robust fuzzy logic controller with self-tuning dead-zone settings for ABS applications. Their major objective was to maintain the optimal wheel slip ratio of 0.2, which would provide consistent and dependable performance even in the face of unanticipated events.

While conventional SMC offers benefits, there are some disadvantages as well, especially in the reaching stage. SMC is only fully robust during the sliding motion itself, which means that it is only guaranteed to be robust against external disturbances and fluctuations in system parameters once the system state reaches the sliding manifold [13]. To address this limitation, the Integral SMC (ISMC) was developed, which eliminates the reaching phase. ISMC ensures robustness from the start by putting the system directly onto the sliding manifold. With this strategy, the system may manage uncertainties and disturbances as effectively as if it were operating under well-defined conditions [14]. Furthermore, ISMC may operate as a perturbation estimator, which reduces the chattering problem that traditional SMC is commonly associated with [13].

Recent study has looked more closely at the advantages of ISMC for ABS systems. For example, on a variety of roads, Abdullah et al.’s [15] ISMC-based ABS controller significantly improved vehicle stability and reduced stopping distances. Their findings demonstrated how ISMC can efficiently deal with external disturbances and model uncertainty while maintaining smooth control. In a different investigation, Li et al. [16] used ISMC with adaptive sliding mode control to improve ABS performance while regulating irregular road friction coefficients. They noticed that the vehicle’s stability and braking efficiency had greatly increased.

Another interesting option is to combine intelligent control approaches with ISMC. For example, Zhou et al. [17] used neural networks with ISMC to develop a self-learning controller for the ABS. This controller allows for real-time adaptability to fluctuating road conditions and vehicle dynamics. This approach combines the flexibility of neural networks with the resilience of ISMC to create a controller that can handle a broad range of conditions while staying robust in the face of disruption.

In addition to these advancements, scientists have been striving to overcome the chattering problem that is characteristic of SMC. To reduce chattering while maintaining robustness, Ahmad et al. [18] developed a boundary layer approach for the ISMC design. Their solution significantly reduces the control procedure while retaining the performance benefits that SMC offers.

ABS control continues to be challenged to new boundaries by ongoing research in advanced SMC technologies such as ISMC and its derivatives. These developments increase vehicle performance and safety while also contributing significantly to the development of strong control systems. They provide essential tools and insights for tackling complex control challenges in a variety of automotive and aerospace applications.

The main goal of the current work is to create an active braking system that maximizes braking force and achieves ideal values by utilizing integral sliding mode control. We consider the uncertainties in system parameters and the disturbances encountered during braking under different load conditions. Two separate brake torques are designed for analysis using a half-vehicle model.

2 Friction model and vehicle dynamic

This section presents the derivation of the dynamical model, focusing specifically on the longitudinal wheel slips – defined as the difference between wheel speed and vehicle speed – for both the front wheel ( λ f ) and the rear wheel ( λ r ). For reference, consider the half-vehicle model illustrated in Figure 1, with the associated nomenclature detailed in Table 1. Only the vertical force, F z , and the traction or braking force F x are considered, the braking force can be expressed as: F x = f ( F z , α t , λ ) , where α t is the tire sideslip angle and λ is the wheel slip in the longitudinal direction as shown in Figure 2.

Figure 1

The vehicle free body diagram.

Table 1

Nomenclature

C G	center of gravity of the vehicle	−
d f	horizontal distance from the center of the front wheel to CG	m
F x f	front longitudinal wheel–road braking force	N
F x r	rear longitudinal wheel–road braking force	N
F z f	front vertical wheel–road contact force	N
F z r	rear vertical wheel–road contact force	N
F z , F x	vertical and braking forces, F z = F z r + F z f , and F x = F x r + F x f	N
h	vertical distance from the road surface to CG	M
o	origin of the coordinate system x,y,z
T f , T r	braking torques	N m
v	longitudinal velocity of CG	m / s
ω f , ω r	wheels’ angular velocity

Figure 2

Wheel sideslip and longitudinal slip.

Physically, λ ∈ [ 0,1 ] , more specifically, λ = 0 represents pure rolling, while λ = 1 represents a locked wheel. And longitudinal wheel slips for the front wheel ( λ f ) and the rear wheel ( λ r ) can be expressed as

(1) λ f = ( v − r ω f ) / v λ r = ( v − r ω r ) / v .

Along the longitudinal axis, the tire–road force is represented as

(2) F x = F z × μ ( λ ) ,

where r is the effective radious and μ ( λ ) is the friction coefficient and can be accurately described by the Burckhardt model [19,20]. The Burckhardt model is widely valued for its accuracy in modeling the complex, nonlinear relationship between the friction coefficient and slip ratio, which is crucial for vehicle dynamics simulations and the development of advanced braking systems. Its ability to realistically represent these interactions makes it an essential tool in designing and testing vehicle safety features and performance enhancements, solidifying its importance in automotive engineering. Using this model as a foundation, the longitudinal friction coefficient takes the following shape:

(3) μ ( λ ; ϑ ) = ϑ 1 ( 1 − e − λ ϑ 2 ) − λ ϑ 3 .

Taking into consideration that the vector ϑ = [ ϑ 1 , ϑ 2 , ϑ 3 ] only has three elements. Numerous different tire–road friction conditions can be modeled by altering the values of these three parameters. The parameter values of ϑ for various road conditions are shown in Table 2.

Table 2

Friction model parameters [1]

Road condition		ϑ 1	ϑ 2	ϑ 3
Dry	Asphalt	1.2801	23.990	0.52
	Concrete	1.1973	25.168	0.5373
	Cobblestone	1.371	6.456	0.669
Wet	Asphalt	0.857	33.822	0.347
Wet	Cobblestone	0.4004	33.708	0.1204
Other	Snow	0.194	94.129	0.064
Other	Ice	0.050	306.39	0

Half-vehicle brake model is the dynamic model utilized in the current work (Figure 1). The analysis does not take into account the yaw and rolling rotations, considering only the bounce and pitching motions of the car shown in Figure 1. This vehicle’s equations of motion could be expressed as

(4) r F xf − T f = J f ω ̇ f r F xr − T r = J r ω ̇ r ,

where the subscripts f and r refer to front and rear, respectively. J f and J r are the wheels’ polar moments of inertia.

The equations of motions for the whole vehicle are

(5) F zf + F zr = mg F zr d r − F zf d f = m v ̇ h m v ̇ = − ( F xf + F xr ) ,

where m is the mass of the vehicle.

By differentiating Equation (1) with respect to time and using Equation (5)

(6) λ ̇ f = r vJ f ( H f v ̇ − R f + T f ) = F f + G f T f λ ̇ r = r vJ r ( H r v ̇ − R r + T r ) = F r + G r T r ,

where

H f = J f r ( 1 − λ f ) + mhr ( d f + d r ) μ ( λ f ) ,

R f = mgr d f ( d f + d r ) ,

H r = J r r ( 1 − λ r ) + mhr ( d f + d r ) μ ( λ r ) ,

R r = mgr d f ( d f + d r ) ,

F f = r vJ f ( H f v ̇ − R f ) ,

F r = r vJ r ( H r v ̇ − R r ) ,

G f = r vJ f > 0 and G r = r vJ r > 0 .

Equation (6) may also be rewritten in terms of the nominal parameter’s value and their variation as follows:

(7) λ ̇ f = F fn + ∆ F f + ( G fn + ∆ G f ) T f λ ̇ r = F rn + ∆ F r + ( G rn + ∆ G r ) T r ,

where where ΔF_f , ΔF_r , ΔG_f , and ΔG_r denote the respective changes in F_f , F_r , G_f , and G_r resulting from variations in the system parameters.

3 Braking control design

The design of an ISMC for the braking systems employing the barrier function is the focus of this section. First, the control input torques are selected as follows:

(8) T = [ T f T r ] T T f = u 1 n + u 1 s T r = u 2 n + u 2 s ,

where u 1 n and u 2 n are the nominal control which is designed to force the nominal system toward asymptotic stability [10], while u 1 s and u 2 s are the barrier function-based integral sliding mode control (BISMC) control components used to reject the uncertainty terms in the braking system model. Substituting Equation (8) in Equation (7) yields

(9) λ ̇ f = F fn + G fn u 1 n + ∆ F f + ∆ G f u 1 n + ( G fn + ∆ G f ) ( u 1 s ) λ ̇ r = F rn + G rn u 2 n + ∆ F r + ∆ G r u 2 n + ( G rn + ∆ G r ) ( u 2 s ) ,

Equation (9) may be stated in both nominal and perturbation form as:

(10) λ ̇ f = F n + G fn u 1 n + G fn u 1 s + δ 1 ( λ , t ) λ ̇ r = F rn + G rn u 2 n + G rn u 2 s + δ 2 ( λ , t ) ,

where u s = u 1 s u 2 s is the control input vector while δ 1 ( λ , t ) = ∆ F f + ∆ G f u 1 n + ∆ G f u 1 s and δ 2 ( λ , t ) = ∆ F r + ∆ G r u 2 n + ∆ G r u 2 s are the perturbation terms .

(11) λ ̇ f = F fn + G fn u 1 n λ ̇ r = F rn + G rn u 2 n .

Let e 1 = λ f − λ fd , e 2 = λ r − λ rd ,where λ fd and λ rd are the desired slip ratio for the front and rear tire, respectively. Then, the nominal error dynamic can be written as

(12) e ̇ = F rf + λ rf ̇ F rn + λ rd ̇ + G fn 0 0 G rn u n ,

where e ̇ = e 1 ̇ e 2 ̇ , G n = G fn 0 0 G rn , u n = u 1 n u 2 n , and F en = F fn + λ rf ̇ F rn + λ rd ̇ .

For the nominal system in Equation (12), the following nominal control is sufficient:

(13) u n = − G n − 1 ( F en + Γ e ) ,

where Γ = c 1 0 0 c 2 , c 1 > 0 and c 2 > 0 .

Now, to eliminate the effects of the perturbation and reduce the system in Equation (9) to the nominal in Equation (10), the sliding variables are defined as

(14) s = λ + z ,

where s = s 1 s 2 T , λ = λ f λ r T , and z = z 1 z 2 T . Let the candidate non-smooth Lyapunov function be

(15) V = | s 1 | + | s 2 | .

And its derivative, V ̇ , ∀ s ≠ 0 , is expressed as [21]

(16) V ̇ = sgn ( s 1 ) sgn ( s 2 ) s ̇ 1 s ̇ 2 = sgn ( s 1 ) sgn ( s 2 ) ( λ ̇ + z ̇ ) ,

where sgn ( ) is the signum function. By substituting Equation (10) in Equation (16), we obtain

(17) V ̇ = sgn ( s 1 ) sgn ( s 2 ) F fn + G fn u 1 n + z ̇ 1 F rn + G rn u 2 n + z ̇ 2 + G fn u 1 s + δ 1 ( λ , t ) G fn u 2 s + δ 2 ( λ , t ) .

Now, imposing the following constraint to z ̇

(18) z ̇ 1 = − F fn − G fn T fn z ̇ 2 = − F rn − G rn T rn .

Then, V ̇ Equation (17) becomes

(19) V ̇ = sgn ( s 1 ) sgn ( s 2 ) G fn u 1 s + δ 1 ( λ , t ) G rn u 2 s + δ 2 ( λ , t ) .

Accordingly, the task of the discontinuous control components u 1 s , u 2 s is to make V ̇ negative definite. This task was accomplished in the study by Al-Samarraie et al. [1] via a discontinuous control given by

(20) u 1 s = − μ f sgn ( s 1 ) u 2 s = − μ r sgn ( s 2 ) ,

where μ f and μ r are the positive control gains and should be chosen such that

(21) μ f > δ 1 ( λ , t ) G fn μ r > δ 1 ( λ , t ) G rn .

As a result, the error dynamic from the first instant, becomes

(22) e ̇ = − Γ e ,

with system characteristics determined according to the matrix Γ , which is freely selected.

However, estimating the perturbation term’s bounds can be time-consuming and frequently results in conservativeness problem. In this study, the BISMC [13] was suggested for the ABS to avoid this issue. The goal is to modify the control gains using a barrier function, which does away with the need to calculate the perturbation terms’ upper and lower bounds and guarantees that the sliding variables (s ₁, s ₂) stay within a constrained, predefined set of invariants [13]. First, a Barrier function is defined [13].

Definition 1: The Barrier function can be defined as an even continuous function

f : x ∈ [ − ε , ε ] → f ( x ) ∈ [ b , ∞ ] strictly increasing on [ 0 , ε ] ,

log | x | → ε f ( x ) = + ∞ ,

f ( x ) has a minimum at zero and f ( 0 ) = b ≥ 0 .

For a fixed ε > 0 .

Now, let f b 1 ( s 1 ) and f b 2 ( s 2 ) be two positive semidefinite barrier functions defined as

(23) f b 1 ( s 1 ) = | s 1 | ε 1 − | s 1 | ,

(24) f b 2 ( s 2 ) = | s 2 | ε 2 − | s 2 | .

And by choosing the control gains as follows:

(25) μ f ( s 1 ) = f b 1 ( s 1 ) ,

(26) μ r ( s 2 ) = f b 2 ( s 2 ) ,

the sliding variables ( s 1 , s 2 ) will remain in the invariant set [21]

∁ = { s 1 | | s 1 | < ε 1 ; s 2 | | s 2 | < ε 2 } .

Then,

(27) u 1 s = − | s 1 | ε 1 − | s 1 | sgn ( s 1 ) = − s 1 ε 1 − | s 1 | u 2 s = − | s 2 | ε 2 − | s 2 | sgn ( s 2 ) = − s 2 ε 2 − | s 2 | .

And the final control algorithm

(28) T = − G n − 1 ( F en + Γ e ) − [ s 1 ε 1 − | s 1 | s 2 ε 2 − | s 2 | ] T z ̇ 1 = − F fn − G fn T fn z ̇ 2 = − F rn − G rn T rn .

In addition, for the states to start at the switching manifold, i.e., the initial values of the switching functions are chosen to be equal to zero. Therefore, for s ( 0 ) = 0 , we have z ( 0 ) = − λ ( 0 ) or z 1 z 2 t = 0 = − λ f λ r t = 0 .

4 Simulation results

This section examines the effectiveness of the suggested strategy using several numerical simulations in MATLAB Ver. 7.9. Different scenarios for various road conditions were simulated for a car model with the nominal specifications listed in Table 3.

Table 3

Nominal vehicle parameters [1,22]

m (kg)	h (mm)	d _f (m)	d _r (m)	r (mm)	J _f (kg m²)	J _r (kg m²)
915	585	1.12	1.24	310	1.2	1.7

The Burckhardt model described in Equation (3) may be used to find the ideal slip ratio for different types of road conditions

λ opt . = 1 ϑ 2 ln ϑ 1 ϑ 2 ϑ 3 ,

where d d λ μ ( λ ) = 0 .

The ideal slip ratio is identified and tabulated in Table 4 for various road conditions (Table 2).

Table 4

Optimal slip ratio for different road conditions

Road condition	Dry asphalt	Wet asphalt	Cobblestone	Snow
λ o p t .	0.17	0.13	0.4	0.06

Additionally, the simulation is run with a starting vehicle speed of 100 km/h, and the controller’s goal is to reduce this speed until it reaches the desired value of 5 km/h. Figures 3–6 show the simulation results for different road conditions, for the control design parameters chosen as c 1 = c 2 = 70 , ε 1 = ε 2 = 0.001 . from these results, it can be seen that, for all cases, the control torques are smooth and free of chattering since BISMC is a smooth continuous controller, unlike the classical discontinuous sliding mode controllers. For dry asphalt, it can be observed, from Figure 3(c), that the slip ratio reaches the optimal value (0.17) within less than 0.07 s and remains at this value thereafter due to the robustness of the proposed approach. In addition, the longitudinal speed will decay smoothly to reach the desired final speed of 5 km/h in less than 2.3 s, as shown in Figure 3(d).

Figure 3

Simulation results for dry asphalt: (a) Front control braking torques, (b) rear control braking torques, (c) longitudinal wheel slip ratio, and (d) longitudinal vehicle speed.

Figure 4

Simulation results for wet asphalt: (a) Front control braking torques, (b) rear control braking torques, (c) longitudinal wheel slip ratio, and (d) longitudinal vehicle speed.

Figure 5

Simulation results for cobblestone: (a) Front control braking torques, (b) rear control braking torques, (c) longitudinal wheel slip ratio, and (d) longitudinal vehicle speed.

Figure 6

Simulation results for snow: (a) Front control braking torques, (b) rear control braking torques, (c) longitudinal wheel slip ratio, and (d) longitudinal vehicle speed.

For wet asphalt, the optimal slip ratio of 0.13 was reached within less than 0.07 s as shown in Figure 4, and the longitudinal speed reached 5 km/h within 3.4 s. Similarly for cobblestone and snow, Figures 5 and 6, the optimal slip ratios were reached within less than 0.07 s in both cases, while the longitudinal vehicle speed reached the desired final value in 2.7 s for cobblestone and 14 s for snow road.

For comparison purposes, the simulation results obtained using BISMC were compared with those obtained with classic integral sliding mode control [1] using the same nominal control. The results are shown in Figure 7, from which it can be seen that both approaches achieved the desired objective of decelerating the vehicle from a speed of 100 to 5 km/h while maintaining an optimal slip ratio. In both cases, the optimal slip ratio was reached relatively fast, within less than 0.07 s. However, with BISMC, the absolute error (ultimate bounds on the sliding surfaces and hence on the slip ratio) never exceeds ε = 0.001 .

Figure 7

Comparison results between BISMC and CISMC for dry asphalt: (a) Front control braking torques, (b) rear control braking torques, (c) longitudinal wheel slip ratio, (d) longitudinal vehicle speed, (e) sliding surface S ₁, and (f) sliding surface S ₂.

Furthermore, the accuracy ( ε ) is a free parameter and can be chosen arbitrarly small with BISMC while in the CISMC with approximation [1,23], the error was around 0.004, and increasing the accuracy further by using sharper approximation could lead to chattering problems.

5 Conclusion

In this work, a robust continuous controller based on integral sliding mode and barrier function was proposed for an ABS taking into account uncertainty in the system model. The suggested controller forces the braking system to behave as an idealized nominal system from the beginning, unaffected by the uncertainty in the system model. It can be inferred from Equation (22) that the braking system behaves as a nominal system from the first instant, with the dynamic properties of the braking system being freely chosen by creating the ideal control law, Equation (13). The stability of the proposed approach was demonstrated via a rigorous stability analysis.

For various road types/conditions, the simulations were performed using MATLAB Ver. 7.9. The results showed that, in all scenarios, the optimal slip ratio was achieved within less than 0.07 s while the time needed for the longitudinal speed to decelerate to the desired final value varied from 2.3 s for dry asphalt to 14 s for roads with snow conditions. The effectiveness of the proposed controller was clear when compared with its CISMC counterpart, achieving a reduction in error from 0.004 to 0.001 and adding the property of freely controlling the accuracy rate as a design parameter. This shows that the chosen dynamic characteristics are actively taken into account by the designed control action to achieve the desired (optimal) slip ratio within 0.02 s. To slow down the vehicle to the desired speed within a specific timeframe, as directed by the controller’s goal, the optimal slip ratio fine-tunes the braking forces.

Funding information: Authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. TAA-G, MAF, and YKA-N designed the controller and performed the simulation. MNH and SAA-S prepared the manuscript with contributions from all co-authors.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: This published article contains all the data collected or analyzed during the course of this research.

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Received: 2024-06-27

Revised: 2024-08-23

Accepted: 2024-09-09

Published Online: 2024-11-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/eng-2024-0090

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sliding mode control; ABS; adaptive systems; simulation; half-vehicle model

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