#### Abstract

The antilock braking system (ABS) is an electromechanical device whose controller is challenging to design because of its nonlinear dynamics and parameter uncertainties. In this paper, an adaptive controller is considered under the assumption that the friction coefficient is unknown. A modified high-order sliding-mode controller is designed to enhance the controller performance. The controller ensures tracking of the desired reference and identifies the unknown parameter, despite parametric variations acting on the real system. The stability proof is done using the Lyapunov approach. Some numerical and experimental tests evaluate the controller on a mechatronic system that represents a quarter-car model.

#### 1. Introduction

The antilock braking system (ABS) in the actual vehicles is a mechatronic system that helps the driver to maintain control of the vehicle during emergency braking by preventing the wheels from lock-up. The ABS is designed to increase the braking efficiency and maintain the vehicle’s maneuverability, reducing the driving instability, obtaining maximum wheel grip on the surface while the vehicle is braking, and decreasing the braking distance.

During the last decade, ABSs were improved considering more advanced technologies and more sophisticated control strategies. However, it is essential to highlight that the tire-road friction coefficient is one of the most critical parameters since friction is the mechanism for transmitting external forces to the vehicle. These friction forces are the primary forces affecting the planar vehicle motion. From a physical point of view, these forces are limited by the road surface coefficient of friction and the instantaneous tire normal forces. The condition (i.e., the value of ) of the road surface, even if regular, could negatively influence the vehicle motion since the road could be dangerously slippery (e.g., due to water or ice). In practical cases, the road condition is one of the most relevant parameters causing the driving control loss. In particular, the knowledge of the real tire-road friction coefficient is critical to apply active control actions properly. Therefore, its precise estimation increases the efficiency of the control system considerably. There is literature regarding the modeling [1] and the estimation [2] of the tire-road friction coefficient. These studies deal mainly with identifying the tire-road friction coefficient to improve the vehicle acceleration or deceleration [3–13]. However, today, the ABS is very important for passenger car safety, so further efforts should be made to continue studying it. In fact, novel techniques can be applied such as those in [14–16].

In this article, the ABS laboratory setup, manufactured by Inteco Ltd., has been used to test the proposed controller. This setup represents a quarter-car model [17], and it consists of two rolling wheels. Earlier works about nonlinear controllers were considered. These works are mainly based on the assumption that the information of all sensors is available for measurement. In [18], an experimental comparison between PID and nonlinear stabilizing controllers is presented, and in [19], an event-triggered control is proposed. Also, the sliding mode control strategies are analyzed in [20, 21]. Other works deal with intelligent control techniques such as adaptive neuro-fuzzy [22, 23], neuro-fuzzy techniques [24], or other fuzzy controllers [25, 26].

In this paper, an adaptive controller using the modified HOSM is designed for the ABS laboratory setup. The controllers ensure tracking of the desired reference, even in uncertainties in the friction coefficient and external perturbations. At the same time, the identification of the friction coefficient parameter is developed. The stability proof using the appropriate function of Lyapunov and the performance of the controller is evaluated by some numerical simulations and experimental tests on the ABS laboratory setup.

The paper is organized as follows. Section 2 introduces the description and the mathematical model of the experimental ABS laboratory setup. In Section 3, the main contributions are presented. Section 4 presents some numerical simulation and real-time tests on the ABS laboratory setup. Some comments conclude the paper.

#### 2. Mathematical Model of the ABS Laboratory Setup

The ABS laboratory setup describes the essential dynamics of a quarter-car model. It consists of two rolling wheels: the lower aluminum wheel emulates the road motion, and the upper plastic wheel simulates the vehicle wheel. In order to accelerate the lower wheel, a DC motor is coupled on it, whereas the upper wheel is equipped with a disk-brake system. Encoders on the wheels allow determining the positions and velocities of the two wheels, using numerical differentiation. This laboratory setup, manufactured by Inteco Ltd., and shown in Figure 1, preserves the fundamental characteristics of an actual ABS system in the range of 0–70 km/h [17].

The dynamic equations of the ABS laboratory setup are obtained from Figure 1 and are currently used in the literature [27–30]. The braking torque is used as a control variable, and it acts on the upper wheel. Additionally, the tangential braking force represents the tractive force generated at the contact between the upper and lower wheel.where and are the angular velocities of the upper and lower wheels, respectively, whose inertia moments are and and whose radii are and . Furthermore, and are the viscous friction coefficients of the upper and lower wheel.

The braking torque is modeled by a first-order equation [17]:where is a constant, is the control input, and describes the relation between and the input applied to the DC motor. This relation can be approximated by an equation similar to the brake pedal model in an automobile [27, 31, 32]:where is the threshold of the brake driving system.

On the other side, the tractive force is proportional, via the tire-road friction coefficient , to the normal load of the vehicle and is a nonlinear function of the longitudinal wheel slip.

Under normal operation conditions, the wheel velocity matches the vehicle forward velocity and . When the braking is applied, tends to be lower than (remaining nonnegative), a slippage occurs, and a tractive force is generated at the contact point, whose magnitude is given bywhere represents the force normalized with respect to and is the force peak value:with being the stiffness factor and being the shape factor. The parameters , , and are determined to match the experimental data.

*Remark 1. *Various models are available in the literature to model the tire behavior, for example, the so-called Pacejka’s “magic formula” [33] which approximates the response curve of the braking process based on experimental data. It is widely used and allows working with a wide range of values, including the linear and nonlinear regions of the tire characteristics.

Hence, considering (5), the dynamic equation of the ABS laboratory setup (1) can be rewritten asTo design the controller, it is assumed that . The output to be controlled is the wheel slip , and the control aim is to design a controller such that tracks in finite time a constant reference in the presence of parameter uncertainties inherent to the ABS laboratory setup.

#### 3. Design of an Adaptive Controller for the ABS Laboratory Setup

In this section, a modified high-order sliding-mode (MHOSM) controller is designed to force the error,to zero in finite time, even in the presence of variations of . The control law needs a control reference . Hence, instead of considering the wheel slip as the control variable, an auxiliary slip velocity will be used [29, 30, 32]. Then, the slip velocity reference is given by . Therefore, the slip velocity error is defined asand the dynamicswith .

However, the friction coefficient is a parameter that, in real cases, may vary considerably, according to the road and tire conditions. Also, the parameter (value of force peak of Pacejka’s magic formula) depends on the tire condition. In this article, a controller in which the parameter is constant and unknown is proposed. The next result solves the control problem in the case of uncertainty of this parameter.

Theorem 1. *Consider the following assumption:*(i)*The slip reference is a constant*(ii)*The angular velocities are measurable*(iii)*The parameter is constant and unknown**Then, the modified high-order sliding mode controller is proposed:with and ensures that the tracking error (9) converges to zero in finite time [34] and the estimation error globally exponentially tends to zero along their derivatives.*

*Proof. *Substituting the control input (11) into the dynamics of the slip velocity error (10), one obtainswith .

Let us consider the following Lyapunov function:withwith andDeriving (15) along the trajectories of the system,withThe derivative of (15) iswithAnalyzing the first term of (19), i.e., can be rewritten asIfthe first term in matrix form is written asSimilarly, the second term of (19) is and can be evaluated asSince and , the second term is presented in the matrix form:Following with the terms of (19), one analyzes the term :and the term :Using (23), (25), (26), and (27), one rewrites (19) asFinally, using (28) and (17), the derivative of the Lyapunov function (14) iswithSince , recalling that is constant, one getsFinally, substituting the equation (30) into (29), one obtainsThen, is negative definite. Therefore, and tend to zero. Since the adaptive controller (12) ensures that tends to in asymptotic time, one concludes that also tends to in asymptotic time.

#### 4. Simulation Results

In this section, some numerical results and real-time experimental results are shown, using the ABS laboratory setup controlled by a PC. The objective is to show the performance of the controller (11).

##### 4.1. Numerical Simulation

To develop the numerical simulations, the coefficients of the ABS laboratory setup are given in Table 1 and the controller implemented in numerical simulations (11) are given in Table 2. Also, the tests are done considering (1700 rpm), as initial conditions for (7). These conditions simulate a vehicle that runs at a speed of 65 km/h, and suddenly, the brake system is activated, sending a control signal to the actuator to start the braking process. It is worth noticing that, in this setup, the nominal value of the friction coefficient between the two wheels, given by the constructor, is . Nevertheless, this coefficient may vary in practice, remaining close to this value.

The numerical simulations are summarized in Figures 2–5, where it can be seen that the proposed controller (11) ensures the performance of the system. Figure 3 shows the wheel velocity and the vehicle longitudinal velocity . The wheel slip and the tracking error are shown in Figure 4. The applied input is shown in Figure 2. Finally, the estimation given by (29) and used in the controller (11) is shown in Figure 5, where the real value is .

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###### 4.1.1. Real-Time Simulation

In this section, some real-time experimental results are shown, using the ABS laboratory setup controlled by a PC. The interested reader can find in [17] the details about the system hardware and the implementation of the proposed controller. The objective is to show the performance of the controller (11). The coefficients of the ABS laboratory setup are given in Table 1, and the controller (11) implemented in real-time simulation is given in Table 3.

The real-time simulation is shown in Figures 6–9. The braking phase of the ABS laboratory setup starts at 5.7 s and finishes at 7 s. It is important to highlight that, after this braking phase, corresponding to the maximum braking efficiency, the performance is no longer relevant. Figure 7 shows the wheel velocity and the vehicle longitudinal velocity . It can be observed that the control input applied to the ABS setup system, shown in Figure 6, reduces the velocities gradually to zero in approximately 1.8 s. Figure 8 shows the behavior of the wheel slip , the wheel slip desired , and the tracking error . Finally, Figure 9 shows the identification of the unknown parameter and the estimation error . Note that, in the real-time simulation, the braking process can be considered concluded after about 1.1 s. The reader can compare these results with those of Section 4.1. It can be noticed that these experimental results differ from the simulation results due to the unmodeled dynamics, parameters variations, etc., affecting the real system. This is particularly evident after 6.5 s, i.e., when the braking process can be considered concluded.

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#### 5. Conclusions

This paper presents a modified high-order sliding mode (HOSM) controller with parameter estimation applied to an ABS laboratory setup. The system emulates a quarter-car model. This controller provides estimations for the friction coefficient acting between the wheels. Once that parameter is estimated, the estimation can be used to determine the modified HOSM controller. This latter ensures tracking of the desired slip reference. The asymptotic stability is proven, and experimental tests show the effectiveness of the proposed controller. For the future, the work will be focused on finite-time sampled-data fuzzy control and the reliable fuzzy control of the ABS considering further dynamics, perturbations acting on the real system and parameter uncertainties.

#### Data Availability

The figures, tables, and other data used to support this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.