A Modified HOSM Controller Applied to an ABS Laboratory Setup with Adaptive Parameter

Departamento de Ciencias Tecnológicas, Universidad de Guadalajara, Centro Universitario de la Ciénega, Av. Universidad 1115, Ocotlán 47820, Jalisco, Mexico Department of Electrical and Electronic Engineering at the National Technological Institute of Mexico Campus Tepic, Av. Tecnológico 2595, Tepic 63175, Nayarit, Mexico Academic Unit of Basic Sciences and Engineering of the Autonomous University of Nayarit, City of Culture “Amado Nervo “, Tepic, Nayarit, Mexico Department of Industrial Engineering at the National Technological Institute of Mexico Campus Ocotlán, Av. Tecnológico S/N, Ocotlán 47829, Jalisco, Mexico Center of Excellence DEWS, University of L’Aquila, Via Vetoio, Loc. Coppito, L’Aquila 67100, Italy


Introduction
e 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. e 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 tireroad friction coefficient is one of the most critical parameters since friction is the mechanism for transmitting external forces to the vehicle. ese 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 μ and the instantaneous tire normal forces. e condition (i.e., the value of μ ) 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. erefore, its precise estimation increases the efficiency of the control system considerably.
In this article, the ABS laboratory setup, manufactured by Inteco Ltd., has been used to test the proposed controller.
is setup represents a quarter-car model [17], and it consists of two rolling wheels. Earlier works about nonlinear controllers were considered. ese 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], neurofuzzy 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. e 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. e 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. e 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.

Mathematical Model of the ABS Laboratory Setup
e 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.
is 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]. e dynamic equations of the ABS laboratory setup are obtained from Figure 1 and are currently used in the literature [27][28][29][30]. e braking torque T b is used as a control variable, and it acts on the upper wheel. Additionally, the tangential braking force F t represents the tractive force generated at the contact between the upper and lower wheel.
where ω 1 and ω 2 are the angular velocities of the upper and lower wheels, respectively, whose inertia moments are J 1 and J 2 and whose radii are r 1 and r 2 . Furthermore, d 1 and d 2 are the viscous friction coefficients of the upper and lower wheel. e braking torque T b is modeled by a first-order equation [17]: where c > 0 is a constant, u ∈ [0, 1] is the control input, and b(u) describes the relation between u and the input applied to the DC motor. is relation can be approximated by an equation similar to the brake pedal model in an automobile [27,31,32]: where u 0 is the threshold of the brake driving system. On the other side, the tractive force F t is proportional, via the tire-road friction coefficient μ ∈ [0, 1], to the normal load of the vehicle and is a nonlinear function of the longitudinal wheel slip.
Under normal operation conditions, the wheel velocity v w matches the vehicle forward velocity v x and λ � 0. When the braking is applied, v w tends to be lower than v x > 0 (remaining nonnegative), a slippage λ > 0 occurs, and a tractive force F t is generated at the contact point, whose magnitude is given by where φ(λ) represents the force F t normalized with respect to θ and D is the force peak value: with B being the stiffness factor and C being the shape factor. e parameters B, C, and D 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 as To design the controller, it is assumed that v x > 0. e output to be controlled is the wheel slip λ, and the control aim is to design a controller such that λ tracks in finite time a constant reference λ ref in the presence of parameter uncertainties inherent to the ABS laboratory setup.

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 θ. e control law needs a control reference λ ref . Hence, instead of considering the wheel slip as the control variable, an auxiliary slip velocity v s � v x − v ω � λv x will be used [29,30,32]. en, the slip velocity reference is given by v s,ref � λ ref v x . erefore, the slip velocity error is defined as and the dynamics . However, the friction coefficient μ is a parameter that, in real cases, may vary considerably, according to the road and tire conditions. Also, the parameter D (value of force peak of Pacejka's magic formula) depends on the tire condition. In this article, a controller in which the parameter θ is constant and unknown is proposed. e next result solves the control problem in the case of uncertainty of this parameter.

Theorem 1. Consider the following assumption:
(i) e slip reference λ ref is a constant (ii) e angular velocities ω 1 , ω 2 are measurable (iii) e parameter θ is constant and unknown en, the modified high-order sliding mode controller is proposed: with and α 11 , α 12 , α 21 , α 22 > 0 ensures that the tracking error (9) converges to zero in finite time [34] and the estimation error θ � θ − θ globally exponentially tends to zero along their derivatives.
Proof. Substituting the control input (11) into the dynamics of the slip velocity error (10), one obtains Let us consider the following Lyapunov function: with with θ � θ − θ and Mathematical Problems in Engineering ξ � Deriving (15) along the trajectories of the system, with with

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

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 ω 1 (0) � ω 2 (0) � 178 (rad/s) (1700 rpm), as initial conditions for (7). ese 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 μ 0 � 1. Nevertheless, this coefficient may vary in practice, remaining close to this value. e 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 v w and the vehicle longitudinal velocity v x . e wheel slip λ and the tracking error e λ � λ − λ ref are shown in Figure 4. e applied input T b is shown in Figure 2. Finally, the estimation θ given by (29) and used in the controller (11) is shown in Figure 5, where the real value is θ � 22.98 N.

Real-Time Simulation.
In this section, some real-time experimental results are shown, using the ABS laboratory setup controlled by a PC. e interested reader can find in [17] the details about the system hardware and the implementation of the proposed controller. e objective is to show the performance of the controller (11). e 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. e real-time simulation is shown in Figures 6-9. e 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 v w and the vehicle longitudinal velocity v x . It can be observed that the control input T b 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 λ, the wheel slip desired λ \r , and the tracking error e λ � λ − λ ref . Finally, Figure 9 shows the identification of the unknown parameter θ and the estimation error e θ � θ − θ. Note that, in the real-time simulation, the braking process can be considered concluded after about 1.1 s. e 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. is is particularly evident after 6.5 s, i.e., when the braking process can be considered concluded.   (11) used in numerical simulations. c 11 Gain of the controllers (11) 50 c 12 Gain of the controllers (11) 15 c 21 Gain of the controllers (11) 50 c 22 Gain of the controllers (11) (11) used in real application. c 11 Gain of the controllers (11) 15 c 12 Gain of the controllers (11) 12 c 21 Gain of the controllers (11) 1.7 c 22 Gain of the controllers (11) 0.5 c Adaptive gain (15) 0.011 k θ Adaptive gain (29)

Conclusions
is paper presents a modified high-order sliding mode (HOSM) controller with parameter estimation applied to an ABS laboratory setup.
e system emulates a quarter-car model. is 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. is latter ensures tracking of the desired slip reference. e 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 H ∞ control of the ABS considering further dynamics, perturbations acting on the real system and parameter uncertainties.
Data Availability e figures, tables, and other data used to support this study are included within the article.  Figure 9: Friction coefficient between the wheels of ABS laboratory setup: (a) real θ (blue) and estimated θ (black) and (b) estimation error θ − θ.