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

Authors: Claudia Carolina Vaca García; Luis Adrián Ferré Covantes; Antonio Navarrete Guzmán; Claudia Verónica Vera Vaca; Cuauhtémoc Acosta Lúa
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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 frictionμ and the instantaneous tire normal forces. The 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. 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. Section2 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 Figure1, preserves the fundamental characteristics of an actual ABS system in the range of 0–70 km/h [17].Figure 1
The ABS laboratory setup and its scheme.The dynamic equations of the ABS laboratory setup are obtained from Figure1 and are currently used in the literature [27–30]. The braking torque Tb is used as a control variable, and it acts on the upper wheel. Additionally, the tangential braking force Ft represents the tractive force generated at the contact between the upper and lower wheel.(1)ω˙1=r1J1Ft−1J1d1ω1+Tb,ω˙2=−r2J2Ft−1J2d2ω2,where ω1 and ω2 are the angular velocities of the upper and lower wheels, respectively, whose inertia moments are J1 and J2 and whose radii are r1 and r2. Furthermore, d1 and d2 are the viscous friction coefficients of the upper and lower wheel.The braking torqueTb is modeled by a first-order equation [17]:(2)T˙b=c−Tb+bu,where c>0 is a constant, u∈0,1 is the control input, and bu describes the relation between u 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]:(3)bu=b1u−b0,ifu≥u0,0,ifu<u0,where u0 is the threshold of the brake driving system.On the other side, the tractive forceFt 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.(4)λ=r2ω2−r1ω1r2ω2=vx−vwvx.Under normal operation conditions, the wheel velocityvw matches the vehicle forward velocity vx and λ=0. When the braking is applied, vw tends to be lower than vx>0 (remaining nonnegative), a slippage λ>0 occurs, and a tractive force Ft is generated at the contact point, whose magnitude is given by(5)Ft=μDφλ=θφλ,θ=μD,where φλ represents the force Ft normalized with respect to θ and D is the force peak value:(6)φλ=sinCarctanBλ,with B being the stiffness factor and C being the shape factor. The 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(7)ω˙1=r1J1θφλ−1J1d1ω1+Tb,ω˙2=−r2J2θφλ−1J2d2ω2.
To design the controller, it is assumed thatvx>0. The 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 λref 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,(8)eλ=λ−λref,to zero in finite time, even in the presence of variations of θ. The control law needs a control reference λref. Hence, instead of considering the wheel slip as the control variable, an auxiliary slip velocity vs=vx−vω=λvx will be used [29, 30, 32]. Then, the slip velocity reference is given by vs,ref=λrefvx. Therefore, the slip velocity error is defined as(9)ev=vs−vs,ref=1−λrefvx−vω,ev=1−λrefr2ω2−r1ω1,and the dynamics(10)e˙v=1−λrefv˙x−v˙ω=1−λrefr2ω˙2−r1ω˙1=−kλrefθφλ+r1J1d1ω1−1−λrefr2J2d2ω2+r1J1Tb,with kλref=r12/J1+1−λrefr22/J2.However, the friction coefficientμ 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. The next result solves the control problem in the case of uncertainty of this parameter.Theorem 1.
Consider the following assumption:(i)
The slip referenceλref is a constant(ii)
The angular velocitiesω1,ω2 are measurable(iii)
The parameterθ is constant and unknown
Then, the modified high-order sliding mode controller is proposed:(11)ω^˙1=−r1J1θ^φλ−1J1d1ω1+Tb,Tb=J1r1χ^,χ^=kλrefθ^φλ−r1J1d1ω1+1−λrefr2J2d2ω2−α11ev1/2−α12ev+x^v,x^˙v=−α2,1ev0−α2,2ev,with kλref=r12/J1+1−λrefr22/J2 and α11,α12,α21,α22>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(12)e˙v=−kλrefθφλ+r1J1d1ω1−1−λrefr2J2d2ω2+r1J1J1r1χ^=α11ev1/2−α12ev+x^v−kλrefφλθ˜,(13)e˙vx^˙v=−α11ev1/2−α12ev+x^v−α21ev0−α22ev+10ϑλθ˜,with ϑλ=−kλrefφλ.
Let us consider the following Lyapunov function:(14)V=Vξ+Vθ,with(15)Vξ=12ξTPξ,Vθ=12γθ˜2,with θ˜=θ−θ^ and(16)ξ=ev1/2evx^v,P=4α21+α112α11α12−α11α11α122α22+α122−α12−α11−α122.
Deriving (15) along the trajectories of the system,(17)V.ξ=ξTPξ˙,V.θ=1γθ˜θ˜˙,with(18)ξ˙=ev.1/2e˙vx^˙v=−12ev1/2Λ1ξ−12Λ2ξ+12ev1/2Θ1θ˜+12Θ2θ˜,Λ1=α110−10002α2100,Λ2=α12002α112α12−202α220,Θ1=ϑλ00,Θ2=02ϑλ0.
The derivative ofVξ (15) is(19)V.ξ=−12ev1/2ξTPΛ1ξ−12ξTPΛ2ξ+12ev1/2ξTPΘ1θ˜+12ξTPΘ2θ˜,with(20)PΛ1=α11α112+2α210−α112α12α112−2α210−α11α12−α1120α11,PΛ2=α124α21+3α1122α11α122−α22−2α11α12α114α22+3α1222α12α122+α22−22α22+α122−3α11α124α22−2α1222α12,PΘ1=4α21+α112α11α12−α11ϑλ,PΘ2=2α11α122α22+α122−α12ϑλ.
Analyzing the first term of (19), i.e., −1/2ev1/2ξTPΛ1ξ can be rewritten as(21)−12ev1/2ξTPΛ1ξ=ev1/2evx^vTα11α112+2α210−α112α12α112−2α210−α11α12−α1120α11ev1/2evx^v=−12ev1/2ev1/2evx^vTα11α112+2α210−α11200−12α11α12−α112−12α11α12α11ev1/2evx^v−12ev1/2α12α112−2α21ev1/2ev.
If(22)−12ev1/2α12α112−2α21ev1/2ev=−12α12α112−2α21ev1/22,the first term in matrix form is written as(23)−12ev1/2ξTPΛ1ξ=−12ev1/2ξTα11α112+2α210−α11200−12α11α12−α112−12α11α12α11ξ−12ξTα12α112−2α2100000000ξ.
Similarly, the second term of (19) is −1/2ξTPΛ2ξ and can be evaluated as(24)−12ξTPΛ2ξ=−12ev1/2evx^vTα124α21+3α1122α11α122−α22−2α11α12α114α22+3α1222α12α22+α122−22α22+α122−3α11α124α22−2α1222α12ev1/2evx^v=−12ξTα124α21+3α1120002α12α22+α122−2α1220−2α1222α12ξ=−12α115α122+2α22ev1/2ev−5α11α12ev1/2x^v.
Sinceev1/2ev=1/ev1/2ev2 and ev1/2x^v=1/ev1/2evx^v, the second term is presented in the matrix form:(25)−12ξTPΛ2ξ=−12ξTα124α21+3α1120002α12α22+α122−2α1220−2α1222α12ξ−12ev1/2ξT0000α115α122+2α22−52α11α120−52α11α120ξ.
Following with the terms of (19), one analyzes the term 1/2ev1/2ξTPΘ1θ˜:(26)12ev1/2ξTPΘ1θ˜=12ev1/2ev1/2evx^vT4α21+α112α11α12−α11ϑλθ˜,and the term 1/2ξTPΘ2θ˜:(27)12ξTPΘ2θ˜=12ev1/2evx^vT2α11α1222α22+α122−2α12ϑλθ˜.
Using (23), (25), (26), and (27), one rewrites (19) as(28)V.ξ=−12ev1/2ξTα11α112+2α210−α1120α115α122+2α22−3α11α12−α112−3α11α12α11ξ−12ξTα124α112+2α210002α12α22+α122−2α1220−2α1222α12ξ+12ev1/2ξT4α21+α112α11α12−α11ϑλθ˜+12ξT2α11α1222α22+α122−2α12ϑλθ˜.
Finally, usingV.ξ (28) and V.θ (17), the derivative of the Lyapunov function (14) is(29)V.=V.ξ+V.θ=−1ev1/2ξTQ1ξ−ξTQ2ξ+1ev1/2ξTR1θ˜+ξTR2θ˜+1γθ˜θ˜.,with(30)Q1=α112α112+2α210−α1105α122+2α223α12−α11−3α121,Q2=α122α112+α21000α22+α122−α120−α121,R1=124α21+α112α11α12−α11,R2=α11α122α22+α122−α12.
Sinceθ˜.=θ˙−θ^., recalling that θ is constant, one gets(31)θ^.=γ1ev1/2ξTR1ϑλ+ξTR2ϑλ+kθJ12r12ω˙1−ω^˙12.
Finally, substituting the equation (30) into (29), one obtains(32)V.=−1ev1/2ξTQ1ξ−ξTQ2ξ−kθφλ2θ˜2≤−1ev1/2λminQ1ξ2−λminQ2ξ2−kθφλmax2θ˜2.
Then,V. is negative definite. Therefore, ev and θ˜ tend to zero. Since the adaptive controller (12) ensures that vs tends to vs,ref in asymptotic time, one concludes that λ also tends to λref 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 Table1 and the controller implemented in numerical simulations (11) are given in Table 2. Also, the tests are done considering ω10=ω20=178rad/s (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 μ0=1. Nevertheless, this coefficient may vary in practice, remaining close to this value.Table 1
Coefficients and system variables for the ABS laboratory setup.
r1Radius of the upper wheel0.0995 mr2Radius of the lower wheel0.0990 mJ1Upper wheel inertia moment7.54×10−3 Kg m2J2Lower wheel inertia moment2.56×10−3 Kg m2β1Upper wheel viscous friction coefficient118.74×10−6 Kgm2/sβ2Lower wheel viscous friction coefficient214.68×10−6 Kgm2/sμFriction coefficient between wheels1b1Constant15.24b0Constant6.21cConstant20.37s−1u0Constant0.415BStiffness factor26.76CShape factor1.13DPeak value22.98Table 2
Coefficients and system variables of controller (11) used in numerical simulations.
γ11Gain of the controllers (11)50γ12Gain of the controllers (11)15γ21Gain of the controllers (11)50γ22Gain of the controllers (11)35.0γAdaptive gain (15)0.025kθAdaptive gain (29)2500The numerical simulations are summarized in Figures2–5, where it can be seen that the proposed controller (11) ensures the performance of the system. Figure 3 shows the wheel velocity vw and the vehicle longitudinal velocity vx. The wheel slip λ and the tracking error eλ=λ−λref are shown in Figure 4. The applied input Tb 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=22.98N.Figure 2
Input controlTb.Figure 3
Angular velocity: (a) upper wheelω1; (b) lower wheel ω2.
(a)(b)Figure 4
(a) Wheel slipλ (black) and wheel slip reference λref (constant, blue). (b) Tracking error eλ=λ−λref.
(a)(b)Figure 5
Friction coefficient between the wheels of the ABS laboratory setup: (a) realθ (blue) and estimated θ^ (black) and (b) estimation error θ−θ^.
(a)(b)


#### 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.Table 3
Coefficients and system variables of the controller (11) used in real application.
γ11Gain of the controllers (11)15γ12Gain of the controllers (11)12γ21Gain of the controllers (11)1.7γ22Gain of the controllers (11)0.5γAdaptive gain (15)0.011kθAdaptive gain (29)5000The real-time simulation is shown in Figures6–9. 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 vw and the vehicle longitudinal velocity vx. It can be observed that the control input Tb 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 eλ=λ−λref. Finally, Figure 9 shows the identification of the unknown parameter θ^ and the estimation error eθ=θ−θ^. 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.Figure 6
Input controlTb.Figure 7
Angular velocity of the upper wheelω1.
(a)(b)Figure 8
(a) Wheel slipλ (black) and wheel slip reference λref (constant, blue). (b) Tracking error eλ=λ−λref.
(a)(b)Figure 9
Friction coefficient between the wheels of ABS laboratory setup: (a) realθ (blue) and estimated θ^ (black) and (b) estimation error θ−θ^.
(a)(b)


## 4.1. Numerical Simulation

To develop the numerical simulations, the coefficients of the ABS laboratory setup are given in Table1 and the controller implemented in numerical simulations (11) are given in Table 2. Also, the tests are done considering ω10=ω20=178rad/s (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 μ0=1. Nevertheless, this coefficient may vary in practice, remaining close to this value.Table 1
Coefficients and system variables for the ABS laboratory setup.
r1Radius of the upper wheel0.0995 mr2Radius of the lower wheel0.0990 mJ1Upper wheel inertia moment7.54×10−3 Kg m2J2Lower wheel inertia moment2.56×10−3 Kg m2β1Upper wheel viscous friction coefficient118.74×10−6 Kgm2/sβ2Lower wheel viscous friction coefficient214.68×10−6 Kgm2/sμFriction coefficient between wheels1b1Constant15.24b0Constant6.21cConstant20.37s−1u0Constant0.415BStiffness factor26.76CShape factor1.13DPeak value22.98Table 2
Coefficients and system variables of controller (11) used in numerical simulations.
γ11Gain of the controllers (11)50γ12Gain of the controllers (11)15γ21Gain of the controllers (11)50γ22Gain of the controllers (11)35.0γAdaptive gain (15)0.025kθAdaptive gain (29)2500The numerical simulations are summarized in Figures2–5, where it can be seen that the proposed controller (11) ensures the performance of the system. Figure 3 shows the wheel velocity vw and the vehicle longitudinal velocity vx. The wheel slip λ and the tracking error eλ=λ−λref are shown in Figure 4. The applied input Tb 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=22.98N.Figure 2
Input controlTb.Figure 3
Angular velocity: (a) upper wheelω1; (b) lower wheel ω2.
(a)(b)Figure 4
(a) Wheel slipλ (black) and wheel slip reference λref (constant, blue). (b) Tracking error eλ=λ−λref.
(a)(b)Figure 5
Friction coefficient between the wheels of the ABS laboratory setup: (a) realθ (blue) and estimated θ^ (black) and (b) estimation error θ−θ^.
(a)(b)


### 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.Table 3
Coefficients and system variables of the controller (11) used in real application.
γ11Gain of the controllers (11)15γ12Gain of the controllers (11)12γ21Gain of the controllers (11)1.7γ22Gain of the controllers (11)0.5γAdaptive gain (15)0.011kθAdaptive gain (29)5000The real-time simulation is shown in Figures6–9. 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 vw and the vehicle longitudinal velocity vx. It can be observed that the control input Tb 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 eλ=λ−λref. Finally, Figure 9 shows the identification of the unknown parameter θ^ and the estimation error eθ=θ−θ^. 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.Figure 6
Input controlTb.Figure 7
Angular velocity of the upper wheelω1.
(a)(b)Figure 8
(a) Wheel slipλ (black) and wheel slip reference λref (constant, blue). (b) Tracking error eλ=λ−λref.
(a)(b)Figure 9
Friction coefficient between the wheels of ABS laboratory setup: (a) realθ (blue) and estimated θ^ (black) and (b) estimation error θ−θ^.
(a)(b)


## 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.Table 3
Coefficients and system variables of the controller (11) used in real application.
γ11Gain of the controllers (11)15γ12Gain of the controllers (11)12γ21Gain of the controllers (11)1.7γ22Gain of the controllers (11)0.5γAdaptive gain (15)0.011kθAdaptive gain (29)5000The real-time simulation is shown in Figures6–9. 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 vw and the vehicle longitudinal velocity vx. It can be observed that the control input Tb 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 eλ=λ−λref. Finally, Figure 9 shows the identification of the unknown parameter θ^ and the estimation error eθ=θ−θ^. 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.Figure 6
Input controlTb.Figure 7
Angular velocity of the upper wheelω1.
(a)(b)Figure 8
(a) Wheel slipλ (black) and wheel slip reference λref (constant, blue). (b) Tracking error eλ=λ−λref.
(a)(b)Figure 9
Friction coefficient between the wheels of ABS laboratory setup: (a) realθ (blue) and estimated θ^ (black) and (b) estimation error θ−θ^.
(a)(b)


## 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 fuzzyH∞ control of the ABS considering further dynamics, perturbations acting on the real system and parameter uncertainties.



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