Robust Control for Anti-lock Braking SystemConsidering Dead Zone
M2015SC018 Ryouhei UKAI
Supervisor:Isao TAKAMI

Abstract
In this study, a method is proposed to guarantee therobust stability of Anti-lock Braking System (ABS) considering the dead zone. The objective of ABS is tocontrol wheel slip to maximize the friction coeﬃcient
between the wheel and the road for any given road surface while the car is controllable. The ABS dynamics
has strong nonlinearity, and it depends on uncertain parameters. These are constantly changing car speed andvariations of the road surface friction such as the wet,
dry and snow road. In addition, the dead zone appears
in the relationship between the brake pad and the brakedisc. These uncertain parameters and the dead zone
have a bad influence on control. The robust stability
for those uncertain parameters are guaranteed by using
the matrix polytopic representation. The dead zone isdealt with a function to cancel the nonlinearity of thedead zone. Finally, the eﬀectiveness of the proposed
method is verified by simulations and experiments.

1Introduction

ABS is an abbreviation for Anti-lock Braking System. ABS is an equipment to prevent the locking of
the wheels in braking operation when sudden braking.
By preventing wheel lock, it is able to prevent the following problems. From wheel slip, worsening of brake andunstableness of car body posture happen. Also, the maneuverability is lost and can’t go around the curve even
if the handle is turned. Namely, the objective of ABS
is to ensure the braking force, car body posture, andmaneuverability. However, there exist uncertain factors
in controlling ABS. These are constantly changing car
speed and variations of the road surface friction such asthe wet, dry and snow road. These uncertain characteristics have a bad influence on control of ABS. The ABS
dynamics depends on the slip rate and road surface friction. Also, this slip rate depends on the wheel speed andthe car speed. So, the ABS dynamics depends on the car
speed and uncertain characteristics such as the road surface friction. From this, the main diﬃculty of designing
ABS control is how to deal with the strong nonlinearity
and uncertain characteristics. For such diﬃculty, many
research exist such as nonlinear PID-type control[1] andFuzzy control[2] approaches. On the other hand, model
based approaches are presented. In several papers, Sliding mode control[3] and Gain scheduling control[4] for
ABS are applied. In this study, the nonlinear system
is linearized[4] by using Taylor expansion around theequilibrium point (λ∗ , τ1∗ ). Here, λ∗ is the reference
slip rate, and τ1∗ is the equilibrium braking torque tokeep λ∗ . From relation of the cornering force and thefriction coeﬃcient, the reference slip rate is 0.2[5]. A
design method is proposed to guarantee the robust stability for uncertain parameters such as the road surface
friction and the car speed. In addition, the dead zone
appears in the relationship between the brake pad andthe brake disc. This dead zone have a bad influence
on control[6][7]. So, a method is proposed to deal withthe dead zone using a function to cancel nonlinearity ofthe dead zone. The matrix polytopic representation forthe system with these parameters are proposed. The
problem is formulated by solving a finite set of linear
matrix inequalities (LMIs). Finally, the eﬀectiveness ofthe proposed method is verified by simulations and experiments.

2Control Target

The model of the simplified ABS experimental device
used in this study is shown in Fig.1. The control target isthe ABS experiment machine which is the quarter modelof the real wheel made by INTECO company. The upper wheel simulates the car wheel, and the lower wheel
simulates the road. The upper wheel has the brake andthe lower wheel has the motor. The motor of the lower
wheel turns to appointed speed and stops. Then, thebrake of the upper wheel works. The objective of thisstudy is to control wheel slip to maximize the friction
coeﬃcient between the wheel and the road for any given
road surface. Here, the value of slip rate that maximize
the friction coeﬃcient between the wheel and the roadfor any given road surface is 0.2[5].
Fnr1
τϭ
µFn

ω1r2
ω2

Figure 1 The simplified ABS experimental device

3Modeling

The physical constants and variables of this study are
shown in Table.1. The dynamical equations of the rotational motion of the upper wheel and the lower wheel
are shown as Eq.(1) and Eq.(2).
J1 ω̇1 = Fn r1 µ(λ) − τ1J2 ω̇2 = −Fn r2 µ(λ)(1)(2)The slip rate is defined by Eq.(3) as the function of thecar velocity and the wheel velocity.
λ=r2 ω2 − r1 ω1r2 ω2(3)The following equation is obtained from Eq.(3).
r1
r1 ω1λ̇ = −ω̇1 +ω̇2
r2 ω2r2 ω22Friction coefficient

Table 1 The physical constants and variables
Radius of the upper wheel r1[m]Radius of the lower wheelr2[m]Moment of inertia
J1[kgm2 ]of the upper wheel
Moment of inertia
J2[kgm2 ]of the lower wheel
Vertical force
Fn[N]Angular velocity
ω1[rad/s]of the upper wheel
Angular velocity
ω2[rad/s]of the lower wheel
Braking torqueτ1[Nm]
Friction coeﬃcientµ[−]between wheels
Slip rateλ[−]Duty rate
ud[−]

Figure 2 The relationship between α and the road conditions
Here, λ∗ is the reference slip rate, and τ1∗ is the equilibrium braking torque to keep the reference slip rate. Theequilibrium braking torque τ1∗ is derived from Eq.(10).
(4)

The dynamical equation of the slip rate is derived fromEq.(1), Eq.(2), Eq.(3) and Eq.(4).
1 r1 2 Fn µ(λ) r2 Fn µ(λ)(1 − λ)λ̇ = −(+)
ω2r2 J1J2
1 r1
τ1+ω2 r2 J1(5)The road friction coeﬃcient is defined by Eq.(6)[8] toconsider variations of the road surface friction such as
the wet road, the dry road and the snow road.
µ(λ) =α arctan(52λ)

1Dry Road
0.9
Wet Road
0.8
Snow Road
0.7
0.6
0.5
0.4
0.3
0.2
0.1
00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5Slip Rate

τ1∗ = −Table 2 The relationship between α and the road conditions
Road condition
αDry road
0.437
Wet road
0.155
Snow road0.070

λ̇ ≃ λ̇(λ∗ , τ1∗ ) =11f (λ) +gτ1
ω2
ω2

∂ λ̇
∂ λ̇
|λ=λ∗ (λ − λ∗ ) +|τ =τ ∗ (τ1 − τ1∗ )∂λ∂τ1 1 1(12)The linearized equation is derived from Eq.(11) andEq.(12).
λ̇ =1(αC1 (λ − λ∗ ) + C2 (τ1 − τ1 ∗ ))
ω2(13)Let β be β = ω12 , Eq.(13) is rewritten as follows.
λ̇ = αβC1 (λ − λ∗ ) + βC2 (τ1 − τ1 ∗ )(14)Here, C1 ,C2 are constants.
3.1

Dead Zone

OutputEq.(7) is obtained from Eq.(5) and Eq.(6).
λ̇ = −(11)Using Taylor expansion around the equilibrium point(λ∗ , τ1∗ ), this nonlinear model is linearized as follows [4].

(6)Here, α is a varying parameter by the road conditions.
The relationship between α and the road conditions areshown in Table.2 and Fig.2.

f (λ∗ )g(7)Here, f (λ) and g are as follows.
r1 2 Fn (α arctan(52λ))r2 J1r2 Fn (α arctan(52λ))(1 − λ)
+J2
r1
g=
r2 J1f (λ) =(8)The behavior around equilibrium point (λ∗ , τ1∗ ) is considered. Eq.(10) is obtained by substituting the equilibrium point in Eq.(7).
1 ∗1gτ
0 = − f (λ∗ ) +
ω2
ω2 10

θ

Inpuƚ

(9)

(10)

Figure 3 The dead zone function
The dead zone appears in the relationship between thebrake pad and the brake disc. The dead zone function isshown in Fig.3. The dead zone have a bad influence oncontrol. The dead zone is dealt with a function to cancel the nonlinearity of the dead zone. The relationshipbetween braking torque τ1 and duty rate ud is shown asEq.(15).
{0(0 ≦ ud ≦ θ)
τ1 (ud ) =(15)a(ud − θ) (ud ≧ θ)

Lemma 1 :If there exist X and Y satisfy following LMI conditions,the system is stabilized by the state feedback u(t) =Kx(t) = Y X −1 x(t). And, by minimizing γ, the costfunction J is also minimized.
minimize γ, subject to
1He[Ai X + Bi Y ] X T (Q 2 )T1
(Q 2 )X−I
1(R 2 )Y0

The new input is defined as follows.
ũd = ud + θ(16)Here, θ is obtained from Eq.(1), (2) and Eq.(15).

The characteristic of the dead zone disappears.
{0(ũd = 0)
τ1 (ũd ) =aũd (ũd ≧ 0)

4Controller Synthesis

4.1

State Space Representation(17)

trace(W ) < γ(29)(18)

In this section, state space representation is derivedfrom Eq.(14). In order to track output of the system tooptimal value without error, an integrator added to thestate variable. The state variable x(t) and input u(t)are defined as follows.
x(t) = [x1 (t) x2 (t)]T∫= [ (λ − λ∗ )dt (λ − λ∗ )]T(19)u(t) = aũd − τ1 ∗(20)

1

Proposed Method
Conventional Method

0.6

0.4

0.2(21)00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4(22)

Time[s]

Polytopic Representation

The polytopic representation is used to guarantee therobust stability for the uncertain parameters such as theroad surface friction and the car velocity. The ranges of
α,β are assumed as follows.
α ∈ [αmin , αmax ]β ∈ [βmin , βmax ]A1 = A(αmin , βmin ), A2 = A(αmin , βmax )A3 = A(αmax , βmin ), A4 = A(αmax , βmax )Figure 4 Slip Rate Simulation at the wet road

1(23)
(24)

Let A1 , A2 , A3 and A4 be the vertex matrices for thevariation range of matrix A.

4.3

In this section, the validity of the proposed method
is illustrated by simulations. The proposed method iscompared with the model not considering the dead zone
by using robust LQ controller. Simulation results of slip
rate in any road conditions are shown in Fig.4, 5 and 6.

Proposed Method
Conventional Method

0.8

Slip Rate

4.2

Simulation

0.8

Then the state space representation is obtained as follows.
ẋ(t) = Ax(t) + Bu(t)
[]
[]
010
A=,B =0 αβC1βC25

Slip Rate1 r1
θ = ud − ( J2 ω̇2 + J ω̇1 )a r2

1Y T (R 2 )T
 < 0(27)0−I(i = 1, 2, 3, 4)[]W I> 0(28)
I X

0.6

0.4(25)

LQ Control Design

In this study, LQ controller is designed based on LMI.
To derives a stabilizing state feedback controller u(t) =Kx(t), the cost function J is defined as follows.
∫ ∞J=(x(t)T Qx(t) + u(t)T Ru(t))dt
(26)

0.2

00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time[s]Figure 5 Slip Rate Simulation at the dry road

0Here, Q ≧ 0, and R > 0 are weight matrices for statevariables, and input. The LMI conditions to minimize
the cost function and to satisfy the stability conditionsare shown as follows.

As can been seen in Fig.4 and Fig.6, the slip rate of proposed method converges more quickly around the optimal value 0.2 and does not oscillate. The simulation result of slip rate at the dry road is not converging around1

Proposed Method
Conventional Method50

Velocity[km/h]

Slip Rate

0.8

0.6

0.40.20040

Car Velocity−experiment
Car Velocity−simulation
Wheel Velocity−experiment
Wheel Velocity−simulation

30
20
100.5

11.5

22.5

33.5

44.5

00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Time[s]Time[s]Figure 6 Slip Rate Simulation at the snow road

Figure 8 Velocity of Car and Wheel

the optimal value 0.2. However, the car speed stopped
at 0.7 seconds. This value is very small. So, it can besaid that the function of ABS is satisfied. Therefore,
the proposed method is better than the conventionalmethod.

the conventional method. The eﬀectiveness of the proposed method is illustrated by experiments. The slip
rate is controlled around the optimal value 0.2 when thecar velocity is 50[km/h] to 18[km/h]. The result meansthat ABS works well when the car velocity is fast, andthe locking of the car wheels can be prevented. Furthermore, the eﬀectiveness of the proposed method isverified.

6Experiment

In this section, the eﬀectiveness of the proposed
method is illustrated by experiments. The experimental result and simulation result of slip rate are shown in
Fig.7. Their result of car and wheel velocity are shownin Fig.8. The slip rate is controlled around the optimalvalue 0.2 when the car velocity is 50[km/h] to 18[km/h].
However, the slip rate and the wheel velocity oscillate
when the car velocity is 18[km/h] to 0[km/h]. The result means that ABS works well when the car velocity is
fast, and the locking of the car wheels can be prevented.
By the result, it can be said that the eﬀectiveness of theproposed method is verified.
1experiment
simulation

Slip Rate

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Time[s]Figure 7 Slip Rate

7Conclusion

In this study, a method to guarantee the robust stability for uncertain parameters such as the road surface
friction and the car speed for anti-lock braking system(ABS) is proposed. Also, a method to deal with thedead zone for ABS is proposed. The validity of theproposed method is illustrated by comparing the model
not considering the dead zone in simulations. In any
road conditions, the proposed method is better thanReferences[1] Fangjun Jiang and Zhiqiang Gao.” An Applicationof Nonlinear PID Control to a Class of Truck ABS
Problems.” IEEE Conference on Decision and Control. Vol. 1, 1998.
[2] Jeﬀery R.Layne and Kevin M Passino.” Fuzzy
Learning Control for Antiskid Braking Systems.”
IEEE Transactions on Control Systems Technology
1.2 ,p122-129, 1993.
[3] Seunghwan Baek, Jeonghoon Song and Duksun
Yun.” Application of a Sliding Mode Control toAnti-lock Brake System.” International Conference
on Control, Automation and Systems, 2008.
[4] Idar Petersen, Tor A. Johansen, Jenis Kalkkuhl andJens Ldemann.” Wheel Slip Control in ABS Brakes
Using Gain Scheduled Constrained LQR.” Control
Conference (ECC), European. IEEE, 2001.
[5] Makoto Yokoyama, Yoshiaki Iwata, Shinji Katayose,
Masamichi Imamura and Makoto Nibe.” Antilock
Braking System Using Sliding Mode Control.” The
Japan Society of Mechanical Engineers, No.96-0678,
1997 (in Japanese).
[6] Leonardo Acho, Faycal Ikhouane, and Gisela Pujol.”
Robust Control Design for Mechanisms with Backlash.”Journal of Control Engineering and Technology (JCET) 3.4 p175-180, 2013.
[7] Nizar J. Ahmad, Mahmud J. Alnaser ,Ebraheem
Sultan and Khuloud A. Alhendi.” Robust Adaptive
Control for a Class of Systems with Deadzone Nonlinearity.” Intelligent Control and Automation 6.01,
2015.
[8] Hiroyuki Inoue, Kenji Mori, Shogo Miyasaka,Takanori Fukao and Norihilo Adachi.” Hybrid Brake
Controller of a Vehicle for Velocity and Slip Servo.”
The Japan Society of Mechanical Engineers, No.038, 2003 (in Japanese).