Robust LQ Control for Anti-lock Braking Systemwith Tracking Velocity Model
M2014SC025 Hisatsugu YAMAZAKI
Supervisor : Isao TAKAMI

Abstract
In this paper, the tracking velocity model for antilock braking system (ABS) is proposed. Furthermore,
the robust stability for car velocity and the coeﬃcientof friction between tire and road is guaranteed by using
polytopic representation. Then the robust controller isobtained by solving a ﬁnite set of Linear Matrix Inequalities (LMIs). Finally, the eﬀectiveness of the proposed
method is illustrated by simulations and experiments.

1

Introduction

Anti-lock Braking System (ABS) prevents cars fromslipping by locks of wheels in brake operations. It is wellknown that friction coeﬃcient for lateral force and longitudinal force between tire and road are high enoughwhen slip rate is nearly around 0.2[1]. By keeping theslip rate 0.2, skidding can be prevented. Since the slip
rate is represented as the function of car velocity andwheel velocity, the ABS dynamics depends on these velocities and the coeﬃcient of friction.
Many contributions for ABS can be found in the literature. Proportional integral derivative (PID) typeapproach[2], nonlinear PID type approach[3] and fuzzycontrol[4] are reported. On the other hand, model based
approach are presented. Sliding mode control is applied[5]. However, this algorithm is quite complicated.
On the other hand, linear control theory enables to design easily a feedback law by describing a plant as linear
state equation. The performance of the designed controller is evaluated easily. In particular, LQ control canyield better performance with smaller input by minimizing a cost function. Hence applying LQ control to ABS
is eﬀective. Here, I consider slip rate follow 0.2 from0. When car velocity is 50[km/h], wheel velocity follows 40[km/h] from 50[km/h], that means, deceleration
is 10[km/h]. When car velocity is 10[km/h], wheel velocity follows 8[km/h] from 10[km/h], that means, deceleration is 2[km/h]. Therefore, the brake torque following
reference slip rate 0.2 is varied by initial value of car
velocity. However, in the model based on slip rate, reference is constant in any car velocity. Therefore, there
is a diﬀerence of control performance by initial value ofcar velocity because control input is not varied by carvelocity.
In this study, the tracking velocity model is proposed.
The tracking velocity model controls slip rate by tracking wheel velocity to reference wheel velocity. In thetracking velocity model, reference wheel velocity is varied by car velocity. Therefore, in high velocity, control
input is large. In low velocity, control input is small.
Therefore, the tracking velocity model is expected to
improve control performance because control input isvaried by car velocity.
Furthermore, the dynamics of ABS depends on theuncertain parameters, which are car velocity and thecoeﬃcient of friction between tire and road. The robust
stability is required to design ABS with respect to these
uncertain parameters. The robust stability for the sys-tem with uncertain parameters is guaranteed theoretically by using matrix polytopic representation. Thenthe robust LQ controller is obtained by solving a ﬁnite
set of Linear Matrix Inequalities (LMIs). Finally, theeﬀectiveness of the proposed method is illustrated bysimulations and experiments.

2Control Target and Modeling

2.1

Equation of Motion

The model of the simpliﬁed ABS experimental device
used in this study is shown in Fig.1. It is one wheel
model that is 1/4 scale of the real vehicle. The upper
wheel simulates the car wheel, and the lower wheel simulates the road. A control law is designed to keep theoptimal slip rate 0.2 by operating the braking torqueapplied to the upper wheel. Table 1 shows physical constants and variables used in this study.

Fig. 1 Schematic Diagram of ABS Experimental Device

Table 1 Physical Parameters
Parameter
SymbolAngular velocity of the upper wheel
ω1Angular velocity of the lower wheel
ω2
Velocity of the upper wheel
Vr
Velocity of the lower wheel
V
Radius of the upper wheel
r1
Radius of the lower wheel
r2
Moment of inertia of the upper wheel
J1
Moment of inertia of the lower wheel
J2
Normal force
FnBraking torqueτb
Slip rate
λCoeﬃcient of friction between wheels
µ(λ)

Unit
[rad/s][rad/s]
[m/s]
[m/s]
[m]
[m]
[kgm2 ]
[kgm2 ][Nm][Nm]The dynamical equations of the rotational motion ofthe upper and lower wheels are shown as Eq.(1) and (2).
J1 ω̇1 = Fn r1 µ(λ) − τbJ2 ω̇2 = −Fn r2 µ(λ)

(1)
(2)

Slip rate is deﬁned as Eq.(3) by car velocity and wheelvelocity.
λ=r2 ω2 − r1 ω1
V − Vr=r2 ω2V(3)Road friction coeﬃcient µ(λ) is given as Eq.(4)[6].
µ(λ) = α tan−1 (80λ)

2.3Extended System

In order to track the output of the system to the optimal value without error, an integrator is added tothe state variable.
Let the state variable be x(t) =∫[Vr − Vr∗
(Vr − Vr∗ )dt]T and control input be u(t) =τb − τb∗ . Then the state space representation is obtainedas Eq.(9).
ẋ(t) = Ax(t) + Bu(t)[ α][ r1 ]− J1− V c1 0
A=,B =
100(4)

Here, α is a varying parameter by road conditions. The
relationship between α and road conditions are shownin Table2 and Fig.2.
Table 2 The relationship between α and road conditionsαroad conditon
0.45
Dry road0.2
Wet road
0.065
Ice road

3Controller Synthsis

3.1

Polytopic Representation(9)

The polytopic representation is used to guarantee therobustness for car velocity and the coeﬃcient of friction.
For Eq.(9), let β be β= V1 . Lower bound and upperbound of uncertain parameters are given as follows.
α ∈ [αmin , αmax ]β ∈ [βmin , βmax ]

(10)
(11)

Let A1 , A2 , A3 and A4 be the vertex matrices for thevariation range of matrix A.
A1 = A(αmax , βmax ), A2 = A(αmin , βmin )A3 = A(αmax , βmin ), A4 = A(αmin , βmax )3.2

LQ Control Design

To derive a stabilizing state feedback u(t) = Kx(t),consider to minimize the following cost function.
∫ ∞J=(x(t)T Qx(t) + u(t)T Ru(t))dt(12)0Fig. 2 The relationship between α and road conditions
2.2

Tracking Velocity Model

The proposed method controls slip rate by tacking
wheel velocity to reference wheel velocity. Therefore,
consider tracking error as Eq.(5).
z = Vr − Vr∗(5)

Here, reference wheel velocity is deﬁned as follows fromEq.(3).
Vr∗ = (1 − λ∗ )V, λ∗ = 0.2(6)Eq.(7) is derived from Eq.(1), (2), (3) and (5).
r2r1r2ż = ( 1 + 2 (1 − λ∗ ))Fn µ(λ) − τb
J1J2J1αr1c1 (Vr − Vr∗ ) − (τb − τb∗ )VJ1Here, c1 is a constant.

trace(W ) < γ (15)

(7)
Here,

To apply linear control theory, nonliner term in Eq.(7) islinearized around equilibrium point (Vr∗ , τb∗ ). Here, τb∗ isthe equilibrium braking torque to keep Vr∗ . Using Taylor
expansion around the equilibrium point, this nonlinear
model is linearized as follows.
ż = −Here Q ≥ 0 is a weight matrix for state variables, andR > 0 is a weight matrix for inputs.
The LMI conditions to minimize the cost function andto satisfy the stability conditions are shown as follows.
Lemma 1 : If there exist X and Y satisf ying
the f ollow LM Is, the system is stabilized by u(t) =Kx(t) = Y X −1 x(t).
minimize :γsubject to X ≻ 011He[Ai X + BY ] X(Q 2 )T Y T (R 2 )T1 ≺ 0(13)Q2 X−I01
R2 Y0−I(i = 1, 2, 3, 4)[]W I≻ 0 (14)
I X(8)

X = P −1 , X −1 ≺ W, Y = KX, J < γ4

Simulation

4.1

Validity of Proposed Method

In this section, the validity of the proposed method isillustrated by simulations. In this study, the range ofroad conditions is assumed as from ice road（α=0.065）
to dry road（α=0.45), and the range of the car velocity
is assumed as from 10[km/h] to 50[km/h]. Simulation
cases are shown in Table.3.

Table 3 Simulation Conditions
Road Condition
DryDry
Ice
Ice

Case1
Case2
Case3
Case4

4.1.1

Initial Value of Car Velocity[km/h]
50205020

Simulation at Case1

The simulation result of slip rate at the case1 is shownin Fig.3. The simulation result of car velocity and wheelvelocity are shown in Fig.4. As can be seen in theseﬁgures, wheel velocity is tracked reference wheel velocity
and slip rate is kept around reference slip rate 0.2.

The simulation result of slip rate at the case3 is shownin Fig.7. The simulation result of car velocity and wheel
velocity are shown in Fig.8. There exist overshoot of slip
rate at ﬁrst but it is controlled to 0.2 from Fig.7.
4.1.4

Simulation at Case4

The simulation result of slip rate at the case4 is shownin Fig.9. The simulation result of car velocity and wheel
velocity are shown in Fig.10. There exist overshoot ofslip rate at ﬁrst but it is controlled to 0.2 as can be seenin Fig.9.

Fig. 9 Case4:Slip Rate
Fig. 3 Case1:Slip Rate
4.1.2

Fig. 10 Case4:Velocity ofCar and Wheel

Fig. 4 Case1:Velocity of From the simulation results, slip rate is stabilized. TheCar and Wheel
robust stability is guaranteed in the range.

Simulation at Case2

The simulation result of slip rate at the case2 is shownin Fig.5. The simulation result of car velocity and wheel
velocity are shown in Fig.6. Slip rate is not kept around0.2. However, the car stops within 0.4 seconds. Since thecar stops at short time, the function of ABS is fulﬁlled.

4.2Eﬀectiveness of Proposed Method

In this section, the eﬀectiveness of the proposed
method is illustrated by simulations. The tracking velocity model is compared with the model based on slip
rate by using robust LQ controller. Simulation cases arecase1,3 and 4. Because the car stops at very short time,Case2 doesn’t compare.
4.2.1

Simulation at Case1

The simulation result of slip rate at the case1 is shownin Fig.11. The simulation result of the braking torque atthe case1 is shown in Fig.12. As can be seen in Fig.11,
both proposed method and conventional method show
similar performances.

Fig. 5 Case2:Slip Rate
4.1.3

Fig. 6 Case2:Velocity ofCar and Wheel

Simulation at Case3

Fig. 11 Case1:Slip Rate
4.2.2
Fig. 7 Case3:Slip Rate

Fig. 12 Case1:Brake Torque

Simulation at Case3

Fig. 8 Case3:Velocity of The simulation result of slip rate at the case3 is shown
Car and Wheelin Fig.13. The simulation result of the braking torqueat the case3 is shown in Fig.12. As can be seen in theseﬁgures, both proposed method and conventional method
show similar performances.

Fig. 17 Slip Rate

Fig. 13 Case1:Slip Rate
4.2.3

Fig. 14 Case1:Brake Torque

Simulation at Case4

The simulation result of slip rate at the case4 is shownin Fig.15. The simulation result of the braking torqueat the case4 is shown in Fig.16. As can be seen in theseﬁgures, slip rate of proposed method becomes smaller
overshoot and converges more quickly on reference sliprate 0.2 than conventional method.

Fig. 18 Velocity of Car and Wheel

6Conclusion

In this study, the tracking velocity model for antilock braking system (ABS) is proposed. Furthermore, amethod to guarantee the robust stability for car velocity
and the friction coeﬃcient is proposed. The eﬀectivenessof the proposed method is illustrated by comparing themodel based on slip rate in simulations. The tracking
velocity model improves the control performance. The
validity of the proposed method is illustrated by simulations and experiments. Slip rate is kept around the optimal value 0.2 when car velocity is more than 15[km/h]
from experiment results. This means ABS works well athigh velocity by the proposed method.

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