Linear Control of ABS without Using Approximation
Around Equilibrium Point
M2015SC009 Yuki MARU
Supervisor: Isao TAKAMI

Abstract
In this paper, linear control of ABS (antilock brakingsystem or antiskid braking system) without using approximation around equilibrium point is proposed. In
conventional study of ABS using linear control, approximation around equilibrium point is used. It makes calculation easier. On the other hand, slip rate and friction
coeﬃcient will become constant. This may occur control
error. Controller is designed to minimize the quadratic
cost function, which is model of LQ controller. Therobust stability for car velocity and friction coeﬃcient
between wheel and road surface are guaranteed by using
polytopic representation. By using descriptor representation and linear fractional transformation(LFT), thesystem which is aﬃne with respect to car velocity andfriction coeﬃcient can be obtained. Then, the problem
is formulated as solving a finite set of Linear Matrix Inequalities (LMIs). Finally, the robust stability and therobust performance are guaranteed. The eﬀectiveness ofthe proposed method is illustrated by simulations.

1

This may occur control error.
In this paper, linear control of ABS without using approximation around equilibrium point is proposed. Therobust stability for car velocity and friction coeﬃcient
between wheel and road surface are 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.

2Control Target and Modeling

2.1

Control Target

The model of the simplified ABS experimental device
used in this study is shown in Fig.1.

Introduction

An objective of ABS is to prevent car from occurring slip by wheel lock in brake operation at low friction
road surface or when sudden baking. ABS was first developed for braking system of aircrafts(antiskid brakingsystem). If aircraft make landing without ABS, wheels
will wear out badly because of friction between road andthe wheels. The wheels have to get changed often because it wears oﬀ quickly, and it cost a lot. If worse, thewheels will burst and an atrocious accident may happen.
Because of these reasons, ABS was developed to give anoptimal braking operation. This is that, until aircraftstops, the brake is operated not to stop the wheels perfectly. ABS(antilock braking system) loaded to cars aredeveloped using aircraft’s ABS. An objective of car’s
ABS is to prevent car from occurring slip by wheel lockin brake operation at low friction road surface or whensudden baking. It is well known that when the slip rateis nearly around 0.2, friction coeﬃcient between wheel
and road surface are high enough[1]. By keeping the slip
rate to 0.2, braking distance and skidding can be prevented. Since the slip rate depends on car velocity andwheel velocity, ABS dynamic model depends on velocity and friction coeﬃcient between wheel and the roadsurface.
Many study have been done about ABS such as nonlinear PID control[2], PID-type fuzzy control[3], and sliding mode control[4]. However, nonlinear control is noteasy to be adapted. On the other hand, by adapting
linear control and represent plant as state equation, it
is able to design feedback system easily. Also, it is easy
to evaluate designed controller. Study of ABS using linear control such as gain scheduling control[5] and LQcontrol[6] are done. In conventional study of ABS using
linear control, approximation around equilibrium point
is used. It makes calculation easier. On the other hand,slip rate and friction coeﬃcient will become constant.

Figure 1 Simplified Diagram of the ABS Experimental
DeviceTable 1 The Physical Parametersparameter
symbolAngular velocity of the upper wheel
ω1Angular velocity of the lower wheel
ω2Radius of the upper wheel
r1
Radius of the lower wheel
r2
Moment of inertia of upper wheel
J1
Moment of inertia of lower wheel
J2
Normal force
FnBrake torqueτ1Slip rate
λFriction coeﬃcient between wheels
µ(λ)

unit
[rad/s][rad/s][m][m][kgm2 ]
[kgm2 ]
[N m]
[N m]

It is the quarter car model. The upper wheel simulates
the car wheel, and the lower wheel simulates the roadsurface. Slip rate λ is taken as an output, and the brake
torque τ1 is taken as an input. By controlling the brake
torque, control law is designed to keep slip rate at theoptimal rate 0.2. To design the control law, diﬀerential
equation expressing the dynamics of slip rate is required.
In this section, the process to derive the linearized differential equation is shown.

2.2

Dynamical Equation

The dynamical equations of the rotational motion of
the upper and lower wheels are shown by Eq(1) andEq(2).
J1 ω˙1 = Fn r1 µ(λ) − τ1J2 ω˙2 = −Fn r2 µ(λ)

(1)
(2)

system(10) can be rewritten as follow.
[][ ∗ ] [ ]A Bx0=yc refC 01u∗A ∗ ∗ ∗Here, yc ref = x∗ , C = 1,u∗ = − Bx .x ,u which satisfy
system(11)is given as follow.

[The slip rate is defined by Eq(3) as the function of car
velocity and wheel velocity.
λ=r2 ω2 − r1 ω1
V − Vw=r2 ω2V1Vw
V̇w + 2 V̇VV

11(d1 + d2 λ)µ(λ) − d3 τ1VVFn r1Fn r2Fn r21d1 =+, d2 = −, d3 = −J1J2J2J1(5)In this study, friction coeﬃcient µ(λ) is given as polynomial approximation Eq(6){p0 + p1 (λ − a) + p2 (λ − a)2 }1 + q1 (λ − a) + q2 (λ − a)2c1 + c2 λ + p2 λ2=αc3 + c4 λ + q2 λ2µ(λ) = α(6)The following equation is given by substituting Eq(6)to Eq(5).State variable x(t) and input u(t) are given asx(t) = λ,u(t) = τ1λ̇ = Ax(t) + Bu(t)(7)
2d1 c2 + d2 c1 + (d1 p2 + d2 c2 )λ + d2 p2 λA = −αV (c3 + c4 λ + q2 λ2 )1
B = − d3
V

2.4Controller Eq(9) which adding feedforward from targetvalue realize tracking control regarding target-value
y ref (t) = yc ref .

ref∗

(9)

(13)

State Space Representation

E x̃˙ e (t) = Ae x̃e (t) + Be ũ(t)
(14)
[]
10
E=
0 V (c3 + c4 λ + q2 λ2 )[]01Ae =0 −α{d1 c2 + d2 c1 + (d1 p2 + d2 c2 )λ + d2 p2 λ2 }[]0Be =−d3 (c3 + c4 λ + q2 λ2 )2.5Descriptor representation

To use polytopic representation, descriptor representation is applied to Eq(14). Let descriptor variable bex̃d (t) = [x̃e (t)T λ̇ u(t)] and derive the following descriptor equation.
x̃˙ d (t) = Ad x̃d (t) + Bd ũ(t)
1 0 0 0
 0 1 0 0 
Ed = 0 0 0 0 0 0 0 00100010  0Ad = 0 Ad32 Ad33 Ad34 000−1Bd = [ 0 001 ](15)

TAd32 = −α{d1 c2 + d2 c1 + (d1 p2 + d2 c2 )λ + d2 p2 λ2 }Ad33 = −V (c3 + c4 λ + q2 λ2 )Ad34 = −d3 (c3 + c4 λ + q2 λ2 )∗Let y (t) = yc ,x(t) = x ,u(t) = u , and consider
steady-state value condition ẋ∗ = 0 in system(8){0 = Ax∗ + Bu∗
yc ref = Cx∗(12)

In order to track the output of the system to theoptimal value without error, one integrator is addedto∫ the state variable. Let state variable be x̃e (t) =[ (λ − λ∗ )dt λ − λ∗ ]T . Then, the state equation isobtained as follows.

In this study, target-value tracking control using feedforward is considered. Let’s consider the following system.
{ẋ(t) = Ax(t) + Bu(t)(8)
y ref (t) = Cx(t)ref]−1 [ ]B
0=yc ref01[]
1=yc refu∗A
C

From Eq.(13), let new state variable and input be x̃(t) =λ − λ∗ ,ũ(t) = τ1 − τ1∗ .

Feedforwardu(t) = Kx(t) + Hy ref (t)[ũ(t) = K x̃(t)u(t) = K x̃(t) + u∗(4)λ̇ = −]

Let define fluctuation from steady-state value x∗ ,u∗ asx̃(t) = x(t) − x∗ ,ũ(t) = u(t) − u∗From Eq(1), Eq(2), Eq(3), and Eq(4), Eq(5) is derived.

2.3x∗u∗(3)

The following equation is obtained from Eq(3).
λ̇ = −(11)

(10)

2.6

Linear Fractional Transformation(LFT)

Only matrix Ad has uncertain parameters. However,high order terms of slip rate λ exist. So, LFT is applied
to transform high order terms of slip rate λ to first orderterm. Matrix Ad can be represented by Eq(16). Here,
An is the matrix which contains first order terms of λ,and Bδ (I − ∆Dδ )−1 ∆Cδ is the matrix which contains
high order term of λ.
Ad = An + Bδ (I − ∆Dδ )−1 ∆Cδ0100010  0An = 0 An32 An33 An34 000−1

3.3

From 3.1 and 3.2, if there exist Xl and Yl satisfying
the following LMI conditions, the system is stabilized
by gain Kl = Yl Xl −1 [6].

(16)minimize : γ
subject to : Xl ≻ 0
1−He{Al (Θi )Xl + Bl Yl } Xl (Q 2 )T1
Q 2 XlIRYl0≻ 0 (i = 1, 2, ..., 8)
[]
W I
≻0I X

TBδ = [ 0 0 1 0 ]Cδ = [ 0 −αd2 p2 λ −V q2 λ −d3 q2 λ ]Dδ = 0, ∆ = λAn32 = −α{d1 c2 + d2 c1 + (d1 p2 + d2 c2 )λ}An33 = −V (c3 + c4 λ)An34 = −d3 (c3 + c4 λ)trace(W ) < γ[X0Xl = X21 X22
X31 X32

Eq(15) can be expressed as Eq(17) by using An , Bδ , Cδ ,Dδ , and ∆.
Ed x̃˙ d = An x̃d + Bδ wδ + Bd ũZδ = Cδ x̃d + Dδ wδwδ = ∆zδ

4(17)Let x̃l (t) = [x̃d (t)T Zδ (t)]T be new descriptor variable,
and the new descriptor equation be Eq.(18).
El x̃˙ l (t) = Al x̃l (t) + Bl ũ(t)[][]Ed 0BdEl =, Bl =0 00[]AnBδ ∆Al =Cδ Dδ ∆ − I

3Controller Design

3.1

LQ controller

(18)

(19)

(20)

close loop system is as follows.
ẋ(t) = (A + BK)x(t)(21)

Following cost function for upper equation is considered∫ ∞J=(x(t)T Qx(t) + u(t)T Ru(t))dt(22)0Here, Q,R are weight matrices as follows.
Q = QT ⪰ 0 , R = R T ≻ 03.2

0R

(24)

(25)
(26)
(27)

]
, Yl = [ Y0 0 ]Simulation

In this section, comparison between proposed modeland conventional model is illustrated by simulation. In
the simulation, robust LQ controller is used. Here, conventional model is given first, then weight matrices forboth models, and finally simulations are illustrated.
State space equation of conventional model is given asbelow. Here, T = V1 .

Parameter box is given as follows.

state feedback controller beu(t) = Kx(t),

0X23
X33

YlT RTx̃˙ e (t) = Ac x̃e (t)Bc ũ(t)
(28)
[]
01
Ac =
d0 −T {(d1 + d2 λ∗ ) dλµ(λ∗ ) + d2 µ(λ∗ )}[]0Bc =−T d3

About linear state equationẋ(t) = Ax(t) + Bu(t),LMI Condition(23)

Polytopic Representation

Slip rate λ and car velocity V are nonlinear variables.
In this study, robust stability is considered. Fluctuation
range of slip rate λ and car velocity V are given as follow.
Also, α is given as polytopic representation.
Θ = {[θ1 , θ2 , θ3 ] : θi ∈ {θi , θi }}(i = 1, 2, 3)
θ1 = λ, θ2 = V, θ3 = α
Θ1 = (θ1 , θ2 , θ3 ), Θ2 = (θ1 , θ2 , θ3 ), · · ·, Θ8 = (θ1 , θ2 , θ3 )

Θ = {[θ1 , θ2 ] : θi ∈ {θi , θi }}(i = 1, 2)(29)
θ1 = T, θ2 = α
Θ1 = (θ1 , θ2 ), Θ2 = (θ1 , θ2 ), Θ3 = (θ1 , θ2 ), Θ4 = (θ1 , θ2 )LMI conditions are given as follows.If there exist Xl andYl satisfying the LMI conditions below, the system isstabilized by gain K = Yc Xc−1 [6].
minimize : γsubject to : X ≻ 0

1−He{Ac (Θi )Xc + Bc YC } Xc (Qc2 )T1
Qc2 XcIRYc
0YcT RT0R(i = 1, 2, 3, 4)
[]
Z I
≻0I X



(30)

≻0(31)(32)

trace(W ) < γc(33)Weight matrices and the upper bound of the cost function of proposed model are as follows.
Q = diag([ 100γ = 33.15654

10000 ]), R = 0.01

Qc = diag([ 100γc = 31.9236910 ]), R = 0.01

slip rate

Weight matrices and the upper bound of the cost function of conventional model are as follows.

0.5
0.4
0.3
0.2
0.1
0Proposed
Conventional

Vr[km]

5040
30
20
100

Proposed
Conventional

00.1

0.2

Vr[km]

Proposed
Conventional

0.2

0.3

t[s]

slip rate

Figure 2 Car Velocity On Dry Road

0.5
0.4
0.3
0.2
0.1
0

Proposed
Conventional

Braking Distance[m]

00.1

0.2

0.3

Vr[km]

22.5

3

3.5

44.555.5t[s]4030
201000123Proposed
Conventional
45t[s]

Since the car velocity and friction coeﬃcient aﬀects
the dynamics, a method is designed to consider variation of these parameters. To linearize the model, descriptor representation and linear fractional transformation (LFT), polytopic representation is applied for ABS.
Friction coeﬃcient is given as function of slip rate λby polynomial approximation. The process to obtain
dynamical model, controller, and LMI conditions areshown in this paper. From simulation, slip rate is controlled around the optimal value 0.2 on frozen road. By
comparing conventional model and proposed model, theresult is almost the same. However, on dry road, braking
distance of conventional model was a little shorter thanproposed model. It is presumed that by using descriptor
representation and LFT to proposed model, conservativeness get higher and readiness decreased.

References

32.5
2
1.5
10.5
000.1

Proposed
Conventional
0.3

0.2

t[s]

50
4030
20
100

Proposed
Conventional

00.5

11.5

22.5

3

3.5

4

4.5

5

5.5

t[s]

Vr[km]

1.5

t[s]Figure 3 Car Velocity On Dry Road50403020100Proposed
Conventional

00.5

11.5

22.5

33.5

44.5

55.5

t[s]

Figure 4 Car Velocity On Dry Road51Figure 5 Car Velocity On Dry Road50
4030
201000.1

0.50.3t[s]0Braking Distance[m]0In this simulation, braking starts from 50[km/h] on dry
road and frozen road.

Conclusion

In this study, linear control of ABS without using
approximation around equilibrium point is proposed.

[1] Yuichi Ikeda, Takashi Nakajima, Yuichi Chida, ”Vehicular Slip Ratio Control Using Nonlinear Control Theory”, Journal of Design and Dynamic Vol.6,No.2, p145-157.,2012[2] Fangjun Jiang, Zhiqiang Gao, ”An Application ofNonlinear PID Control to a Class of Truck ABSProblems”, Decision and Control, 2001. Proceedingsof the 40th IEEE conference on 2001.
[3] Chih-Keng Chen, Ming-Chang Shih, ”PID-Type
Fuzzy Control for Anti-Lock Brake Systems with Parameter Adaptation”, JSME International Journal
Series C Mechanical System, Machine Elements, andManufacturing, Vol.47, No.2, Special Issue on International Symposium on Sped-up and Service Technology for Railway and MAGLEV Systems, p.675685, 2004
[4] Nilanjan Patra, Kalyankumar Datta, ”Sliding mode
Controller for Wheel-slip Control of Anti-lock Braking System”, IEEE International Conference on
Advanced Communication Control and ComputingTechnologies, 2012[5] Idar Petersen, Tor A.Johansen, Jens Kalkkuhl andJens Ludemann, ”Wheel Slip Control in ABS Brakes
Using Gain Scheduled Constrained LQR” ,Proc. Europian Contr.Conf..,Porto, pp, 606-611.,[6] Hiroshi Kataoka, Tatuya mizuno, Hisatsugu Yamazaki, Gan Chen, Isao Takami, Robust Stabilization of Antilock Braking System with LQ Control ,MOVIC2014