Gain Scheduled Control of ABS Based on Friction
Coeﬃcient Estimation
M2015SC002 Kazuki GOTO
Supervisor Isao TAKAMI

Abstract
FnAnti-lock Brake System(ABS) prevents cars from slip
by locks of wheels in brake operation at low friction roadsurface. The ABS are designed to optimize braking effectiveness while maintaining car controllability. TheABS is controlled by slip rate. In this paper, a friction
coeﬃcient between a road surface and the wheels of avehicle is estimated and a reference slip rate is set asa function of the friction coeﬃcient based on the estimation method. The gain scheduling control based onlyapunov function is applied to the ABS. Friction coefficient is estimated by unscented Kalman filter. Finally,
validity of the proposal method is illustrated by simulations.

1Introduction

In the case of breaking such as hitting brakes, theAnti-lock Brake system which is a device prevents thelock of the wheel. It is controlled by slip rate, and it is
set to follow the reference slip rate. A lot of studies haveconsidered to improve ABS control methods. Examples
include sliding mode control[1] and nonlinear control[2].
The slip rate of the control target is set at 0.2[3]. But
the slip rate depends on the friction coeﬃcient. So it
is considered that the reference slip rate has the mostsuitable value depending on friction coeﬃcient. In thispaper, a friction coeﬃcient between a road surface andthe wheels of a vehicle is estimated and a reference slip
rate is set as a function of the friction coeﬃcient basedon the estimation method. The state of the road surface is assumed to be first frozen, then wet and the lastdry. This paper design the controller in consideration ofan reference slip rate depending on these road surfaces
state. Unscented Kalman filter(UKF) is used to estimate the friction coeﬃcient. Kalman Filter is a filter toestimate the state of a certain dynamic system using anobserved value with the noise. In this paper, the friction coeﬃcient is estimated by UKF. Then schedulingparameter of GS controller set friction coeﬃcient, andthe reference slp rate is set as a function of the friction
coeﬃcient the estimated by UKF. The eﬀectiveness ofthe proposed method is illustrated by simulations.

2r1µFnω2

r2

Figure 1 Simplified diagram of the ABS experimented
deviceTable 1 parameter
Radius of the upper wheel
Radius of the lower wheel
Vertical force
Moment of inertiaof the upper wheel
Moment of inertiaof the lower wheel
Angular Velocityof the upper wheel
Angular Velocityof the lower wheel
Braking torque
Slip rate
Friction coeﬃcient
between wheels

and Eq.(2).

r1 = 0.0995
r2 = 0.099
Fn = 58.234
J1 = 7.53 × 10−3[m][m][N][kgm2 ]J2 = 25.60 × 10−3

[kgm2 ]

ω1[rad/s]

ω2[rad/s]

τ1
λµ[Nm](−)[−]

J1 ω˙1 = Fn r1 µ − τ1(1)J2 ω˙2 = −Fn r2 µ(2)The slip rate is defined in Eq.(3).
λ=Modeling

The model of the simplified ABS experimental device
used in this study is shown in Figure1[4]. Table1 shows
the physical constants and variables used in this study.
The upper wheel simulates the car wheel, and the lower
wheel simulates the road surface. The car velocity assumes to be between 10 to 50[km], and designs the control system that a friction coeﬃcient let a slip rate follow
the reference value within 0.1 to 0.7. It is assumed herethat µ =0.7 is for dry road and µ =0.1 is frozen road.
The reference slip rate can be obtained by friction coefficient.
The dynamical equations of the rotational motion of
the upper wheel and lower wheel are shown in Eq.(1)ω1r2 ω2 − r1 ω1r2 ω2(3)The following equation is obtained from Eq.(1), Eq.(2)and Eq.(3).
λ̇ = −1 r2 Fn µr1
1 r1((1 − λ) +)+)
ω2J2J1 r2
ω2 J1 r2(4)The behavior around equilibrium point (λ∗ , τ1 ∗ ) is considered. Here, λ∗ is the reference slip rate, and τ1 ∗ isthe equilibrium brake torque. Using Taylor expansion
around the equilibrium point, nonlinear model Eq.(4)can be linearized.
λ̇ ≃ λ̇(λ∗ , τ1 ∗ ) +∂∂λ̇(λ − λ∗ ) +λ̇(τ − τ ∗ )
∂λ∂τ

(5)

The following expression are provided from Eq.(5).

4.1

λ̇ = −σ(C1 λ∗ + C2 − C1 µ)(λ − λ∗ ) + σC3 (τ1 − τ1∗ ) (6)This paper try to design the gain scheduling controller.
The following equation is about the linear matrix inequallity.
ẋ(t) = Ax(t) + Bu(t)
(9)

σ=

3

1r 2 Fn
r2 Fnr 1 2 Fn
r1, C1 =, C2 =+, C3 =
ω2
J2J2
J1 r2
J1 r2

Handling of the Slip Rate

It is widely accepted that a reference slip rate of 0.2 isconvenient in conventional method. However it is shownin figure2[4] that the reference slip rate is suitable foreach friction coeﬃcient. Reference slip rate is diﬀerent
depending on the stare of the wheel. The slip rate(λ)
corresponding to the maximum friction coeﬃcient iscalled λopt . Slip rate has close relation to a braking friction coeﬃcientµb and a slide slip friction coeﬃcientµs .
By keeping slip rate to optimal value, µs keeps very highvalue. Therefore, directional stability and maneuverability are both assured. Here, 0(%)≦ λ ≦ λopt is calledstability region, λopt < λ ≦ 100(%) is called unstableregion[5]. In this paper, a friction coeﬃcient between
road surface and the wheels of a vehicle is estimatedand a reference slip rate is set as a function of the friction coeﬃcient based on the estimation method. The
reference slip rate can refer to the relations of the friction coeﬃcient between a road surface and the wheels
as follows. (α, β are constant)λ∗ = αµ + β(7)GS Control Design

The parameter box Θ is defined by vertexes which areupper bound and lower bound of µ. The schedulingparameter θ is the friction coeﬃcient.
Θ = θ = µ : θ ∈ {θ, θ}(10)

To derive a stabilizing state feedback controller u = Kx,
the following cost function is considered.
∫ ∞J=(x(t))T Qx(t) + u(t)T Ru(t))dt(11)0Here Q > 0 is a weight matrix for state variables, andR > 0 is a weight matrix for inputs. By minimizing γ,
the cost function J is minimized.
P (A + BK) + (A + BK)T P + K T RK + Q < 0 (12)xT0 P x0 ≺ γ
LMI condition to derive the stabilizing state feedback
GS controller is as follows. Let X := P −1 , Y := KX.
Theorem1If there exists X, Y (Θ) satisfying the following LMI, thesystem is stable.
minimize : γ, subject to X > 01He[AX + BY (θ)] X T (Q 2 )T1
Q2 X−I
1R 2 Y (θ)
0[WI]
I>0X

1Y (θ)T (R 2 )T<00−I(13)trace(W ) < γ

Figure 2 Comparison of slip rate

5Control Design

In this paper, the state variable assumes it the deviation of the slip rate and the integral calculus. Stateequation is obtained from∫ Eq.(8). Let state variablex(t) = [x1 (t) x2 (t)]T = [ (λ − λ∗ ) λ − λ∗ ]T and inputu(t) = τ1 −τ1∗ . The state equation is obtained as follows.
ẋ(t) = Ax(t) + Bu(t)[01A=0 −σ(C1 λ∗ + C2 − C1 µ)[]0
B=σC3(15)Finally, the gain scheduling controller K is given as follow.
K = Y (θ)X −1(16)14(14)(8)]

Kalman Filter

A friction coeﬃcient between a road surface and thewheels of a vehicle can be measured in real time. Inthis paper, a friction coeﬃcient between a road surface
and the wheels of a vehicle is estimated and a target slip
rate is set as a function of the friction coeﬃcient basedon the estimation method. So, in this study, Unscented
Kalman filter is used to estimate the friction coeﬃcient.
This paper designs UKF based on the theory of the referenced paper[6]. UKF can be applied for nonlinear system. Therefore, UKF can be applied to the estimate
friction coeﬃcient. The following equations can be used
to design the UKF. (τg is Torque acting on the balancelever)J2 ω˙2 = −r2 τg S − r2 Sτ1(17)S=µL(sinϕ − µcosϕ)

(18)

50

S is function of µ. So to estimate the friction coeﬃcient,
S is added to state variable. Let state variable x2 (t) =[ω2 (t) S(t)]T , and input u2 (t) = τ1 .

[A2 =y(t) = C2 x2 (t)[ r2 ][ ]τg r2 ]
1JJ22, B2 =, C2 =
0000040
Velocity[km/h]

ẋ(t) = A2 x2 (t) + B2 S(t)u2 (t)Proposal method
Conventional method(19)(20)30

20

10µ is varing parameter, and S is varying parameter, too.
S is defined as follows.
S(k + 1) = S(k) + v(k)(21)v(k) is process noise. Let state variable x2d (t) =[ω2 (k) S(k)]T and input u2d (t) = τ1 . The discretized
nonlinear state space representation is as follows.

001

2345678Time[s]Figure 3 Comparison of the car velocity at the frozen
road
1
Proposal method
Conventional method

x2d (k + 1) = A2 dx(k) + B2 dS(k)u2 d(k) + Bv v(k) (22)0.8y(k) = C2 x2d (k) + ω(k)∫ TsA2 T s
A2 d = e, B2d = (eA2 λ dλ)B20.6

Slip Rate(23)0.40

Bv = [01]0.2T0ω(k) is observation noise. T s is sampling time. let x̂ isa state estimate, and g(k) is a Kalman gain. Optimalvalue of x̂(k) and g(k) are as follows.
x̂(k) = x̂− (k) + g(k)y(k) − ŷ − (k)g(k) =−Pxy(k)−Pyy(k) + σω2012345Time[s]678Figure 4 Slip Rate at the frozen road
(24)
0.3

muestimate mu(25)x̂− (k) is a priori state estimate. ŷ − (k) is a priori output−estimate. Pxy(k) is a priori state, output error covari−ance matrix. Pyy(k) is a priori output error covariance2matrix. σω is of covariance of observation noise ω(k).
The optimal value x̂(k) is decided by the Kalman gain
g(k). Therefore, the friction coeﬃcient between a roadsurface and the wheels of a vehicle can be estimated byEq.(24).

mu0.2

0.1

000.2

0.4

0.6

0.8

1

Time[s]6Simulation

In this section, the eﬀectiveness of the designed controller is verified by simulations. Simulations are conducted on frozen road, wet road and dry road. Simulations are shown the slip rate, comparison of the car
velocity and comparison of the friction coeﬃcient. The
relations of reference slip rate and friction coeﬃcientis shown in Table2. The initial speed of the car is50[km/h].
Table 2 Relations of a friction coeﬃcient and the reference slip rate
RoadFriction coeﬃcient Reference slip rate
Frozen
0.1
0.3
Wet
0.4
0.25
Dry0.70.2

Figure 5 Comparison of the friction coeﬃcient

Figure.3,6,9 show the proposal method can stop thanconventional method. From the simulation result, proposal method has a shorter time to stop than conventional method. Figure.4,7,10 show the reference slip rate
can change by the friction coeﬃcient. In Figure5,8,11
shows the comparison of the friction coeﬃcient. Solidline shows the friction coeﬃcient value that really togives, and dot line shows friction coeﬃcient value thatto estimate by UKF.

7Conclusion

In this paper, a friction coeﬃcient between a road surface and the wheels of a vehicle is estimated and a reference slip rate is set as a function of the friction coeﬃcient
based on the estimation method on ABS. Therefore, the50

Proposal method
Conventional method40
Velocity[km/h]

Velocity[km/h]

4030

20

100Car Velocity

50

Proposal method
Conventional method

3020

1000.20.4

0.6

0.80100.2

Time[s]Figure 6 Comparison of the car velocity at the wet road

Proposal method
Conventional method

0.8

0.8

0.6

0.6

Slip Rate

Slip Rate

Slip Rate

1

Proposal method
Conventional method

0.4

0.4
0.2

0.2

000.2

0.4
0.6
Time[s]

0.8

1

0.1

0.2

0.3

0.4

mu
estimate mu

0.7

0.6

0.6

mu

0.8

0.7

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1
0.4

0.6

0.6

0.8

mu
estimate mu

0.9

0.8

0.2

0.5

Figure 10 Slip rate at the dry road
110.9

000

Time[s]Figure 7 Slip rate at the wet roadmu

0.6Figure 9 Comparison of the car velocity at the dry road

100.4
Time[s]1Time[s]Figure 8 Comparison of the friction coeﬃcienteﬀectiveness of the proposed method is verified.

References[1] Makoto Yokoyama, Yosiaki Iwata, Sinji Katayori,
Masamiti Imamura, Makoto Niibe. ”Sliding ModeControl of Anti-lock Brake System.”, The Japan Society of Mechanical Engineers. Vol.63. No611. 1997.
(in Japanese).
[2] Tor A.Johasen, Idar Petrersen, Jens Kalkkuhl andJens Ludemann. ” Gain-scheduled Wheel Slip Control in Automotive Brake Systems.”, IEEE Transaction on Control System Technology, Vol11, 2003.
[3] Yuichi IKEDA,Takashi NAKAJIMA,Yuichi CHIDA.
”Vehicular Slip Ratio Control Using Nonlinear Control Theory. ”, Journal of System Design and Dy-000.1

0.2

0.3

0.40.5

Time[s]Figure 11 Comparison of the friction coeﬃcientnamics Vol6,2012.
[4] R.Khatun, C. M. Bingham, N. Schofield, and P. H.
Mellor. ”Application of Fuzzy Control Algorithms
for Electric Vehicle Antilock Braking/Traction Control Systems.” IEEE Transactions on VehicularTechnology, Vol.52, No.5, September 2003.
[5] NihonABScorporation ”Study of ABS for car use.”
Sankaishacorporation. 1993.
[6] Hiroshi Kataoka. ”Gain Scheduling Control for ABS
with Estimation of Friction Coeﬃcient by Unscented
Kalman Filter.”, 2015.