Gain Scheduling Control for ABS with Estimation ofFriction Coeﬃcient by Unscented Kalman Filter
M2014SC008 Hiroshi KATAOKA
supervisor：Isao TAKAMI

Abstract
Anti-lock Brake System(ABS) is strongly nonlinear.
In my study, the system is linearized and state equation
is derived. However, the system has rational terms. Using descriptor representation and linear fractional transformation, equivalent system that has only linear terms
is derived. The gain scheduling control based on Lyapunov function is applied for the ABS. Friction coeﬃcient that is one of scheduling parameters is measured
by unscented kalman filter since the friction coeﬃcient
can not be measured on real time. The eﬀectiveness ofthe proposed method is illustrated by simulations.

1

Introduction

Anti-lock braking system (ABS) prevents cars fromslip by locking wheels in brake operation at low friction
road surface, or when suddenly braking. It is well known
that when slip rate is nearly around 0.2, friction coeﬃcient for lateral force and longitudinal force between tireand road are high enough[2] . By keeping the slip rate
0.2, braking distance can be shortened and skidding canbe prevented. Since the slip rate depends on car velocity
and wheel velocity, the ABS dynamics depends on themand the friction coeﬃcient of road. The main diﬃcultyin the design of ABS is caused by its strongly nonlinear
and uncertain characteristics. For such diﬃculty, many
contributions for ABS can be found in the literature. For
example, PID-type approaches[1] and composite control[2] are reported. Extremum seeking control basedon numerical optimization is presented in [3], [4]. The
fuzzy control which has learning ability to compensate
for adverse road condition is proposed[5] . On the other
hand, model based approaches are presented. In several
papers, sliding mode control for ABS is applied[6],[7],[8] .
Gain scheduling control is proposed[9],[10] . It is wellknown that GS control has a potential to deal with large
variation range and to improve the control performance.
Recently, unscented kalman filter (UKF) is proposed
to estimate uncertain parameters[11] . It is nonlinear
kalman filter. Extend kalman filter is one of the nonlinear kalman filter. However, Extend kalman filter isrequired to linearize nonlinear system. Since linearized
system doesn’t require in algorithm of UKF, UKF canbe applied for nonlinear system as that.
In this paper, a design method is proposed to guarantee the robust stability for ABS. Since the car velocity and friction coeﬃcient aﬀects the dynamics, thevariations of these parameters have to be considered.
The polytopic representation for the system with theseparameters is proposed. Since the system described inframework of state space equation has rational termsof car velocity and friction coeﬃcient, guaranteeing therobust stability for the system with car velocity and friction coeﬃcient by using the polytopic representation isdiﬃcult. By using descriptor representation and linear
fractional transformation (LFT), the system which isaﬃne with respect to car velocity and friction coeﬃcient
can be obtained. Then, the GS controller based on lin-

ear control theory can be designed. The robust stability
and the robust performance are guaranteed theoreticallyby solving LMIs. The GS controller whose schedulingparameters are car velocity and friction coeﬃcient is designed. However, friction coeﬃcient can not be measured on real time. To measure the friction coeﬃcient,
UKF is proposed. The measured friction coeﬃcient isassumed as scheduling parameters. Then, GS controller
is applied for ABS and the eﬀectiveness of the proposed
method is illustrated by simulations.

2

Modeling and Control Target

The model of the simplified ABS experimental device
used in this study is shown in Fig. 1. It is the one wheel
model that is 1/4 scale of the real car. The upper wheel
simulates the car wheel, and the lower wheel simulates
the road. The point A is the axis of rotation of the balance lever, which supports the axis of the upper wheel.
Table 1 shows the physical constants and variables usedin this study.

Fig. 1 Simplified Diagram of the ABS Experimental
DeviceTable 1 The Physical Constants and Variables
ω1
Angular velocity of the upper wheel
ω2Angular velocity of the lower wheel
τ1Braking torquer1
Radius of the upper wheel
r2
Radius of the lower wheel
J1
Moment of inertia of the upper wheel
J2
Moment of inertia of the lower wheel
Fn
Normal forceµCoeﬃcient of friction between wheels
τg
Torque acting on the balance lever
L
Distance to the contact between wheels
from the axis of rotation of the balance leverφAngle between L and the normalof contact at wheels
λSlip rate

The dynamical equations of the rotational motion of theupper wheel and lower wheel are shown by Eq.(1) andEq.(2).
J1 ω̇1 = Fn r1 µ − τ1J2 ω̇2 = −Fn r2 µ

(1)
(2)

The slip rate is defined by Eq.(3) as the function of car
velocity and wheel velocity.
λ=r2 ω2 − r1 ω1
r2 ω2

3.1

State equation is obtained from Eq.(10). In order totrack the output of the system to the optimal value without error, I add one integrator to the state∫ variable. Let
state variable x(t) = [x1 (t) x2 (t)]T =[ (λ − λ∗ )dt λ −λ∗ ]T and input u(t) = τ1 − τ1∗ . Then the state equation
is obtained as follows.
ẋ(t) = Ax(t) + Bu(t)y(t) = Cx(t)
[
]
[
]
0
0 1
A= 0 a, B = b ,c = [ 1
e
e

(3)

The sum of torques corresponding to the point A is given
as follows.
Fn L(sin φ − µ cos φ) = τg + τ1
The following equation is obtained from Eq.(3).
r1 ω1
r1
ω̇1 +
ω̇2
λ̇ = −r2 ω2
r2 ω22

1
1
f (λ) +g(λ)τ1 , ω2 ̸= 0
ω2
ω2

(5)
Here Eq.(11) has rational terms of µ and ω2 . It is difficult to use polytopic representation. Eq.(11) is transformed to descriptor form to transform from rational topolynomial. Descriptor equation is obtained as follows.

(6)

E ẋ(t) = Ax(t) + Bu(t)The behavior around equilibrium point (λ , τ1∗ ) is considered. Here, λ∗ is the reference slip rate, and τ1∗ is the∗[E=equilibrium braking torque to keep λ .
f (λ∗ )g(λ∗ )(7)

Eq.(8) is obtained by substituting the equilibrium point
in Eq.(6).
11f (λ∗ ) +g(λ∗ )τ1∗ = 0
ω2
ω2

(8)

Using Taylor expansion around the equilibrium point,
nonlinear model Eq.(6) can be linearized[9] .
λ̇ ≃ λ̇(λ∗ , τ1∗ ) +

∂ λ̇
∂ λ̇
|λ=λ∗ (λ − λ∗ ) +|τ =τ1∗ (τ1 − τ1∗ )
∂λ
∂τ1
(9)

The linearized equation is derived by substituting Eq.(7)into Eq.(9).
c1 µ2 + c2 µλ̇ = ()(λ − λ∗ )
ω2 (c6 µ2 + c7 µ + c8 )c3 µ2 + c4 µ + c5+()(τ1 − τ1∗ )ω2 (c6 µ2 + c7 µ + c8 )

1
0

0
e

]

[
,A =

0 1
0 a

(12)

]

[
,B =

3.2

Polytopic Representation

In this section, I show the process to obtain ABS system which is multi-aﬃne for uncertain parameters µ andω2 . The robust stability for these parameters is guaranteed by using polytopic representation. The systemdescribed in the framework of the state space representation has rational terms of these parameters. It is diﬃcult
to use polytopic representation for the system which hasrational terms of uncertain parameters. Therefore, thesystem which is multi-aﬃne for uncertain parameters µand ω2 is obtained by using the descriptor representation and LFT.

]

Putting All Uncertain Parameter and Varying Parameter into One Matrix

Eq.(12) becomes polynomial for uncertain parameters µ and ω2 . However, there exists these parameters in matrices E, A and B. If Eq.(12) is transformed to an ordinary state space representation ẋ(t) =E −1 Ax(t) + E −1 Bu(t), matrices E −1 A and E −1 B haverational terms of µ and ω2 . It is diﬃcult to use polytopic
representation. I put all uncertain parameters into one
matrix by expanding dimension of matrices. Uncertain
parameters µ and ω2 are put into matrix Ad by defining
xd (t) = [x(t) λ̇ u]T .



(10)

0
b

Note that Eq.(12) is polynomial for uncertain parameters µ and ω2 .

Ed ẋd (t) = Ad xd (t) + Bd u(t)Here, c1 , c2 , · · ·, c8 are constants. Note that the dynamics of λ̇ depends on ω2 and µ.

30 ]a = c1 µ2 + c2 µ
b = c3 µ2 + c4 µ + c5∗

τ1∗ = −(11)e = ω2 (c6 µ2 + c7 µ + c8 )(4)Finally, the dynamical equation of slip rate is derived
from Eq.(1), Eq.(2), Eq.(4) and Eq.(5) as follows.
λ̇ =Transformation to Polynomial1 0
Ed = 
0
0

0
1
0
0

0
0
0
0



0
0 1
0 
 0 0
,A =
0  d  0 a
0
0 0

Bd = [ 0

0

0

(13)


0
0
1
0 
−e b 
0 −1

T

1 ]

Note that there exist the all uncertain parameters andµ and ω2 in only Ad .
3.3

Transformation to Malti-aﬃneOnly the matrix Ad has uncertain parameters. However, there are higher order terms of µ in it. In this case,
it is still diﬃcult for the matrix to use the polytopic representation. LFT is applied to transform the high orderterms of µ to first order terms of µ.
Let x̃d be new descriptor variable x̃d (t) =[xTd (t) zδ (t)]T , and the descriptor equation which istransformed by LFT is the following equation. Then,the system becomes multi-aﬃne for µ and ω2 .
Ẽd x̃˙ d (t) = A˜d x̃d (t) + B˜d u(t)[tildeEd =


Ed
0

0
0

0 1
0 0
An = 
 0 c µ
2
0 0

]

[

, Ãd =

An
Cδ

Bδ ∆
−I

0
1
−(c7 µ + c8 )ω20][, B̃d =

Bd
0

]




0
0
0
 0 

, Bδ =  1 
c4 µ + c5 
−10Cδ = [ 0 c1 µ −c6 µω2 c3 µ ]∆=µAn , Cδ and Bδ ∆ don’t have polynomial but only multiaﬃne terms of µ and ω2 . The ABS model which is multiaﬃne for ω2 and µ can finally be obtained.

4

GS Control Design

I design a gain scheduling controller. The parameterbox Θ is defined by vertexes which are upper bound andlower bound of µ and ω2 . Let scheduling parameters θ1 ,
θ2 be θ1 =ω2 , θ2 =µ. Since I use parameter Lyapunov
dependent function, parameter box includes derivativeof its.
Θ = {[θ1 , θ2 , θ3 , θ4 ] : θi ∈ {θi , θ̄i }}(i = 1, 2) (14)
θ1 = ω2 , θ2 = µ, θ3 = ω̇2 , θ4 = µ̇
To derive a stabilizing state feedback u(t) = K̃d (θ)x(t),I consider minimizing the following quadratic cost function.
∫ ∞J=(x̃d (t)T Qx̃d (t) + u(t)T Ru(t))dt(15)0Here Q ≥ 0 is a weight matrix for state variables, and
R > 0 is a weight matrix for inputs. From Lyapunov’s
stability theorem, if there exists P(θ) satisfying the following matrix inequalities, the descriptor system is stable and J ≺ trace(Ẽd P (θ)) is guaranteed[12].
Ẽd P (θ) = (Ẽd P (θ))T ⪰ 0
(16)
T
˜
˜
˜
˜
P (θ)(Ad (θ) + Bd K̃d (θ)) + (Ad (θ) + Bd K̃d (θ)) P (θ)+ Q + K̃d (θ)T RK̃d (θ) ≺ 0(17)

Let X̃d (θ) := P (θ)−1 . Then, let Ỹd (θ) := K̃d (θ)X̃d (θ).
Although I extended the dimension of the system byusing descriptor variables, I can design the controllerthat uses original states without extended variables bygiving these restrictions. In view of structure of Ẽd , I
restrict X̃d (θ) and variable matrix Ỹd (θ) as follows.
[X̃d (θ) =X11 (θ)X21 (θ)X31 (θ)0
X22 (θ)X32 (θ)0
X23 (θ)X33 (θ)], Ỹd (θ) = [ Y (θ) 00 ](18)The stability is guaranteed by solving LMIs with thescheduling parameters θ at vertexes of the parameterbox. Let matrices X̃d (θ), X11 (θ) Ỹd (θ), Y (θ) be as follows, the matrices are represented by Θi (i=1,...,16).
X̃d (θ) = X̃d0 + θ1 X̃d1 + θ2 X̃d2

(19)

X̃11 (θ) = X̃11,0 + θ1 X̃11,1 + θ2 X̃11,2(20)

Ỹd (θ) = Ỹd0 + θ1 Ỹd1 + θ2 Ỹd2
(21)
Y (θ) = Y0 + θ1 Y1 + θ2 Y2(22)
Θ1 = (θ1 , θ2 , θ3 , θ4 ), Θ2 = (θ̄1 , θ2 , θ3 , θ4 )...
Θ15 = (θ̄1 , θ̄2 , θ̄3 , θ4 ), Θ16 = (θ̄1 , θ̄2 , θ̄3 , θ̄4 ) (23)The cost function J shown in Eq.(15) is adopted to derive the stabilizing state feedback GS controller. Then,
multi-aﬃne restriction is applied for Eq.(17). LMI condition to derive the stabilizing state feedback GS controller is as follows.
Lemma 1 If there exists X11 (Θi )≻0, X̃d (Θi ),Ỹd (Θi ) satisf ying the f ollowing LM I, the system isstable.
minimize :γsubject to X11 (Θi ) ≻ 0M (Θi ) − Ẽd X̃˙ d (Θi )1Q 2 X̃d (Θi )1
R 2 Ỹd (Θi )X̃d (Θi )T (Q 2 )T
−I
0


1
Ỹd (Θi )T (R 2 )T


0
−I

[

]

1

W
I

I
X11 (Θi )≺ 0 (24)≻ 0 (25)

trace(W ) < γ (26)M (Θi ) := He[Ãd (Θi )X̃d (Θi ) + B̃d Ỹd (Θi )] (i = 1, ..., 16)

The gain scheduling controller K̃d (θ) with framework of
the descriptor representation is given as follow.
K̃d (θ) = [Y (θ)X11 (θ)−1 0

0]

(27)

Finally, the controller K(θ) = Y (θ)X11 (θ)−1 withframework of state space representation is obtained.

5

Estimation of Friction Coeﬃcient

In this paper, the GS controller whose scheduling parameters are friction coeﬃcient and car velocity is designed. However, friction coeﬃcient can not be measured on real time. UKF is able to apply for nonlinear
system. Therefore, UKF is applied to estimate friction
coeﬃcient. To apply the UKF, system to estimate thefriction coeﬃcient is considered. From Eq.(2), (3), thefollowing equations is derived.
r2 Mg S
r2 SM1−J2J2µS=
L(sinϕ − µcosϕ)ω̇2 = −

(28)
(29)

From Eq.(28), nonlinear state space representation toestimate the friction coeﬃcient is built and S is functionof µ. To estimate the friction coeﬃcient, S is added tostate variable. Let state variable xz (t) be xz (t)=[ω2 (t)S(t) ]T and input uz (t) =τ1 .
ẋz (t) = Axz (t) + Bz S(t)uz (t)y(t) = Cz (t)xz (t)[]M g r20r2
J2
Az =, Bz = [ J200Cz = [ 1

(30)

0 ]

T

T

0 ]

The nonlinear state space representation (30) is discretized. Since µ is varying parameter, S is also varying
parameter. S is defined as follows.
S(k + 1) = S(k) + v(k)(31)

v is process noise. The discretized nonlinear state space
representation (32) is as follows. Let state variablexzd (t) be xzd (t)=[ω2 (k) S(k) ]T and input uzd (t) =τ1 .
xzd (k + 1) = Azd x(k) + Bzd S(k)uzd (k) + Bv v(k) (32)y(k) = Cz xzd (k) + w(k)
∫ Ts
Az Ts
Azd = e, Bzd = (eAz λ dλ)Bz0

Bv = [ 0 1 ]Tv(k) is process noise, w(k) is observation noise. Ts issampling time. UKF can be applied for nonlinear system (32). let x̂(k) be a state estimate. By the following
equation, optimal value of x̂(k) is decided.
x̂(k) = x̂− (k) + g(k){y(k) − ŷ − (k)}(33)

x̂− (k) is a priori state estimate. g(k) is a kalman gain.
ŷ − (k) is a priori output estimate. Optimal value of g(k)is decided by the following equation.
g(k) =

−
Pxy
(k)
−
2
Pyy
(k) + σw

7

Conclusion

In this paper, a method to guarantee the robust stability for Anti-lock Braking System is shown. Since the
car velocity and friction coeﬃcient aﬀect the dynamics, a method is designed to consider variation of theseparameters. From the simulation result, the slip rateis controlled to optimal value in snowy road. Friction
coeﬃcient can be estimated accurately by unscented
kalman filter. Therefore, the eﬀectiveness of the proposed method is verified.

References
(34)

−
Pxy
(k) is a priori state, output error covariance matrix.
−
2(k) is a priori output error covariance matrix. σw
isPyyof covariance of observation noise w(k). Since decisionof optimal value of g(k) determines the optimal valueof x̂(k), friction coeﬃcient can be estimated by Eq.(33),
(34).

6

In Fig.4, 5, dot line shows friction coeﬃcient value andsolid line shows estimated value. If UKF is applied for
ABS, slip rate can be controlled at optimal value insnowy road. However, slip rate doesn’t reach to optimal value 0.2 in dry road. The car stops at 0.81 seconds
in dry road. Since stopping time is short, it can be said
that function of ABS is fulfilled. Friction coeﬃcient canbe estimated in both dry and snowy roads.

Simulation

I conducted the simulations of µ = 0.7 and µ = 0.1
when slip rate is 0.2. The range that robust stabilityof car velocity guaranteed in my study is from 10 to
50[km/h]. Here assuming that µ = 0.7 is dry road andµ = 0.1 is snowy road.

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Fig. 2 Slip Rate at snowy

Fig. 3 Slip Rate at dry[9] Idar Petersen, Tor A. Johansen, Jens Kalkkuhl and Jens
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Fig. 4 Friction coeﬃcient at Fig. 5 Friction coeﬃcient at
snowy
dry

[12] Izumi MASUMUCHI and Etsujiro SHIMEMURA, “On Application of the Descriptor From to Design of Gain Scheduling
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