Guarantee of the Robustness for the changes of roadcondition and vehicle velocity in ABS
M2013SC013 Hiroshi Sasaki
Supervisor：Isao Takami

1Introduction

ABS was born to prevent the flat phenomenon of railways or the burst of the tire at landing. Recently, almost
vehicles are equipped with ABS as a safety device.
The control of ABS is required to prevent wheel lockand to short the braking distance keeping braking power
maximum at braking in all road conditions. The dynamic behavior of ABS depends on the vehicle velocity
and the friction coeﬃcient of the road surface. Becauseof this, some research papers of the robust control design for the vehicle velocity or the friction coeﬃcient of
the road surface are reported [5]. As well, the dynamic
behavior of ABS has strong non-linearity. Because ofthis, some research papers of the PID control designor SMC (Sliding Mode Control) design for the strong
non-linearity of ABS are reported [9]-[14]. The vehicle
velocity can be measured from the wheel velocity, but
the friction coeﬃcient of the road surface can’t be measured in ABS.
Therefore, in this research, the method of the control
design for the changes of the vehicle velocity and the friction coeﬃcient of the road surface is proposed to satisfy
the two conditions required for ABS ”all road conditions” and ”to prevent wheel lock at braking”. The firstcharacteristic of this research is to design the observer
to estimate the friction coeﬃcient of the road surface.
The equation of the friction coeﬃcient of the road surface is derived from the motion equation which obtainedin the formulation of the ABS model. The second characteristics of this research are to implement the optimum descriptor variables and to extract the fluctuating
parameters usefully with LFT (Linear Fractional Transformation). Because of this, polytopic LPV (Linear Parameter Varying) system is composed without ignoring
the non-linearity of the fluctuating parameter. In thecase of doing the GS control design in a framework of
the state space expression, the design result tends to beconservative because it is diﬃcult to polytypically express strictly the fluctuating parameter which appears
non-linearly in the state equation. Some research papers that the conservatism of the design result is reducedor the treatment of Lyapunov function which depends
on parameters is made easy by implementing descriptor
variables are reported [2],[7],[8]. The third characteristic of this research is to propose the method of theGS control design based on Lyapunov function which
depends on parameters considering the time varying of
the vehicle velocity and the friction coeﬃcient of theroad surface while estimating the friction coeﬃcient of
the road surface with designed observer. Because ofthis, to be able to construct the stable control system
for all changes of the vehicle velocity and the friction
coeﬃcient of the road surface is proved in theory. Aswell, the eﬀectiveness of the proposed method is verified
through simulation and experiment.

2Controlled Object

In this research, the brake control is done with theABS experimental unit like Figure 1. This unit has twowheels. The upper wheel simulates the vehicle wheel,
and the lower wheel simulates the road surface. As well,
the upper wheel has a hydraulic disk brake. This unit
can measure the wheel velocity and the vehicle velocity.

Figure 1 ABS Experimental Unit
2.1

Formulation of the model

The physical model of this unit is shown in Figure 2.

Figure 2 Physical Model of ABS
The definition of the parameters are shown as follows.
x1 (t):angular velocity of the upper wheel[rad/s]
x2 (t):angular velocity of the lower wheel[rad/s]
r1 :radius of the upper wheel[m]r2 :radius of the lower wheel[m]J1 :moment of inertia of the upper wheel[kg · m2 ]J2 :moment of inertia of the lower wheel[kg · m2 ]Fn :total force generated by the upper wheel and pressing
on the lower wheel[N]M1 :braking torque[Nm]Mg :gravitational and shock absorber torques acting on
the balance lever[Nm]µ:friction coeﬃcient of the road surface

L:distance between the contact point of the wheels andthe rotational axis of the balance lever[m]φ:angle between the normal in the contact point and theline L[rad]The physical expressions of the upper wheel and thelower wheel are given as eq.(1),(2) by Newton’s motion
equation.
J1 ẋ1 (t) = Fn r1 µ − M1
J2 ẋ2 (t) = −Fn r2 µ

2.3State Equation

In this research, the error of the slip ratio and theintegration of the error are added to a state variable
to let the slip ratio follow the targetvalue. The state∫variable x(t) is given as x(t) = [ (λ − λ∗ )dt λ − λ∗ ],and the input of the system is given as u(t) = M1 − M1∗ .
E ẋ = Ax + Bu

(1)
(2)

A=

[00As well, by the sum of torques around the point A,
Mg + M1
Fn =L(sin φ − µ cos φ)where,
S=µL(sin φ − µ cos φ)r2 x2 (t) − r1 x1 (t)r2 x2 (t)3(4)(5)(6)(7)

Next, the diﬀerential equation of the slip ratio λ is derived.
λ̇ = f (λ, x2 (t), S) + g(λ, x2 (t), S)M1]
10
E=0 x2 (c22 S − β1 )(3)Here, the slip ratio λ is defined as a control indicator.
λ=Control Design

The GS control design is done regarding the vehicle
velocity and the friction coeﬃcient of the road surface
as the scheduling parameters. The friction coeﬃcientof the road surface is expressed with S which has µ.
The roles of ABS are to prevent wheel lock and reducethe vehicle velocity while letting the braking force keep
maximum at braking. To perform this, the slip ratio isneeded to follow the ideal slip ratio which can maximize
the braking force.
3.1

GS Control Design

Parameter boxes that upper and lower bounds of thescheduling parameter θ and its changing speed θ̇ are tops
are defined.
Θ = {θ = [θ1 , θ2 , θ3 : θi ∈ {θi , θi }}(8)

Θd = {θ̇ = [θ̇1 , θ̇2 , θ̇3 : θ̇i ∈ {v i , v i }}where,θ1 = S, θ2 = S 2 , θ3 = x2 (t) (i = 1, 2, 3)

1r1f (λ, x2 (t), S) ={− c11 S + (1 − λ)c14 S}
x2 (t)r2
1
r1g(λ, x2 (t), S) ={− (c12 S + c13 ) + (1 − λ)c15 S}
x2 (t)
r2(c11 , c12 , c13 , c14 , c15 are constants.)
Here, linearization is done around the equilibrium point(λ∗ , M1∗ ).
λ̇ − λ̇∗ ∼=

∂ λ̇
∂ λ̇
(λ − λ∗ ) +(M1 − M1∗ )∂λ∂M1(9)(c22 S − β1 )x2 (t)λ̇ ∼= (α11 S 2 + β2 S)(λ − λ∗ )+ (α21 S 2 + α22 S + β12 )(M1 − M1∗ )(10)(α11 , α21 , α22 , β1 , β2 are constants.)
2.2]
[]
10,B
=
α11 S 2 + β2 S
α21 S 2 + α22 S + β12[Here, eq.(3) is substituted into eq.(1),(2),J1 ẋ1 (t) = (Mg + M1 )r1 S − M1
J2 ẋ2 (t) = −(Mg + M1 )r2 S(12)

Derivation of the friction coeﬃcient of theroad surface µµ is derived to estimate the friction coeﬃcient of theroad surface. By eq.(2),µ=−J2ẋ2 (t)Fn r 2(11)In eq.(12), the matrixes A, B, E have the scheduling parameters. Being able to derive the computable
LMI condition by implementing the redundant descriptor variable for the system like this is known [2]. In
this research, the scheduling parameters are gathered
into the matrix
AG by giving a descriptor variable as∫xG (t) = [ (λ − λ∗ )dt λ − λ∗ λ̇ M1 − M1∗ ]T .
EG ẋG = AG xG + BG u00
AG = 0010α11 θ2 + β2 θ1010EG = 0001(−c22 θ1 + β1 )θ3001000000(13)00α21 θ2 + α22 θ1 + β12 1 0000, BG =  000−1The scheduling parameters have been gathered into thematrix AG from eq.(13). There is a polynominal expression of the scheduling parameters in the∫ matrix AG .
Therefore, a descriptor variable xǴ (t) = [ (λ−λ∗ )dt λ−λ∗ λ̇ M1 − M1∗ λ − λ∗ λ̇ M1 − M1∗ M1 − M1∗ ]T isgiven to express polynominal expressions of the scheduling parameters monominally.
EǴ ẋǴ = AǴ xǴ + BǴ u(14)0000AǴ = 000010α11 θ20000001−c22 θ1 θ300000[EGEǴ =000α21 θ21000000β2 θ10000000β1 θ300000

03.2 Condition to discriminate stability0β12  The condition to discriminate stability for the descrip0 tor system which expressed by eq.(20) is shown as fol00  lows[7]. Considering the structure of the matrix Ed in0  this research, the candidates of the Lyapunov matrix

00α22 θ1000000][ ]0BG, BǴ =00The matrix AǴ has a bilinear term of the schedulingparameters θ1 θ3 . Here, the matrix AǴ is transformed
equally to the form which doesn’t have θ1 θ3 to derive thecomputable LMI condition in the case of using Lyapunov
function which depends on parameters. The method toextract the scheduling parameters with LFT in order
to transform like this is adopted [7]. In this research,a scheduling parameter ∆ is given as eq.(16) and thematrix AǴ (θ) is given as eq.(15) with ∆ considering theform of the scheduling parameters in the matrix AǴ .
AǴ (θ) = An + Bδ ∆(I − Dδ ∆)−1 Cδ
∆=

3
∑

θi ∆i

(15)
(16)

i=1

∆1 = diag(1, 1, 1, 1, 0, 0, 0, 0)∆2 = diag(0, 0, 0, 0, 1, 1, 0, 0)∆3 = diag(0, 0, 0, 0, 0, 0, 1, 1)

EǴ ẋǴ = An xǴ + Bδ wδ + BǴ uzδ = Cδ xǴ + Dδ wδ
wδ = ∆zδ
An11An =An21

(17)
(18)
(19)Cδ2 ]Finally, the coeﬃcient matrix Ad of xd is transformed to the form which doesn’t have the term (I −Dδ ∆)−1 by giving a new descriptor variable xd (t) =[xǴ (t)T zδ (t)T ]T . From the above, the computable LMI
condition can be derived. The system after this transformation is given as eq.(20).
Ed ẋd = Ad (θ)xd + Bdu u[AnAd =Cδ] [An11Bδ ∆= An21Dδ ∆ − ICδ1[EǴ
Ed =0An12An22Cδ2][ ]0BǴ, Bdu =00He{Ad (θ)Xd (θ) + Bdu Yd (θ)} − Ed Ẋd (θ) < 0 (21)
Eq.(22) is obtained by multiplying [I Bδ ∆(I −Dδ ∆)−1 ]and its transposition into the left side and the right sideof eq.(21).
(22)

There is the product of the scheduling parameter ∆and the Lyapunov matrix which depends on parameters X(θ) in eq.(22). But there is not this productin eq.(21). In eq.(21), there is the product of ∆ andX31 (θ), X32 (θ), X33 (θ). By giving Xd (θ) as eq.(23) andgiving limitation as eq.(24) for X31 (θ), X32 (θ), X33 (θ),
eq.(21) becomes multi aﬃne for θ. This means that thestability is guaranteed by resolving the LMI conditionat the endpoint nodes of the scheduling parameter θ andits changing speed θ̇.
Xd (θ) = X̂d0 +3∑

θi X̂di(23)∆i [X31i X32i X33i ] = 0

(24)

i=1]
[]
An12
Bδ1, Bδ =An22
Bδ2Cδ = [Cδ1

By the reference [7], the suﬃcient condition tomake the system of eq.(20) stable is to be X(θ) >0, X21 (θ), X22 (θ), X23 (θ), X31 (θ), X32 (θ), X33 (θ), Y (θ)
which satisfied with eq.(21). Here, He{M } = M + M T
is adopted.

He{AǴ (θ)X(θ) + BǴ Y (θ)} − Ẋ(θ) < 0Then, a system which equals to eq.(14) is expressed aseq.(17)-(19). In this research, considering the design of
the GS controller which based on Lyapunov function
which depends on parameters, the matrixes An , Cδ aretransformed without including the scheduling parameters.

[Xd (θ) and the variable matrix Yd (θ) are limitted as follows.
[]X(θ)
00Xd (θ) = X21 (θ) X22 (θ) X23 (θ) , X(θ) > 0X31 (θ) X32 (θ) X33 (θ)
Ẋ(θ) 0 0Ed Ẋd (θ) =  00 0 , Yd (θ) = [Y (θ) 0 0]
00 0(20)Bδ1 ∆Bδ2 ∆Dδ ∆ − I]

In the case of considering H2 control specification foreq.(20), the generalized controlled system is given aseq.(25)-(26).
Ed ẋd = Ad (θ)xd + Bdw w + Bdu uz = Cd xd + Dd u
[ ]Bw
Bdw = 0 , Cd = [C0(25)
(26)

]
[ ]
102
Q0 0] , C =1, Dd =0R2
[Here, w means a disturbance input, z means a evaluation output, Q means a weighting matrix for the statevariable xG (t), R means a weight for a control input.
The matrixes Ad , Xd , Yd are expressed as eq.(27).
[Ad (θ) Xd (θ) Yd (θ)] =8∑i=1βi (θ) [Adi Xdi Ydi ] (27)βi ≥ 0,8∑βi = 1i=1

The LMI conditions are given as eq.(28)-(33) to seek
the state feedback style GS controller in order to makethe system which expressed by eq.(25)-(26) stable in thevariation ranges of the scheduling parameter θ and itschanging speed θ̇. The problem is to minimize H2 norm
from w to z.
minimize : γs.t. : X(θ) > 0[]
He{Ad (θ)Xd (θ) + Bdu Yd (θ)} − Sk Bdw<0TBdw−I[]
X(θ)(CX(θ) + DY (θ))T>0CX(θ) + DY (θ)
W

Sk =(30)
(31)

2(32)

θ̇i Ed Xd (θi )(33)

T race(W ) < γ3
∑

(28)
(29)

i=1

The GS controller Kd (θ) in Descriptor Expression is
given as eq.(34) by obtaining X(θ), Y (θ) which satisfied
with eq.(28)-(33).
Kd (θ) = Y (θ)X(θ)−1(34)Moreover, if X(θ), Y (θ), W are single and Sk = 0 ineq.(28)-(33), the robust controller of a fixed gain considering the time varying of the scheduling parameter isobtained.

4Simulation

Whether designed GS controller with designed observer can control correctly is tested on the following
conditions.
• The vehicle velocity slows from 50 km/h.
• Simulation is done with designed observer.

• The GS controller for the friction coeﬃcient of theroad surface µ and the vehicle velocity x2 (t) withdesigned observer has been designed.
Problems are shown as follows.
• To follow the ideal slip ratio more perfectly.
• To do simulation and experiment considering thefriction behavior of the hydraulic disk brake system.
• Only S and x2 (t) are adopted as the schedulingparameters.

References[1] A. Watanabe, K. Watanabe, Z. Luo and S. Endo,”Gain Scheduling Control of Nonlinear System,”
T he Society of Instrument and Control Engineers,No. 197-16, 2001(in Japanese).
[2] I. Masabuchi and E. Shimemura, ”On Application of theDescriptor Form to Design of Gain Scheduling systems,”
T he Institute of Systems, Control and Inf ormation Engineers,Vol. 12, No. 7, pp. 390-394, 1999.
[3] I. Takami and S. Ishida, ”Gain Scheduling Control using Descriptor Expression for Jib Crane,”
N anzan U niversity Bulletin, pp. 67-76, 2012.
[4] I. Petersen, T. A. Johansen, J. Kalkkuhl and J. Ludemann, ”WHEEL SLIP CONTROL IN ABS BRAKESUSING GAIN SCHEDULED CONSTRAINED LQR,”
EU ROP EAN CON T ROL CON F EREN CE, pp. 146-155,
2001.
Kataoka,T.
Mizuno,H.
Yamazaki,G.
[5] H.
Chen and I. Takami,”Robust Stabilization ofAntilock
BrakingSystemwithLQ
Control,”
T he 12th International Conf erence on M otion and V ibration,
1B24, 2014.

[6] T. Kanada,
Y. Watanabe and G. Chen,”Robust
H2
control
fortwo-wheeled
inverted
pendulumrobotusing
LEGO
Mindstorms,”
T he Society of Signal P rocessing Applications and T echnology of Japan
Vol. 14, No. 1, 2011.
[7] G. Chen and H. Shibata, ”System Analysis using
theredundancyof
Descriptor
Expression,”
T he Institute of Systems, Control and Inf ormation Engineers,Vol. 47, No. 5, pp. 211-216, 2003.
[8] G. Chen and T. Sugie, ”Gain Scheduling for Descriptor
Systems via Parameter Dependent Lyapunov Functions,”
T he Society of Instrument and Control Engineers, Vol.
34, No. 9, 1159/1163, 1998.
[9] F. Jiang, ”An Application of Nonlinear PID Control to a Class of Truck ABS Problems,” inP roceedings of the 40th IEEE Conf erence on Decision and Control,
pp. 516-521, 2001.
[10] M. Yokoyama, Y. Iwata, S. Katayose, M. Imamura and M.
Nibe, ”Antilock Braking System Using Sliding Mode Control,” T he Japan Society of M echanical Engineers, Vol.
63, No. 611, 1997.

Figure 3 Vehicle Velocity and Wheel Velocity[11] E. Kayacan,
Y. Oniz,and O. Kaynak,”AGreySystem
Modeling
Approach
forSlidingModeControlofAntilock
Braking
System,”IEEE T RAN SACT ION S ON IN DU ST RIAL ELECT RON ICS,
Vol. 56, No. 8, 2009.
[12] S. Baek, J. Song, D. Yun, H. Kim and K. Boo, ”Application of a Sliding Mode Control to Anti-lock Brake System,”International Conf erence on Control, Automation and Systems,
pp. 307-311, 2008.

Figure 4 Slip Ratio

5Conclusion
Results are shown as follows.
• The observer to estimate µ correctly has been designed.

[13] M. Kasahara, Y. Kanai and Y. Mori, ”Vehicle Braking Control Using Sliding Mode Control - Switching Control for Speed and Slip Ratio -,” ICROS −SICE International Joint Conf erence, pp. 4047-4052,
2009.
[14] D. Antic, V. Nikolic, D. Mitic, M. Milojkovic andS. Peric, ”SLIDING MODE CONTROL OF ANTILOCKBRAKINGSYSTEM:ANOVERVIEW,”F ACT A U N IV ERSIT AT IS Series : Automatic Control and Robotics,
Vol. 9, No. 1, pp. 41-58, 2010.