Investigation of sliding surface design

An 8-degree-of-freedom dof nonlinear vehicle model was developed for this paper, and the effects of brake-system parameter varia-tions, such as a brake actuator time constant, target sl

Investigation of Sliding-Surface Design on the

Performance of Sliding Mode Controller in

Antilock Braking Systems

Taehyun Shim, Sehyun Chang, and Seok Lee

Abstract—Sliding mode control (SMC) has widely been

em-ployed in the development of a wheel-slip controller because of its

effectiveness in applications for nonlinear systems as well as its

performance robustness on parametric and modeling

uncertain-ties The design of a sliding surface strongly influences the overall

behavior of the SMC system due to the discontinuous switching

of control force in the vicinity of a sliding surface that produces

chattering This paper investigates the effects of sliding-surface

design on the performance of an SMC-based antilock braking

system (ABS), including a brake-torque limitation, an actuator

time delay, and a tire-force buildup Different sliding-surface

de-signs commonly used in ABS were compared, and an alternative

sliding-surface design that improves convergence speed and

os-cillation damping around the target slip has been proposed An

8-degree-of-freedom (dof) nonlinear vehicle model was developed

for this paper, and the effects of brake-system parameter

varia-tions, such as a brake actuator time constant, target slip ratios, an

abrupt road friction change, and road friction noises, were also

assessed.

Index Terms—Antilock braking system (ABS), sliding mode

control (SMC), sliding-surface design.

NOMENCLATURE Vehicle

mtot Vehicle total mass (in kilograms)

ms Vehicle sprung mass (in kilograms)

Jroll Roll inertia (in kilograms meter square)

Jyaw Yaw inertia (in kilograms meter square)

L Length of wheel base (in meters)

La Distance of center of gravity (c.g.) of sprung mass from

front axle (in meters)

Lb Distance of c.g of sprung mass from rear axle

(in meters)

tf Track width at front axle (in meters)

tr Track width at rear axle (in meters)

hcg c.g height of sprung mass (in meters)

ho Average roll center distance below sprung mass c.g

(in meters)

Manuscript received January 25, 2006; revised November 29, 2006,

April 15, 2007, and April 19, 2007 This work was supported by the Institute for

Advanced Vehicle Systems (IAVS), University of Michigan—Dearborn The

review of this paper was coordinated by Dr M Abul Masrur.

T Shim and S Lee are with the Department of Mechanical Engineering,

Uni-versity of Michigan—Dearborn, MI 48128 USA (e-mail: tshim@umich.edu;

eoklee@umd.umich.edu).

S Chang is with the Department of Mechanical Engineering, University of

Michigan, Ann Arbor, MI 48109 USA (e-mail: sehyun@umich.edu).

Digital Object Identifier 10.1109/TVT.2007.905391

hl Distance of sprung mass c.g to ho(in meters)

hf Front roll center height at front axle (in meters)

hr Rear roll center height at rear axle (in meters)

U/V Longitudinal/lateral velocities of c.g in body-fixed coordinate (in meters per second)

uo Initial longitudinal velocity of c.g (in meters per second)

ϕ Roll angle (in radians)

˙

ϕ Roll angular velocity of c.g in body-fixed coordinate (in radians per second)

Ω Yaw angular velocity of c.g in body-fixed coordinate (in radians per second)

˙

Ω Yaw acceleration of c.g in body-fixed coordinate (in radians per second square)

a x Longitudinal acceleration at c.g (in meters per second square)

a y Lateral acceleration at c.g (in meters per second square)

Suspension/tire

muf Vehicle unsprung mass at front axle (in kilograms)

mur Vehicle unsprung mass at rear axle (in kilograms)

huf Unsprung mass c.g height at front axle (in meters)

hur Unsprung mass c.g height at rear axle (in meters)

κrf Suspension roll stiffness at front axle (in newton meters per degree)

κrr Suspension roll stiffness at rear axle (in newton meters per degree)

κroll Roll stiffness (in newtons per meter)

Broll Roll damping coefficient (in newton seconds per meter)

Jw Rotational inertia of each wheel (in kilograms meter square)

Rw Tire radius (in meters)

λ Tire longitudinal slip

α Tire lateral slip (in radians)

ω Angular velocity of wheel rotation (in radians per second)

δ Road wheel steer angle (in radians)

Tb External torque applied at wheel (in newton meters)

I INTRODUCTION

IN RECENT years, the use of electronic control systems has increasingly become popular in passenger vehicles, result-ing in a significant improvement in driver satisfaction, comfort, and safety These systems combine the existing hardware with 0018-9545/$25.00 © 2008 IEEE

new electronic components and maximize their performance.

An antilock braking system (ABS) was the first system used,

but now, the systems also include traction control, vehicle

dynamic control, active steering control, direct yaw moment

controls, etc These systems are activated when the vehicle is

at the physical limit of adhesion between the tires and the road,

allowing the driver to keep control of the vehicle

For the ABS, various control methodologies, such as

feed-back linearization [1], fuzzy logic [2], neural network [3],

sliding mode control (SMC) [4]–[7], and hybrid control, which

uses two different controls in a combined way [8]–[12], have

been developed since its introduction in the 1950s All of

these methods try to accurately control the wheel slip to

maximize the effectiveness of the tire force in the

longitu-dinal direction, reducing the stopping distance by preventing

wheel lockup as well as providing a steering capability in the

lateral direction An ABS has been proven to be extremely

effective during braking on a slippery road condition Among

the different control methodologies used in ABS, the SMC

has widely been investigated [4]–[7] because of its robust

characteristics for model uncertainties as well as its

effective-ness for applying control to nonlinear systems Reference [4]

showed the SMC for a wheel-slip control using a one-wheel

model with an inclusion of tire relaxation length Reference

[5] developed the wheel-slip controller for a full-vehicle model

using the SMC in which an extended Kalman filter (EKF)

[20] was used to estimate the brake pad friction and the tire

braking force Reference [6] demonstrated an

eddy-current-braking system for a hybrid electric vehicle using a nonlinear

8-degree-of-freedom (dof) vehicle model and an SMC-based

wheel-slip controller Other approaches to obtain a controlled

braking torque using the SMC can be found in [7]–[10] In

[7], it is assumed that the tire force is available using a

smart tire concept The hybridization of SMC and a fuzzy

method to improve the braking performance were presented

in [8]–[10] The SMC with a pulsewidth modulation (PWM),

which is useful to control the conventional hydraulic brake

system, was demonstrated by vehicle tests on an in-door test

bench in [27] Reference [28] presented an extremum seeking

control via sliding mode approach to maximize the braking

force

In the development of SMC, the design of the sliding surface

dictates the behavior of the overall control system Due to

the discontinuous switching of control force in the vicinity of

the sliding surface, it produces a chattering which can cause

system instability and damage to both actuators and plants

During the application to actual brake systems, the performance

of theoretically determined switching control is degraded due

to the brake-torque limitation, brake actuator time delay, and

tire-force buildup, resulting in an oscillation near the sliding

surface Thus, a reduction of a chattering and a rapid

conver-gence to the sliding surface in the actual brake systems are of

importance In order to reduce the chattering, a higher order

SMC [13] and a boundary-layer method [14] with moderate

tuning of a saturation function have extensively been used In

addition, the brake-by-wire concept was investigated in recent

years to overcome the chattering by precisely applying the

op-timum brake pressure to each wheel using the electrohydraulic

Fig 1 Schematic of the 8-dof vehicle model and the forces at a wheel during the braking.

and the electromechanical brakes [15] However, the effects

of sliding-surface design on the overall controller performance (minimizing the chattering, improving the tracking speed, etc.) have not closely been investigated

This paper studies the effects of sliding-surface design on the overall performance of the ABS Different sliding-surface designs introduced in [4]–[7], [27], and [28] for the wheel-slip control on the ABS have been compared through simu-lation An alternative sliding-surface design that improves the convergence speed to the sliding surface has been proposed and simulated using an 8 degree of freedom (8-dof) nonlinear vehicle model During the simulation, the effects of brake-system parameter variations, such as a brake actuator time constant, target slip ratios, an abrupt road friction change, and road friction noises, were also assessed In the following section, a vehicle dynamic model is described and validated

by comparing the simulation results with those of the CarSim vehicle model [16] The sliding mode controller used for the wheel-slip control with different sliding surfaces is presented

in Section IV The simulations and discussion are given in Section V

II VEHICLE-MODELDEVELOPMENT Fig 1 shows the schematic of an 8-dof vehicle model used for this paper This model has 4 dof for the chassis velocities and 1 dof at each of the four wheels representing the wheel spin dynamics The chassis velocities at the c.g include the

longitudinal velocity U , the lateral velocity V , the roll angular

velocity ˙ϕ, and the yaw angular velocity Ω The model neglects

the pitch and heave motions

The application of Newton’s second law to the lumped vehi-cle mass longitudinally, laterally, and about the longitudinal and vertical axes through the center of mass produces the following equations of motion

Longitudinal motion

F x =mtot ( ˙U −V Ω)=

2

(F xi cos δ −F yi sin δ)+

4

F xi (1)

Lateral motion

F y =mtot ( ˙V +U Ω)=

2

i=1 (F xi sin δ+F yi cos δ)+

4

i=3

F yi (2)

Yaw motion

M z =JyawΩ=L˙ a

2

i=1 (F xi sin δ+F yi cos δ) −Lb

4

i=3

F yi

+tf2

2

i=1

(−1) i−1 (F

xi cos δ −F yi sin δ)+ tr2

4

i=3

(−1) i−1 F

xi (3)

Roll motion

Jrollϕ + B¨ rollϕ + κ˙ roll ϕ = msgh1sin ϕ + ms(a y )h1cos ϕ

(4)

where i in (1)–(4) is 1, 2, 3, and 4, and it represents the left

front, right front, left rear, and right rear wheels, respectively

The resulting accelerations affect the distribution of vertical

tire forces at each of the wheels Although this model does not

have a suspension between the body and the wheels, the vertical

load transfer between front/rear and inside/outside due to the

vehicle longitudinal and lateral accelerations has been included

The vertical forces Fz can be derived from their moments

about the center of mass in equilibrium with the corresponding

moments due to the static weight and the longitudinal and

lateral accelerations (ax and ay, respectively)

F zrf =−(ay cos ϕ+g sin ϕ) tf (κrf +κrr) κrf h1ms − msLbhf ay

tf L −mufa y huf

tf

−1(mufhuf+mshcg+murhur )ax L+ 1mtotg Lb L (5)

F zrf =(a y cos ϕ+g sin ϕ) tf (κrf +κrr) κrf h1ms +msLbhf ay tf L +mufa y huf

tf

−1(mufhuf+mshcg+murhur )ax L+ 1mtotg Lb L (6)

F zrr=−(ay cos ϕ+g sin ϕ) tr(κrf +κrr) κrrh1ms − msLahray

trL −mura y hur

tr

+ 1(mufhuf+mshcg+murhur )ax L+ 1mtotg La L (7)

F zlr =(a y cos ϕ+g sin ϕ) tr(κrf +κrr) κrrh1ms +msLahray trL +mura y hur

tr

+ 1(mufhuf+mshcg+murhur )ax L+ 1mtotg La L . (8)

In the aforementioned vertical-force equations, the first term

represents the load transfer due to the sprung-mass roll moment,

and the second term is the contribution from the sprung-mass

lateral acceleration The load-transfer effect due to the lateral

acceleration is shown in the third term The fourth and fifth

terms indicate the load transfer due to the longitudinal

accelera-tion and the static loading condiaccelera-tion, respectively The detailed

derivation can be found in [6] It should be noted that the same

spring properties are used at the left and right of the front and

rear axles, respectively

For each of the wheels, a separate equation of motion must

be derived relating the angular acceleration ˙ω to the respective

wheel torque Tband longitudinal tire force Fx

Fig 2 Block diagram of a vehicle model.

The longitudinal and lateral forces on each tire are calculated using the Milliken tire model [18] by taking into account the slip angle, the longitudinal slip, and the vertical force This tire model is slightly modified from the Magic Formula tire model [19] by normalizing the variables according to the vertical load

Equation (10) shows the longitudinal F x and the lateral F ytire forces

F x=µ x · F zp · R(s R) · s x

s2

x + η(s R)2tan2α ,

F y =µ y· F zp · R(s R)· η tan α

s2

where

R(s R) =D sin(C arctan (B(1 −E)s R +E arctan(BsR)))

(11)

s R=



C x s x

µ x F zp

2 +



C y tan α

µ y F zp

2

(12)

η(s R) =







0.5(1 + η0)

−0.5(1 − η0) cos(0.5sR), for sR ≤ 2π

η0=C y µ x

C x µ y

µ x and µy are the tire longitudinal and lateral friction

coeffi-cients, and Cx and Cy represent the longitudinal and lateral

stiffness coefficients The peak vertical force Fzp is approx-imated by the following function according to the vertical

load Fz

1 + (1.5F z /mtot· g)3. (14) These equations are combined in the Matlab/Simulink envi-ronment, as shown in Fig 2 In order to reflect realistic wheel and brake systems, first-order dynamics of the tire- and brake-force buildups as well as the brake-torque limit have been implemented in the wheel dynamics

III VEHICLE-MODELVALIDATION The responses of the vehicle model have been compared with those of the CarSim vehicle model for a fishhook maneuver A

Fig 3 Comparison of the vehicle responses between the 8-dof vehicle

model and the CarSim for a fishhook maneuver during a vehicle speed of

u0 = 50 km/h (a) Hand wheel steering input-gear ratio (16) (b) Trajectory.

(c) Yaw rate Φ (d) Lateral acceleration ay

steering wheel input is applied at 1 s while a vehicle is moving

at a speed of 50 km/h The steering wheel input was increased

from 0◦ to 200◦ for the first 0.4083 s, which is maintained

for 0.1617 s, and then applied in the opposite direction at the

same magnitude within 0.817 s The same tire model [18] has

been used for both vehicle models during the simulation Fig 3

compares the responses of lateral acceleration, yaw rate, and

vehicle trajectory between the 8-dof and the CarSim models

Vehicle parameters of a midsize sedan, as shown in Table I,

were used for this simulation As shown in Fig 3, the 8-dof

vehicle-model responses closely matched those of the CarSim

vehicle model The slight discrepancies shown in Fig 3 come

from the differences in complexities of the vehicle models The

CarSim vehicle model has more dof compared with the 8-dof

model used in this paper

IV DEVELOPMENT OF AWHEEL-SLIP

CONTROLLERUSINGSMC

A wheel-slip controller is typically designed to achieve a

target (desired) slip ratio for a given driving condition A wheel

slip ratio for a braking can be defined as follows:

where u xw represents the longitudinal velocity of the wheel

center along the longitudinal tire-force direction For brake

application, the target slip ratio of ABS can be set to the peak

slip ratio of λ = λpeak, which satisfies dFx /dλ = 0, in order to

minimize the stopping distance The target slip ratio that

pro-duces a maximum longitudinal tire force is not fixed and instead

varies with the road-condition changes, as shown in Fig 4

Thus, the wheel-slip controller must track the different wheel

slip ratios for ABS application as well as the arbitrary target

slip ratio for an application of vehicle stability control (VSC)

The following section shows a development of the

wheel-slip controller based on the SMC for the ABS application

Three different sliding-surface designs were used to assess the

tracking performance of the target slip ratio

TABLE I

V EHICLE P ARAMETERS

Fig 4. Tire longitudinal force F x versus slip ratio λ on various road

conditions.

A Conventional SMC Design

For a brake-system application, the controlled brake torque

Tbin the SMC consists of an equivalent control torque T b,eq

and a switching control torque T b,sw , i.e., Tb= T b,eq + T b,sw The equivalent control torque can be interpreted as a control

that makes the system states (λ, ˙λ) move along the desired

sliding surface It is determined by the wheel dynamics in (9) and the slip ratio in (15) The switching control torque ensures that the trajectory of the system is reached at the desired sliding surface and its magnitude can analytically be obtained by using the Lyapunov stability condition

In previous research, two types of sliding-surface design,

namely, σ = ˜ x = λ − λd [4], [5], [7] and σ = ˜ x + γ t

0xdτ˜ [6], [8], were mainly used for the brake-system applications

In these designs, the controlled torque appears in the first

derivative of the slip-ratio tracking error, which is defined as

the difference between a current slip ratio and a target slip ratio,

i.e., ˜x = λ − λd

1) Equivalent Control Torque T b,eq : The equivalent

con-trol for these sliding-surface designs is determined from the

condition of ˙σ = 0 in which a target slip ratio λd is assumed

constant ( ˙λd= 0)

In the first sliding-surface design σ = ˜ x, the first derivative

of σ can be expressed as ˙σ = ˙˜ x = ˙λ = 0 By using the wheel

dynamics in (9), the first derivative of a slip ratio becomes

˙˜x = ˙λ = Rw

u xw Jw

(Tb− F x · Rw)− (1 + λ)

u xw

˙u xw = 0. (16)

From (16), the equivalent control torque T b,eqcan be

deter-mined as

T b,eq = Fx · Rw+Jw(1 + λ)

Rw

For the second sliding-surface design σ = ˜ x + γ t

0xdτ , the˜ derivative of the sliding surface and the equivalent control can

be written as

˙σ = ˙˜ x + γ ˜ x = ˙λ − ˙λd + γ(λ − λd)

u xw Jw(Tb− F x Rw)− (1 + λ)

u xw ˙uxw + γ (λ − λd) = 0

(18)

where γ is strictly positive constant

T b,eq = F x Rw+(1+λ) ˙u xw Jw

Rw−γ(λ−λd)u xw Jw

Rw

. (19)

In the aforementioned equations, the value of the equivalent

brake torque T b,eq is difficult to determine from direct

mea-surement Thus, it is replaced with an approximated equivalent

control torque ˆT b,eqin which the estimation of vehicle states

and tire forces is used The estimation of vehicle states and tire

forces is achieved using the EKF technique presented in [20],

and a brief summary of its procedures for the 8-dof nonlinear

vehicle model is shown next The state and measurement

equa-tions are

˙xa= f ( xa, y = h ( xa, u)

The augmented nonlinear state equation consists of the

tire-force term ( ˙F x,y = ˙F x,y and ¨F x,y = 0) and the 8-dof vehicle

dynamics given in (1)–(4) and (9) Thus, the augmented state

vector with approximation symbol “∧” is composed of nine

vehicle states, six estimated tire forces, and six first derivatives

of estimated forces

xa(1 : 9) = U , ˆˆ V , ˆ Ω, ˆ ϕ, ˆ˙ ϕ, ˆ ω1, ˆ ω2, ˆ ω3, ˆ ω4

T

xa(10 : 15) = Fˆx1 , ˆ F x2 , ˆ F x3 , ˆ F x4 ,

ˆ

F y1+ ˆF y2 ,

ˆ

F y3+ ˆF y4

T

xa(16 : 21) = Fˆ˙x1 , Fˆ˙x2 , Fˆ˙x3 , Fˆ˙x4 ,

ˆ˙

F y1+Fˆ˙y2 ,

ˆ˙

F y3+Fˆ˙y4 T.

(20)

TABLE II

S ENSOR N OISE C HARACTERISTICS IN T ERMS OF S TANDARD D EVIATIONσ

Input vectors are composed of the steering angles and the brake torques at each wheel as follows:

where u1= [δ1, δ2, δ3, δ4]T, and u2= [Tb1, Tb2, Tb3, Tb4]T For the output equation, the measurements of the longitudinal and lateral accelerations, the yaw rate, and the roll angle at the c.g., and each wheel angular velocity were used

y = [a x , a y , Ω, ϕ, ω1, ω2, ω3, ω4]T. (22)

The system noise Q was accordingly chosen by comparing

its relative magnitude order with the corresponding noise co-variance in order to obtain estimation accuracy and robustness under model uncertainty In particular, the covariance for tire-force term was set with large values to adopt a fast tire-tire-force change during transient motion [20], [21] Noise covariance in Table II was determined by assuming a uniform noise distri-bution with a standard deviation given in [23]–[25] Although the estimation performance using the EKF is quite dependent

on system model accuracy as well as sensor accuracy, this drawback of EKF can be addressed using vehicle-parameter identification [22] or the integration of inertial sensors with GPS by compensating sensor noise and bias [25]

The longitudinal acceleration at each wheel ˙u xwi (i = 1 −4)

is approximated using the following kinematic equations based

on the vehicle geometry:

ˆ˙u wi= ˆ˙u i cos δi+ ˆ˙ν i sin δi , (i = 1 −4) (23) where

ˆ˙ui= ˙U + l u,i · Ω2

= (a x + V · Ω) + l u,i · Ω2, l u,i ∈ [−La, −La, Lb, Lb]

ˆ˙ν i= ˙V + l ν,i · Ω2

= (ay − U · Ω) + l ν,i · Ω2, l ν,i ∈



tf

2, − tf

2,

tr

2, − tr

2



.

For simplicity, the estimation of other terms in (19), such

as longitudinal wheel speed, wheel slip ratio, Rw, and Jw, was not considered to determine the approximated equivalent torque By using the estimated tire forces and the longitudinal acceleration, the approximated equivalent control torque ˆT b,eq

can be used for the following two different sliding-surface designs:

Case 1) σ = ˜ x = λ − λd

ˆ

T b,eq= ˆF x Rw+Jw(1 + λ)

Case 2) σ = ˜ x + γ t

0xdτ˜ ˆ

T b,eq= ˆF x Rw+(1+λ)ˆ˙ u xw

Jw

Rw−γ(λ−λd)u xw Jw

Rw .

(25)

2) Switching Control Torque T b,sw : The role of a switching

control torque is to drive the system states to the sliding surface

(σ = 0) By defining the switching control torque as T b,sw =

−Ksgn(σ), where K is a switching control gain and sgn(σ) is

a sign function, a controlled brake torque becomes

Tb= ˆT b,eq + T b,sw = ˆT b,eq − Ksgn(σ). (26)

The switching control gain K shown in (26) can be chosen

by considering a stability condition and the limitation of the

actuator The stability condition can be determined by the

Lyapunov stability criteria For the given sliding-surface design,

the stability condition can be expressed as

σ ˙σ ≤ −η|σ|, where η is a strictly positive constant. (27)

For the first sliding-surface design σ = ˜ x, by using (24) and

(27), the following condition must be met for stability:

σ



Rw

u xw Jw

ˆ

T b,eq − Ksgn(σ) − F x Rw − (1 + λ)

u xw

˙u xw



< −η|σ| (28)

In order to obtain the switching control gain, let K =

(u xw Jw/Rw)(F + η) so that it will satisfy the stability

con-dition in (27), and substitute K into (28) After

apply-ing triangular inequality, the switchapply-ing control gain can be

written as

K ≥ Rw ˆF

x −F x+J

w

(1+λ)

Rw



ˆ˙uxw − ˙u xw+u xw Jw

Rw

η. (29)

For the second sliding-surface design σ = ˜ x + γ t

0xdτ ,˜ the switching control gain that satisfies the Lyapunov

stabil-ity criteria can be determined similar to (28) and (29) and

expressed as

σ



Rw

u xw Jw

ˆ

T b,eq − Ksgn(σ) − F x Rw − (1 + λ)

u xw ˙u xw



K ≥ Rw ˆF

x − F x + J

w

(1 + λ)

Rw



ˆ˙uxw − ˙u xw

+ γ u xw Jw

Rw |λ − λd| + u xw Jw

In (29) and (31), it is assumed that the approximation errors

of Fx and ˙uxw are bounded within A1and A2as follows:



 ˆF x − F x ≤ A

1, andˆ˙uxw − ˙u

xw ≤ A

2. (32)

In general, A1 and A2 are considered as the design

pa-rameters, and the smaller boundaries mean more expensive

estimations for the exact values Thus, in the robust design

viewpoint, it is assumed that these approximation boundaries

are proportional to the maximum error percentage of the esti-mation values as follows:

A1= C1 ˆF

x , and A

2= C2ˆ˙uxw (33)

where C1 and C2 represent the maximum error percentages

of a longitudinal tire force and a wheel center acceleration, respectively

In addition, in order to avoid the chattering problem due to the imperfect switching control under the physical limits of the actuator or model uncertainty, the sign function is substituted

by the following saturation function with the boundary-layer thickness Φ around the sliding surface:

sat

σ



sgn(σ), if|σ| ≥ Φ

σ

B Alternative Sliding-Surface Design

Among the two sliding-surface designs previously

intro-duced, the first sliding-surface design σ = ˜ x corresponds to

the bang-bang control [26] In this design, an error dynamics between the tracking error and its derivative, such as an expo-nential convergence of tracking error, is not incorporated For

the second sliding-surface design σ = ˜ x + γ t

0xdτ , the ex-˜

ponential error convergence can be found in ˙σ = ˙˜ x + γ ˜ x = 0.

However, there is a drawback for this design since the accumu-lated error of the sliding surface makes the switching control ac-tive after the tracking performance, i.e., ˜x = ˙˜ x = 0, is achieved.

In order to improve the convergence rate, the following sliding-surface design had been adopted in [26] and [27], i.e.,

σ = ˙˜ x + γ ˜ x However, the SMC approaches in [26] and [27]

were not designed by analytical design procedure to determine

an equilibrium control and a switching control; for instance, the different control rule for positive and negative sides of sliding surface was applied in [26], and the SMC approach

in [27] was limited to the wheel-slip control using the PWM

In this paper, we propose an analytical design procedure for

the sliding surface σ = ˙˜ x + γ ˜ x The equivalent brake torque

can be determined from the sliding condition of σ = 0 and the

Lyapunov stability condition for the slip-ratio error

The equivalent torque for the proposed sliding surface can be derived similar to that of the second sliding-surface design, i.e.,

˙σ = ˙˜ x + γ ˜ x = 0, as shown in (18).

In order to determine the magnitude of the switching control gain, the Lyapunov method is applied on the slip-ratio error domain using the candidate of the Lyapunov function as

V = 1

2x˜

For stability, the time derivative of V (˜ x) should be less than

or equal to zero By differentiating (35), we obtain

˙

From the sliding-surface equation, (36) can be rewritten as

˙

V = 1

γ (σ − ˙˜x) ˙˜x = 1

γ σ ˙˜ x −1

γ | ˙˜x|2≤ 0.

Since− (1/γ)| ˙˜x|2≤ 0, (36) can be reduced as

From (37)

σ ˙˜ x = σ



Rw

u xw Jw

(Tb−RwF x) − (1+λ)

u xw

˙uxw



= σ



Rw

u xw Jw

ˆ

T b,eq −Ksgn(σ)−RwF x − (1+λ)

u xw

˙uxw



= σ



Rw

u xw Jw



RwFˆx+(1+λ)ˆ˙ u xw Jw

Rw−γ (λ−λd)uˆxw Jw

Rw

− Ksgn(σ)−RwF x



− (1+λ)

u xw

˙uxw



= σ



R2w

u xw Jw

ˆ

F x −F x +(1+λ)

u xw

ˆ˙u xw − ˙u xw −γ(λ−λd)

u xw Jw

sgn(σ)



≤ 0.

Let K = (uxw Jw/Rw)F , and substitute it into the

aforemen-tioned equation, yielding

σ ˙˜ x = σ



R2

w

u xw Jw

ˆ

F x − F x +(1 + λ)

u xw

ˆ˙u xw − ˙u xw

− γ(λ − λd)− F sgn(σ)



≤ 0.

Then

σ



R2w

u xw Jw

ˆ

F x − F x +(1 + λ)

u xw

ˆ˙u xw − ˙u xw

− γ(λ − λd)



≤ F |σ| (38)

F can be obtained by using the triangle inequality

R2w

u xw Jw



 ˆF x − F x +(1 + λ)

u xw



ˆ˙uxw − ˙u xw + γ |λ − λ

d| ≤ F.

(39)

By using the error boundary given in (32), the switching gain

can be written as

K ≥ RwA1+ Jw

(1 + λ)

Rw

A2+ γ u xw Jw

Rw |λ − λd| (40)

In summary, the control torque input with respect to each

sliding-surface design is as follows:

Case 1) σ1= ˜x

Tb= ˆF x · Rw+Jw(1 + λ)

Rw

ˆ˙u xw − K1sat

σ

1

Φ

K1= RwA1+ Jw

(1 + λ)

u xw Jw

Case 2) σ2= ˜x + γ t

0xdτ˜

Tb= ˆF x Rw+ (1 + λ)ˆ˙ u xw

Jw

Rw

− γ(λ − λd)u xw Jw

Rw − K2sat

σ

2

Φ

K2= RwA1+ Jw(1 + λ)

Rw

A2+ γ u xw Jw

Rw |λ − λd|

+u xw Jw

Rw

Case 3) σ3= ˙˜x + γ ˜ x

Tb= ˆF x Rw+(1+λ)ˆ˙ u xw

Jw

Rw−γ(λ−λd)u xw Jw

Rw

− K3sat

σ

3

Φ

K3=RwA1+Jw(1+λ)

Rw A2+γ

u xw Jw

Rw |λ−λd| (43)

where A1= C1| ˆ F x | ≥ | ˆ F x − F x |, A2= C2|ˆ˙u xw | ≥ |ˆ˙u xw −

˙u xw |, and η, γ, Φ, C1, and C2are the design parameters From (41)–(43), in addition to the difference of the sliding surface for each case, the switching term of Case 2) has one

more term (γ) than Case 1) and one more term (η) than Case 3).

Regarding Case 3), if the Lyapunov stability condition is

de-fined using η as ˙ V = ˜ x ˙˜ x ≤ −η instead of (36), the switching

term of Case 3) is the same as that of Case 2) This additional

term (η) of Case 3) can be interpreted as the accelerating

switching force at the beginning of controller activation How-ever, in this paper, this term was not considered because it tends

to attenuate the performance of slip-ratio tracking by causing a relatively large oscillation around a desired slip ratio

V SIMULATION/DISCUSSION

In order to assess the effect of controller performance due

to the different sliding-surface designs, the three designs pre-sented in the previous section have been compared through sim-ulation For each case, the design parameters of the wheel-slip controller are tuned offline using a commercial optimization software, iSIGHT [17], for various target slip ratios and road conditions A sequentiquadratic-programming (NLPQL) al-gorithm in iSIGHT was used in order to minimize the perfor-mance index that is defined as the summation of the absolute error between the target slip ratio and the actual slip ratio during the simulation The smaller performance index can be inter-preted as less chatter (oscillation) around the target slip ratio that reduces the brake load as well as increases the durability of the brake hardware

Among the design parameters needed for a robust wheel-slip

controller design, a sliding-surface design parameter γ and a switching-gain design parameter η are optimized to minimize

the slip-ratio tracking error with preselected values for the rest

of the design parameters The reason for only using γ and η as

the tuning parameters is to reduce computation load for lots of

simulation case studies, because even though other parameters

are fixed with an optimized parameter, tuning tends to lead to

the similar result of the tracking performance regardless of the

change of other parameters This point will be discussed later in

the case study of the effect of the approximation error boundary

As an example of the preselected design parameters, the

maximum error boundaries for the estimated longitudinal tire

force and the longitudinal wheel acceleration are each set at

50%, and the boundary-layer thickness Φ is set as the value

which can prevent the chattering problem In particular, the

boundary-layer thickness Φ affects on a guaranteed tracking

precision ε for Case I) (ε = Φ = ˜ xboundary) and Case III)

(ε = Φ = γ ˜ xboundary) [14] so that a preselected value, i.e.,

˜

xboundary= 0.025, was used in order to keep the tracking

precision consistent for Cases I) and III) However, the integral

sliding surface in Case II) has a freedom to determine the

boundary-layer thickness because the tracking error can

theoretically be zero as follows even though the boundary

layer may have a steady-state value within the boundary layer

This steady-state boundary-layer value σss can be derived by

letting x0= t

0xdt and considering the dynamics ˙x˜ 0= ˜x and

˙˜x from (16) and (42) Then, inside the boundary thickness, i.e.,

0 < σ = ˜ x + γx0< Φ, the equilibrium points are

˙x0= ˜x = 0

˙˜x = Rw

u xw Jw

ˆ

F x − F x · Rw− K2

σss

Φ

− (1 + λ)

u xw

ˆ˙u xw − ˙u xw = 0

→ σss=

t



0

˜

xdt

= ΦJw

K2Rw



Rw

Jw

ˆ

F x −F x ·Rw−(1+λ)ˆ˙u xw − ˙u xw

Therefore, the boundary-layer thickness of Case II) was set as

Φ = 1 in this paper

The design-parameter optimization and the simulation

analy-sis are performed by using the ideal full-state feedback model

shown in Fig 5(a) with varying conditions such as different

actuator time constants, target slip ratios, and road frictions In

addition, as shown in Fig 5(b), the effect of road uncertainty

on the controller performance, along with the pretuned design

parameters, was also investigated by including the road

uncer-tainty as a road input noise on a vehicle model An additional

estimation module for the vehicle states and the tire forces has

been added for this paper

In the simulation, a midsize sedan was used, and the

ref-erence vehicle parameters in the 8-dof model are based on

those of the CarSim “big sedan” model The detailed vehicle

parameters are listed in Table I Pure braking tests in a

straight-line maneuver were performed at two different initial velocities,

namely, 120 km/h (µH= 0.9) and 60 km/h (µH= 0.1) The

maximum available brake torque at each wheel is limited to

2000 Nm, and a time constant for the tire-force lag of 0.01 s is

used

Fig 5 Block diagram of the SMC system (a) Ideal full-feedback model (b) Road noise and estimated feedback control.

Fig 6 Comparisons of the total tracking error for target slip ratio with varying brake time constants among three different sliding-surface designs

(λ d,f=−0.12, λ d,r=−0.1, and µH= 0.9).

A Effect of Brake Actuator Time Constants τb

The actuator dynamics generally affects the performance of the SMC by delaying responses or increasing the oscillation in transient responses due to the physical limit of actuators For the brake actuator modeled as a first-order system, the effect of its time constant on the target slip-ratio tracking performance was investigated with offline-tuned optimal design parameters Fig 6 shows the comparison of a total slip-ratio tracking error among the three different sliding-surface designs with various braking actuator time constants During the simulation,

Fig 7 Comparisons of the target slip-ratio tracking with varying brake time

constants among three different sliding-surface designs (λ d,f=−0.12,

λ d,r=−0.1, and µH= 0.9) (a) Front slip ratio (λd =−0.12, τb =

0.1) (b) Magnified front slip ratio (λd =−0.12, τb= 0.1) (c) Front

slip ratio (λd =−0.12, τb= 0.005) (d) Magnified front slip ratio (λd =

−0.12, τb= 0.005).

the road condition of µH= 0.9 is used, and the slip ratios

of 12% and 10% have been used as the target slip ratio for

the front and rear, respectively As shown in Fig 6, the total

slip-ratio tracking error for the proposed sliding-surface design

is smaller than those of the other sliding-surface designs for all

the time constants being used in the brake actuators

Fig 7 shows the time responses of the slip-ratio tracking

for various brake actuator time constants It is shown that the

oscillation in the slip-ratio response tends to increase around

the target slip due to the physical limit of a brake actuator as its

time constant increases

For an actuator time constant equal to 0.1 s, the second

sliding-surface design shows the largest initial oscillation, but

this oscillation tends to be eliminated, as shown in Fig 7(a)

However, even though the first sliding-surface design has

slightly less total tracking error in Fig 6, its oscillation

con-tinues during the simulation Thus, we can expect that the

performance of the second sliding surface at τb= 0.1 is still

better than that of the first sliding-surface design as being

consistent with the other actuator time constant cases

Fig 8 Comparisons of the total tracking error for target slip ratio with varying approximation error boundaries among three different sliding-surface designs

(λ d,f=−0.12, λ d,r=−0.1, µH= 0.9, and τb= 0.05).

B Effect of Approximation Error Boundary

in Switching Controls

The speed at which the system states reach the sliding surface

is strongly influenced by the switching control The effect of the approximation error boundaries of the longitudinal tire force and the longitudinal wheel acceleration in the switching control has been compared for three different sliding-surface designs

In the simulation, the approximation error boundaries of C1and

C2 shown in (43) are set as C1= C2= C Then, the offline

tuning was performed to find the other optimal design

parame-ters, including a sliding-surface design γ, a switching gain η,

and a boundary-layer thickness Φ for different approximation error boundaries

Fig 8 compares the total slip-ratio tracking errors for different approximation error boundaries For the first and second sliding-surface designs, the slip-ratio tracking perfor-mance is slightly degraded as the approximation error bound-ary increases, whereas the proposed sliding-surface design

σ3= ˙˜x + γ ˜ x tends to slightly decrease the slip-ratio tracking

error as the approximation error boundary increases However, the effect of the approximation error boundary is relatively small compared with that of the brake actuator time constant, which means that its effect can be compensated by optimizing other design parameters

C Effect of Target Slip Ratios λd

The target slip ratio used in the wheel-slip control can be changed depending on the application of the control systems as

Fig 9 Comparisons of the total tracking error for target slip ratio with varying

target slip ratios among three different sliding-surface designs (τb= 0.05).

well as the road-condition variations For instance, a target slip

ratio often used for ABS is the slip ratio that generates the peak

longitudinal force, and it varies with road-condition changes

For an application of VSC, the target slip ratio that produces a

corrective yaw moment for stabilizing the vehicle attitude can

be arbitrary

Fig 9 shows the total slip-ratio tracking errors among

the three different sliding-surface designs for different road

conditions (µH= 0.9 and 0.1) and target slip ratios (λd=

0.12 and 0.22) Initial vehicle speeds of 120 and 60 km/h

were used for µH= 0.9 and µH= 0.1, respectively During the

simulation, a fixed actuator time constant of τb= 0.05 is used,

and the rest of the control design parameters were optimized

to generate minimum tracking errors The proposed

sliding-surface design shows smaller tracking errors than the other

sliding-surface designs

As shown in Figs 6, 8, and 9, the optimal design parameters

vary according to the tuning conditions such as brake time

con-stants, target slip ratios, road frictions, and preselected design

parameters From the overall performance point of view, these

results indicate that the optimal design parameters should be

chosen by a tradeoff approach, considering the dominant brake

operating conditions or by utilizing a lookup table according to

the vehicle driving conditions

The responses of the slip-ratio tracking and the brake control

torques for different sliding-surface designs are compared in

Figs 10 and 11 In Fig 10, the brake-torque command is

the summation of the equivalent and switching control torques

with the brake actuator limitation (saturation) The actual brake

torque indicates that the brake torque passes through the

first-order brake actuator system Although the slip tracking

re-Fig 10 Comparisons of the target slip-ratio tracking among three

differ-ent sliding-surface designs (λ d,f=−0.12, λ d,r=−0.1, µH= 0.9, and

τb= 0.05).

Fig 11 Comparisons of the controlled brake torque among three different

sliding-surface designs (λ d,f=−0.12, λ d,r=−0.1, µH= 0.9, and τb=

0.05) (a) Equivalent brake torque—front (b) Switching brake torque—front.

(c) Brake-torque command with saturation—front (d) Actual brake torque with

τb —front.

sponses for all controllers show an initial oscillation tendency, the proposed sliding-surface design can attenuate the oscillation faster than the other sliding-surface designs do, as shown in Fig 10

σ

B Alternative Sliding- Surface Design< /i>

Among the two sliding- surface designs previously

intro-duced, the first sliding- surface design σ = ˜ x corresponds to... proposed sliding surface can be derived similar to that of the second sliding- surface design, i.e.,

˙σ = ˙˜ x + γ ˜ x = 0, as shown in (18).

In order to determine the magnitude of. .. durability of the brake hardware

Among the design parameters needed for a robust wheel-slip

controller design, a sliding- surface design parameter γ and a switching-gain design parameter