MSC_2011_Conf_Meritor_Inc

Anti-Lock Braking Systems for Commercial
Vehicles Using Sliding Mode Control and an
Adaptive Nonlinear Observer
Ragnar Ledesma, PE, PhD
Principal Engineer
Meritor, Inc.

2
Presentation Outline
• Introduction
• Sliding Mode Control for Wheel Slip Regulation
• Adaptive Nonlinear Observer
• ADAMS Modeling and Simulation
• Summary

Background
• Changes to FMVSS 105 and 121 resulted in 30% reduction in required stopping
distances for fully laden tractor-trailer combinations
• For a panic stop from 60 mph required stopping distance has been reduced from
108 meters to 75 meters
• 2 ways to satisfy the new requirement:
– Design heavier brakes to resist higher peak torques
– Change in ABS control algorithm to sustain higher steady-state brake torques

Conventional ABS Control Algorithm
• Discrete on-off control results in a
fluctuation of the tire slip ratio
• Does not maximize the braking force that
can be produced at the tire-ground
interface
• Can be improved by designing an
algorithm that holds the slip ratio at a
steady value, near the peak of the µ-slip
curve

Brake Force and Tire Slip Ratio
xvr /1 ×−= ωλ

Sliding Mode Control for Wheel Slip Regulation
• Sliding mode control (also known as variable structure control)
• Robust with respect to uncertainties in selected model parameters or system
states
• Suitable for nonlinear systems (µ-slip curve is nonlinear)
• Independent braking of each wheel
• Start with the equation of rotational motion of one wheel
bb TrFJ −×=
•
ω
bb TrFJ −×=
•
ω
= moment of inertia of the wheel and tire
bF = braking force at the contact patch
J
bT = controlled brake torque

• Define switching function (sliding
manifold)
• Decompose controlled brake torque into
“equivalent control” and “reaching control”
• The equivalent control torque is the brake
torque that is required once the system
has entered the sliding regime
• Applying the equivalent torque allows the
system states to “slide” along the sliding
manifold, i.e., the value of the switching
function is maintained at zero, in the
absence of external disturbances
dxvs λλω −=),(
)sgn(sTTT reqb ×−=
rJarFT xbeq /ˆ)ˆ1(ˆ ××−−×= λ
xvr /1 ×−= ωλ

• The reaching control torque is the control torque that is required when the system is not
in the sliding regime
• This term in the control torque serves to drive the system states toward the sliding
manifold
• The inequality relation imposes a constraint on the magnitude of the reaching control
torque
• The control system is designed to be robust with respect to errors in estimating the
braking force
])ˆ/([/ˆ 2
max
η+××∆××≥ JvrFrJvT xbxr
0>η
η

• An estimate of the braking force at each wheel end can be obtained by
using an adaptive nonlinear observer
• Consider the equations of motion for a single wheel and tire model with
traction forces between the tire and ground (LuGre tire friction model)
)( 210 rnxeq vzzFvm σσσ ++= 
τωωσσσσω uvzzFrJ rn +−++×−= )( 210 
),,,( 210 κσσσ tire model parameters
viscous rotational friction between the wheel and the spindleωσ
)( xr vrv −= ω the relative velocity between the tire carcass and the tip of the tread

• z is an internal state variable describing the shear deformation between
the tire tread and carcass
• The function g(vr) describes the friction forces between the tire tread and
the ground
Lzrz
vg
v
vz
r
r
r /
)(
0
×××−×−= ωκ
σ
θ
)/exp()()(
δ
µµµ SrCSCr vvvg −×−+=

• Recast the LuGre tire friction model into the following standard form of
nonlinear differential equations:
0
)],(),([
=
=
++++=
θ
ψθφ


Cxy
EyRuxyxyBAxx
u = plant input = the controlled brake/traction torque
y = plant output = the wheel angular velocity

Transformation to Standard Form
• Start with
)( 210 rnxeq vzzFvm σσσ ++= 
τωωσσσσω uvzzFrJ rn +−++×−= )( 210 
Lzrz
vg
v
vz
r
r
r /
)(
0
⋅××−×−= ωκ
σ
θ

• Define new variables and new functions
ωη ⋅+⋅⋅= Jvmr xeq
zFrJ n ⋅⋅⋅+⋅= 1σωχ
z
vg
v
zv
r
r
r
)(
),( 0σ
ϕ =
Lzrz /),( ⋅××= ωκωψ

• Resulting differential equations in standard form
ωωψϕθη ⋅+⋅+−⋅−⋅−= )]1([),(),()/1( 2
rm
J
rzzvrmz
e
re
τω ωσσηση u
rm
J
FrmF
e
nen +⋅−+⋅⋅⋅+⋅⋅−= ])1([)/( 22
2
2

τω ωσσσχσσχ uJ +⋅−⋅+⋅−= ])/[()/( 1010


• Structure for nonlinear adaptive observer:
]~)ˆ,(ˆ[~ˆ
ˆˆ
~]~)ˆ1()ˆ,(ˆ)ˆ,(ˆˆ[ˆˆ
yxyy
xCy
yKEyRuyxyxyBxAx
αφγθ
θαψφθ
+×=
=
+++++++=


T
kkkK ),,( 321= the output feedback gain vector
yyy ˆ~ −= the error in the output estimate
γα, observer design parameters

ADAMS Model of Tractor-Trailer Combination
Gross Combined Weight = 76,500 lbs

Simulation Results: Panic Stop from 60 MPH
• Vehicle speed and distance travelled
• Stopping distance = 54 meters, road friction coefficient = 0.9

• Acceleration response at driver’s seat

• Controlled brake torque at front axle and rear drive axles

• Controlled brake torque at trailer axles

• Wheel angular velocities

• Tire slip ratio at front axle and rear drive axles

• Tire slip ratio at trailer axles

• Actual versus estimated brake forces at front axle

• Actual versus estimated brake forces at forward-rear drive axle

• Actual versus estimated brake forces at rearward-rear drive axle

• Actual versus estimated brake forces at trailer axles

Summary
• Sliding mode control for wheel slip regulation has been presented
• Adaptive nonlinear observer for estimating brake forces was developed using the
LuGre tire frction model
• ADAMS modeling and simulation results were presented to demonstrate the
effectiveness of the new control algorithm
• Shorter stopping distances can be achieved with current disc brake systems in
conjunction with improved ABS control systems

MSC_2011_Conf_Meritor_Inc

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