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 -slip curve

Brake Force and Tire Slip Ratio

xvr /1

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 (-slip curve is nonlinear)

• Independent braking of each wheel

• Start with the equation of rotational motion of one wheel

bb TrFJ

bb TrFJ

= moment of inertia of the wheel and tire

bF = braking force at the contact patch

J

bT = controlled brake torque

Sliding Mode Control for Wheel Slip Regulation

• 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 ),(

)sgn(sTTT reqb

rJarFT xbeq /ˆ)ˆ1(ˆ

xvr /1

Sliding Mode Control for Wheel Slip Regulation

• 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 JvrFrJvT xbxr

0

Adaptive Nonlinear Observer

• 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

uvzzFrJ rn )( 210

),,,( 210 tire model parameters

viscous rotational friction between the wheel and the spindle

)( xr vrv the relative velocity between the tire carcass and the tip of the tread

Adaptive Nonlinear Observer

• 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

Lzrzvg

vvz

r

rr /

)(0

)/exp()()( SrCSCr vvvg

Adaptive Nonlinear Observer

• Recast the LuGre tire friction model into the following standard form of nonlinear differential equations:

0

)],(),([

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

uvzzFrJ rn )( 210

Lzrzvg

vvz

r

rr /

)(0

Transformation to Standard Form

• Define new variables and new functions

Jvmr xeq

zFrJ n 1

zvg

vzv

r

rr )(),( 0

Lzrz /),(

Transformation to Standard Form

• Resulting differential equations in standard form

)]1([),(),()/1(2rm

Jrzzvrmz

ere

urm

JFrmF

enen ])1([)/(

222

2

uJ ])/[()/( 1010

Adaptive Nonlinear Observer

• Structure for nonlinear adaptive observer:

]~)ˆ,(ˆ[~ˆ

ˆˆ

~]~)ˆ1()ˆ,(ˆ)ˆ,(ˆˆ[ˆˆ

yxyy

xCy

yKEyRuyxyxyBxAx

TkkkK ),,( 321 the output feedback gain vector

yyy ˆ~ the error in the output estimate

, 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

Simulation Results: Panic Stop from 60 MPH

• Acceleration response at driver’s seat

Simulation Results: Panic Stop from 60 MPH

• Controlled brake torque at front axle and rear drive axles

Simulation Results: Panic Stop from 60 MPH

• Controlled brake torque at trailer axles

Simulation Results: Panic Stop from 60 MPH

• Wheel angular velocities

Simulation Results: Panic Stop from 60 MPH

• Tire slip ratio at front axle and rear drive axles

Simulation Results: Panic Stop from 60 MPH

• Tire slip ratio at trailer axles

Simulation Results: Panic Stop from 60 MPH

• Actual versus estimated brake forces at front axle

Simulation Results: Panic Stop from 60 MPH

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

Simulation Results: Panic Stop from 60 MPH

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

Simulation Results: Panic Stop from 60 MPH

• 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