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Non-linear control of wheel slip in

anti-lock braking systems

Presented by,

Saurabh Gupta - 123014014

Darshan Bang - 123100036

Kajal Khan - 123100046

based on

Hossein Mirzaeinejad, Mehdi Mirzaei*, A novel method for non-linear control of wheel

slip in anti-lock braking systems , Control Engineering Practice 18 (2010) 918926.

Guided by,

Prof. Abhishek Gupta

Prof. V. Kartik

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Contents

Anti-lock Braking System

Schematic of ABS

Modeling

Control system design

Control law without integral feedback

Control law with integral feedback

Simulation

Conclusions

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Anti-lock braking system

Prevent wheels from

locking during heavy

braking

Modulates the brake line

pressure independent of

the pedal force

Reduce stopping

distances

Improve stability Improve steer-ability

during braking

Objectives

Antilock braking systems (ABSs) are electronic systems that

monitor and control wheel slip during vehicle braking. Principle:- A skidding wheel has less traction than a rotating

wheel.

Features

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Schematic of ABS

(http://www.google.co.in/imgres?q=abs+system&sa=X&hl=en&biw=1366&bih=667&tbm=isch&tbnid=DVlx205u0VvitM:&imgrefurl=http://www.cvel.clemson.

edu/auto/systems/braking.html&docid=7RQ57SQgNULEmM&imgurl=http://www.cvel.clemson.edu/auto/systems/images/ABS-architecture.png&w=396&h=272

&ei=IzhoUeb-FIWGrAfgtoHgBA&zoom=1&ved=1t:3588,r:12,s:0,i:145&iact=rc&dur=1066&page=1&tbnh=174&tbnw=254&start=0&ndsp=15&tx=147&ty=35)

http://www.google.co.in/imgres?q=abs+system&sa=X&hl=en&biw=1366&bih=667&tbm=isch&tbnid=DVlx205u0VvitM:&imgrefurl=http://www.cvel.clemsonhttp://www.google.co.in/imgres?q=abs+system&sa=X&hl=en&biw=1366&bih=667&tbm=isch&tbnid=DVlx205u0VvitM:&imgrefurl=http://www.cvel.clemsonhttp://www.google.co.in/imgres?q=abs+system&sa=X&hl=en&biw=1366&bih=667&tbm=isch&tbnid=DVlx205u0VvitM:&imgrefurl=http://www.cvel.clemsonhttp://www.google.co.in/imgres?q=abs+system&sa=X&hl=en&biw=1366&bih=667&tbm=isch&tbnid=DVlx205u0VvitM:&imgrefurl=http://www.cvel.clemson

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ABS - Modeling

Wheel free body diagram during modeling

( )

Governing Equations

V = longitudinal velocity of vehicle

= rotational speed ofwheel= longitudinal tire forceTb = braking torque

= longitudinal slip ratio

= friction coefficient

R = wheel radius

= total mass of the model= wheel mass () + of sprung mass ()

1

+ +

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Control System Design

Purpose of controller is to maintain

wheel slip x2 =

and its integral 3 , closed totheir desired responses +

3

[ ] are the state vectors ( , ) are the outputs of system

State space form

Predictive approach with integral feedback is used

Reference model for wheel slip: 1 Where, 0 . 1 5 , 2 0

1

+

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Control System Design Cont.

12 + + 3

=

0

11+0.25 1+0.25

+ 0.5 3 + 1 + 0.25

+

3 3 3 3

0

Without integral feedback

Performance index

Ratio of weighting factors

Necessary Condition for optimality

h: prediction time interval

Current tracking errors

With Integral feedback

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Control Law without integral

feedback

+

1

1 [ + ]

Tracking error dynamics

of the wheel slip

corresponds to nominal model which differsfrom the actual model due to vehicle mass

Uncertainty and uncertainty in road condition

Applying control law to the actual model

Error in estimating is due to the error in estimation of friction force So if error on is bounded then error inwill also be bounded by a constant F>0 ( ) where = Fh So to control tracking error, h should be decreased, but with decreasing h, control

energy becomes large and oscillatory

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Control Law with integral feedback

+ 3 + ( )Where,

11+0.25 1+0.5

3 0.5

Error dynamics of the integral

variable 3:

3

3

3

1 0

3 +

( )

0

Applying control law to the actual model:

3 and ( ) have the same sign So integral variable error reduces the

effect of model uncertainties

So wheel sleep tracking error is muchless the previous case

Steady state tracking error ( 3 0) 0 , 3

Wheel slip tracking error in steady state will be zero

Effect of model uncertainties is transferred to the tracking errorof the integral variable

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Simulation

Simulation results during braking with and without control: (a)

wheel and vehicle speed (uncontrolled), (b) wheel and vehiclespeed (controlled), (c) wheel slip, and (d) wheel slip tracking error

Parameter Value

R 0.326 m

L 2.5 m

hcg 0.5 m

mw 40 kg

mvs 415 kg

I 1.7 kgm-2

Flat dry road, =0.8

Velocity 20 m/s (72 km/hr)

There is no modelinguncertainty

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Simulation Cont

Comparison of performances of the controller with and without integral

feedback: (a) wheel slip tracking error and (b) braking torque.

Uncertainties in vehicle mass = 15 %

Uncertainties in Coefficient of friction = 10 %Time of prediction (h) = 0.003 s

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Conclusions

Controller with integral feedback control is more robust,

can handle nonlinearity and uncertainty of the model in

a better way

Controller without integral feedback As time ofprediction (h), error in tracking slip , energy input and torque input ~

Controller with integral feedback reduces the oscillation,

with better tracking error

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13

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Appendix A

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Appendix B